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--- abstract: 'It is important to identify the change point of a system’s health status, which usually signifies an incipient fault under development. The One-Class Support Vector Machine (OC-SVM) is a popular machine learning model for anomaly detection and hence could be used for identifying change points; however, it is sometimes difficult to obtain a good OC-SVM model that can be used on sensor measurement time series to identify the change points in system health status. In this paper, we propose a novel approach for calibrating OC-SVM models. The approach uses a heuristic search method to find a good set of input data and hyperparameters that yield a well-performing model. Our results on the C-MAPSS dataset demonstrate that OC-SVM can also achieve satisfactory accuracy in detecting change point in time series with fewer training data, compared to state-of-the-art deep learning approaches. In our case study, the OC-SVM calibrated by the proposed model is shown to be useful especially in scenarios with limited amount of training data.' author: - \| Baihong Jin$^1$ Yuxin Chen$^2$ Dan Li$^3$ [Kameshwar Poolla]{}$^{1}$ [Alberto Sangiovanni-Vincentelli]{}$^{1}$\ $^{1}$Department of EECS, University of California, Berkeley\ [`{bjin,poolla,alberto}@eecs.berkeley.edu`]{}\ $^{2}$California Institute of Technology\ [`chenyux@caltech.edu`]{}\ $^{3}$Institute of Data Science, National University of Singapore\ [`idsld@nus.edu.sg`]{} bibliography: - 'refs.bib' title: 'A One-Class Support Vector Machine Calibration Method for Time Series Change Point Detection ' --- Support vector machine, change point detection Acknowledgment {#acknowledgment .unnumbered} ============== This work is supported in part by the National Research Foundation of Singapore through a grant to the Berkeley Education Alliance for Research in Singapore (BEARS) for the Singapore-Berkeley Building Efficiency and Sustainability in the Tropics (SinBerBEST) program, and by the National Science Foundation under Grant No. 1645964.
--- abstract: 'We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary classes), and how they allow nontrivial “element by element” constructions. We conclude with a new proof of the equivalence between saturated and homogeneous which does not use the coherence axiom of abstract elementary classes.' address: \| Department of Mathematics\ Harvard University\ Cambridge, Massachusetts, USA author: - Sebastien Vasey bibliography: - 'bfo-accessible.bib' date: \| \ AMS 2010 Subject Classification: Primary: 18C35. Secondary: 03C45, 03C48, 03C52, 03C55, 03C75, 03E05, 03E55. title: 'Accessible categories, set theory, and model theory: an invitation' --- Introduction ============ Category theory, model theory, and set theory are all foundational branches of mathematics. In this paper, I will attempt to give a taste of the food one obtains when mixing the three of them together. I do not really know how to name this dish but, for the present paper at least and at the risk of scaring researchers from all three fields, I will call it categorical model theory. This terminology appears in the title of Makkai and Paré’s book [@makkai-pare], but unfortunately leaves out the set-theoretic aspect. Let me then reassure the set theorists: set theory is very much part of categorical model theory (see Section \[set-sec\]). So what is categorical model theory? To start with, a model (or structure) consists of a set (the “universe”) together with relation and functions on that set. For example the set of integers together with addition (seen as a binary function) is a model. Part of model theory studies the definable subsets of such structures: sets that can be expressed from the relations and functions using simple set operations such as union, intersection, complement, and projection. This approach has had quite a bit of success (one of the easiest outcomes to describe is the theory of o-minimality [@vandendries; @pila-wilkie]). It is moreover possible to study definable sets in a categorical setup (a field named categorical logic, see e.g. [@makkai-reyes]). Nevertheless, this is not at all what we will do here. To make a loose analogy, categorical model theory relates to studying definable sets of a fixed structure in roughly the same way that thermodynamics relates to quantum physics: category theory looks at an entire class of structures, together with morphisms between them. Thus instead of only looking at $(\mathbb{Z}, +)$, we will look at the class of abelian groups. What should be the morphisms in that class? The classical answer is that it should be the group homomorphisms – the maps preserving the additive structure – yielding the well known category ${\operatorname{\bf Ab}}$. There are however other possible choice of morphisms. For example, an abelian group is a structure $(A, +)$ satisfying certain axioms that can be expressed using first-order sentences (including for example $(\forall x \forall y) (x + y = y + x)$). In model-theoretic terminology, it is a model of the theory (i.e. set of sentences) $T_{ab}$ of abelian groups. When working abstractly with a class of models of a given first-order theory $T$, what should be the “right” notion of morphism? One answer that is well studied in model theory is the notion of an elementary embedding: an embedding preserving all formulas, possibly with parameters. The elementary embeddings preserve a lot more than the homomorphisms (in particular, they preserve the definable sets) and yield another category, ${\operatorname{\bf Elem}}(T)$, whose objects are the models of $T$ and whose morphisms are elementary embeddings. Tarski and Vaught showed that ${\operatorname{\bf Elem}}(T)$ always has directed colimits (the class ${\operatorname{\bf Elem}}(T)$ is closed under unions of chains ordered by elementary substructures), while this is not always true for the category ${\operatorname{\textbf{Mod}}}(T)$ of models of $T$ with homomorphisms. An elementary embedding of abelian groups is quite difficult to describe, and for a category theorist the category ${\operatorname{\bf Elem}}(T_{ab})$ of abelian groups with elementary embeddings is quite poorly behaved (all morphisms are monomorphisms, so it lacks a lot of limits and colimits; it does not have any notion of quotient for example). On the other hand, from the set-theoretic point of view the category ${\operatorname{\bf Elem}}(T_{ab})$ is quite interesting and easy to work with, precisely because all morphisms are monomorphisms (for many purposes, one can think of the maps as inclusions, and think of this category as a kind of concrete poset, with certain isomorphisms between the elements). Still, there may be other interesting types of monomorphisms (for abelian groups, the injective homomorphisms, or the pure morphisms, are obvious choices). Another problem that comes up with the traditional model-theoretic approach is that interesting classes may (provably) not be models of a first-order theory: consider torsion abelian groups, Banach spaces, Zilber’s pseudo-exponential fields [@zilber-pseudoexp], etc. Moving to infinitary logics may partly fix this second problem but one is still somewhat tied to notions of elementary embeddings. There are other issues with the traditional “Tarskian” definition of a model as a set with functions and relations on it (see Macintyre’s essay [@macintyre-bsl]). Several natural categories, such as arrow categories and more generally categories of functors, cannot easily be described within the Tarskian frame, for example. Let us then agree to take a categorical approach: categorical model theory will study categories that, in some sense, look or behave like the categories of models studied in classical model theory. What kind of “classical model-theoretic” behavior are we looking for? Two basic results of model theory are the aforementioned Tarski-Vaught chain theorem (closure under chains of elementary substructures) and the downward Löwenheim-Skolem-Tarski theorem (every structure has a “small” elementary substructure). Category-theoretically, the first can be described by the existence of certain colimits (of chains, or more generally of directed diagrams). The second needs a notion of “smallness”, i.e. really a notion of size. In a category, objects don’t have a “universe”, and even if they do, the cardinality of the universe may not tell us much. Nevertheless, it is possible to define a notion of size, the presentability rank, by looking at how an object embeds into sufficiently directed colimits (as a simple example, a finite set contained inside an infinite union will be contained inside a component of the union; this property could serve as a definition of finiteness – thus presentability ranks generalize cardinalities to other categories than the category of sets). After making this idea precise, we arrive at the definition of an accessible category, one of the main frameworks of categorical model theory. Roughly, it is a category with all sufficiently directed colimits so that any object is a directed colimits of a fixed set of “small” subobjects (see Definition \[acc-def\] here). For example, a set is a union of its finite subsets, an abelian group is a union of its finitely generated subgroups, etc. Accessible categories were first defined by Lair [@lair-accessible], their theory was created by Makkai-Paré [@makkai-pare] and further developed in Adámek-Rosický [@adamek-rosicky]. I invite the reader to consult these references for more on the history. There are a lot of examples of accessible categories, including all those discussed before, ${\operatorname{\bf Ab}}$, Banach spaces with contractions (or isometries), ${\operatorname{\bf Elem}}(T)$ for any first-order theory $T$, and more generally ${\operatorname{\bf Elem}}(\phi)$ for $\phi$ an ${\mathbb{L}}_{\infty, \infty}$-formula[^1] and a suitable notion of elementary embeddings. In passing, let us note that the fact that these examples encompass those studied in continuous model theory may well make accessible categories an interesting framework to reunify continuous with discrete model theory (see e.g. [@lrcaec-jsl]). Before going further, however, let’s address a question a logically-inclined reader may have: what about the (first-order) compactness theorem? Well, you cannot make an omelet without breaking eggs: in view of Lindström’s theorem [@chang-keisler 2.5.4], going significantly beyond ${\operatorname{\bf Elem}}(T)$ will imply losing the compactness theorem. In fact, the situation is worse than this: when passing to a category we “forget” a lot of the logical structures on the objects. For example, we cannot really study definable sets anymore: starting with a category of models with elementary embeddings, we can form a new category by “Morleyizing” – adding a relations for each formula. The Morleyized category is (even concretely) isomorphic to the old one, but model theorists studying quantifier elimination would not want to identify the two categories[^2]... Why in the world, then, would one want to forget the logical structure? We have already argued that many mathematical categories of interest cannot be studied model-theoretically. Even in model theory, it can be helpful to distinguish between internal properties (visible at the level of a single structure, such as quantifier elimination) and external properties (visible by looking at the structure of the category). More precisely, let us define an external property to be a property that is invariant under equivalence of categories. One example of an interesting external property is the existence of a universal object in a given cardinality. Such properties show up in Shelah-style model theory [@shelahfobook]. In fact, Shelah has observed [@shelahaecbook p. 23] that what he calls dividing lines (e.g. model-theoretic stability, simplicity, NIP, etc.) can be characterized by both internal and external properties[^3]. Stability, for example, is equivalent to failure of the order property (internal), or to the existence of saturated models in certain cardinals (external). Thus, while dividing lines do have a logical characterization, they are also invariant under equivalence of categories. This is interesting insight suggests it may be possible to still do Shelah-style model theory categorically (and this is indeed the case, as exemplified in the large body of work on classification theory for AECs, see the references in Section \[reading-sec\]). At a broader level, working with accessible categories entails a higher level of generality. This has downsides but also benefits: more categories of interest are accessible, and accessible categories are closed under more operations. This can make the theory easier to develop in some cases. For example, starting from the class of models of a first-order $T$, one can form its category of $\lambda$-saturated models (for some $\lambda$). One can also form its class of models omitting some type. One could even just look at the class of models of $T$ of cardinality at least $\lambda$. None of these examples are classes of models of a first-order but they are still accessible categories. As another example, there is a very general definition of stability, using forking independence, that can be given in any accessible category and makes no mention of logic. This definition specializes to (and is arguably much simpler than) the usual first-order one. See Section \[indep-sec\]. Coming back to the failure of the compactness theorem, another counterpoint is that there are many different types and levels of compactness[^4]. Some compactness can be recovered using large cardinals (e.g. compactness for ${\mathbb{L}}_{\kappa, \kappa}$), other types of compactness are implied by the “complexity” of the class or category under consideration (for example universal classes, see Definition \[univ-def\], do satisfy a weak version of the compactness theorem [@abv-categ-multi-apal 3.8]; classes with low descriptive set-theoretic complexity also behave better than the general case [@almost-galois-stable]). Still other types of compactness are implied by the stability-theoretic properties of the class (e.g. in a first-order stable theory, any long-enough sequences contains an indiscernible subsequence, c.f. [@shelahfobook I.2.8] or Theorem \[indisc-extraction\], but this is not the case in general without large cardinals [@jechbook 18.18]). This interplay between set-theoretic, model-theoretic, and stability-theoretic compactness is a fascinating aspect of categorical model theory, which is harder to see and study in setups where the compactness theorem applies. At this point a still skeptical reader may say that, while all these philosophical points are interesting, the setup of accessible categories seems too general or difficult for an interesting theory. There are several answers to this objection. First, category theory itself is very general, but it still is an interesting framework in which to present and understand many different branches of mathematics. There are interesting theorems in category theory (see the epilogue of [@ct-context]), but of course they often inform and supplement rather than completely supersede results in more specialized branches. This does not make the importance of category theory in doubt. Similarly, I think of accessible category as a framework rather than as an all-encompassing object of study. There are less general frameworks (abstract elementary classes, universal classes, first-order model theory, universal algebra, ${\mathbb{L}}_{\omega_1, \omega}$, locally presentable categories, Grothendieck abelian categories, etc.) in which one will be able to say more, but may also sometimes be more limited by the lower generality. The “best” framework depends on the question; finding it is a hard part of the mathematician’s job. Second, there are nontrivial theorems about accessible categories. Let’s just mention a basic one for now: any accessible category is equivalent to the category of models of an ${\mathbb{L}}_{\infty, \infty}$ sentence, with morphisms the homomorphisms (Corollary \[acc-to-log\]). For example, we have already seen that we can Morleyize the models of a first-order theory to obtain a category where the elementary embeddings are simply the injective homomorphisms. Since injective homomorphisms are simply homomorphisms that preserve the non-equality relation, we can add this non-equality relation to the models to get an instance of the theorem. This result gives a surprising correspondence between categorical and “logical” model theory (taken in the broad sense of studying classes of models of ${\mathbb{L}}_{\infty, \infty}$). Thus studying categories of models of ${\mathbb{L}}_{\infty, \infty}$ theories is just as hard as studying accessible categories generally. The correspondence also shows that, for the purpose of categorical model theory, the logics with generalized quantifiers described for example in [@model-theoretic-logics] are not necessary: if we study the category of models of a certain sentence expressed in a complicated logic with quantifiers such as “there exists uncountably many”, together with a certain logical notion of morphism, then as long as this category is an accessible category, we can find an equivalent category that will be axiomatized simply in ${\mathbb{L}}_{\infty, \infty}$. See Example \[acc-to-log-ex\](\[toy-quasi-ex\]), the “toy quasiminimal class”, for an instance of this phenomenon. An intermediate type of setup is abstract model theory, which studies frameworks such as abstract elementary classes (AECs) [@sh88]. There we are still studying “Tarskian” classes of models, but the notion of embedding is axiomatically specified rather than defined to be elementarity. Any AEC can naturally be seen as an accessible category, and any class of models of an ${\mathbb{L}}_{\infty, \infty}$ theory can naturally be seen as an $\infty$-AEC (Definition \[infty-aec-def\]). Thus there is a three way correspondence[^5] between categorical model theory (accessible categories), abstract model theory ($\infty$-abstract elementary classes), and logical model theory (${\mathbb{L}}_{\infty, \infty}$). Such a correspondence should be compared with Tarski’s presentation theorem [@tarski-th-models-i] (see Theorem \[tarski\] here): classes closed under substructure, isomorphisms, and unions of chains (abstract model theory) are the same as classes of models of universal theories (logical model theory). Another example of this phenomenon is given by the Birkhoff variety theorem [@birkhoff-variety 10]: a class of algebras is a variety (logical model theory) if and only it is closed under products, subobjects, and quotients (categorical model theory), see [@adamek-rosicky 3.9]. In fact it turns out that many types of accessible categories have natural logical and abstract classes characterizations [@adamek-rosicky; @multipres-pams]. This is useful, because one can often take advantage of the concreteness of abstract model theory to use certain set-theoretic ideas. “Element by element” constructions in abstract elementary classes are a case in point (Section \[aec-sec\]). Third, accessible categories are already well connected to the rest of mathematics. While Makkai and Paré had a model-theoretic motivation, one of the first use of accessible categories was in algebraic topology: a model category is a category endowed with three distinguished classes of morphisms, called fibrations, cofibrations and weak equivalences, satisfying axioms that are properties of the category of topological spaces (with the weak equivalences being the weak homotopy equivalences). In particular, weak equivalences should satisfy a “two out of three” property, and the morphisms should form certain weak factorization systems (e.g. any morphism should factor, in a somewhat canonical way, as a cofibration followed by a fibration that is also a weak equivalence). The reader can think of fibrations as “nice surjections” and cofibrations as “nice inclusions”. See [@hess-model-survey] for a survey of the use of model categories in algebraic topology, and e.g. [@hoveybook] for the general theory. It turns out that the most convenient model categories to work with are the locally presentable ones (i.e. those that are bicomplete and accessible). This is because, roughly speaking, in such categories we can often find set-sized families of “small” cofibrations that generate the rest. Such model categories are called cofibrantly generated. For example, the category of topological spaces is not accessible (Example \[pres-ex\](\[pres-ex-top\])), but the category of simplicial sets is. The model category induced on it will, it turns out, be cofibrantly generated, and in some homotopical sense equivalent to the one on topological spaces. An application of the theory of accessible categories to model categories is the existence, assuming Vopěnka’s principle, of certain localizations [@localization-vopenka; @left-det-rt]. See also [@beke-sheafifiable] for more on the connections between accessible categories, logic, and model categories. Accessible categories have also been used in homological algebra. An only recently solved problem, the flat cover conjecture, asks whether every module has a flat cover [@enochs-flat-cover-orig]. This was proven by Bican, El Bashir, and Enochs [@flat-cover]. Two proofs were given, one using set-theoretic tools of Eklof-Trlifaj [@ext-vanish]. This proof was recognized by Rosický [@flat-covers-factorizations] to really be an instance of a “small object argument”: a way to build cofibrantly generated weak factorization systems in any locally presentable category. I believe these connections are interesting, especially because logic and model theory have not historically threaded too much into algebraic topology and homological algebra. If accessible categories are really tied to model theory, then these results should have also model-theoretic meaning. Recently, joint work with Lieberman and Rosický [@more-indep-v2] showed that in fact the notion of a model category (or more generally a weak factorization system) being cofibrantly generated is closely tied to the theory of model-theoretic forking. Essentially, in a locally presentable model category, a weak factorization system is cofibrantly generated exactly when the category obtained by restricting to only to cofibrations is stable, in the sense of having a forking-like independence notion. See Section \[indep-sec\] for an overview of this result. In conclusion, categorical model theory is a fascinating mix of set theory, category theory, and model theory that sheds light on all these topics and on some other parts of mathematics. This paper is meant to survey some basic results, as well as discuss some topics of current research. I will survey set-theoretic aspects (Section \[set-sec\]), as well as questions on stable independence (Section \[indep-sec\]), a recent research development that may be of interest to model theorists, category theorists, and mathematicians at large. Some methods applicable to concrete setups such as AECs, Section \[aec-sec\] will also be discussed. I will end with some open problems (Section \[problem-sec\]), as well as a short list of helpful resources to learn more (Section \[reading-sec\]). A first appendix gives a streamlined method for handling the bookkeeping in point by point constructions, and the second appendix gives a very short introduction to first-order stability theory. I will assume that the reader is familiar with very basic logic and model theory [@chang-keisler], set theory [@hrbacek-jech], and category theory [@maclane; @joy-of-cats]. Still, an effort has been made to repeat many standard definitions and provide examples and intuitions behind proofs for those that are not necessarily proficient in all three fields. Acknowledgments {#acknowledgments .unnumbered} --------------- I would like to thank Jiří Rosický and Michael Lieberman for introducing me to accessible categories and taking an early look at this survey. The work on category-theoretic sizes and stable independence presented here comes from our joint collaboration. I thank Marcos Mazari-Armida and the referee for helpful comments. I also thank Justin Cavitt, Rebecca Coulson, and Rehana Patel, for encouraging me to write Appendix \[fo-sec\]. Main definitions and examples ============================= Set theory ---------- We assume familiarity with ordinals and cardinals. We think of ordinals as transitive sets ordered by membership, and identify cardinals with the corresponding ordinals. We denote by $\omega$ the first infinite ordinal, i.e. the set of natural numbers. For a cardinal $\lambda$, we write $\lambda^+$ for the successor of $\lambda$, the minimal cardinal strictly bigger than $\lambda$. Cardinals of the form $\lambda^+$ are called successor cardinals, and cardinals that are not successors are called limit. We write ${{}^{Y}X}$ for the set of functions from $Y$ to $X$ and, for an ordinal $\alpha$, ${{}^{<\alpha}X}$ for $\bigcup_{\beta < \alpha} {{}^{\beta}X}$. For $\lambda$ and $\mu$ cardinal, $\lambda^{<\mu} = \|{{}^{<\mu}\lambda}\|$ and $\lambda^\mu = \|{{}^{\mu}\lambda}\|$. A partially ordered set (or poset for short) is a transitive, reflexive, and antisymmetric relation. A subset $I_0$ of a poset $I$ is cofinal if for all $i \in I$ there is $i_0 \in I_0$ so that $i \le i_0$. The cofinality, ${\text{cf} (I)}$, of $I$ is the minimal cardinality of a cofinal subset of $I$. Of course, we identify an ordinal $\alpha$ with the corresponding linear order $(\alpha, \in)$. A cardinal $\lambda$ is regular if it is infinite and ${\text{cf} (\lambda)} = \lambda$. Category theory --------------- A category is called small if it has only a set of objects, and it is called large if it has a proper class of non-isomorphic objects. All the categories in this paper will be locally small (i.e. the collection ${\operatorname{Hom}}(A, B)$ of morphisms from the object $A$ to the object $B$ is always a set, never a proper class). Our categorical notation and conventions will mostly be those of [@joy-of-cats] and [@adamek-rosicky]. In particular, a monomorphism (or mono for short) is a morphism $f$ such that $f g_1 = fg_2$ implies $g_1 = g_2$ for any two morphisms $g_1$ and $g_2$. The notion of an epimorphism (epi for short) is defined dually. A diagram in a category ${\mathcal {K}}$ is a functor $D: I \to {\mathcal {K}}$, where $I$ is a (here always small) category, called the index of the diagram. In this paper, the index $I$ will usually be a partially ordered set, which will be identified with the corresponding category. A cocone[^6] for a diagram $D: I \to {\mathcal {K}}$ consist in an object $A$ together with morphism $(D_i \xrightarrow{f_i} A)_{i \in I}$ such that whenever $i \xrightarrow{d} j$ is a morphism of $D$, we have that $f_i = f_j d$. The cocones for $D$ form a category, ${\mathcal {K}}_D$, where a morphism from $(D_i \xrightarrow{f_i} A)_{i \in I}$ to $(D_i \xrightarrow{g_i} B)_{i \in I}$ is a ${\mathcal {K}}$-morphism $A \xrightarrow{h} B$ so that $h f_i = g_i$ for all $i \in I$. A colimit for $D$ is an initial object in the category ${\mathcal {K}}_D$ (recall that an initial object in a category is one so that there is a unique morphism from it to any other object). Presentability and accessible categories ---------------------------------------- For $\lambda$ a cardinal, a partially ordered set is called $\lambda$-directed if every subset of cardinality strictly less than $\lambda$ has an upper bound. Note that any non-empty poset is $\lambda$-directed for $\lambda \le 2$ (an empty poset is $0$-directed but not $1$-directed), $3$-directed is equivalent to $\aleph_0$-directed, and for a singular cardinal $\lambda$, $\lambda$-directed is equivalent to $\lambda^+$-directed. Thus we usually assume that $\lambda$ is a regular cardinal. We will say directed instead of $\aleph_0$-directed. For $\lambda$ regular, an example of a $\lambda$-directed poset that the reader can keep in mind is $\lambda$ itself, seen as a chain (i.e. a linear order) of order type $\lambda$. Another typical example is the poset $[A]^{<\lambda}$ of all subsets of a fixed set $A$ which have cardinality strictly less than $\lambda$, ordered by containment. A $\lambda$-directed diagram is a diagram indexed by a $\lambda$-directed poset. A category has $\lambda$-directed colimits if any $\lambda$-directed diagram has a colimit. Note that for $\lambda_1 < \lambda_2$, $\lambda_2$-directed implies $\lambda_1$-directed, hence having $\lambda_1$-directed colimits is stronger than having $\lambda_2$-directed colimits. The following observation will be used without comments: If a category has colimits of all chains indexed by regular cardinals, then it has directed colimits. It is immediate using the definition of cofinality that a category with all colimits of chains indexed by regular cardinals has colimits of all chains indexed by ordinals. Now any finite directed poset has a maximum, so colimits of such diagrams are trivial. Furthermore, any infinite directed partially ordered set $I$ can be written as $\bigcup_{\alpha < \|I\|} I_\alpha$, where each $I_\alpha$ has strictly smaller cardinality than $I$, is directed, and $I_\alpha \subseteq I_\beta$ for $\alpha < \beta$. We can therefore proceed by induction on $\|I\|$ to show that the category has all colimits indexed by $I$. A similar result no longer holds in the uncountable case [@adamek-rosicky Exercise 1.c(2)]: if $\lambda$ is uncountable, having colimits indexed by ordinals of cofinality at least $\lambda$ is strictly weaker than having all $\lambda$-directed colimits. In many concrete algebraic cases, directed colimits exist and are essentially computed by taking unions. For example, the category ${\operatorname{\bf Ab}}$ of abelian groups has directed colimits, and they can essentially be computed by taking unions of the resulting system of groups. However in the category of Banach spaces with contractions, colimits of increasing chains are given by taking the completion of their union. Of course, some categories don’t have directed colimits at all. For example, the category of well-orderings, with morphisms the order-preserving maps, does not have directed colimits. It does however have $\aleph_1$-directed colimits. The category of complete Boolean algebras (with homomorphisms), on the other hand, does not have $\lambda$-directed colimits for any $\lambda$. The notion of a $\lambda$-presentable object is key to the definition of an accessible category. \[pres-def\] For $\lambda$ a regular cardinal, an object $A$ of a category ${\mathcal {K}}$ is $\lambda$-presentable if the functor ${\operatorname{Hom}}(A, -)$ preserves $\lambda$-directed colimits. Said more transparently, $A$ is $\lambda$-presentable if for any $\lambda$-directed diagram $D: I \to {\mathcal {K}}$ with colimit $(D_i \xrightarrow{g_i} {\operatorname{colim}}D)_{i \in I}$, any map $A \xrightarrow{f} {\operatorname{colim}}D$ factors essentially uniquely through $D$: there exists $i \in I$ and $A \xrightarrow{f_i} D_i$ such that $f_i g_i = f$. Essentially uniqueness means that, moreover, if $j \in I$ and $A \xrightarrow{f_j} D_j$ are such that $f_j g_j = f$, then there exists $k \ge i,j$ such that $g_{i,k} f_i = g_{j, k} f_j$ (here, $g_{i, j}$ and $g_{i, k}$ are the diagram maps from $i$ to $j$ and $i$ to $k$ respectively). $$\xymatrix@=3pc{ A \ar[r]^f \ar@{.>}[dr]_{f_i} & {\operatorname{colim}}D \\ & D_i \ar[u]_{g_i} \\ }$$ When $\lambda = \aleph_0$, we will say that $A$ is finitely presentable. We say that $A$ is presentable if it is $\lambda$-presentable for some $\lambda$. The presentability rank of a presentable object $A$ is the least regular cardinal $\lambda$ such that $A$ is $\lambda$-presentable. Note that a $\lambda$-presentable object is also $\mu$-presentable for all regular $\mu > \lambda$. \[pres-ex\] 1. \[pres-ex-set\] In the category of sets (with morphisms all functions), a set is $\lambda$-presentable if and only if it has cardinality strictly less than $\lambda$. Indeed, directed colimits are essentially unions, and a set $A$ has cardinality strictly less than $\lambda$ if and only if whenever $A \subseteq \bigcup_{i < \lambda} A_i$, there exists $i < \lambda$ such that $A \subseteq A_i$. Thus the presentability rank of an infinite set is the successor of its cardinality. Similarly, in many algebraic categories (in fact in any abstract elementary classes – see Definition \[infty-aec-def\] and Example \[unif-ex\](\[unif-ex-aec\])), the presentability rank of a big-enough object will be the successor of the cardinality of its universe. 2. In very algebraic categories, such as ${\operatorname{\bf Ab}}$, an object is $\lambda$-presentable if and only if it can be presented using strictly less than $\lambda$-many objects and equations [@adamek-rosicky 3.12]. This is the motivation for the term “presentable”. 3. \[pres-ex-wo\] [@internal-sizes-jpaa 6.2] Consider the category of well-orderings, with morphisms the initial segment embeddings. Note that (as opposed to what would happen if the maps were just order-preserving maps) this category has all directed colimits. An infinite well-order is $\lambda$-presentable precisely when it has cofinality strictly less than $\lambda$. 4. Consider the category of all algebras with one $\omega$-ary operation, and morphisms the homomorphisms. Then an object is $\lambda$-presentable if and only if it is generated by a set of cardinality strictly less than $\lambda$. Here, the free algebra on $\lambda$-many generators is $\lambda^+$-presentable but has cardinality $\lambda^{\aleph_0}$ (which could be strictly bigger than $\lambda$). 5. \[pres-ex-hilb\] Consider the category of complete metric spaces, with isometries as morphisms. For $\lambda$ regular uncountable, a space $A$ is $\lambda$-presentable if and only if its it has density character strictly less than $\lambda$ (the density character of a topological space is defined as the least cardinality of a dense subset). Thus the presentability rank is the successor of the density character. This holds more generally in all the “continuous” examples, including Banach and Hilbert spaces [@lrcaec-jsl 3.1]. Note that in general the density character is different from the cardinality. For example, there are no Hilbert spaces of cardinality $\lambda$ when $\lambda < \lambda^{\aleph_0}$, but there are no problems finding a Hilbert space generated by an orthonormal basis of cardinality $\lambda$. Finite sums from such a basis will give the desired dense subset. 6. \[pres-ex-met\] [@internal-sizes-jpaa 6.1][^7] Consider the one point metric space $\{0\}$ in the category of complete metric spaces with isometries. This space is $\aleph_1$-presentable (by the preceding discussion) but it is not $\aleph_0$-presentable. Indeed, the inclusion of $\{0\}$ into $\{0\} \cup \{\frac{1}{n} \mid 0 < n < \omega\}$ does not factor through $\{\frac{1}{n} \mid 0 < n \le m\}$, for any $m < \omega$. In fact, the empty metric space is the only $\aleph_0$-presentable object of the category. 7. \[pres-ex-top\] [@adamek-rosicky 1.2(10)] In the category of topological spaces (with morphisms the continuous functions), a discrete spaces (i.e. where every set is open) is $\lambda$-presentable exactly when it has cardinality strictly less than $\lambda$. On the other hand, a non-discrete space $A$ is never presentable. To see this, fix a regular $\lambda$ and a non-open set $X \subseteq A$. Without loss of generality, $A \cap \lambda = \emptyset$. For $\alpha <\lambda$, let $D_\alpha$ be the space $A \cup \lambda$, where the non-trivial open sets in $D_\alpha$ are of the form $X \cup [\beta, \lambda)$ for $\beta \in [\alpha, \lambda)$. For $\alpha \le \beta$, the inclusion of $D_\alpha$ into $D_\beta$ is continuous ($D_\alpha$ has more open sets than $D_\beta$), and these inclusions give a $\lambda$-directed diagram. The colimit $D$ of this diagram is the indiscrete space on the set $A \cup \lambda$. Because $D$ is indiscrete, the inclusion of $A$ into $D$ is continuous, but it cannot factor through any $D_\alpha$ (because the inverse image of $X \cup [\alpha, \lambda)$, an open set of $D_\alpha$, is $X$, which is not open in $A$). In view of the many examples above where the presentability rank was the successor of some natural notion of size (see Section \[set-size-sec\] on what is true in general), it is convenient to have a name for this predecessor. We will call it the internal, or category-theoretic, size: \[size-def\] The (internal – or category-theoretic) size of an object $A$ is the predecessor (if it exists) of its presentability rank. We can now precisely say what an accessible category is: \[acc-def\] For a regular cardinal $\lambda$, a category ${\mathcal {K}}$ is $\lambda$-accessible if: 1. ${\mathcal {K}}$ has $\lambda$-directed colimits. 2. (Smallness condition) There is a set $S$ of $\lambda$-presentable objects such that every object of ${\mathcal {K}}$ is a $\lambda$-directed colimits of elements of $S$. If ${\mathcal {K}}$ is $\lambda$-accessible and bicomplete (i.e. complete and cocomplete: it has all limits and colimits), we say that it is locally $\lambda$-presentable. When $\lambda = \aleph_0$, we will talk about a finitely accessible or locally finitely presentable category. We say that ${\mathcal {K}}$ is accessible \[locally presentable\] if it is $\lambda$-accessible \[locally $\lambda$-presentable\] for some regular cardinal $\lambda$. There are three occurrences of the parameter $\lambda$ in Definition \[acc-def\], and there are no real reasons why these occurrences should all be the same cardinal. Thus we could parameterize the definition further into three cardinals, see [@internal-sizes-jpaa 3.6]. In fact, we can be even more precise and allow singular cardinals in the definition. For example, an $(\aleph_0, \aleph_1, <\aleph_{\omega_1})$-accessible category would be a category with directed colimits where every object is an $\aleph_1$-directed colimits of a fixed set of $(<\aleph_{\omega_1})$-presentable objects, where $(<\aleph_{\omega_1})$-presentable means $\lambda_0$-presentable for some regular $\lambda_0 < \aleph_{\omega_1}$. Thus we want each object in the diagram to have presentability rank some $\aleph_\alpha$, $\alpha < \omega_1$, but there may not be a single $\aleph_\alpha$ that bounds the rank of each object. For simplicity, we will not take this approach here but will often discuss, for example, accessible categories with all directed colimits. \[acc-ex\] 1. The category ${\operatorname{\bf Set}}$ of sets with functions as morphisms is locally finitely presentable: it is bicomplete and every set is a directed colimit of finite sets. 2. Any complete lattice is (when seen as a category) a locally presentable category. 3. The category ${\operatorname{\bf Set}}_{mono}$ of sets with injective functions as morphisms, is finitely accessible but not bicomplete (for example, it does not have coequalizers; intuitively it is impossible to quotient). There are two general phenomenons at play here: first [@indep-categ-advances 6.2], given any accessible category ${\mathcal {K}}$, the category ${\mathcal {K}}_{mono}$ obtained by restricting to its monomorphisms will be accessible as well. Second, if ${\mathcal {K}}$ is an accessible category where all morphisms are monomorphisms (this is typically the case in model-theoretic setups), then ${\mathcal {K}}$ cannot be locally presentable, unless it is a complete lattice (see the previous example), so in particular small. 4. The category ${\operatorname{\bf Ab}}$ of a abelian groups is locally finitely presentable. The finitely generated abelian groups are the finitely presentable objects and any other abelian group is a directed colimit of those. More generally, for a fixed ring $R$, the category ${R\text{-}\operatorname{\bf {Mod}}}$ of $R$-modules is locally finitely presentable. 5. The category ${\operatorname{\bf Gra}}$ of graphs (symmetric reflexive binary relations) with morphisms the graph homomorphisms is locally finitely presentable (finitely presentable objects are finite graphs). 6. The category ${\operatorname{\bf Ban}}$ of Banach spaces (with morphisms the contractions) is locally $\aleph_1$-presentable, but not locally $\aleph_0$-presentable (for reasons connected to Example \[pres-ex\](\[pres-ex-met\])). 7. The category ${\operatorname{\bf Bool}}$ of boolean algebras (with morphisms the boolean algebra homomorphisms) is locally finitely presentable. On the other hand, the category of complete boolean algebras is not even accessible: it does not have $\lambda$-directed colimits for any $\lambda$. For a fixed regular cardinal $\lambda$, the category of $\lambda$-complete Boolean algebras is however $\lambda$-accessible. 8. The category of well-orderings with morphisms the order-preserving maps is $\aleph_1$-accessible but not finitely accessible (it does not have directed colimits). On the other hand the category of well-orderings with morphisms the initial segment maps is not accessible, even though it has directed colimits: for any regular cardinal $\lambda$, there is a proper class of non-isomorphic well-orderings of cofinality $\lambda$, see Example \[pres-ex\](\[pres-ex-wo\]). We could still look at the full subcategory of well-orderings of order type $\lambda^+$ or less. It turns out that this will be a $\lambda^+$-accessible category. 9. The category of all well-founded models of (a sufficiently-big fragment of) ZFC, or of all well-founded models of ZFC + V = L, with morphisms the elementary embeddings, is $\aleph_1$-accessible. A variation of this example is studied in [@internal-sizes-jpaa §6.1]. 10. Any Fraïssé class can be seen as generating a finitely accessible category. In fact, there exists a general categorical theory of Fraïssé constructions [@fraisse-kubis]. 11. \[free-ex\] The category of all free abelian groups (with group homomorphisms) is not accessible if $V = L$, but is $\kappa$-accessible for $\kappa$ a strongly compact cardinal [@makkai-pare §5.5]. Thus whether a category is accessible can, in certain cases, be a set-theoretic question. See Section \[set-func-sec\] for more on this phenomenon. Categories of structures {#struct-sec} ------------------------ In order to get more examples of accessible categories and introduce related frameworks, let us move toward logic. For completeness and because we will work with infinitary languages, we start by repeating the basic definitions. The reader will not lose much by skipping them. More details can be found in [@dickmann-book] or [@adamek-rosicky Chapter 5]. For $\kappa$ an infinite cardinal, a $\kappa$-ary vocabulary (or signature) is a set $\tau$ containing[^8] relations and functions symbols of arity strictly less than $\kappa$. When $\kappa = \aleph_0$, we call $\tau$ a finitary vocabulary, and when $\kappa \ge \aleph_1$ an infinitary vocabulary. By default, a vocabulary means a $\kappa$-ary vocabulary for some $\kappa$ (so possibly infinitary). More precisely, a $\kappa$-ary vocabulary $\tau$ is a a set ${\langle n_i : i \in I_R, m_j: j \in I_F \rangle}$, where $I_R$ and $I_F$ are disjoint index sets and $n_i, m_j$ are cardinals strictly less than $\kappa$. For such a vocabulary, a $\tau$-structure $M$ consists of a set $A = U M$ (the universe), for each $n_i \in \tau$, an $n_i$-ary relation $R_i$ on $A$, and for each $m_j \in \tau$ an $m_j$-ary function $f_j$ on $A$. A term in the vocabulary $\tau$, over a given set of variables $V$ (disjoint from any other object that we consider, and of cardinality $\kappa$ – we will never mention $V$ again), is defined inductively as either a variable $x$, or as $f_j (\rho_\alpha)_{\alpha < m_j}$, for each $\rho_\alpha$ a term. An atomic formula is an expression of the form $\top$, $\bot$, $\rho = \rho'$ or $R_i (\rho_\alpha)_{\alpha < n_i}$ for terms $\rho$ ,$\rho'$, $\rho_\alpha$. For $\lambda \ge \kappa$ an infinite cardinal, an ${\mathbb{L}}_{\lambda, \kappa}$ formula is defined inductively as either an atomic formula, $\bigwedge_{\alpha \in S} \phi_\alpha$, $\bigvee_{\alpha \in S} \phi_\alpha$, $\phi \rightarrow \psi$, $\neg \phi$, $(\forall {\bar{x}}) \phi$, $(\exists {\bar{x}}) \phi$, where $\phi_\alpha$, $\phi$, $\psi$, are formulas, $\|S\| < \lambda$, ${\bar{x}}$ is a sequence of variables of length strictly less than $\kappa$, and we require that the formulas obtained from conjunctions and disjunctions still have fewer than $\kappa$-many variables. We also define ${\mathbb{L}}_{\infty, \kappa} = \bigcup_{\lambda} {\mathbb{L}}_{\lambda, \kappa}$ and ${\mathbb{L}}_{\infty, \infty} = \bigcup_{\kappa} {\mathbb{L}}_{\infty, \kappa}$ as expected. The free variables of a formula are the ones that appear in it and are not bound by any quantifier. We write $\phi ({\bar{x}})$ for a formula $\phi$ with free variables among ${\bar{x}}$. A sentence is a formula without free variables, and a theory is a set of sentences. For a $\tau$-structure $M$, ${\bar{a}}$ a sequence of elements from $M$, and $\phi ({\bar{x}})$ a formula (${\bar{x}}$ and ${\bar{a}}$ of the same length), we define what it means for $M$ to satisfy (or be a model of) $\phi$, with ${\bar{a}}$ standing for ${\bar{x}}$, $M \models \phi ({\bar{a}})$ for short, as expected. A $\tau$-structure satisfies (or is a model of) a theory if it satisfies all sentences of the theory. For a vocabulary $\tau$, a homomorphism from a $\tau$-structure $M$ to a $\tau$-structure $N$ is a function $f$ from $U M$ to $U N$ that preserves all atomic formulas: if $\phi ({\bar{x}})$ is atomic, ${\bar{a}}$ is a sequence in $M$, and $M \models \phi ({\bar{a}})$, then $N \models \phi (f ({\bar{a}}))$. We let ${\operatorname{\bf Str}}(\tau)$ denote the category of all $\tau$-structures with homomorphisms. For $T$ a theory in ${\mathbb{L}}_{\infty, \infty}$, we let ${\operatorname{\textbf{Mod}}}(T)$ denote the full subcategory of ${\operatorname{\bf Str}}(\tau)$ consisting of all models of $T$. When $\phi$ is a sentence, ${\operatorname{\textbf{Mod}}}(\phi)$ will denote ${\operatorname{\textbf{Mod}}}(\{\phi\})$. Note that ${\operatorname{\textbf{Mod}}}(\phi)$ is not always an accessible category, essentially because the homomorphisms are not the right notion of embedding when $\phi$ is too complex. Thus we more generally define, for $\Phi \subseteq {\mathbb{L}}_{\infty, \infty}$, a $\Phi$-elementary map from $M$ to $N$ to be a function $f$ from $U M$ to $U N$ that preserves all formulas in $\Phi$. We let ${\operatorname{\bf Elem}}_\Phi (\tau)$ denote the category of all $\tau$-structures with homomorphism, and ${\operatorname{\bf Elem}}_\Phi (T)$ denote the full subcategory of ${\operatorname{\bf Elem}}_\Phi (\tau)$ consisting of models of $T$. One can check that for $\lambda$ a regular cardinal and $T$ an ${\mathbb{L}}_{\lambda, \lambda}$ theory, ${\operatorname{\bf Elem}}_{{\mathbb{L}}_{\lambda, \lambda}} (T)$ is $\lambda$-accessible [@makkai-pare §3.4]. We will focus on the following simple type of formulas: A formula of ${\mathbb{L}}_{\infty, \infty}$ is positive existential if it can be built from atomic formulas using only conjunctions, disjunctions and existential quantifications. A formula is basic if it is of the form $(\forall {\bar{x}})(\phi \rightarrow \psi)$, where $\phi$ and $\psi$ are positive existential. A basic theory is a set of basic sentences. Note that any theory in ${\mathbb{L}}_{\infty, \infty}$ can be “Morleyized” to a basic theory, by adding a relation symbol for each formula [@makkai-pare 3.2.8]. In fact, for $\kappa \le \lambda$ and $T$ a theory in ${\mathbb{L}}_{\lambda, \kappa}$, ${\operatorname{\bf Elem}}_{{\mathbb{L}}_{\lambda, \kappa}} (T)$ is isomorphic to ${\operatorname{\textbf{Mod}}}(T')$, for some basic ${\mathbb{L}}_{\lambda, \kappa}$-theory $T'$. This fact is sometimes called Chang’s presentation theorem by model theorists and is the reason why we can restrict ourselves to basic formulas without losing generality. \[logical-ex\] Let $\kappa$ and $\lambda$ be regular cardinals. 1. [@adamek-rosicky §5.1] If $\tau$ is a $\kappa$-ary vocabulary then ${\operatorname{\bf Str}}(\tau)$ is a locally $\kappa$-presentable category. 2. [@adamek-rosicky 5.35] For any basic theory $T$ in ${\mathbb{L}}_{\infty, \lambda}$, ${\operatorname{\textbf{Mod}}}(T)$ is accessible with $\lambda$-directed colimits. In the second result, existence of $\lambda$-directed colimits can be proven directly. The smallness condition in the definition of an accessible category follows, for example, from the infinitary downward Löwenheim-Skolem theorem. It turns out that Example \[logical-ex\] is sharp, in the sense that any accessible category is equivalent to a category of the form ${\operatorname{\textbf{Mod}}}(T)$, for $T$ a basic theory (Corollary \[acc-to-log\]). Similarly, it is possible to characterize the locally presentable categories as the categories of models of certain theories (called limit). See [@adamek-rosicky 5.30]. $\infty$-abstract elementary classes ------------------------------------ We now introduce $\infty$-abstract elementary classes ($\infty$-AECs). They are in some sense a compromise: less abstract than accessible categories, but still more abstract than categories of models of basic theories. The definition of an AEC (the case $\mu = \aleph_0$ below) is due to Shelah [@sh88]. It was generalized to the case of a $\mu$-AEC in [@mu-aec-jpaa]. We introduce essentially the same notion, but to be precise we make $\mu$ (and the vocabulary) part of the data. The reader should first read it with the case $\mu = \aleph_0$ in mind. \[infty-aec-def\] An $\infty$-abstract elementary class (or $\infty$-AEC for short) ${\mathbf{K}}$ consists of: 1. A regular cardinal $\mu = \mu ({\mathbf{K}})$. 2. A $\mu$-ary vocabulary $\tau = \tau ({\mathbf{K}})$. 3. A class $K$ of $\tau$-structures. 4. A partial order ${{\le_{{\mathbf{K}}}}}$ on $K$. satisfying the following four axioms: - : $K$ is closed under isomorphisms, $M {{\le_{{\mathbf{K}}}}}N$ implies that $M$ is a $\tau$-substructure of $N$ (i.e. $U M \subseteq U N$ and the inclusion $M \to N$ is a homomorphism; we write $M \subseteq N$), and ${{\le_{{\mathbf{K}}}}}$ respects isomorphisms in the sense that if $M, N \in K$, $M {{\le_{{\mathbf{K}}}}}N$, and $f: N \cong N'$, then $f[M] {{\le_{{\mathbf{K}}}}}N'$. - : if $M_0, M_1, M_2 \in K$, $U M_0 \subseteq UM_1$, $M_1 {{\le_{{\mathbf{K}}}}}M_2$, and $M_0 {{\le_{{\mathbf{K}}}}}M_2$, then $M_0 {{\le_{{\mathbf{K}}}}}M_1$. - : if ${\langle M_i : i \in I \rangle}$ is a $\mu$-directed system in ${\mathbf{K}}$ (i.e. $I$ is a $\mu$-directed poset, all the $M_i$’s are in $K$, and $i \le j$ implies $M_i {{\le_{{\mathbf{K}}}}}M_j$), then letting $M := \bigcup_{i \in I} M_i$ (defined as expected), we have: - $M \in K$. - $M_i {{\le_{{\mathbf{K}}}}}M$ for all $i \in I$. - If $N \in K$ and $M_i {{\le_{{\mathbf{K}}}}}N$ for all $i \in I$, then $M {{\le_{{\mathbf{K}}}}}N$. - : there exists a cardinal $\lambda = \lambda^{<\mu} \ge \|\tau\| + \mu$ such that for any $M \in K$ and any $A \subseteq U M$, there exists $M_0 \in K$ with $M_0 {{\le_{{\mathbf{K}}}}}M$, $A \subseteq U M_0$, and $\|U M_0\| \le \|A\|^{<\mu} + \lambda$. We write ${\text{LS}}({\mathbf{K}})$ (the Löwenheim-Skolem-Tarski number of ${\mathbf{K}}$) for the minimal such $\lambda$. Unless $K$ is empty, the vocabulary $\tau ({\mathbf{K}})$ can always be recovered from $K$. Thus we usually just write ${\mathbf{K}}= (K, {{\le_{{\mathbf{K}}}}})$ and say that ${\mathbf{K}}$ is a $\mu$-AEC to make the $\mu$ associated with it clear. When $\mu = \aleph_0$, we omit it and just say that ${\mathbf{K}}$ is an AEC. We also will not distinguish between $K$ and ${\mathbf{K}}$, writing for example $M \in {\mathbf{K}}$ instead of $M \in K$. Another convention: if we write $M {{\le_{{\mathbf{K}}}}}N$, we will automatically mean that also $M, N \in K$. In any $\infty$-AEC, there is a natural notion of morphism. Let ${\mathbf{K}}$ be an $\infty$-AEC. For $M, N \in {\mathbf{K}}$, a ${\mathbf{K}}$-embedding from $M$ to $N$ is an injective $\tau ({\mathbf{K}})$-homomorphism $f$ from $M$ to $N$ such that $f[M] {{\le_{{\mathbf{K}}}}}N$. When ${\mathbf{K}}$ is clear from context, we write $f: M \to N$ to mean that $f$ is a ${\mathbf{K}}$-embedding from $M$ to $N$. We will often identify ${\mathbf{K}}$ with the category whose objects are the structures in ${\mathbf{K}}$ and morphisms the ${\mathbf{K}}$-embeddings. \[aec-categ-prop\] If ${\mathbf{K}}$ is a $\mu$-AEC, then it has the following properties as a category: - It is a subcategory of ${\operatorname{\bf Str}}(\tau ({\mathbf{K}}))$. Further, it is isomorphism-closed in ${\operatorname{\bf Str}}(\tau ({\mathbf{K}}))$ (a subcategory ${\mathcal {L}}$ of a category ${\mathcal {K}}$ is isomorphism-closed – or replete – if whenever $A$ is an object of ${\mathcal {L}}$ and $f: A \to B$ is an isomorphism of ${\mathcal {K}}$, then both $f$ and $B$ are in ${\mathcal {L}}$). This is the essential content of the abstract class axiom. - It is a concrete category, as witnessed by the universe functor $U : {\mathbf{K}}\to {\operatorname{\bf Set}}$. - All its morphisms are monomorphisms, and in fact concrete monomorphisms (i.e. they are also monomorphisms in the category of sets – injective functions). More is true: it is noticed in [@joy-of-cats §8] that the “right” notion of monomorphism in many examples ends up being the notion of a concrete embedding [@joy-of-cats 8.6] whose definition mirrors the coherence axiom of AECs. In fact, what the coherence axiom says is exactly that the morphisms of ${\mathbf{K}}$ are concrete embeddings in the sense of [@joy-of-cats 8.6]. - It has $\mu$-directed colimits (this is the essential content of the chain axiom). In fact these $\mu$-directed colimits are concrete in the sense that the functor $U$ preserves them: they are computed the same way as in ${\operatorname{\bf Set}}$, by taking unions. - It is an ${\text{LS}}({\mathbf{K}})^+$-accessible category. Indeed, the objects of cardinality at most ${\text{LS}}({\mathbf{K}})$ are all ${\text{LS}}({\mathbf{K}})^+$-presentable, and there is only a set of them up to isomorphism. Moreover, the coherence and smallness axioms together imply that any other object $M \in {\mathbf{K}}$ can be written as the ${\text{LS}}({\mathbf{K}})^+$-directed union of $\{M_0 \in {\mathbf{K}}\mid M_0 {{\le_{{\mathbf{K}}}}}M, \|U M_0\| \le {\text{LS}}({\mathbf{K}})\}$ (we think of it as a ${{\le_{{\mathbf{K}}}}}$-system indexed by itself). Note however that ${\mathbf{K}}$ need not be $\mu$-accessible. In fact it is easy to see that for any regular cardinal $\lambda$, the ($\aleph_0$-)AEC of all sets of cardinality at least $\lambda$ (ordered with subset) is not $\lambda$-accessible. We will see later (Theorem \[aec-acc\]) that the smallness axiom is, modulo the other axioms, equivalent to the smallness condition in the definition of an accessible category. In fact, one will not loose much by forgetting about logic and simply thinking of an $\mu$-AEC as a concrete category $({\mathcal {K}}, U)$ with concrete $\mu$-directed colimits and where all morphisms are concrete embeddings. Examples of AECs can be found in [@bv-survey-bfo §3.2], and some more examples of $\infty$-AECs can be found in [@mu-aec-jpaa §2]. Some other recently studied examples of interests include the category of flat modules with flat monomorphisms (see Example \[indep-ex\](\[indep-ex-mod\])), a special case of AECs on class of modules of the form ${{}^{\perp}N}$ [@bet] and an example of what module theorists call a Kaplansky classes, see e.g. [@kaplansky-finite]. For now, we note the following general fact: The class of models of a basic ${\mathbb{L}}_{\infty, \lambda}$-theory is a $\lambda$-AEC, when ordered with substructure (see Example \[logical-ex\]). In fact, given any basic ${\mathbb{L}}_{\infty, \lambda}$-theory $T$, if the vocabulary contains a binary relation $R$ and $T$ contains the two basic sentences - $(\forall x \forall y)((R (x, y) \land x = y) \rightarrow \bot)$ - $(\forall x \forall y)(x = y \lor R (x, y))$ then all the morphisms of ${\operatorname{\textbf{Mod}}}(T)$ will be monomorphisms (the two sentences above say that $R$ should be interpreted as the non-equality relation in any model of $T$), and the category ${\operatorname{\textbf{Mod}}}(T)$ will essentially be a $\lambda$-AEC. Closing the loop, we will see later that any accessible category where all morphisms are monomorphisms is equivalent to an $\infty$-AEC (Theorem \[acc-to-aec\]). This also follows from the already mentioned equivalence between accessible categories and categories of models of ${\mathbb{L}}_{\infty, \infty}$ sentences. Two remarks are in order. First, the reader might wonder about the hypothesis that all morphisms are monomorphisms. There is a generalization of the notion of an AEC, called an abstract elementary category which removes this restriction [@beke-rosicky 5.3]. In fact, let us define a $\mu$-abstract elementary category ${\mathbf{K}}$ to be a subcategory of ${\operatorname{\bf Str}}(\tau)$, $\tau = \tau ({\mathbf{K}})$ a $\mu$-ary vocabulary, satisfying all the properties listed in Remark \[aec-categ-prop\], except that the morphisms are only required to be initial (in the sense of [@joy-of-cats 8.6]), but no longer required to be concrete monomorphisms. We will not discuss $\mu$-abstract elementary categories further after this section. Second, and more importantly, the correspondences between ${\mathbb{L}}_{\infty, \infty}$, accessible categories, and $\infty$-AECs do not hold cardinal by cardinal: as we have seen in Remark \[aec-categ-prop\] it is not true that any $\lambda$-AEC is $\lambda$-accessible (and it is similarly not true that a category of models of a basic ${\mathbb{L}}_{\infty, \lambda}$ theory is $\lambda$-accessible). It is however true that a $\lambda$-accessible category is a $\lambda$-abstract elementary category as well as a category of models of a basic ${\mathbb{L}}_{\infty, \lambda}$ theory. In fact, we have, for a fixed regular $\lambda$, the following hierarchy, where each level has more categories than the next: 1. $\lambda$-accessible categories. 2. Categories of models of a basic ${\mathbb{L}}_{\infty, \lambda}$ theory. 3. $\lambda$-abstract elementary categories. 4. Accessible categories with $\lambda$-directed colimits. 5. $\infty$-abstract elementary categories = accessible categories = categories of models of a basic ${\mathbb{L}}_{\infty, \infty}$ sentence. This is relevant, especially because of the big differences between an $\aleph_1$-AEC and an $(\aleph_0$-)AEC: a lot more can be done in the latter setup (see Section \[aec-sec\]). \[ban-ex\] Consider the category ${\operatorname{\bf Ban}}_{mono}$ of Banach spaces with isometries. This is an $\aleph_1$-accessible category with all directed colimits and all morphisms monos. However, these directed colimits are not concrete (they are not unions, but completions of unions), so ${\operatorname{\bf Ban}}_{mono}$ does not seem to obviously be an AEC. One can show [@hilb-mono-v3] that in fact no faithful functor from ${\operatorname{\bf Ban}}_{mono}$ into ${\operatorname{\bf Set}}$ preserves directed colimits. Thus ${\operatorname{\bf Ban}}_{mono}$ is indeed not equivalent to an AEC. It will however be an $\aleph_1$-AEC. Fundamental results =================== We start developing the theory of accessible categories from scratch, proving some basic results and ending by sketching the equivalence of the three frameworks described above. Most of the results of this section are well known and appear in [@adamek-rosicky] or [@mu-aec-jpaa]. First, we recall some more category-theoretic terminology. Monomorphisms are in a sense a very weak generalization of the notion of an injection[^9]. A much stronger one is given by the following definition: suppose we have $A \xrightarrow{i} B \xrightarrow{r} A$ which compose to the identity ($ri = {\operatorname{id}}_A$). Then we call $i$ a section (or split monomorphism) and $r$ a retraction (or split epimorphism). In such a situation, we say that $A$ is a retract of $B$. It is easily checked that sections and retractions are indeed monomorphisms and epimorphisms respectively. The canonical inclusions $A \to A \oplus B$ and projections $A \oplus B \to A$ in the category of $R$-modules are good examples of sections and retractions. In case we also have that $ir = {\operatorname{id}}_B$, then we write $r = i^{-1}$, $i = r^{-1}$, and call them isomorphisms. Note that a retraction which is also a mono is an isomorphism [@joy-of-cats 7.36]. Thus when all morphisms are monos (e.g. in an $\infty$-AEC), any retraction (and thus any section) is an isomorphism. Our first goal will be to show that any object in an accessible category is presentable. This follows from the following result, which essentially says that a small union (colimit) of small objects is small (a diagram is called $\lambda$-small if its indexing category has strictly less than $\lambda$-many objects): \[colimit-size\] For $\lambda$ a regular cardinal, a colimit of a $\lambda$-small diagram of $\lambda$-presentable objects is $\lambda$-presentable. In particular all the objects of an accessible category are presentable. Let $D: I \to {\mathcal {K}}$ be a $\lambda$-small diagram consisting of $\lambda$-presentable objects, with colimit $(D_i \xrightarrow{d_i} A)_{i \in I}$. Let $B$ be a $\lambda$-directed colimit of another diagram $E: J \to {\mathcal {K}}$, and let $A \xrightarrow{f} B$. For each $i \in I$, the map $f d_i$ factors through some $E_{j_i}$, $j_i \in J$, by $\lambda$-presentability of $D_i$. Since $J$ is $\lambda$-directed and $\|I\| < \lambda$, there exists $j \in J$ so that $j \ge j_i$ for all $i \in I$. Then the universal property of the colimit implies that $f$ must factor through $E_j$. The “in particular” part follows from the rest because in a $\lambda$-accessible category, any object is (by the smallness condition) a colimit of a (set-sized) diagram consisting of $\lambda$-presentable objects. We now work toward proving that an accessible category will, for each $\lambda$, have only a set (up to isomorphism) of $\lambda$-presentable objects. We start with the following technical observations: \[retract-rmk\] Work in a category ${\mathcal {K}}$. 1. If $A_1 \xrightarrow{i_1} B \xrightarrow{r_1} A_1$ and $A_2 \xrightarrow{i_2} B \xrightarrow{r_2} A_2$ are such that $r_\ell i_\ell = {\operatorname{id}}_{A_\ell}$ (i.e. they are section/retraction pairs) and $i_1 r_1 = i_2 r_2$, then $A_1$ and $A_2$ are isomorphic (as witnessed by $r_2 i_1$ and $r_1 i_2$). Since $B$ has only a set of endomorphisms (i.e. morphisms from and to itself), there is only a set (up to isomorphism) of retracts of any given object. 2. \[retract-rmk-2\] A retract of a $\lambda$-presentable object is $\lambda$-presentable (straightforward diagram chase from the definition of $\lambda$-presentability). 3. \[retract-rmk-3\] If a $\lambda$-presentable object $A$ is a $\lambda$-directed colimit of a diagram $D: I \to {\mathcal {K}}$, with colimit cocone $(D_i \xrightarrow{d_i} A)_{i \in I}$, then $d_i$ is a retraction for some $i \in I$, so $A$ is a retract of $D_i$ (by $\lambda$-presentability, the identity map on $A$ must factor through one of the components of the diagram: ${\operatorname{id}}_A = d_i f$ for some $i \in I$). We obtain: \[pres-set\] A $\lambda$-accessible category has, up to isomorphism, only a set of $\lambda$-presentable objects. Let $S$ be the set of $\lambda$-presentable objects given by the smallness condition. By Remark \[retract-rmk\], any $\lambda$-presentable objects will be a retract of members of $S$, and there is only a set of such retracts. In order to say more, we try to understand when a $\lambda$-accessible will be $\mu$-accessible for $\mu > \lambda$. This is not a trivial consequence of the definition because a $\lambda$-directed poset may not be $\mu$-directed. In fact, as we will see, it is not true in general that a $\lambda$-accessible category is $\mu$-accessible for all $\mu > \lambda$ (it turns out that the accessibility spectrum – the class of cardinals $\lambda$ such that a category is $\lambda$-accessible – is an interesting measure of the complexity of the category, see Section \[set-size-sec\]). Before looking at counterexamples, let us state a positive result. For a regular cardinal $\mu$, an infinite cardinal $\lambda$ is called $\mu$-closed[^10] if $\theta^{<\mu} < \lambda$ for all $\theta < \lambda$. Note that any uncountable cardinal is $\aleph_0$-closed and in general for any fixed $\mu$ there is a proper class of regular $\mu$-closed cardinal (given any infinite cardinal $\lambda_0$, the cardinal $\left(\lambda_0^{<\mu}\right)^+$ is always $\mu$-closed). \[raise-index\] Let $\mu \le \theta \le \lambda$ be regular cardinals and let ${\mathcal {K}}$ be a $\theta$-accessible category with $\mu$-directed colimits. If $\lambda$ is $\mu$-closed, then ${\mathcal {K}}$ is $\lambda$-accessible. Given an object $A$ of ${\mathcal {K}}$, we have to write $A$ as a $\lambda$-directed colimit of $\lambda$-presentable objects. First, by $\theta$-accessibility we know we can write $A$ as a $\theta$-directed colimit of a diagram $D: I \to {\mathcal {K}}$ of $\theta$-presentable objects. Now let $J$ be the poset of all $\mu$-directed subsets of $I$ of cardinality strictly less than $\lambda$, ordered by containment. For each $I_0 \in J$, we can use that ${\mathcal {K}}$ has $\mu$-directed colimits to look at the colimit ${\operatorname{colim}}(D {\upharpoonright}I_0)$ of the diagram $D$ restricted to $I_0$. This process induces a new diagram $E: J \to {\mathcal {K}}$, where $E_{I_0} = {\operatorname{colim}}(D {\upharpoonright}I_0)$. Notice that by Theorem \[colimit-size\], $E$ consists of $\lambda$-presentable objects. Further, because $\lambda$ is $\mu$-closed, any subset of $I$ of cardinality strictly less than $\lambda$ will be contained in some member of $J$. In particular, $J$ is $\lambda$-directed and ${\operatorname{colim}}E = {\operatorname{colim}}D = A$. Thus any object is a $\lambda$-directed colimits of $\lambda$-presentable objects. The argument also shows that each of these $\lambda$-presentable object is a colimit of $\mu$-presentable objects indexed by a poset of cardinality strictly less than $\lambda$. By Lemma \[pres-set\], there is only a set of $\mu$-presentable objects, hence only a set of such diagrams up to isomorphism, so there is only a set of $\lambda$-presentable objects. Any accessible category is $\lambda$-accessible for a proper class of cardinals $\lambda$. Moreover, any $\theta$-accessible category with directed colimits (in particular any finitely accessible or locally $\theta$-presentable category) is $\lambda$-accessible for all regular cardinals $\lambda > \theta$. For any regular cardinal $\lambda$, an accessible category has only a set, up to isomorphism, of $\lambda$-presentable objects. Let ${\mathcal {K}}$ be an accessible category and fix $\theta \ge \lambda$ regular such that ${\mathcal {K}}$ is $\theta$-accessible. By Lemma \[pres-set\], ${\mathcal {K}}$ has only a set of $\theta$-presentable objects, hence (because $\lambda$-presentable implies $\theta$-presentable) a set of $\lambda$-presentable objects. In the proof of Theorem \[raise-index\], the hypothesis that $\lambda$ is $\mu$-closed was used to show that for any $\mu$-directed poset $I$, any subset of $I$ of cardinality strictly less than $\lambda$ can be completed to a $\mu$-directed subset of cardinality strictly less than $\lambda$. It more generally suffices to assume that for any $\theta < \lambda$, ${\text{cf} ([\theta]^{<\mu})} < \lambda$ (recall that $[\theta]^{<\mu}$ is the set of all subsets of $\theta$ of cardinality strictly less than $\mu$, ordered by containment). Following [@makkai-pare 2.3.1], we will write $\mu {\triangleleft}\lambda$ when this holds (one can check the relation ${\triangleleft}$ indeed gives a partial order on the regular cardinals). Since we always have that $\lambda {\triangleleft}\lambda^+$, it follows, for example, that any $\lambda$-accessible category is also $\lambda^+$-accessible [@adamek-rosicky 2.13(2)]. However, when $\lambda > 2^{<\mu}$, $\lambda$ is $\mu$-closed if and only if $\mu {\triangleleft}\lambda$ (because of the equation $\lambda^{<\mu} = 2^{<\mu} \cdot {\text{cf} ([\lambda]^{<\mu})}$, well known to set theorists, see [@internal-sizes-jpaa 2.5]). Below $2^{<\mu}$, the behavior of ${\text{cf} ([\lambda]^{<\mu})}$ can be somewhat understood through the lens of Shelah’s PCF theory [@shg; @cardarithm]. For $\mu$ a regular uncountable cardinal, let ${\mathcal {K}}$ be the category of $\mu$-directed posets, with morphisms the order-preserving maps. One can check that ${\mathcal {K}}$ is $\mu$-accessible. Let $\lambda > \mu$ be a regular cardinal such that there exists a $\theta < \lambda$ with ${\text{cf} ([\theta]^{<\mu})} \ge \lambda$ (so $\theta$ witnesses that $\mu {\ntriangleleft}\lambda$; take for example $\theta = \beth_{\omega} (\mu)$, $\lambda = \theta^+$, see discussion above). Then ${\mathcal {K}}$ is not $\lambda$-accessible because the poset $[\theta]^{<\mu}$ is $\mu$-directed, hence an object of ${\mathcal {K}}$, but cannot be written as a $\lambda$-directed colimit of $\lambda$-presentable objects. Indeed, suppose it can, and let $D: I \to {\mathcal {K}}$ be the corresponding diagram. The images of the colimit maps form a collection $(Y_i)_{i \in I}$ of $\lambda$-presentable subsets of $[\theta]^{<\mu}$, and the $\lambda$-presentable objects are those of cardinality strictly less than $\lambda$, so each $Y_i$ has cardinality strictly less than $\lambda$. Each $\{\alpha\}$, for $\alpha < \theta$, lies in some $Y_i$, and because $I$ is $\lambda$-directed, there must exist $i^\ast \in I$ such that each $\{\alpha\}$ lies in $Y_{i^\ast}$. Because $Y_{i^\ast}$ is $\mu$-directed, it must in fact be cofinal in $[\theta]^{<\mu}$, but we know that $Y_{i^\ast}$ has cardinality strictly less than $\lambda$, contradiction. Different types of categories have different appropriate notions of functors. For accessible categories, an accessible functors play an important role: A functor $F: {\mathcal {K}}\to {\mathcal {L}}$ is $\lambda$-accessible if both ${\mathcal {K}}$ and ${\mathcal {L}}$ are $\lambda$-accessible categories and $F$ preserves $\lambda$-directed colimits. We say that $F$ is accessible if it is $\lambda$-accessible for some $\lambda$. One reason accessible functors are important is the adjoint functor theorem. In general category theory, Freyd’s adjoint functor theorem [@joy-of-cats 18.12] tells us that a functor between complete categories is (left) adjoint if and only if it preserves limits and satisfies a technical “solution set condition” that can be quite difficult to check. The statement simplifies when looking at accessible categories (recall that a bicomplete accessible category is just a locally presentable category). We will not look into this direction much further, so we omit the proof. A functor between two locally presentable categories is adjoint if and only if it preserves limits and is accessible. It is also true that any left or right adjoint functor between accessible categories is accessible [@adamek-rosicky 2.23]. A question we will be interested in is how a given functor interacts with sizes: for a regular cardinal $\lambda$, we say that a functor $F$ preserve $\lambda$-presentable objects if $F A$ is $\lambda$-presentable whenever $A$ is $\lambda$-presentable. Accessible functors are useful because they preserve certain sizes: \[unif-thm\] For any accessible functor $F$, there exists a proper class of regular cardinals $\lambda$ such that $F$ is $\lambda$-accessible and preserves $\lambda$-presentable objects. Let $F: {\mathcal {K}}\to {\mathcal {L}}$ be a $\mu$-accessible functor. Up to isomorphism, there is only a set of $\mu$-presentable objects in ${\mathcal {K}}$, so there exists a regular cardinal $\mu' \ge \mu$ such that $F A$ is $\mu'$-presentable for every $\mu$-presentable objects $A$. Now let $\lambda \ge \mu'$ be a $\mu$-closed cardinal. Let $A$ be a $\lambda$-presentable object in ${\mathcal {K}}$. By the proof of Theorem \[raise-index\], $A$ can be written as a $\lambda$-directed colimit of objects that are each $\lambda$-small $\mu$-directed colimits of $\mu$-presentable objects. Since $A$ is $\lambda$-presentable, it must be a retract of an object of this diagram (Remark \[retract-rmk\](\[retract-rmk-3\])): hence $A$ is a retract of a $\lambda$-small $\mu$-directed colimits of $\mu$-presentable objects. Since by hypothesis $F$ preserves such colimits and any functor preserves retractions, $F A$ is a retract of a $\lambda$-small $\mu$-directed colimit of $\mu'$-presentable objects in ${\mathcal {L}}$. By Theorem \[colimit-size\], $F A$ is $\lambda$-presentable. \[unif-thm-more\] The proof gives more: if $\mu \le \lambda_0 \le \lambda_1 \le \lambda$ are all regular, $F: {\mathcal {K}}\to {\mathcal {L}}$, ${\mathcal {K}}$ and ${\mathcal {L}}$ are both $\lambda_0$-accessible, $F$ preserves $\mu$-directed colimits, $\lambda$ is $\mu$-closed, and $F A$ is $\lambda_1$-presentable whenever $A$ is $\lambda_0$-presentable, then $F$ preserves $\lambda$-presentable objects. In particular, an accessible functor preserving directed colimits will preserve $\lambda$-presentable objects for all high-enough regular $\lambda$. Dually, it is natural to ask when a functor reflects $\lambda$-presentable objects, i.e. when $F A$ $\lambda$-presentable implies $A$ $\lambda$-presentable. A sufficient condition is for $F$ to reflect split epimorphisms (if $F f$ is a split epi – i.e. a retraction – then $f$ is a split epi), see [@beke-rosicky 3.6]. \[reflect-pres\] If $F: {\mathcal {K}}\to {\mathcal {L}}$ is a $\lambda$-accessible functor reflecting split epimorphisms, then $F$ reflects $\lambda$-presentable objects. Assume that $F A$ is $\lambda$-presentable. Since ${\mathcal {K}}$ is $\lambda$-accessible, $A$ is a the colimit of a $\lambda$-directed diagram $D: I \to {\mathcal {K}}$ consisting of $\lambda$-presentable objects, with colimit cocone $(D_i \xrightarrow{d_i} A)_{i \in I}$. Since $F$ preserves $\lambda$-directed colimits, $F A$ is the colimit of $F D$, so as $F A$ is $\lambda$-presentable, Remark \[retract-rmk\](\[retract-rmk-3\]) implies $F d_i$ is a retraction for some $i \in I$. Because $F$ reflects split epis, $d_i$ is a retraction, so $A$ is a retract of the $\lambda$-presentable object $D_i$, hence is $\lambda$-presentable (Remark \[retract-rmk\](\[retract-rmk-2\])). \[reflect-pres-rmk\] A functor that reflects isomorphism and whose image contains only monomorphisms will automatically reflect split epimorphisms. \[unif-ex\] 1. Let ${\mathcal {K}}$ be a locally presentable category. The binary product functor $F: {\mathcal {K}}\to {\mathcal {K}}$ sending $A$ to the category-theoretic product $A \times A$ is accessible, because it is adjoint to the diagonal functor. Thus by the uniformization theorem, $F$ preserves $\lambda$-presentable objects for a proper class of $\lambda$. That is, $F$ does not make the product “too much bigger”. 2. [@beke-rosicky 3.2(4)] For $\mu$ a regular cardinal, let $F: {\operatorname{\bf Set}}\to {\operatorname{\bf Set}}$ send the set $X$ to the set $[X]^{<\mu}$ of subsets of $X$ of cardinality strictly less than $\mu$. This is an accessible functor but, if $\mu$ is uncountable and $\lambda < \lambda^{<\mu}$, $F$ will not preserve $\lambda^+$-presentable objects. 3. [@beke-rosicky 3.3] Let $F: {\operatorname{\bf Grp}}\to {\operatorname{\bf Ab}}$ be the abelianization functor from the category of group to the category of abelian groups. This is a right adjoint functor, so it preserves colimits, and hence by Remark \[unif-thm-more\] preserves $\lambda$-presentable objects for all regular cardinals $\lambda$. However, if $G$ is a simple group then its abelianization $F (G)$ is the zero group. Thus (since there exists simple groups in all infinite cardinalities) $F$ can make sizes drop. Indeed, one can check that $F$ does not reflect split epimorphisms. 4. If ${\mathcal {K}}$ is locally presentable and $U: {\mathcal {K}}\to {\operatorname{\bf Set}}$ is an accessible functor preserving limits, then by the adjoint functor theorem, $U$ is left adjoint. If we think of $U$ as a forgetful functor, the right adjoint will be the free functor. Since ${\mathcal {K}}$ has directed colimits, $U$ will preserve $\lambda$-presentable objects for all high-enough regular cardinals $\lambda$. 5. \[unif-ex-aec\] Let ${\mathbf{K}}$ be a $\mu$-AEC, and let $U: {\mathbf{K}}\to {\operatorname{\bf Set}}$ be the forgetful universe functor. By definition of a $\mu$-AEC, $U$ preserves $\mu$-directed colimits, hence is accessible. The abstract class axiom ensures that $U$ reflects isomorphisms. Moreover, any morphism in the image of $U$ must be a monomorphism so by Remark \[reflect-pres-rmk\], $U$ reflects split epis. Finally, it is easy to check that any ${\text{LS}}({\mathbf{K}})^+$-presentable object in ${\mathbf{K}}$ will have cardinality at most ${\text{LS}}({\mathbf{K}})$ (write the object as the ${\text{LS}}({\mathbf{K}})^+$-directed colimit of its subobject of cardinality at most ${\text{LS}}({\mathbf{K}})$). By the uniformization theorem, if $\lambda > {\text{LS}}({\mathbf{K}})$ is a regular $\mu$-closed cardinal, then $F$ preserves and reflects $\lambda$-presentable objects. This means that $A$ is $\lambda$-presentable in ${\mathbf{K}}$ if and only if it its universe has cardinality strictly less than $\lambda$. In particular (taking $\mu = \aleph_0$) in an AEC, category-theoretic sizes correspond exactly to cardinalities (above ${\text{LS}}({\mathbf{K}})$). See [@internal-sizes-jpaa §4] for more on such results. 6. In the $\aleph_1$-AEC ${\mathbf{K}}$ of Banach spaces (with subspace inclusions), the universe functor $U$ does not preserve $\aleph_1$-presentable objects: an $\aleph_1$-presentable Banach space will not have countable cardinality. This is because $U$ does not preserve $\aleph_0$-directed colimits (even though ${\mathbf{K}}$ does have those colimits, they are not concrete: one cannot compute them by taking unions). Let’s use the uniformization theorem to better understand the relationship between the smallness axiom in $\infty$-AEC (Definition \[infty-aec-def\]) and the smallness condition in the definition of an accessible category (Definition \[acc-def\]). \[aec-acc\] Assume that ${\mathbf{K}}$ satisfies all the axioms of a $\mu$-AEC (Definition \[infty-aec-def\]), except perhaps for the LST smallness axiom. The following are equivalent: 1. \[aec-acc-1\] ${\mathbf{K}}$ is accessible. 2. \[aec-acc-2\] ${\mathbf{K}}$ satisfies the LST smallness axiom, and hence is a $\mu$-AEC. We have seen already (Remark \[aec-categ-prop\]) that (\[aec-acc-2\]) implies (\[aec-acc-1\]). Assume now that (\[aec-acc-1\]) holds: ${\mathbf{K}}$ is accessible. Then the universe functor $U: {\mathbf{K}}\to {\operatorname{\bf Set}}$ is accessible (preservation of $\mu$-directed colimits is just because they are computed the same way in ${\mathbf{K}}$ and ${\operatorname{\bf Set}}$ – this is what the chain axiom says). By the uniformization theorem, we can pick a regular cardinal $\theta \ge \mu + \|\tau ({\mathbf{K}})\|$ such that $U$ is $\theta$-accessible and preserves $\theta$-presentable objects. We will show that ${\text{LS}}({\mathbf{K}}) \le \theta^{<\mu}$. Let $M \in {\mathbf{K}}$, and let $A \subseteq U M$. Set $\lambda := \left(\left(\theta + \|A\|\right)^{<\mu}\right)^+$. By Remark \[unif-thm-more\], $U$ is $\lambda$-accessible and preserves $\lambda$-presentable objects. In particular, ${\mathbf{K}}$ is $\lambda$-accessible so we can write $M$ as a $\lambda$-directed colimit (union) of $\lambda$-presentable objects: $M = \bigcup_{i \in I} M_i$. Now $\|A\| < \lambda$, so by $\lambda$-directedness there must exist $i \in I$ such that $A \subseteq U M_i$. Since $M_i$ is $\lambda$-presentable in ${\mathbf{K}}$, $U M_i = M_i$ must be $\lambda$-presentable in ${\operatorname{\bf Set}}$, hence it must have cardinality strictly less than $\lambda$ (see Example \[pres-ex\](\[pres-ex-set\])), as desired. We can now sketch part of the proof that any accessible category can be presented as the class of models of an ${\mathbb{L}}_{\infty, \infty}$ sentence. Let us make such a statement precise first: recall that two categories ${\mathcal {K}}$ and ${\mathcal {L}}$ are equivalent if there exists a functor $F: {\mathcal {K}}\to {\mathcal {L}}$ that is full (surjective on morphisms), faithful (injective on morphisms), and essentially surjective on objects (any object of ${\mathcal {L}}$ is isomorphic to an object in the image of $F$). Essential surjectivity (rather than surjectivity) on objects makes equivalence of categories weaker than isomorphism of categories, but it is typically the former notion of “being the same” that is used in category theory[^11]. For $\infty \ge \lambda \ge \mu$, let us then define a category to be $(\lambda, \mu)$-elementary if it is equivalent to a category of the form ${\operatorname{\textbf{Mod}}}(T)$, for $T$ a theory in ${\mathbb{L}}_{\lambda, \mu}$. We will see that any $\lambda$-accessible category is $(\infty, \lambda)$-elementary. The proof proceeds in two steps. First, the category is embedded into a category of structures, and second this category of structures is axiomatized. We will only look at the first step (which is sufficient if one only cares about $\lambda$-AECs). The main idea is to represent each object using its Hom functor. The second step will be proven in the very special case of universal classes. \[acc-to-aec\] If ${\mathcal {K}}$ is a $\lambda$-accessible category, then there is a (finitary) vocabulary $\tau$ and an embedding[^12] $E: {\mathcal {K}}\rightarrow {\operatorname{\bf Str}}(\tau)$ which is full and preserves $\lambda$-directed colimits. In particular, if in addition all the morphisms of ${\mathcal {K}}$ are monomorphisms, ${\mathcal {K}}$ is equivalent to a $\lambda$-AEC. Let ${\mathcal {K}}_\lambda$ be a small full subcategory of ${\mathcal {K}}$ containing representatives of each $\lambda$-presentable object. For each $M \in {\mathcal {K}}_\lambda$, let $S_M$ be a unary relation symbol and for each morphism $f$ in ${\mathcal {K}}_\lambda$, let $\underline{f}$ be a binary function symbol. The vocabulary $\tau$ will consist of all such $S_M$ and $\underline{f}$. Now map each $M \in {\mathcal {K}}$ to the following $\tau$-structure $E M$: 1. Its universe are the morphisms $g: M_0 \rightarrow M$, where $M_0 \in {\mathcal {K}}_\lambda$. 2. For each $M_0 \in {\mathcal {K}}_\lambda$, $S_{M_0}^{EM}$ is the set of morphisms $g: M_0 \rightarrow M$. 3. For each morphism $f: M_0 \rightarrow M_1$ of ${\mathcal {K}}_\lambda$, and each $g: M_1 \rightarrow M$, $\underline{f}^{E M} (g) = g f$. When $g \notin S_{M_1}^{EM}$, just let $\underline{f}^{EM} (g) = g$. Map each morphism $f: M \rightarrow N$ to the function $\bar{f} : E M \rightarrow E N$ given by $\bar{f} (g) = fg$. Essentially, $E$ is the functor $E (M) = {\operatorname{Hom}}(-, M)$ from ${\mathcal {K}}$ to ${{}^{{\mathcal {K}}_\lambda^{op}}{\operatorname{\bf Set}}}$. The Yoneda embedding lemma tells us that $E {\upharpoonright}{\mathcal {K}}_\lambda$ is full and faithful [@joy-of-cats 6.20]. The definition of a $\lambda$-presentable object also ensures that $E$ preserves $\lambda$-directed colimits. By writing any object as a $\lambda$-directed colimit of $\lambda$-presentable objects, we get that $E$ is full and faithful. To see the “in particular” part, consider the smallest isomorphism-closed subcategory ${\mathcal {L}}$ of ${\operatorname{\bf Str}}(\tau)$ that contains $E[{\mathcal {K}}]$. This category is equivalent to ${\mathcal {L}}$, satisfies the abstract class axiom, has concrete $\lambda$-directed colimits, and (trivially, because the morphisms are homomorphisms) satisfies the coherence axiom. Since ${\mathcal {K}}$ is accessible and $E$ is full and faithful, $E[{\mathcal {K}}]$ also is accessible, and hence ${\mathcal {L}}$ is accessible. Now apply Theorem \[aec-acc\]. \[acc-to-log\] Any $\lambda$-accessible category is $(\infty, \lambda)$-elementary. In particular, a category is accessible if and only if it is $(\infty, \infty)$-elementary. We first apply Theorem \[acc-to-aec\] to reduce the problem to axiomatizing a class of structures, and then axiomatize this class (we will not explain how here). \[acc-to-log-ex\] 1. For $\lambda$ a regular cardinal, we have seen that the AEC of all sets of cardinality at least $\lambda$ is not $\lambda$-accessible. In fact, this AEC is not even $(\infty, \lambda)$-elementary but the proof is not trivial, see [@henry-aec-uncountable-jsl]. 2. \[toy-quasi-ex\] [@logic-intersection-bpas §4] Consider the following AEC, ${\mathbf{K}}$, sometimes called the toy quasiminimal class: the vocabulary contains a single binary relation. The objects of ${\mathbf{K}}$ are the equivalence relations with countably infinite classes. The ordering says that equivalence classes do not grow. It is not difficult to see that ${\mathbf{K}}$ is finitely accessible: an object is $\lambda$-presentable if and only if the relation has fewer than $\lambda$-many classes, and every equivalence relation is the directed colimits of its restrictions to finitely-many classes. By Corollary \[acc-to-log\], ${\mathbf{K}}$ is $(\infty, \omega)$-elementary, hence is equivalent to the category of models of an ${\mathbb{L}}_{\infty, \omega}$-theory. This latter category is obtained, roughly speaking, by collapsing each equivalence class to a point. However it is well known that ${\mathbf{K}}$ itself is not the category of models of a basic ${\mathbb{L}}_{\infty, \omega}$ theory. In fact, it is not finitary in the sense of Hyttinen-Kesälä [@finitary-aec]: roughly, one cannot figure out whether a map is a morphism by checking finitely-many points at a time (like we would be able to do for homomorphisms in a finitary vocabulary). This shows that the concept of being a finitary AEC is not invariant under equivalence of categories. 3. \[inter-ex\] An $\infty$-AEC ${\mathbf{K}}$ has intersections if for any $M \in {\mathbf{K}}$, and any non-empty collection $\{M_i : i \in I\}$ of ${{\le_{{\mathbf{K}}}}}$-substructures of $M$, $\bigcap_{i \in I} M_i$ induces a ${{\le_{{\mathbf{K}}}}}$-substructure of $M$. Generalizing the previous example, one can show that any $\lambda$-AEC with intersections is a $\lambda$-accessible category [@multipres-pams 3.3], hence $(\infty, \lambda)$-elementary. In fact, as the definition makes apparent, $\lambda$-AECs with intersections correspond exactly to the $\lambda$-accessible categories with wide pullbacks (and all morphisms monos), see [@multipres-pams 5.7]. This gives a clear sense in which this class of AECs is natural, and less complex than general AECs. See [@ap-universal-apal §2], [@logic-intersection-bpas], or [@internal-sizes-jpaa §5] for more on AECs with intersections. As a consolation prize for not proving the axiomatizability part of Corollary \[acc-to-log\], let us prove it for a simpler framework: that of universal classes. \[univ-def\] For $\mu$ a regular cardinal, a $\mu$-universal class is a $\mu$-AEC ${\mathbf{K}}$ such that ${{\le_{{\mathbf{K}}}}}$ is just the $\tau ({\mathbf{K}})$-substructure relation, and moreover if $M \in {\mathbf{K}}$ and $M_0 \subseteq M$, then $M_0 \in {\mathbf{K}}$. Said another way, a $\mu$-universal class is simply a class of structures in a fixed $\mu$-ary vocabulary that is closed under isomorphisms, substructures, and $\mu$-directed unions (we identify the class with the $\mu$-AEC). When $\mu = \aleph_0$, we omit it. \[tarski\] Let $\mu$ be a regular cardinal and let $K$ be a class of structures in a fixed $\mu$-ary vocabulary $\tau$. The following are equivalent: 1. \[tarski-1\] $K$ is the class of models of a universal ${\mathbb{L}}_{\infty, \mu}$ theory (i.e. a theory where each sentence is of the form $(\forall {\bar{x}}) \psi$, with $\psi$ quantifier-free). 2. \[tarski-2\] $K$ is a $\mu$-universal class. The implication (\[tarski-1\]) implies (\[tarski-2\]) is easy to check. Assume now that (\[tarski-2\]) holds. Call $M \in K$ $\mu$-generated if $M = {\operatorname{cl}}^M (A)$, for some $A \in [U M]^{<\mu}$ (here, ${\operatorname{cl}}^M$ denotes the closure of $A$ under the functions of $M$ – note that such a closure is always a substructure of $M$, hence in $K$ by definition of a $\mu$-universal class). Note that $\mu$-generated is equivalent to $\mu$-presentable [@internal-sizes-jpaa 5.7], but this will not be needed. Now by listing all the isomorphism types of the $\mu$-generated models in $K$, form a quantifier-free formula $\phi ({\bar{x}})$ of ${\mathbb{L}}_{\infty, \mu}$ such that $N \models \phi[{\bar{a}}]$ if and only if ${\operatorname{cl}}^N ({\operatorname{ran}}({\bar{a}}))$ is in $K$. The formula $\psi := (\forall {\bar{x}}) \phi ({\bar{x}})$ axiomatizes $K$. Indeed, if $N \in K$ then $N \models \psi$. Conversely, if $N \models \psi$ then $\{{\operatorname{cl}}^N ({\operatorname{ran}}({\bar{a}})) \mid {\bar{a}}\in {{}^{<\mu}U N}\}$ is a $\mu$-directed system in $K$ whose union is $N$, hence $N$ is in $K$. This proves (\[tarski-1\]). There is also a category-theoretic characterization of $\mu$-universal classes: they are the $\mu$-accessible categories with all morphisms monos and all connected limits [@multipres-pams 5.9]. \[univ-ex\] 1. The class of all vector spaces over a fixed field $F$ is a universal class. 2. The class of locally finite groups is a universal class (it is a good exercise to try to write down the universal sentence of \[tarski\](\[tarski-1\]) for this case). 3. \[univ-skol-ex\] If $T$ is an ${\mathbb{L}}_{\infty, \infty}$ theory, we can “Skolemize it” (add functions to pick witnesses of existential sentences) to get a universal class in an expanded vocabulary, whose restriction to the original vocabulary is the class of models of $T$. 4. Any $\mu$-universal class is a $\mu$-AEC with intersections (see Example \[acc-to-log-ex\](\[inter-ex\])). 5. The AEC of algebraically closed fields (with subfield) is not universal: the rationals form a non-algebraically closed subfield of the complex numbers. This AEC does have intersections, however. 6. The class of all linear orders, or the class of all graphs, is a universal class. Universal classes are in a sense the simplest type of AECs. In fact, a key result is Shelah’s presentation theorem [@shelahaecbook I.1.9], which says that, given an AEC ${\mathbf{K}}$, we can find a universal class[^13] ${\mathbf{K}}^\ast$ in an expanded vocabulary whose reduct (i.e. restriction to the original vocabulary) to ${\mathbf{K}}$ is ${\mathbf{K}}$. In fact, the reduct gives a functor from ${\mathbf{K}}^\ast$ to ${\mathbf{K}}$ which is faithful (injective on morphisms), surjective, and preserves directed colimits. This is the motivation for the following generalization of Shelah’s presentation theorem to accessible categories with all morphisms monos. \[shelah-pres\] If ${\mathcal {L}}$ is an accessible category with $\mu$-directed colimits and all morphisms monos, then there exists a $\mu$-universal class ${\mathbf{K}}$ and a faithful essentially surjective functor $H: {\mathbf{K}}\to {\mathcal {L}}$ preserving $\mu$-directed colimits. First, let ${\mathcal {L}}^\ast$ be the category obtained by taking free $\mu$-directed colimits of the $\mu$-presentable objects of ${\mathcal {L}}$. This is a $\mu$-accessible category [@adamek-rosicky 2.26], and the natural “projection” functor $F: {\mathcal {L}}^\ast \to {\mathcal {L}}$ is essentially surjective and preserves directed colimits. Moreover, $F$ is also faithful: given two distinct morphisms $f, g: A \to B$ in ${\mathcal {L}}^\ast$, separate them on $\lambda$-presentable objects by finding $f_0, g_0: A_0 \to B_0$ distinct with $A_0$ and $B_0$ $\lambda$-presentable and maps $u: A_0 \to A$, $v: B_0 \to B$ such that $fu = v f_0$, $g u = v g_0$: $$\xymatrix@=3pc{ A \ar@<-.5ex>[r]_f \ar@<.5ex>[r]^g & B \\ A_0 \ar[u]_{u} \ar@<-.5ex>[r]_{f_0} \ar@<.5ex>[r]^{g_0} & B_0 \ar[u]_v \\ }$$ Since $F$ is faithful on $\lambda$-presentable objects, $F f_0 \neq F g_0$. Moreover, $F v$ is a mono, so it follows that $F f \neq F g$. By Corollary \[acc-to-log\], ${\mathcal {L}}^\ast$ is $(\infty, \mu)$-elementary, hence without loss of generality is the category of models of a basic ${\mathbb{L}}_{\infty, \mu}$ theory $T$. Skolemizing $T$ (see Example \[univ-ex\](\[univ-skol-ex\])), one obtains a $\mu$-universal class ${\mathbf{K}}$ so that the reduct functor $G: {\mathbf{K}}\to {\mathcal {L}}^\ast$ is faithful, essentially surjective, and preserves $\mu$-directed colimits. Let $H := F \circ G$. \[pres-rmk\] 1. Any $\mu$-AEC is an accessible category with $\mu$-directed colimits and all morphisms monos (Remark \[aec-categ-prop\]), so when $\mu = \aleph_0$ we in particular recover a version of Shelah’s original presentation theorem. Note that, when $\mu > \aleph_0$, it is not so easy to imitate Shelah’s proof, see [@internal-improved-v3-toappear §3] for a discussion. 2. \[pres-morley-rmk\] In the case $\mu = \aleph_0$, a classical result of model theory, Morley’s omitting type theorem tells us in particular (in categorical language, see [@makkai-pare 3.4.1]) that for any $(\infty, \omega)$-elementary category ${\mathcal {K}}$ (hence by Tarski’s presentation theorem in particular for a universal class), there is a faithful functor ${\operatorname{\bf Lin}}\to {\mathbf{K}}$ preserving directed colimits (where ${\operatorname{\bf Lin}}$ is the category of linear orders and order-preserving maps). The functor constructed by the model-theoretic process is called the Ehrenfeucht-Mostowski (EM) functor, and its images are called Ehrenfeucht-Mostowski models. They can be described as the models generated by order-indiscernibles satisfying a fixed collection of quantifier-free types (an EM blueprint). In fact, from any faithful functor ${\operatorname{\bf Lin}}\to {\mathcal {K}}$ preserving directed colimits, one can recover such a blueprint [@boney-er-v2 5.5]. Combining Morley’s theorem with Tarski’s presentation theorem and the categorical version of Shelah’s presentation theorem, we obtain that for any large accessible category ${\mathcal {L}}$ with directed colimits and all morphisms monos there is a faithful functor ${\operatorname{\bf Lin}}\to {\mathcal {L}}$ preserving directed colimits. Notice that this works not only for AECs but also for continuous classes (such as Banach spaces). Set-theoretic topics {#set-sec} ==================== Cardinality vs presentability {#set-size-sec} ----------------------------- Is the behavior of presentability, the notion of size defined in Definition \[pres-def\], similar to the behavior of cardinality in concrete classes? There are at least two questions one can consider in this direction. First, in the category of sets, a set is $\lambda$-presentable if and only if it has cardinality strictly less than $\lambda$. In particular, the presentability rank of a set is a successor. This happens in all the examples listed in \[pres-ex\]. Thus one can ask: in an arbitrary accessible category, are high-enough presentability ranks always successors? Second, the Löwenheim-Skolem theorem for first-order logic says that any theory with an infinite model has models of all high-enough cardinalities. Is there a similar version for accessible categories? Since we do not have the compactness theorem, let us restrict ourselves to large categories, and let us consider only eventual behavior. Cardinality here is again not the right notion (consider Example \[pres-ex\](\[pres-ex-hilb\]): there are no Hilbert spaces in certain cardinalities), but we can still ask: does every large accessible category have objects of each high-enough size? (see Definition \[size-def\]). Following [@beke-rosicky 2.4], let us call an accessible category ${\mathcal {K}}$ LS-accessible if there exists a cardinal $\theta$ such that ${\mathcal {K}}$ has an object of size $\lambda$ for all cardinals $\lambda \ge \theta$. We are then asking whether every large accessible category is LS-accessible. Both questions turn out to be connected to the accessibility spectrum: the set of regular cardinals $\lambda$ such that a given category is $\lambda$-accessible. Regarding the first question, we have: \[succ-rank-thm\] Let ${\mathcal {K}}$ be a category and let $\lambda$ be a weakly inaccessible cardinal[^14]. If ${\mathcal {K}}$ is $\mu$-accessible for unboundedly-many regular $\mu < \lambda$, then no object of ${\mathcal {K}}$ has presentability rank $\lambda$. We show more generally that every object of ${\mathcal {K}}$ is a $\lambda$-directed colimits of objects of presentability rank strictly less than $\lambda$. This will imply the result by Remark \[retract-rmk\]. Let $S$ be the set of regular cardinals $\mu < \lambda$ such that ${\mathcal {K}}$ is $\mu$-accessible. Given an object $A$, we know that for each $\mu \in S$, we can write $A$ as a $\mu$-directed colimit of a diagram $D^\mu : I^\mu \to {\mathcal {K}}$ consisting of $\mu$-presentable objects. One can then put together the $D^\mu$’s to obtain a new $\lambda$-directed diagram with colimit $A$ and consisting of objects of presentability rank strictly less than $\lambda$. \[succ-cor\] 1. \[succ-cor-1\] [@beke-rosicky 4.2] In an accessible category with directed colimits, all high-enough presentability ranks are successors. 2. \[suc-cor-2\] [@internal-sizes-jpaa 3.11] If the singular cardinal hypothesis (SCH) holds[^15], then in every accessible category, all high-enough presentability ranks are successors. <!-- --> 1. By Theorem \[raise-index\], ${\mathcal {K}}$ is $\mu$-accessible for all high-enough regular $\mu$, so the result is immediate from Theorem \[succ-rank-thm\]. 2. Assume that ${\mathcal {K}}$ is $\mu$-accessible, and let $\lambda > 2^{<\mu}$ be a weakly inaccessible cardinal. Let $\theta_0 < \lambda$ be infinite and let $\theta := \left(\theta_0^{<\mu}\right)^+$. By the SCH hypothesis, $\theta < \lambda$ (see [@jechbook 5.22]) and by Theorem \[raise-index\], ${\mathcal {K}}$ is $\theta$-accessible. We have shown that ${\mathcal {K}}$ is accessible in unboundedly-many regular cardinals below $\lambda$, hence by Theorem \[succ-rank-thm\] ${\mathcal {K}}$ has no objects of presentability rank $\lambda$. Since SCH holds above a strongly compact cardinal [@jechbook 20.8], we can replace the SCH assumption by a large cardinal axiom. The rough idea here is that if an accessible category is of “low-enough” complexity (as measured by its accessibility spectrum), then automatically its presentability ranks will have good behavior. Often additional cardinal arithmetic hypotheses can lower the complexity of the accessibility spectrum, to the point that we can prove results about general accessible categories. Let us now give an example (without proof) of this behavior for the second question above, whether every large accessible category has objects of all high-enough sizes (see Definition \[size-def\]): \[ls-acc-thm\] Assume SCH. If ${\mathcal {K}}$ is a large accessible category with all morphisms monos, and $\lambda$ is a high-enough successor cardinal such that ${\mathcal {K}}$ is $\lambda$-accessible, then ${\mathcal {K}}$ has an object of presentability rank $\lambda$. \[ls-sch\] Assume SCH. 1. \[ls-sch-1\] Any large accessible category with directed colimits and all morphisms monos is LS-accessible. 2. \[ls-sch-2\] [@internal-improved-v3-toappear 7.12] Any large accessible category has objects in all sizes of high-enough cofinality. <!-- --> 1. Immediate from Theorems \[raise-index\] and \[ls-acc-thm\]. 2. This can be obtained from Theorem \[ls-acc-thm\] by combining careful use of the proofs of the uniformization theorem and Theorem \[reflect-pres\], together with the (nontrivial) fact that the inclusion ${\mathcal {K}}_{mono} \to {\mathcal {K}}$ is an accessible functor [@indep-categ-advances 6.2]. Corollary \[ls-sch\](\[ls-sch-1\]) is due to Lieberman and Rosický, and can be proven without SCH [@ct-accessible-jsl 2.7]: let $E: {\operatorname{\bf Lin}}\to {\mathcal {K}}$ be faithful and preserving directed colimits (see Remark \[pres-rmk\](\[pres-morley-rmk\])). By the uniformization theorem, $E$ preserves $\lambda$-presentable objects for all high-enough regular $\lambda$, and by Theorem \[reflect-pres\], $E$ also reflects them (faithful functors reflect epimorphisms, and epimorphisms in ${\operatorname{\bf Lin}}$ are isomorphisms). Thus for $\lambda$ a big-enough cardinal and $I$ a linear order of cardinality $\lambda$, $E (I)$ will have size exactly $\lambda$. Large cardinals and images of accessible functors {#set-func-sec} ------------------------------------------------- Consider the functor $F: {\operatorname{\bf Set}}\to {\operatorname{\bf Ab}}$ that associates to each set the free abelian group on that set. It is easily checked that $F$ is an accessible functor but, as noticed before (Example \[acc-ex\](\[free-ex\])) the question of whether the image of $F$ (i.e. the full subcategory of ${\operatorname{\bf Ab}}$ consisting of free abelian groups) is accessible is set-theoretic. One can ask this question generally: when is the image of an accessible functor accessible? The problem of course lies in proving existence of sufficiently directed colimits for this image. For technical reasons, we will close the image under subobjects: in the example of the free abelian group functor, subgroup of free groups are free, so the image is already closed under subobjects the only challenge is to check that a sufficiently directed diagram consisting of free groups has a cocone. More precisely, define the powerful image of an accessible functor $F: {\mathcal {K}}\to {\mathcal {L}}$ to be the smallest full subcategory $P$ of ${\mathcal {L}}$ that contains $F[{\mathcal {K}}]$ and is closed under subobjects (i.e. if $A \to B$ is a monomorphism and $B \in P$, then $A \in P$). The question becomes: when is the powerful image of an accessible functor accessible? The following result is due to Makkai and Paré: \[mp-thm\] If there is a proper class of strongly compact cardinals[^16], then the powerful image of any accessible functor is accessible. Fix a $\lambda$-accessible functor $F$ with powerful image $P$. By the uniformization theorem, we can assume without loss of generality that $F$ preserves $\lambda$-presentable objects. Let $\kappa > \lambda$ be strongly compact. We show that $P$ has $\kappa$-directed colimits. Using ideas around Corollary \[acc-to-log\], we can reduce the problem of finding a cocone to the consistency of a certain ${\mathbb{L}}_{\kappa, \kappa}$-theory. This theory can be shown to have all subsets of size strictly less than $\kappa$ consistent hence, by the compactness theorem for ${\mathbb{L}}_{\kappa, \kappa}$, to be consistent. The large cardinal assumption can be slightly weakened to a proper class of almost strongly compact cardinals [@btr-almost-compact-tams] but this is best possible [@lc-tame-pams]: the powerful image of every accessible functor is accessible if and only if there is a proper class of almost strongly compact cardinals. However, weaker statements can be proven from weaker large cardinal axioms (e.g. measurable or weakly compacts), see [@lieberman-almost-measurable-v4; @bl-powerful-images-v4]. We will even mention a ZFC theorem about images of accessible functors (Theorem \[sing-compact\]). Questions about the image of accessible functors can be used to study various kinds of compactness. One example is tameness in AECs ([@ct-accessible-jsl 5.2]). As a simpler example, we consider the following property: \[ap-def\] An object $A$ in a category is an amalgamation base if any span $C \leftarrow A \rightarrow B$ can be completed (not necessarily canonically) to a commutative square: $$\xymatrix@=3pc{ C \ar@{.>}[r] & D \\ A \ar[r] \ar[u] & B \ar@{.>}[u] }$$ A category has the amalgamation property (or has amalgamation) if every object is an amalgamation base. A question one might ask is whether amalgamation up to a certain level (e.g. for all $\lambda$-presentable objects for some big $\lambda$) implies amalgamation the rest of the way. Large cardinals imply a simple answer (earlier results used model-theoretic techniques, see e.g. [@baldwin-boney]): If ${\mathcal {K}}$ is a $\lambda$-accessible category, $\kappa > \lambda$ is strongly compact, and the full subcategory of ${\mathcal {K}}$ consisting of $\kappa$-presentable objects has amalgamation, then ${\mathcal {K}}$ has amalgamation. Let ${\mathcal {K}}^{\square}$ be the category of commutative squares in ${\mathcal {K}}$, and let ${\mathcal {K}}^{sp}$ be the category of spans $B \leftarrow A \rightarrow C$, with the morphisms in each category defined as expected. Consider the functor $F: {\mathcal {K}}^\square \to {\mathcal {K}}^{sp}$ that “forgets” the top corner of each square. One can check that this is a $\lambda$-accessible functor preserving $\lambda$-presentable objects, and moreover its image $P$ is closed under subobjects, hence is equal to its powerful image. This image is, by definition of $F$, the category of spans that can be amalgamated. By Theorem \[mp-thm\], $P$ is $\kappa$-accessible. Now any span $S$ of ${\mathcal {K}}^{sp}$ is a $\kappa$-directed colimit of $\kappa$-presentable objects, and each of these objects can by assumption be amalgamated, hence are in $P$. Because $P$ has $\kappa$-directed colimits, $S$ is also in $P$. This shows that any span can be amalgamated, hence that ${\mathcal {K}}$ has amalgamation. As a final application, we mention how Shelah’s singular compactness theorem [@shelah-singular] can be restated as a theorem about image of accessible functors. One of the most well known statement of the singular compactness theorem is that an abelian group of singular cardinality all of whose subgroups of lower cardinality are free is itself free. The proof can be axiomatized to apply to other kinds of objects than groups: modules, well-coloring in graphs, transversals, etc. In [@cellular-singular-jpaa], Beke and Rosický state the following general form: \[sing-compact\] Let $F: {\mathcal {K}}\to {\mathcal {L}}$ be an $\aleph_0$-accessible functor. Assume that $F$-structures extend along morphisms. Let $A \in {\mathcal {L}}$ be an object whose size is a singular cardinal. If all subobjects of $A$ of lower size are in the image of $F$, then $A$ itself is in the image of $F$. Here, we say that $F$-structure extend along morphisms if for any ${\mathcal {K}}$-morphism $g: A \to B$, any object $A'$ of ${\mathcal {K}}$, and any isomorphism $i: F A' \cong F A$, there exists $f: A' \to B'$ and an isomorphism $j: F B' \cong F B$ such that the following diagram commutes: $$\xymatrix@=3pc{ F A' \ar[d]_i\ar@{.>}[r]_{F f} & F B' \ar@{.>}[d]_{j} \\ F A \ar[r]_{F g} & F B }$$ This can be thought of as a generalization of the Steinitz exchange property in vector spaces and fields. Let $F: {\operatorname{\bf Set}}_{mono} \to {\operatorname{\bf Ab}}$ be the restriction of the free abelian group functor to ${\operatorname{\bf Set}}_{mono}$. This is an $\aleph_0$-accessible functor and $F$-structures extend along morphisms (we can rename to the case where $g$ is the inclusion of $A$ into a superset $B$; then if $i: F A' \cong F A$, we know that both free groups have the same number of generators, and one can add $\|B \backslash A\|$-many elements to $A'$ to obtain a superset $B'$ so that $j$ extends $i$ to an isomorphism of $FB'$ with $FB$). Thus we recover Shelah’s original application of the singular compactness theorem: if $A$ is an abelian group of singular cardinality, all of whose subobjects of lower cardinality are free, then all these subobjects lie in the image of $F$, hence by Theorem \[sing-compact\] $A$ must also be in this image, i.e. be free. Vopěnka’s principle ------------------- is a large cardinal axiom, whose consistency strength is between huge and extendible. A thorough introduction to the category-theoretic implications of Vopěnka’s principle is in [@adamek-rosicky §6]. Students of set theory may be familiar with Vopěnka’s principle as the statement that in any proper class of structures in the same vocabulary, there exists an elementary embedding between two distinct members of the class (see [@jechbook p. 380]). Another logical characterization of Vopěnka’s principle, due to Stavi, is that every logic has a Löwenheim-Skolem-Tarski number (see [@magidor-van2011 Theorem 6]). A purely combinatorial characterization of Vopěnka’s principle — and the one that Vopěnka first stated — is that there are no rigid proper classes of graphs. Stated in category-theoretic terms, no large full subcategory of the category of graphs is rigid, where a category is rigid if all of its morphisms are identities. In fact, there is a more general category-theoretic formulation of Vopěnka’s principle: any locally presentable category can be fully embedded into the category of graphs [@adamek-rosicky 2.65]. Thus Vopěnka’s principle is equivalent to the statement that no locally presentable category has a large rigid full subcategory. Even more strongly (because by Theorem \[acc-to-aec\] any accessible category can be fully embedded into ${\operatorname{\bf Str}}(\tau)$, a locally presentable category), no large full subcategory of an accessible category can be rigid. Thus if $C$ is a proper class of objects from an accessible category, Vopěnka’s principle tells us there must be a morphism between two distinct objects of $C$ (the first logical version mentioned above is the special case of the accessible category of $\tau$-structures with elementary embeddings). The following criteria makes it easy to check that a category is accessible [@adamek-rosicky 6.9, 6.17]. It can be seen as a category-theoretic version of the fact that every logic has a Löwenheim-Skolem-Tarski number. Assume Vopěnka’s principle. If ${\mathcal {L}}$ is a full subcategory of an accessible category ${\mathcal {K}}$ and ${\mathcal {L}}$ has $\mu$-directed colimits for some $\mu$, then the inclusion ${\mathcal {L}}\to {\mathcal {K}}$ preserves $\lambda$-directed colimits for some $\lambda$ and ${\mathcal {L}}$ is accessible. This result has been used to prove existence of certain homotopy localizations [@localization-vopenka; @left-det-rt]. There is also an accessible functor characterization of Vopěnka’s principle: every subfunctor of an accessible functor is accessible [@adamek-rosicky 6.31]. Weak diamond and amalgamation bases ----------------------------------- In complicated categories where pushouts are not available (e.g. when all morphisms are monos), the amalgamation property can play a key role. In this subsection, we look at a set-theoretic way to obtain it in a general class of concrete categories. More generally, we will look at amalgamation bases: objects $A$ such that any span with base $A$ can be completed to a commutative square (see Definition \[ap-def\]). We will study them in AECs (Definition \[infty-aec-def\]), although several of the concepts are category-theoretic and the result can be generalized to $\mu$-AECs [@mu-aec-jpaa 6.12], or other kinds of concrete categories [@multidim-v2 5.8]. The key set-theoretic component is the weak diamond, a combinatorial principle introduced by Devlin and Shelah [@dvsh65]. We will use it in the following form (see [@dvsh65 6.1,7.1]): \[wd-thm\] Let $\lambda$ be an infinite cardinal and let ${\langle f_\eta : \lambda \to \lambda \mid \eta \in {{}^{\lambda}2} \rangle}$ be a sequence of functions. If there exists $\theta < \lambda$ such that $2^{\theta} = 2^{<\lambda} < 2^\lambda$, then ($\lambda$ is regular uncountable and) there exists $\eta \in {{}^{\lambda}2}$ such that the set $S_\eta$ defined below is stationary[^17]. $$S_\eta := \{\delta < \lambda \mid \exists \nu \in {{}^{\lambda}2} : \eta {\upharpoonright}\delta = \nu {\upharpoonright}\delta, \eta {\upharpoonright}(\delta + 1) \neq \nu {\upharpoonright}(\delta + 1), \text{ and }f_\eta {\upharpoonright}\delta = f_\nu {\upharpoonright}\delta\}$$ \[wd-rmk\] Given any fixed cardinal $\theta$, there is a unique cardinal $\lambda$ such that $2^{\theta} = 2^{<\lambda} < 2^\lambda$, which can equivalently be described as the minimal cardinal $\lambda$ such that $2^{\theta} < 2^{\lambda}$. Note that $\theta < \lambda \le 2^{\theta}$. If $\theta$ is finite, $\lambda = \theta^+ = \theta + 1$ but if $\theta$ is infinite, $\lambda$ is uncountable and moreover regular (because of the formula $2^\lambda = \left(2^{<\lambda}\right)^{{\text{cf} (\lambda)}}$, see [@jechbook 5.16(iii)]). If the generalized continuum hypothesis[^18] (GCH) holds, then $\lambda = \theta^+$, but in general it could be that $\lambda = \theta^+$ even if GCH fails (e.g. if $\theta = \aleph_0$, $2^{\aleph_0} = \aleph_2$, and $2^{\aleph_1} = \aleph_3$). Nevertheless, it is also consistent with the axioms of set theory that $\lambda > \theta^+$. We will say that the weak generalized continuum hypothesis (WGCH) holds if $2^\theta < 2^{\theta^+}$ for all infinite cardinals $\theta$. In this case, the hypothesis of the previous theorem holds exactly when $\lambda$ is an infinite successor cardinal. Note also that, when $\lambda$ is regular uncountable, the conclusion of the Devlin-Shelah theorem implies that $2^{\lambda_0} < 2^{\lambda}$ for all $\lambda_0 < \lambda$. Indeed, given $F: {{}^{\lambda}2} \to {{}^{\lambda_0}2}$, we can let $f_\eta (\alpha)$ be $F (\eta) (\alpha)$ if $\alpha < \lambda_0$, or $0$ otherwise. Fixing the given $\eta \in {{}^{\lambda}2}$, and $\delta$ in $S_\eta$ bigger than $\lambda_0$, we obtain $\nu \neq \eta$ with $F (\nu) = F (\eta)$, so $F$ is not injective [@proper-and-imp Appendix, 1.B(3)]. The essence of the conclusion of the Devlin-Shelah Theorem (\[wd-thm\]) is that if we think of the $f_\eta$’s as being indexed by branches of a binary splitting tree of height $\lambda$, then there exists two branches (i.e. some $\eta$ and some $\nu$) that split at some big height $\delta < \lambda$ and where moreover the corresponding functions are equal up to $\delta$. If we think of the $f_\eta$’s as embedding structures into a common codomain, them being equal up to $\delta$ will mean that a certain diagram commutes. To state the promised application to amalgamation bases in AECs, we first give some terminology: for ${\mathbf{K}}$ an AEC and $\lambda$ a cardinal, we write ${\mathbf{K}}_\lambda$ (resp. ${\mathbf{K}}_{<\lambda}$) for the class of objects in ${\mathbf{K}}$ of cardinality $\lambda$ (resp. strictly less than $\lambda$). Of course, we identify it with the corresponding category. An object $N \in {\mathbf{K}}_\lambda$ is called universal if any $M \in {\mathbf{K}}_\lambda$ embeds into $N$. We will show that, if $\lambda$ satisfies the hypotheses of the Devlin-Shelah theorem and ${\mathbf{K}}$ has a universal model in ${\mathbf{K}}_\lambda$, then amalgamation bases are cofinal in ${\mathbf{K}}_{<\lambda}$. The result is due to Shelah [@sh88]. The proof proceeds by contradiction: if amalgamation bases are not cofinal, we can build a tree of failure, and embed each branch of this tree into the universal model. Applying the weak diamond to this tree will yield enough commutativity to get that the tree has a lot of amalgamation bases. One reason Theorem \[ap-thm\] is interesting is that it turns a one-dimensional property in $\lambda$ (existence of a universal model) into a two-dimensional property below $\lambda$ (existence of amalgamation bases). There is in fact a higher-dimensional generalization [@multidim-v2 11.16], which is much harder to state (but see Section \[higher-dim-sec\]). \[ap-thm\] Let ${\mathbf{K}}$ be an AEC and let $\lambda > {\text{LS}}({\mathbf{K}})$ be such that there exists $\theta < \lambda$ with $2^{\theta} = 2^{<\lambda} < 2^\lambda$. If there exists a universal model in ${\mathbf{K}}_\lambda$, then for any $M \in {\mathbf{K}}_{<\lambda}$, there exists $N \in {\mathbf{K}}_{<\lambda}$ such that $M {{\le_{{\mathbf{K}}}}}N$ and $N$ is an amalgamation base in ${\mathbf{K}}_{<\lambda}$. Suppose not. First recall that $\lambda$ is regular uncountable (Remark \[wd-rmk\]). Fix $M \in {\mathbf{K}}_{<\lambda}$ with no amalgamation base extending it in ${\mathbf{K}}_{<\lambda}$. Because ${\mathbf{K}}$ is isomorphism-closed in ${\operatorname{\bf Str}}(\tau ({\mathbf{K}}))$, we may do some renaming to assume without loss of generality that $U M \subseteq \lambda$. We build ${\langle M_\eta \mid \eta \in {{}^{<\lambda}2} \rangle}$ such that for all $\beta < \lambda$ and all $\eta \in {{}^{\beta}2}$: 1. $M_{{\langle \rangle}} = M$. 2. \[req-2\] $M_\eta \in {\mathbf{K}}_{<\lambda}$. 3. \[req-3\] $U M_\eta \subseteq \lambda$. 4. $M_{\eta {\upharpoonright}\alpha} {{\le_{{\mathbf{K}}}}}M_\eta$ for all $\alpha < \beta$. 5. \[req-5\] If $\beta$ is limit, $M_\eta = \bigcup_{\alpha < \beta} M_\alpha$. 6. \[req-6\] The span $M_{\eta \smallfrown 0} \leftarrow M_\eta \rightarrow M_{\eta \smallfrown 1}$ cannot be completed to a commutative square (where the maps are inclusion embeddings). This is possible: the construction proceeds by transfinite induction on the length of $\eta$. The base case is given, at successors we use that $M_\eta$ cannot be an amalgamation base in ${\mathbf{K}}_{<\lambda}$ by assumption (and do some renaming to implement (\[req-3\])). At limit stages, we take unions (and use the axioms of AECs). This is enough: for each $\eta \in {{}^{\lambda}2}$, define $M_\eta := \bigcup_{\alpha < \lambda} M_{\eta {\upharpoonright}\alpha}$. The axioms of AECs imply, of course, that $M_{\eta {\upharpoonright}\alpha} {{\le_{{\mathbf{K}}}}}M_\eta$ for all $\alpha < \lambda$. Moreover, $M_\eta \in {\mathbf{K}}_\lambda$. Indeed, $M_{\eta {\upharpoonright}\alpha} \neq M_{\eta {\upharpoonright}(\alpha + 1)}$, as otherwise we would trivially have been able to amalgamate the span $M_{\eta {\upharpoonright}\alpha \smallfrown 0} \leftarrow M_{\eta {\upharpoonright}\alpha} \rightarrow M_{\eta {\upharpoonright}\alpha \smallfrown 1}$. Finally, requirement (\[req-3\]) ensures that $UM_\eta \subseteq \lambda$. Fix a universal model $N \in {\mathbf{K}}_\lambda$. By doing some renaming again, we can assume without loss of generality that $U N \subseteq \lambda$. For each $\eta \in {{}^{\lambda}2}$, fix a ${\mathbf{K}}$-embedding $g_\eta: M_\eta \rightarrow N$. Our goal will be to get a contradiction to requirement (\[req-6\]). We are not there yet, because even if $\eta {\upharpoonright}\delta = \nu {\upharpoonright}\delta$, the maps $g_\eta$ and $g_{\nu}$ will not necessarily agree on $M_{\eta {\upharpoonright}\delta}$. This is where the weak diamond will come in. Define $f_\eta : \lambda \to \lambda$ by $f_\eta (\alpha) = g_\eta (\alpha)$ if $\alpha \in U M_\eta$, and $f_\eta (\alpha) = 0$ if $\alpha \notin U M_\eta$. We are now in the setup of the Devlin-Shelah theorem: fix $\eta \in {{}^{\lambda}2}$ such that the set $S_\eta$ defined there is stationary. Consider the set $C := \{\delta < \lambda \mid U M_{\eta {\upharpoonright}\delta} \subseteq \delta\}$. The set $C$ is closed (because of requirement (\[req-5\])) and unbounded. To see the latter, we run a standard “catching your tail” argument[^19]: fix $\alpha < \lambda$. We inductively build an increasing sequence of ordinals ${\langle \alpha_n : n < \omega \rangle}$ as follows: take $\alpha_0 = \alpha$, and given $\alpha_n$, we know $U M_{\eta {\upharpoonright}\alpha_n} \subseteq \lambda$ (requirement (\[req-3\])), and $\|U M_{\eta {\upharpoonright}\alpha_n}\| < \lambda$ (requirement (\[req-2\])), so use regularity of $\lambda$ to pick $\alpha_{n + 1} \in [\alpha, \lambda)$ with $U M_{\eta {\upharpoonright}\alpha_n} \subseteq \alpha_{n + 1}$. At the end, we let $\delta := \sup_{n < \omega} \alpha_n$. The construction, together with requirement (\[req-5\]), implies that $\delta \ge \alpha$ and $\delta \in C$. Because $C$ is closed unbounded and $S_\eta$ is stationary, we can pick $\delta \in C \cap S_\eta$. By definition of $S_\eta$, there exists $\nu \in {{}^{\lambda}2}$ such that $\eta {\upharpoonright}\delta = \nu {\upharpoonright}\delta$, $\eta (\delta) \neq \nu (\delta)$, and $f_\eta {\upharpoonright}\delta = f_\nu {\upharpoonright}\delta$. Let $\rho := \eta {\upharpoonright}\delta$. By definition of $C$, $f_\eta {\upharpoonright}M_\rho = f_\nu {\upharpoonright}M_\rho$. This implies that $f_\eta$, $f_\nu$, and $N$ witness the span $M_{\eta {\upharpoonright}(\delta + 1)} \leftarrow M_{\rho} \rightarrow M_{\nu {\upharpoonright}(\delta + 1)}$ can be completed to a commutative square, contradicting requirement (\[req-6\]). If we know weak GCH holds, and moreover we know that ${\mathbf{K}}$ has a single object (up to isomorphism) in two successive cardinalities[^20], the theorem simplifies and we get the amalgamation property locally: Let ${\mathbf{K}}$ be an AEC, and let $\mu \ge {\text{LS}}({\mathbf{K}})$ be such that $2^{\mu} < 2^{\mu^+}$. If ${\mathbf{K}}$ has a single model (up to isomorphism) in cardinalities $\mu$ and $\mu^+$, then ${\mathbf{K}}_\mu$ has the amalgamation property. By its uniqueness, the object of cardinality $\mu^+$ must be universal. Applying Theorem \[ap-thm\] with $\lambda = \mu^+$, we get that there exists an amalgamation base in ${\mathbf{K}}_\mu$, and this amalgamation base must be isomorphic to any other object of cardinality $\mu$, hence ${\mathbf{K}}_\mu$ has amalgamation. Generalized pushouts and stable independence {#indep-sec} ============================================ Many of the “classical” categories of mainstream mathematics are bicomplete: they have all limits and colimits. However, problems will occur if we want to study such categories set-theoretically, and specifically if we want to restrict ourselves to certain classes of monomorphisms. It is clear quotients (coequalizers) will be lost, but one will also lose pushouts: any category with pushouts, all morphisms monos, and an initial object must be thin, hence essentially just a poset [@indep-categ-advances 3.30(3)]. Still, it is natural to ask for approximations to pushouts. The results of this section are a survey of the work of Lieberman, Rosický, and the author on this question [@indep-categ-advances; @more-indep-v2]. For the purpose of the discussion to follow, let’s briefly repeat that for a given a diagram $D: I \to {\mathcal {K}}$, a cocone for that diagram is an object $A$ together with maps $(D_i \xrightarrow{f_i} A)_{i \in I}$ commuting with the diagram. One can look at the category ${\mathcal {K}}_D$ of cocones for $D$, where the morphisms are defined as expected. The colimit of a diagram $D$ is then simply an initial object (one that has a unique morphism to every other object) in the category of cocones. In case $D$ is a span $B \xleftarrow{f} A \xrightarrow{g} C$, a cocone is simply an amalgam of this span, and an initial object in the category ${\mathcal {K}}_D$ of cocones would be a pushout. The amalgamation property (Definition \[ap-def\]) simply says that ${\mathcal {K}}_D$ is non-empty, i.e. that there is some cocone, maybe satisfying no universal property whatsoever. A much stronger approximation is the existence of weak pushouts: a weak pushout of a given span $D$ is a cocone that is weakly initial in the category of cocones for $D$: there is a morphism to every other cocone, but that morphism is not required to be unique. In categories where all morphisms are monos, weak pushouts are still too strong of a requirement: Consider the category ${\operatorname{\bf Set}}_{mono}$ of sets with injections. Consider the inclusion of $A = \emptyset$ into $B = \{0,1 \}$ and $C = \{0, 2\}$. Let $D = \{0,1,2\}$. Then $D$, together with the corresponding inclusions, is a cocone/amalgam for the span: $B \leftarrow A \rightarrow C$. On the other hand, consider $D' = \{1,2,3,4\}$ and $f_1: B \to D'$, $f_2: C \to D'$ defined by $f_1 (0) = 3$, $f_1 (1) = 1$, $f_2 (0) = 4$, $f_2 (2) = 2$. $(f_1, f_2)$ is also an amalgam of $B \leftarrow A \rightarrow C$, but $D$ has no morphisms to $D'$ (in the category of cocones for the span $B \leftarrow A \rightarrow C$), and by cardinality considerations $D'$ has no morphisms to $D$ either. Thus ${\operatorname{\bf Set}}_{mono}$ does not have weak pushouts. What is happening in the example is that we had two choices for amalgamating $B$ and $C$: either sending $0$ to the same place, or sending it two different elements. These two choices are then incompatible, in the sense that no amalgam of one type will ever have a morphism into an amalgam of the other type (in the appropriate category of amalgams of a fixed span). In other words, the category of amalgams is not connected. Let us make this explicit, first in complete generality, then for the specific case of amalgams: Two objects $A$ and $B$ are called comparable[^21] if either ${\operatorname{Hom}}(A, B) \neq \emptyset$ or ${\operatorname{Hom}}(B, A) \neq \emptyset$. We say that $A$ and $B$ are connected if there exists $A = A_0, A_1, \ldots, A_n = B$ such that $A_{i}$ and $A_{i + 1}$ are comparable for all $i < n$. This is an equivalence relation, and the equivalence class is called a connected component of the category. The category is called connected if any two of its objects are connected. We say that $A$ and $B$ are jointly connected if there exists an object $C$ with morphisms $A \to C$, $B \to C$. Of course, the connected components of a category are exactly the connected component of the undirected graph whose vertices are objects and where there is an edge between $A$ and $B$ exactly when $A$ and $B$ are comparable. Assuming amalgamation, joint connectedness coincides with connectedness: \[compat-lem\] In a category with the amalgamation property, two objects are connected if and only if they are jointly connected. Let ${\mathcal {K}}$ be a category with the amalgamation property. First note that if two objects $A$ and $B$ are comparable, then they are jointly connected (for example, if $A \xrightarrow{f} B$ then $A \xrightarrow{f} B = C \xleftarrow{{\operatorname{id}}_B} B$ witness the joint connectivity). Also, if $A$ and $B$ are jointly connected, as witnessed by $A \to C$, $B \to C$, then $A$ and $C$ are comparable, and $C$ and $B$ are comparable, so $A$ and $B$ are connected. We now show that being jointly connected is an equivalence relation. This will be enough because being connected is the smallest equivalence relation extending comparability. So assume that $A_0$, $A_1$ are jointly connected, as witnessed by $A_0 \rightarrow B \leftarrow A_1$, and $A_1$, $A_2$ are jointly connected, as witnessed by $A_1 \rightarrow C \leftarrow A_2$. Amalgamate $B$ and $C$ over $A_1$, forming the diagram below: $$\xymatrix@=1pc{ & & D & & \\ & B \ar@{.>}[ru] & & C \ar@{.>}[lu] & \\ A_0 \ar[ru] & & A_1 \ar[lu] \ar[ru] & & A_2 \ar[lu] }$$ Then $D$ and the expected compositions witness that $A_0$ and $A_2$ are jointly connected. Two amalgams $(B \xrightarrow{f^a} D^a, C \xrightarrow{g^a} D^a)$ and $(B \xrightarrow{f^b} D^b, C \xrightarrow{g^b} D^b)$ of a span $B \leftarrow A \rightarrow C$ are jointly connected if they are jointly connected in the category of cocones for the appropriate span. Explicitly, there exists $D$ and morphisms into $D$ making the following diagram commute: $$\xymatrix{ & D^b \ar@{.>}[r] & D \\ B \ar[ru]^{f^b} \ar[rr]\|>>>>>{f^a} & & D^a \ar@{.>}[u] \\ A \ar[u] \ar[r] & C \ar[uu]\|>>>>>{g^b} \ar[ur]_{g^a} & \\ }$$ We say that two amalgams $D^a$ and $D^b$ are connected if they are connected in the appropriate category of cocones, i.e. there exists a chain of amalgams $D^0, D^1, \ldots, D^n$ (along with respective morphisms) such that $D^0 = D^a$, $D^n = D^b$, and $D^{i}$ is jointly connected to $D^{i + 1}$ for all $i < n$. Note that if ${\mathcal {K}}$ has the amalgamation property, then the category of cocones over a diagram in ${\mathcal {K}}$ also has the amalgamation property, so in this case connectedness already implies joint connectedness. The replacement for pushouts we are looking for will therefore consist in a choice of connected component for each span. Eventually, we may want to look for a weakly initial object for this specific component class (called a prime object by model theorists), but this seems to be too strong a requirement to start with: we want to prove it, not assume it.[^22] Thus we instead impose a transitivity conditions on the choice of connected component that make the resulting squares into the morphisms of an arrow category (and also holds of pushouts). Still by analogy with pushouts, and for reasons to be discussed later, we require this new category to be accessible and call the result a stable independence notion. \[indep-def\] A stable independence notion in a category ${\mathcal {K}}$ is a class of squares (called independent squares and marked here with the anchor symbol ${\unionstick}$) such that: 1. Independent squares are closed under connectedness: if one amalgam of a given span is independent, then all the connected amalgams are also independent. 2. Existence: any span can be amalgamated to an independent square. 3. Uniqueness: any two independent amalgams of the same span are connected. 4. Transitivity: $$\xymatrix@=1pc{ B \ar[r]\ar@{}[dr]\|{{\unionstick}} & D\ar@{}[dr]\|{{\unionstick}} \ar[r] & F \ar@{}[dr]\|{\Rightarrow} & B \ar[r]\ar@{}[dr]\|{{\unionstick}} & F \\ A \ar [u] \ar [r] & C \ar[u] \ar[r] & E \ar[u] & A \ar [u] \ar [r] & E \ar[u] }$$ 5. Symmetry: “swapping the ears” $B$ and $C$ preserves independence. 6. Accessibility: the arrow category whose objects are morphisms of ${\mathcal {K}}$ and whose morphisms are independent squares is accessible. We let ${\mathcal {K}}_{{\downarrow}}$ denote this category. The existence property implies the amalgamation property. Hence any two connected amalgams are, in fact, jointly connected. Note also that any two amalgams of a span which contains an isomorphism will be connected, hence (by the existence property) independent squares [@indep-categ-advances 3.12]. In particular, the identity morphism in the arrow category ${\mathcal {K}}^2$ will indeed be an independent square, hence a morphism of ${\mathcal {K}}_{{\downarrow}}$. The transitivity property ensures that ${\mathcal {K}}_{{\downarrow}}$ is closed under composition, and hence is indeed a category. Most of the examples below are listed in [@indep-categ-advances 3.31], though of course many are trivial or date back to the beginning of stable first-order theories. See there for more references. \[indep-ex-0\] 1. In an accessible category with weak pushouts, there is a stable independence notion given by all commutative squares. This holds more generally in any accessible category with the amalgamation property where any two amalgams are always connected, for example in accessible categories with a terminal object. 2. The category ${\operatorname{\bf Set}}_{mono}$ has a stable independence notion: identifying the morphisms with inclusions, given $A \subseteq B \subseteq D$, $A \subseteq C \subseteq D$, we say that the resulting square $A, B, C, D$ is independent exactly when $B \cap C = A$. That is, the amalgam must be disjoint. In general, the ears of an independent squares “do not interact”: independent squares are a notion of “free” amalgam. 3. The category of vector spaces over a fixed field, with morphisms the injective linear transformations, has a stable independence notion, again given by disjoint amalgamation. This holds more generally for the category of modules over a fixed ring. 4. The category of all algebraically closed fields of a fixed characteristic (with morphisms field homomorphisms) has a stable independence notion, essentially given by algebraic independence: again identifying the morphisms with inclusions, if we have a square of fields $F_0 \subseteq F_1 \subseteq F_3$, $F_0 \subseteq F_2 \subseteq F_3$, we say it is independent if for any subset $A \subseteq F_1$ and any $b \in F_1$, if $b$ is in the algebraic closure (computed in $F_3$) of $A \cup F_2$, then it is in the algebraic closure of $A \cup F_0$. Here, it does not suffice to require that $F_1 \cap F_2 = F_0$ (because the pregeometry induced by algebraic closure is not modular). 5. \[diff-field-ex\] A differential field is a field together with an operator $D$ that preserves sums and satisfies Leibnitz’ law: $D fg = g D f + f D g$. A differentially closed field is, roughly, a differential field in which every system of linear differential equations that could possibly have a solution has a solution. Model-theoretic methods were used to obtain the first proof that every differential field of characteristic zero has, in some sense, a differential closure. Uniqueness of that differential closure was proven by Shelah, using the fact that the category of differentially closed fields of characteristic zero has a stable independence notion. See [@sacks-diff] for a short overview. 6. \[indep-stable-ex-0\] Generalizing the last four examples, let $T$ be a stable first-order theory ($T$ is stable if it does not have the order property, and $T$ has the order property if there exists a formula $\phi ({\bar{x}}, {\bar{y}})$, a model $M$ of $T$, and a sequence ${\langle a_i : i < \omega \rangle}$ of elements in $M$ such that $M \models \phi[{\bar{a}}_i, {\bar{a}}_j]$ if and only if $i < j$, see Definition \[op-def\]). Consider the category ${\mathcal {K}}= {\operatorname{\bf Elem}}(T)$ of models of $T$ with elementary embeddings. Then ${\mathcal {K}}$ has a stable independence notion. The proof (originally due to Shelah), is not trivial, see Appendix \[fo-sec\] for a short exposition. The definition of an independent square mirrors that of fields (the reader should think of the formula $\psi$ in the next sentence as a polynomial): a square $M_0 \preceq M_1 \preceq M_3$, $M_0 \preceq M_2 \preceq M_3$ is independent exactly when for any finite sequences ${\bar{a}}$ from $M_1$ and ${\bar{b}}$ from $M_2$, and any formula $\psi ({\bar{x}}, {\bar{y}})$, if $M_3 \models \psi[{\bar{a}}, {\bar{b}}]$, then there exists a sequence ${\bar{b}}_0$ in $M_0$ so that $M_3 \models \psi[{\bar{a}}, {\bar{b}}_0]$. In fact, independent squares are the same as what model theorists call nonforking squares. Thus we have recovered the important model-theoretic notion of forking from simple category-theoretic considerations! In fact, the original motivation behind the definition of stable independence was to generalize forking. 7. Conversely, if $T$ is not stable then ${\operatorname{\bf Elem}}(T)$ does not have a stable independence notion [@indep-categ-advances 9.9]. 8. The previous item implies, in particular, that the category of graphs, with morphisms the full subgraph embeddings does not have a stable independence notion. Another proof will be given in Remark \[canon-rmk\]. The category ${\operatorname{\bf Lin}}$ of linear orders with order-preserving maps similarly does not have a stable independence notion. 9. The category of graphs with morphisms the (not necessarily full) subgraph embeddings does have a stable independence notion: two graphs are independent over a base graph (inside an ambient graph) if all the cross-edges between the two are inside the base graph. See Example \[indep-ex\](\[indep-gr-2-ex\]). 10. [@indep-categ-advances 4.8(5)] The category of Hilbert spaces with isometries has a stable independence notion, given by pullback squares. These correspond roughly to squares where everything is “as orthogonal as possible”. 11. We will give later general constructions of a stable independence notion, giving many other nontrivial examples. For example, for a fixed ring, the category of all flat modules with morphisms the flat monomorphisms has a stable independence notion (Example \[indep-ex\](\[indep-ex-mod\])). For model theorists, we note that in concrete cases (e.g. inside an $\infty$-AEC ${\mathbf{K}}$), the independence notion can be extended to a relation of the form ${A {\unionstick}_{M}^{N} B}$, where $M {{\le_{{\mathbf{K}}}}}N$ and $A, B \subseteq U N$, satisfying generalizations of the properties of a stable independence notion. The accessibility axiom can then be shown to be equivalent to the conjunction of the two classical properties of forking: the witness property (failure to be independent is witnessed by small subsets of the ears) and the local character property (every type is independent over a small set). The proof of this fact is essentially a generalization of the proof of Theorem \[aec-acc\], see [@indep-categ-advances 8.14]. We deduce, in particular, that an $\infty$-AEC has quite a bit of structure when it has a stable independence notion (e.g. it is tame and stable [@indep-categ-advances 8.16]). One of the most important fact about stable independence (and a strong justification for the definition) is that it leads to a canonical notion: if a category has a stable independence notion, then under very reasonable conditions it can have only one. For first-order forking, this is well known [@hh84], but in fact it holds even in the very general categorical setup. The proof for AECs in [@bgkv-apal] was adapted to $\infty$-AECs with chain bounds in [@indep-categ-advances 9.1], and finally to any category with chain bounds in [@more-indep-v2 A.6]. We say a category has chain bounds if any ordinal-indexed chain has a cocone (this holds, of course, anytime the category has directed colimits). \[canon-thm\] A category with chain bounds has at most one stable independence notion. In fact, given a stable independence notion ${\unionstick}$, any other relation satisfying all the axioms of stable independence notion except perhaps accessibility will have to be ${\unionstick}$. First, existence of a stable independence notion implies that the category itself is accessible [@indep-categ-advances 3.27]. Let ${\unionstick}^1$ be a stable independence notion and let ${\unionstick}^2$ satisfy all the axioms of stable independence, except perhaps accessibility. Given a span $B \leftarrow A \rightarrow C$, it suffices to show that it has an amalgam which is independent in the sense of both ${\unionstick}^1$ and ${\unionstick}^2$ (then uniqueness and invariance under connectedness show that ${\unionstick}^1$ and ${\unionstick}^2$ must coincide for any amalgam of this span). The idea of the construction is perhaps best described by the following property of a vector space: given any sequence $I$ of vectors and any other vector $a$, there exists a finite subset $I_0$ of $I$ such that $(I - I_0) \cup \{a\}$ is independent. Thus the idea is to first complete the span $B \leftarrow A \rightarrow C$ to a ${\unionstick}^2$-independent square, then build many ${\unionstick}^2$-independent “copies” of that square. Analogously to the fact mentioned for vector spaces, all except a few of these copies must also be ${\unionstick}^1$-independent. \[canon-rmk\] The canonicity theorem provides us with a tool to prove the non-existence of stable independence notions. Indeed, it suffices for this purpose to find two different notions satisfying all the axioms of stable independence except perhaps for accessibility. For example, the category of graph with full subgraph embeddings does not have a stable independence notion. Indeed, the relations “all cross-edges are contained inside the base” and “all cross-edges outside those in the base are present” satisfy all the axioms of stable independence except for accessibility [@indep-categ-advances 3.31(6)]. We now survey two different general constructions of a stable independence notion. In both cases, the hypotheses are provably optimal in the sense that the existence of stable independence implies them. The first construction works in any $\infty$-AEC with chain bounds, but uses large cardinals and only obtains an independence notion on a cofinal subclass (consisting of “sufficiently homogeneous” objects). \[vopenka-constr\] Assume Vopěnka’s principle. Let ${\mathbf{K}}$ be an $\infty$-AEC. The following are equivalent: 1. ${\mathbf{K}}$ does not have the order property[^23]. 2. ${\mathbf{K}}$ has a cofinal subclass of “sufficiently homogeneous” objects which has a stable independence notion. If ${\mathbf{K}}$ has the order property, as witnessed by a long sequence ${\langle {\bar{a}}_i : i < \lambda \rangle}$ ($\lambda$ here is a big regular cardinal), we can use the accessibility and uniqueness properties of stable independence to get a subsequence ${\langle {\bar{a}}_i : i \in S \rangle}$ with $\|S\| = \lambda$ that is independent and indiscernible (essentially, this means that the sequence is “very homogeneous” – it looks like a sequence of mutually transcendental elements in an algebraically closed fields). The symmetry axiom can then be use to show that the elements of the subsequence can be permuted without impacting their properties. In particular, the sequence cannot serve as a witness for the order property, contradiction. Going from no order property to stable independence, the very rough idea is to imitate the standard construction given Appendix \[fo-sec\], but replace $\aleph_0$ by a sufficiently-big strongly compact, and look only at the locally $\kappa$-homogeneous models (see [@indep-categ-advances 7.3]). Vopěnka’s principle is used to prove that this subclass is an $\infty$-AEC, and also provides enough compactness to prove the existence property. For the second method, we start with a “classical” category ${\mathcal {K}}$: a locally presentable category. We single out a certain class of morphisms ${\mathcal{M}}$ (usually some class of nice monomorphisms), and we want to build a stable independence notion on the category ${\mathcal {K}}_{{\mathcal{M}}}$ obtained by restricting the morphisms in ${\mathcal {K}}$ to be those of ${\mathcal{M}}$. To make this setup more precise, we will have to make some assumptions on ${\mathcal{M}}$. Of course, we want at minimum ${\mathcal{M}}$ to contain all isomorphisms and be closed under compositions. It turns out it is very convenient to assume that ${\mathcal{M}}$ satisfies a coherence property, similar to the coherence axiom of $\infty$-AECs: for morphisms $f, g$ of ${\mathcal {K}}$, if $gf \in {\mathcal{M}}$ and $g \in {\mathcal{M}}$, then $f \in {\mathcal{M}}$. We will say that ${\mathcal{M}}$ is coherent. For technical reasons, we will also want ${\mathcal{M}}$ to be closed under retracts (in the arrow category ${\mathcal {K}}^2$). This holds under very mild conditions. For example, if ${\mathcal{M}}$ is closed under compositions, contains all split monos, and is coherent, then it is closed under retracts. Since we want to use pushouts to build our stable independence notion, we will require ${\mathcal{M}}$ to be closed under pushouts: a pushout of a morphism in ${\mathcal{M}}$ (not necessarily along a morphism in ${\mathcal{M}}$) should be in ${\mathcal{M}}$. Finally, to perform iterative constructions, we will need ${\mathcal{M}}$ to be closed under transfinite compositions. Under all these conditions, we get that ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence notion if and only if it is accessible and ${\mathcal{M}}$ is cofibrantly generated. Here, ${\mathcal{M}}$ is cofibrantly generated if there is a subset ${\mathcal{X}}$ of ${\mathcal{M}}$ (i.e. not a proper class) such that closing ${\mathcal{X}}$ under retracts, pushouts, and transfinite compositions gives back ${\mathcal{M}}$. This notion originated in algebraic topology, where the cell complexes are precisely the objects that can be obtained from finitely-many pushouts and composition by starting from inclusions $\partial D^n \to D^n$ of the boundary of the $n$-ball. As we will see, similar notions have been used in homological algebra as well. \[cofib\] Let ${\mathcal {K}}$ be a locally presentable category and let ${\mathcal{M}}$ be a coherent class of morphisms containing all isomorphisms, closed under retracts, transfinite compositions, and pushouts. The following are equivalent: 1. \[cofib-1\] ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence notion. 2. \[cofib-2\] ${\mathcal {K}}_{{\mathcal{M}}}$ is accessible and ${\mathcal{M}}$ is cofibrantly generated. Neither direction is easy. First, we need a candidate definition for a stable independence notion. Let us say that a commutative square with morphisms in ${\mathcal{M}}$ is effective if the induced map from the pushout is in ${\mathcal{M}}$. That is, the outer square in the diagram below is effective if all its maps are in ${\mathcal{M}}$ and the induced map $f$ from the pushout $P$ is in ${\mathcal{M}}$. $$\xymatrix@=1pc{ B \ar[dr] \ar[rr] & & D \\ & P \ar@{.>}[ur]_{f} & \\ A \ar[rr] \ar[uu] & & C \ar[uu] \ar[lu] }$$ Without any additional hypotheses, it is not too difficult to show that effective squares will satisfy all the axioms of stable independence, except perhaps for accessibility. Let ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ be the category whose objects are morphisms in ${\mathcal{M}}$ and morphisms are effective squares. We can also show that ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ has directed colimits (and they are computed as in ${\mathcal {K}}^2$). - : Starting from a stable independence notion on ${\mathcal {K}}_{{\mathcal{M}}}$, it is not too difficult to show that ${\mathcal {K}}_{{\mathcal{M}}}$ must be accessible [@indep-categ-advances 3.27]. Moreover, the stable independence notion must be given by effective squares, by the canonicity theorem (\[canon-thm\]). Let $\lambda$ be a big-enough regular uncountable cardinal such that all the relevant categories are $\lambda$-accessible. For $\mu \ge \lambda$ regular, let ${\mathcal{M}}_\mu$ denote the class of morphisms in ${\mathcal{M}}$ with $\mu$-presentable domain and codomain (in ${\mathcal {K}}$). For a class ${\mathcal{H}}$ of morphisms, let ${\operatorname{cof}}({\mathcal{H}})$, the cofibrant closure of ${\mathcal{H}}$, denote the closure of ${\mathcal{H}}$ under retracts, pushouts, and transfinite compositions. We will show that ${\mathcal{M}}= {\operatorname{cof}}({\mathcal{M}}_\lambda)$. For this, we prove by induction that for all regular cardinals $\mu$, ${\mathcal{M}}_\mu \subseteq {\operatorname{cof}}({\mathcal{M}}_\lambda)$. For $\mu \le \lambda$ this is trivial and if $\mu$ is limit, ${\mathcal{M}}_{\mu} = \bigcup_{\theta < \mu} {\mathcal{M}}_{\theta}$ by Corollary \[succ-cor\](\[succ-cor-1\]). Thus we can assume $\mu = \mu_0^+$. Let $\delta := {\text{cf} (\mu_0)}$, and fix a morphism $A \xrightarrow{f} B$ in ${\mathcal{M}}_{\mu}$. We write $f$ as the directed colimit of a chain ${\langle f_i : i < \delta \rangle}$ in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$, where each $f_i$ is in ${\mathcal{M}}_{\theta}$ for a regular $\theta \le \mu_0$. Finding such a chain is nontrivial (in accessible categories, objects can be written as directed colimits of directed diagrams of small objects, it is not clear you can find chains like this): assuming for simplicity all morphisms are monos, we use that it is possible in AECs, together with the fact every $\lambda$-accessible category ${\mathcal {L}}$ with directed colimits is a reflexive full subcategory of a finitely accessible category (which is given by taking free directed colimits of the $\lambda$-presentable objects of ${\mathcal {L}}$), and finitely accessible categories with all morphisms monos are AECs (Theorem \[acc-to-aec\]). Once we have the $f_i$’s, say $A_i \xrightarrow{f_i} B_i$, they are each by the induction hypothesis part of ${\operatorname{cof}}({\mathcal{M}}_\lambda)$. It suffices to use them to generate $f$. Let $(A_i \xrightarrow{g_i} A, B_i \xrightarrow{h_i} B)_{i < \delta}$ be the colimit maps. $$\xymatrix@=1pc{ A \ar@{}[dr]\|{{\unionstick}} \ar[r]^f & B \\ A_i \ar[u]^{g_i} \ar[r]_{f_i} & B_i \ar[u]_{h_i} }$$ Take the pushout $P_0$ of $f_0$ along $g_0$. We get that $f = p_0 \bar{f}_0$, where $p_0$ is the induced map from the pushout, which is also in ${\mathcal{M}}$ by assumption. We have managed to generate $\bar{f}_0$, and we now repeat what we did for $f$ but for $p_0$ instead (write it as the colimit of an increasing chain of morphisms that are above $f_1$, take the pushout of the first morphism, etc.). In the end, we will have written $f$ as a $\delta$-length transfinite compositions of pushouts of members of ${\operatorname{cof}}({\mathcal{M}}_\lambda)$, as desired. - (\[cofib-2\]) implies (\[cofib-1\]): As before, fix $\lambda$ an uncountable regular cardinal so that all relevant categories are $\lambda$-accessible and ${\mathcal{M}}= {\operatorname{cof}}({\mathcal{M}}_\lambda)$. By using what is called “good colimits” [@fat-small-obj B.1], we can in fact show that ${\mathcal{M}}$ is generated from ${\mathcal{M}}_\lambda$ by just using pushouts and transfinite compositions (no retracts). Let ${\mathcal{H}}$ be the class of all morphisms that, in ${\mathcal {K}}_{{\mathcal{M}}, {\downarrow}}$ are $\lambda$-directed colimits of morphisms of ${\mathcal{M}}_\lambda$. It suffices to show that ${\mathcal{H}}= {\mathcal{M}}$, and for this it suffices to see that ${\mathcal{H}}$ is closed under pushouts and transfinite compositions. This can readily be done, using the definition of an effective square. In Theorem \[cofib\], if ${\mathcal {K}}$ is $\lambda$-accessible and all morphisms of ${\mathcal{M}}$ are monos, then ${\mathcal {K}}_{{\mathcal{M}}}$ will be (equivalent to) a $\lambda$-AEC [@more-indep-v2 3.10]. This is especially interesting when $\lambda = \aleph_0$ (i.e. ${\mathcal {K}}$ is locally finitely presentable), in which case we get an AEC. Compared to Theorem \[vopenka-constr\], Theorem \[cofib\] does not use large cardinal axioms and more importantly gives a stable independence notion on the whole category. The hypotheses seem to hold often-enough in practice (all the examples below and more are in [@more-indep-v2 §6]): \[indep-ex\] 1. Let ${\mathcal {K}}$ be the category of graphs (reflexive and symmetric binary relations) with homomorphisms. This is locally finitely presentable. Let ${\mathcal{M}}$ be the full subgraph embeddings. We have seen (Remark \[canon-rmk\]) that ${\mathcal {K}}_{{\mathcal{M}}}$ does not have a stable independence notion. This automatically tells us that ${\mathcal{M}}$ is not cofibrantly generated. 2. \[indep-gr-2-ex\] Let ${\mathcal {K}}$ again be the category of graphs with homomorphisms. This time, let ${\mathcal{M}}$ be the subgraph embeddings (corresponding to monomorphisms in ${\mathcal {K}}$ – the full subgraph embeddings correspond to the regular monomorphisms: those that are equalizers of some pair of morphism). Then ${\mathcal{M}}$ is cofibrantly generated by $\emptyset \to 1$ and $1 + 1 \to 2$, where $1$ is a vertex, $2$ is an edge, and $1 + 1$ is an empty graph on two vertices. Therefore ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence notion. 3. An orthogonal factorization system in a category ${\mathcal {K}}$ consist of two classes of morphisms $({\mathcal{M}}, {\mathcal{N}})$ such that both contain all the isomorphisms, both are closed under composition, and every map factors as $gf$ with $f \in {\mathcal{M}}$ and $g \in {\mathcal{N}}$ in a way that is unique up to unique isomorphism. For example, in the category of sets (epi, mono) is an orthogonal factorization system. A weak factorization system is defined in a somewhat similar way, except the uniqueness condition is considerably relaxed. An example is (mono, epi) in the category of sets. It turns out that (in a cocomplete category) the left part of a weak factorization system is always closed under pushouts, retracts, and transfinite compositions. Thus we can call a weak factorization system $({\mathcal{M}}, {\mathcal{N}})$ cofibrantly generated if ${\mathcal{M}}$ is cofibrantly generated. Thus if ${\mathcal{M}}$ is the left part of a weak factorization system in a locally presentable category ${\mathcal {K}}$, ${\mathcal{M}}$ is coherent, and ${\mathcal {K}}_{{\mathcal{M}}}$ is accessible, then ${\mathcal {K}}_{{\mathcal{M}}}$ has a stable independence notion if and only if the weak factorization system is cofibrantly generated. In fact, the small object argument tells us that if ${\mathcal{X}}$ is any set of morphisms in a locally presentable category, then ${\operatorname{cof}}({\mathcal{X}})$ will form the left part of a weak factorization system [@beke-sheafifiable 1.3]. Thus in the setup of Theorem \[cofib\], existence of a stable independence notion on ${\mathcal {K}}_{{\mathcal{M}}}$ automatically implies that ${\mathcal{M}}$ forms the left part of a weak factorization system. 4. In algebraic topology, a model category is a bicomplete category together with three classes of morphisms, $F$ (fibrations), $C$ (cofibrations), and $W$ (weak equivalences), such that the weak equivalences satisfy the two out of three property (if two of $g, f$ and $gf$ are weak equivalences, so is the third), and both $(C, F \cap W)$ and $(C \cap W, F)$ form weak factorization systems. A model category is called cofibrantly generated when both weak factorization systems are cofibrantly generated. The typical example includes the usual fibrations, cofibrations, and weak homotopy equivalences in the category of topological spaces, or the monomorphisms, Kan fibrations, and weak homotopy equivalences in the category of simplicial sets. There are however model category of a more algebraic flavor, including a model category on chain complexes of modules [@hoveybook Chapter 2]. Thus the previous example also describes a two way connection between model categories and stable independence. 5. \[indep-ex-mod\] Let $R$ be a (associative and unital) ring and let ${\mathcal {K}}$ be the category of $R$-modules with homomorphisms. One can check that ${\mathcal {K}}$ is locally finitely presentable. We want to study the modules that are flat (i.e. directed colimits of free modules – this is the easiest equivalent definition for the purpose of this discussion). We will do this through the class ${\mathcal{M}}$ of flat monomorphisms: monomorphisms whose cokernel is flat. In particular, an inclusion $A \subseteq B$ is flat if and only if $B / A$ is flat. One can check that ${\mathcal{M}}$ contains all isomorphisms, is coherent, and closed under pushouts, retracts, and transfinite compositions. Now the class ${\mathcal {K}}_{{\mathcal{M}}}$ is not quite the right category, we really want to study the category ${\mathcal{F}}_{{\mathcal{M}}}$, where ${\mathcal{F}}$ is the full subcategory of ${\mathcal {K}}$ consisting of flat modules. Note that ${\mathcal{F}}$ can be described in terms of ${\mathcal{M}}$: an object $A$ is in ${\mathcal{F}}$ if and only if the initial morphism $0 \to A$ is in ${\mathcal{M}}$. We will say that $A$ is a cofibrant object (with respect to ${\mathcal{M}}$). Before even worrying about stable independence, is ${\mathcal{F}}_{{\mathcal{M}}}$ an accessible category? Using Theorem \[acc-to-aec\], one can see that this is equivalent to asking whether it is an AEC. Similar examples were studied by Baldwin-Eklof-Trlifaj [@bet], where it is shown that ${\mathcal{F}}_{{\mathcal{M}}}$ is an AEC if and only if ${\mathcal{F}}$ has refinements: there is a regular cardinal $\lambda$ such that any flat module $A$ is the colimit of an increasing chain ${\langle A_i : i < \delta \rangle}$ with $A_0 = 0$ and $A_{i + 1} / A_i$ flat and $\lambda$-presentable. Earlier, Rosický had essentially shown [@flat-covers-factorizations 4.5] that having refinements is equivalent to ${\mathcal{M}}$ being cofibrantly generated (even by a subset of ${\mathcal{M}}\cap {\mathcal{F}}$)! Thus having refinements is yet another disguise for being cofibrantly generated. Using a slight variation on Theorem \[cofib\], one can then close the loop to deduce that if ${\mathcal{F}}$ has refinements then ${\mathcal{F}}_{{\mathcal{M}}}$ has a stable independence notion. It turns out that having refinement is closely connected to the (now resolved) flat cover conjecture [@enochs-flat-cover-orig] (a dualization of the fact that every module has an injective envelope). An argument of Bican, El Bashir, and Enochs [@flat-cover] (which also easily could have been deduced from existing results on accessible categories, see [@flat-covers-factorizations 3.2] or [@more-indep-v2 6.21]) establishes that indeed, ${\mathcal{F}}$ has refinements, and thus ${\mathcal{F}}_{{\mathcal{M}}}$ indeed is an AEC with a stable independence notion. Higher dimensional independence and categoricity {#higher-dim-sec} ------------------------------------------------ Recall that stable independence was defined as a class of squares satisfying certain properties, the most important of which was that the arrow category ${\mathcal {K}}_{{\downarrow}}$ (whose objects are the arrows of ${\mathcal {K}}$ and whose morphisms are the independent squares) should be accessible. Now, given any accessible category, it makes sense to ask whether it has a stable independence notion. Does ${\mathcal {K}}_{{\downarrow}}$ have a stable independence notion? If it does (call this stable independence notion ${\unionstick}^\ast$), then what do the objects and arrow look like in the corresponding category $({\mathcal {K}}_{\downarrow})_{{\downarrow}^\ast}$? Well, the objects are the morphisms of ${\unionstick}^\ast$, i.e. independent squares. The morphisms, in turn, are ${\unionstick}^\ast$-independent “squares” in ${\mathcal {K}}_{{\downarrow}}$, but they really are morphisms between two independent squares of ${\mathcal {K}}$, hence it makes sense to call them independent cubes. Continuing in this way, one can define when a category has an $n$-dimensional stable independence notion, for any $n < \omega$ (the original definition of stable independence is the case $n = 2$). Very nice, but what are (possibly multidimensional) stable independence notions good for? Very roughly, they are useful to prove the existence and uniqueness of certain objects in a category. As a simple example, let ${\mathbf{K}}$ be an AEC with ${\text{LS}}({\mathbf{K}}) = \aleph_0$ (e.g. the AEC of abelian groups, ordered with subgroup). Let’s assume that there is an object of cardinality $\aleph_0$, and suppose we want to establish that ${\mathbf{K}}$ has an object of cardinality $\aleph_1$. A simple way would be to first show that for every countable $A \in {\mathbf{K}}$, there exists a countable $B \in {\mathbf{K}}$ with $A {{<_{{\mathbf{K}}}}}B$. Equivalently, there is a morphism with domain $A$ that is not an isomorphism. If this last property holds, then we can build a strictly increasing chain ${\langle A_i : i < \omega_1 \rangle}$ of countable objects (taking unions at limits) and the union of this chain will be the desired object of cardinality $\aleph_1$. One can think of this construction as establishing a $0$-dimensional property in $\aleph_1$ (existence of an object) by using a $1$-dimensional property in $\aleph_0$ (existence of extensions). There was nothing special about $\aleph_0$ and $\aleph_1$ in this example, we could have replaced them with $\lambda$ and $\lambda^+$ for an arbitrary infinite cardinal $\lambda$. In particular, existence in $\aleph_2$ is implied by extensions in $\aleph_1$. Now how do we get existence of extensions in $\aleph_1$? Well, it is a $1$-dimensional property, so it seems reasonable that we should look for a $2$-dimensional property in $\aleph_0$. Such a property is the disjoint amalgamation property: any span can be completed to a pullback square. Note that in many cases, an independent square will be a pullback square [@indep-categ-advances 10.6]. Still, existence is a purely combinatorial property. Stable independence notions become really useful to prove uniqueness properties. The simplest uniqueness property is what model theorists call categoricity (this is somewhat unfortunate terminology, dating back from before the invention of category theory [@veblen-geom-categ; @los-conjecture]). \[categ-def\] For an infinite cardinal $\lambda$, a category is called $\lambda$-categorical (or categorical in $\lambda$) if it has exactly one object of size $\lambda$ (up to isomorphism). The following are classical examples of the occurrence of categoricity: 1. The category of abelian groups is not categorical in any infinite cardinal (for a cardinal $\lambda$, take $\lambda$ copies of $\mathbb{Z}$ and $\lambda$ copies of $\mathbb{Z} / 2 \mathbb{Z}$: the first is torsion-free the other is not – they are not isomorphic). 2. The category of all dense linear orders without endpoints is categorical in $\aleph_0$, but not in any uncountable cardinal. 3. The category of sets is categorical in every infinite cardinal. 4. The category of vector spaces over $\mathbb{Q}$ is categorical in every uncountable cardinal (but not in $\aleph_0$: consider the spaces of dimension 1 and 2). 5. The category of all algebraically closed fields of a fixed characteristic is categorical in every uncountable cardinal (but again not in $\aleph_0$). 6. [@internal-sizes-jpaa 6.3] The category of all Hilbert spaces is categorical in every uncountable size (but not in every uncountable cardinal). A classical theorem of Morley [@morley-cip] says that for a countable first-order theory $T$, if ${\operatorname{\bf Elem}}(T)$ is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals. A proof can be sketched as follows: derive the existence of a stable independence notion from categoricity, then use it to build enough of a dimension theory to transfer categoricity. While it seems that having a single object of a given size is a relatively rare occurrence, it seems to be a useful test case as tools developed in proofs of categoricity transfers are often much more general. For example, the category of abelian groups with monomorphisms still has a stable independence notion [@indep-categ-advances 5.3]. For this reason, many generalizations of Morley’s theorem have been conjectured. One variation is: \[categ-conj\] If an AEC is categorical in a proper class of cardinals, then it is categorical on a tail of cardinals. Note that we have strengthened the starting assumption to categoricity on a proper class. Often, assuming categoricity in a single “high-enough” cardinal, where “high-enough” has some effective meaning, is enough. Note also that the conjecture is open even for accessible categories (although I suspect it should be wrong in full generality there). The eventual categoricity conjecture for AECs implies the eventual categoricity conjecture for finitely accessible categories [@beke-rosicky 5.8(2)]. It should be noted that multidimensional stable independence notions were introduced by Shelah to make progress on categoricity questions in ${\mathbb{L}}_{\omega_1, \omega}$ (in a very different form and a much more special situation than described here), see [@sh87a; @sh87b]. Very recently, multidimensional independence was used by Shelah and myself to prove the eventual categoricity conjecture in AECs, assuming the existence of a proper class of strongly compact cardinals [@multidim-v2]. So how exactly is multidimensional stable independence used? It is a complicated story, that I do not have space to tell here. One idea is that if we have some version of multidimensional stable independence notion just for objects of size $\lambda$, then we can lift it up to a multidimensional stable independence notion on the whole category. Another key result is that once we have a multidimensional stable independence notion, we can get a much better approximation to existence of pushouts. In particular, among the independent amalgams of a given span, there will be a weakly initial one, and it will be unique up to (not necessarily unique) isomorphism. Such an object is called a prime object (over the span) by model theorists. In this sense, every span has a “very weak” colimit. This is enough to obtain a notion of generation that helps build a dimension theory. Finally, let’s note that categoricity is a $0$-dimensional uniqueness property, and we may want to know higher dimensional versions (even before proving existence of prime objects). For the two-dimensional case, instead of building amalgam that are, in the sense above, minimal, we will want to get amalgams that are maximal, in the sense of being very homogeneous (i.e. with a lot of injectivity). This is given by the notion of a limit object, which we survey (in the one dimensional case) in the next section. Element by element constructions in concrete categories {#aec-sec} ======================================================= The framework of abstract elementary classes is very useful to perform element by element constructions: we can use the concreteness to build certain objects “point by point”. We really will just use that we work in an accessible category with concrete directed colimits and all morphisms embedding (so nothing about the vocabulary will be needed). The reader uncomfortable with logic can think about this more general case. We first need a definition of a “point” and what it means for two points to be the same. This depends of course on a base set, e.g. in the category of fields, the elements $e^{1/2}$ and $e^{1/4}$ are the same over $\mathbb{Q}$ but not the same over $\mathbb{Q} (e)$. The equivalence class of a point over a given base will be called a type[^24]. The definition for AECs is due to Shelah, but in the wider setup of concrete categories it was first explored by Lieberman and Rosický [@ct-accessible-jsl 4.1]: \[point-def\] Let ${\mathbf{K}}= ({\mathcal {K}}, U)$ be a concrete category and let $A$ be an object of ${\mathcal {K}}$. The category of points over $A$, ${\mathbf{K}}_A^\ast$, is defined as follows: - Its objects are pairs $(f, a)$, where $f: A \to B$ and $a \in U B$. - A morphism from $(A \xrightarrow{f} B, a)$ to $(A \xrightarrow{g} C, b)$ is a ${\mathcal {K}}$-morphism $B \xrightarrow{h} C$ such that $h f = g$ and $h (a) = b$. This is simply a pointed version of the category of cocones over the diagram consisting of only $A$. Morphisms between two cocones must respect points. Note that we wrote $h (a)$ instead of the more pedantic $(U h) (a)$. \[type-def\] Let ${\mathbf{K}}= ({\mathcal {K}}, U)$ be a concrete category and let $A$ be an object of ${\mathcal {K}}$. Two points $(f, a)$, $(g, b)$ over $A$ have the same type if they are connected in the category of points over $A$ (Definition \[point-def\]). The type of a point over $A$ is just its equivalence class under the relation of having the same type (in the category of points over $A$). If ${\mathbf{K}}$ has amalgamation, then the category of points over $A$ also has amalgamation. Thus two points $(A \xrightarrow{f} B, a)$ and $(A \xrightarrow{g} C, b)$ have the same type exactly when they are jointly connected in the category of points over $A$ (see Lemma \[compat-lem\]). Explicitly, this means there exists $B \xrightarrow{h_1} D$ and $C \xrightarrow{h_2} D$ such that the diagram below commutes and moreover $h_1 (a) = h_2 (b)$: $$\xymatrix@=3pc{ B \ar@{.>}[r]^{h_1} & D \\ A \ar[u]^{f} \ar[r]_{g} & C \ar@{.>}[u]_{h_2} }$$ Note that if $(A \xrightarrow{f} B, a)$ is a point in an AEC, then after some “renaming”, it has the same type as some point $(A \xrightarrow{i} C, b)$, where $i$ is an inclusion (i.e. $A {{\le_{{\mathbf{K}}}}}C$). It can sometimes be convenient (to avoid keeping track of morphisms) to just look at points induced by inclusions, and this gives a slightly simpler notation for types. Nevertheless, we will not follow this approach here. Let ${\mathbf{K}}$ be an AEC and let[^25] $M \in {\mathbf{K}}$. 1. Let ${\textbf{S}}(M)$ be the collection of all types over $M$. 2. For $M {{\le_{{\mathbf{K}}}}}N$, a type $p \in {\textbf{S}}(M)$ is realized in $N$ if there exists a point $(M \xrightarrow{i} N, b)$ whose type over $M$ is $p$ (here, $i$ is the inclusion). \[type-monot\] The following are easy exercises from the definitions. Let ${\mathbf{K}}$ be an AEC, $M {{\le_{{\mathbf{K}}}}}N$. 1. If $p \in {\textbf{S}}(M)$ is realized in $N$ and $N {{\le_{{\mathbf{K}}}}}N'$, then $p$ is realized in $N$’. 2. If $p \in {\textbf{S}}(M)$, then there exists an extension of $M$ in which $p$ is realized. Moreover, by the smallness axiom, we can ensure that extension has size at most ${\text{LS}}({\mathbf{K}}) + \| U M\|$. In particular, ${\textbf{S}}(M)$ is a set. 3. If $q \in {\textbf{S}}(M_0)$, $M_0 {{\le_{{\mathbf{K}}}}}M$, and ${\mathbf{K}}$ has the amalgamation property, then there exists an extension of $M$ in which $q$ is realized (and we can similarly ensure the extension has size at most ${\text{LS}}({\mathbf{K}}) + \| U M\|$). How can types allow us to construct category-theoretically interesting objects? The following is an interesting definition: Let ${\mathbf{K}}$ be an AEC, let $\lambda$ be an infinite cardinal, and let $M {{\le_{{\mathbf{K}}}}}N$ both be in ${\mathbf{K}}$. We say that $N$ is $\lambda$-universal over $M$ if for any given $M \xrightarrow{f} M'$ such that $M'$ has size strictly less than $\lambda$, there exists $M' \xrightarrow{g} N$ such that the following diagram commutes (by convention, we will not label inclusions): $$\xymatrix@=3pc{ N & \\ M \ar[u] \ar[r]_{f} & M' \ar@{.>}[ul]_{g} }$$ When $\lambda = \|U M\|^+$, we omit it and say that $N$ is universal over $M$. By renaming, we can assume without loss of generality that $f$ is also an inclusion. Recall also that “size” in AECs means the same as “cardinality of the universe” (see \[unif-ex\](\[unif-ex-aec\])). In an AEC with amalgamation, we can build universal objects categorically by a simple exhaustion argument. However, in typical cases, $N$ above will be much bigger in size than $M$. For example, if $M$ has size $\lambda$ and $\|{\textbf{S}}(M)\| > \lambda$, then because a universal object over $M$ must realize all types over $M$, it must have at least $\lambda^+$-many elements. What if, on the other hand, $\|{\textbf{S}}(M)\| = \lambda$? This condition is given a name (recall that we use ${\mathbf{K}}_\lambda$ to denote the class of objects of size $\lambda$): \[stable-def\] An AEC is said to be $\lambda$-stable (or stable in $\lambda$) if $\|{\textbf{S}}(M)\| = \lambda$ for any $M \in {\mathbf{K}}_\lambda$. The name “stable” refers to the same kind of model-theoretic stability as “stable independence notion”. In fact, in the first-order case, being stable on a proper class of cardinals is equivalent to the existence of a stable independence notion. The reader can think of stability as saying that “most” elements over a given base are transcendentals (i.e. all have the same type). In general, the existence of a stable independence notion in an AEC implies stability in certain cardinals [@indep-categ-advances 8.16]. We will show that, assuming stability in $\lambda$, we can build universal extensions of the same cardinality. The result is due to Shelah [@shelahaecbook II.1.16] but the proof we give here is new and, as opposed to previous proofs (see [@ct-accessible-jsl 6.2]) does not use the coherence axiom. \[univ-ext\] Let ${\mathbf{K}}$ be an AEC and let $\lambda \ge {\text{LS}}({\mathbf{K}})$. Assume that ${\mathbf{K}}$ is stable in $\lambda$ and ${\mathbf{K}}_\lambda$ has amalgamation. For any $M \in {\mathbf{K}}_\lambda$, there exists $N \in {\mathbf{K}}_\lambda$ such that $M {{\le_{{\mathbf{K}}}}}N$ and $N$ is universal over $M$. We first build ${\langle M_i : i \le \lambda \rangle}$ increasing continuous in ${\mathbf{K}}_\lambda$ (i.e. $i < j$ implies $M_i {{\le_{{\mathbf{K}}}}}M_j$, all objects are in ${\mathbf{K}}_\lambda$, and at limit ordinals we take unions) such that $M_0 = M$ and $M_{i + 1}$ realizes all types over $M_i$ for all $i < \lambda$. This is possible: the only step to implement is the successor step, so assume that $M_i$ is given and we have to build $M_{i + 1}$. We know that $\|{\textbf{S}}(M_i)\| = \lambda$ by stability, so list the types as ${\langle p_j : j < \lambda \rangle}$. We build ${\langle M_{i, j} : j \le \lambda \rangle}$ increasing continuous in ${\mathbf{K}}_\lambda$ such that $M_{i, 0} = M_i$ and $M_{i, j + 1} \in {\mathbf{K}}_\lambda$ realizes $p_j$ for all $j < \lambda$. This is possible by Remark \[type-monot\] and at the end we can set $M_{i + 1} = M_{i, \lambda}$. We now prove that $M_\lambda$ is universal over $M_0$. This is the hard part of the argument. Fix $M_0 \xrightarrow{f_0} M_0'$, with $M_0' \in {\mathbf{K}}_\lambda$. We want to find $g: M' \to M_\lambda$ with $gf_0$ fixing $M_0$. For each $i \le j \le \lambda$, we will build $M_i \xrightarrow{f_i} M_i'$ and $M_i' \xrightarrow{h_{ij}} M_j'$ such that for $i \le j \le k \le \lambda$, $M_i' \in {\mathbf{K}}_\lambda$, $h_{ik} = h_{jk}h_{ij}$, $h_{ii} = {\operatorname{id}}_{M_i'}$, and $h_{ij} f_i = f_j {\upharpoonright}M_i$. That is, the following commutes: $$\xymatrix@=3pc{ M_i' \ar[r]_{h_{ij}} & M_j' \\ M_i \ar[u]_{f_i} \ar[r] & M_j \ar[u]_{f_j} }$$ We further require this diagram to be smooth (i.e. at limit ordinals $k$, $f_k$ is given by the colimit of $(f_i, h_{ij})_{i \le j < k}$). To go from $i$ to $i + 1$, we make sure that the type of a certain point $(M_i \xrightarrow{f_i} M_i', a_i)$ is realized in $M_{i + 1}$: there is $b_i \in U M_{i + 1}$ so that $(M_i \to M_{i + 1}, b_i)$ has the same type as $(f_i, a_i)$. This type equality is witnessed by $f_{i + 1}$ and $h_{i, i + 1}$. In particular, $h_{i, i + 1} (a_i) = f_{i + 1} (b_i)$, so $h_{i, i + 1} (a_i)$ is in the range of $f_{i + 1}$. We make sure to choose the $a_i$’s in such a way that, in the end, we catch our tail: $\{h_{j, \lambda} (a_j) : j < \lambda\} = U M_\lambda'$. This is possible by some bookkeeping (essentially just using that $\|\lambda \times \lambda\| = \lambda$). We give the details (for a more general setup) in the appendix, see Theorem \[univ-ext-pf\]. The construction we just described ensures that $f_\lambda$ is a surjection, hence an isomorphism! Thus $g := f_\lambda^{-1} h_{0 \lambda}$ is the desired embedding of $M_0'$ into $M_\lambda$. \[mh-sat-rmk\] A similar argument (see Theorem \[mh-sat-pf\]) establishes the model-homogeneous = saturated lemma [@shelahaecbook II.1.14]: when $\lambda \ge {\text{LS}}({\mathbf{K}})$, objects that realize all types over every substructure of size $\lambda$ will in fact be universal over every substructure of size $\lambda$. Once we know we can build universal extensions of the same cardinality, we can consider iterating them, and we get to the notion of a limit object: Let ${\mathbf{K}}$ be an AEC, let $M {{\le_{{\mathbf{K}}}}}N$, let $\lambda \ge {\text{LS}}({\mathbf{K}})$, and let $\delta < \lambda^+$ be a limit ordinal. We say that $N$ is $(\lambda, \delta)$-limit over $M$ if there exists an increasing continuous chain ${\langle M_i : i \le \delta \rangle}$ in ${\mathbf{K}}_\lambda$ such that $M_0 = M$, $M_\delta = N$, and $M_{i + 1}$ is universal over $M_i$ for all $i < \delta$. Theorem \[univ-ext\] establishes that, if ${\mathbf{K}}$ is stable in $\lambda$, there exists limit models. It turns out that limit models have uniqueness properties as well. For example, a back and forth argument establishes that for $\delta_1, \delta_2 < \lambda^+$, if $M_\ell$ is $(\lambda, \delta_\ell)$-limit over $M$ for $\ell = 1,2$ and ${\text{cf} (\delta_1)} = {\text{cf} (\delta_2)}$, then there exists an isomorphism from $M_1$ to $M_2$ which fixes $M$. What if ${\text{cf} (\delta_1)} \neq {\text{cf} (\delta_2)}$? The question of uniqueness of limit objects asks whether the above is still true in this case (this can be thought of as a one-dimensional version of categoricity). For model theorists, we note that a positive answer is equivalent to superstability for first-order theories (thus limit objects provide a good way to define superstability categorically). More generally, in a stable theory $T$, limit models will be isomorphic exactly when their cofinality is at least $\kappa (T)$. In any case, proving uniqueness of limit models requires a good notion of stable independence for elements, allowing one to build very independent objects point by points. This is another (much harder than in Theorem \[univ-ext\]) example of the use of “points” in the theory of AECs. See [@gvv-mlq] for an exposition of some result in the theory of limit objects in AECs, though stronger theorems have now been found, e.g. [@vandieren-symmetry-apal]. A state of the art result from categoricity is [@categ-saturated-afml 5.7(2)] (an AEC ${\mathbf{K}}$ has no maximal objects if for any $M \in {\mathbf{K}}$ there exists $N \in {\mathbf{K}}$ with $M {{\le_{{\mathbf{K}}}}}N$ and $M \neq N$): Let ${\mathbf{K}}$ be an AEC with amalgamation and no maximal objects. Let $\mu > {\text{LS}}({\mathbf{K}})$. If ${\mathbf{K}}$ is categorical in $\mu$, then for any $\lambda \in [{\text{LS}}({\mathbf{K}}), \mu)$, any two limit objects of cardinality $\lambda$ are isomorphic. The reader may ask what limit models look like in specific categories. This is studied in recent work of Kucera and Mazari-Armida [@limit-abelian-apal; @univ-modules-pp-v4] for certain categories of modules. We end this section by noting that stable independence notions themselves can be built “element by element”. For this, one starts with a good notion of stable independence for types (what Shelah calls a good frame), and tries to “paste the points together”. See [@jrsh875] for an introduction to the theory of good frames and [@downward-categ-tame-apal; @categ-primes-mlq; @categ-amalg-selecta] for recent applications to the categoricity spectrum problem. We state in particular the following result [@categ-amalg-selecta]: Let ${\mathbf{K}}$ be an AEC with amalgamation. Assume the weak generalized continuum hypothesis: $2^\lambda < 2^{\lambda^+}$ for all infinite cardinals $\lambda$. If ${\mathbf{K}}$ is categorical in some cardinal $\mu \ge {\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}$, then ${\mathbf{K}}$ is categorical in all cardinals $\mu' \ge {\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}$, and moreover there is a stable independence notion on the category ${\mathbf{K}}_{\ge {\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}}$. Note that the statements of the last two theorems are purely category-theoretic, in the sense that their statement does not use concreteness, points, etc (if the reader is worried about whether the definition of an AEC is category-theoretic, they can look at the results for the special case of a finitely accessible category with all morphisms monos, see Theorem \[aec-acc\]). I am not aware of “purely category-theoretic” proofs of any statements like this, so I suspect that the element by element methods used to study AECs can be useful even if one is interested only in categorical problems. Some known results and open questions {#problem-sec} ===================================== Questions on categoricity ------------------------- Shelah’s eventual categoricity conjecture for AECs (Conjecture \[categ-conj\]) is still open, in ZFC, but is known to hold from many different types of (quite mild) assumptions. In many cases, we can say more about the “high-enough” bound and even (in (\[categ-3\]) below) list exactly what the possibly categoricity spectrums are. For example: 1. (Large cardinal axioms, [@multidim-v2]) Assuming there is a proper class of strongly compact cardinals, an AEC categorical in a proper class of cardinals is categorical on a tail of cardinals. 2. (Large cardinal axioms plus cardinal arithmetic, [@multidim-v2]) Assuming there is a proper class of measurable cardinals and WGCH[^26], an AEC categorical in a proper class of cardinals is categorical on a tail of cardinals. 3. \[categ-3\] (Amalgamation plus cardinal arithmetic [@categ-amalg-selecta 9.7]) Assume WGCH. Let ${\mathbf{K}}$ be a large AEC with amalgamation (and ${\mathbf{K}}_{<{\text{LS}}({\mathbf{K}})} = \emptyset$). Exactly one of the following holds: 1. ${\mathbf{K}}$ is not categorical in any cardinal above ${\text{LS}}({\mathbf{K}})$. 2. There exists $n \le m < \omega$ such that ${\mathbf{K}}$ is categorical in all cardinals in $[{\text{LS}}({\mathbf{K}})^{+n}, {\text{LS}}({\mathbf{K}})^{+m}]$ and no other cardinals. 3. There exists $\chi < {\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}$ such that ${\mathbf{K}}$ is categorical in all $\mu \ge \chi$ (and no other cardinals). It is known that examples exist of all three types. 4. (No maximal objects plus strong cardinal arithmetic [@categ-amalg-selecta 10.14]) Assume $\Diamond_S$ for every stationary set $S$ (this holds for example if V = L). An AEC with no maximal objects categorical in a proper class of cardinals is categorical on a tail of cardinals. 5. (Universal class [@ap-universal-apal; @categ-universal-2-selecta]) If a universal class ${\mathbf{K}}$ is categorical in some $\mu \ge \beth_{{\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}}$, then it is categorical in all $\mu' \ge \beth_{{\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}}$. This holds more generally for multiuniversal classes [@abv-categ-multi-apal] Other approximations to categoricity (for example from tameness, a locality property of types that has not been discussed here) are in [@sh394; @tamenesstwo; @tamenessthree]. Note that given any finitely accessible category ${\mathcal {K}}$, we have that ${\mathcal {K}}_{mono}$ is an AEC and the embedding ${\mathcal {K}}_{mono} \to {\mathcal {K}}$ preserves and reflects presentability ranks, hence categoricity. Thus the results above are, in particular, valid for any finitely accessible category. 1. Is the eventual categoricity conjecture for AECs provable in, or at least consistent with, ZFC? 2. Is there a counterexample to eventual categoricity for accessible categories? Is it true at least for accessible categories with directed colimits? Can we generalize the proof of eventual categoricity for continuous first-order logic [@shus837] to category-theoretic setups? 3. (Diliberti) Can one give “purely category-theoretic” proofs of categoricity transfers, at least in simple cases (for example, for eventual categoricity in locally finitely presentable categories)? 4. Is eventual categoricity true for locally presentable categories? Questions on set-theoretic aspects ---------------------------------- 1. Is the presentability rank of every high-enough object in an accessible category always a successor? 2. What is an example of a large accessible category that is not LS-accessible[^27]? 3. If a category is $\lambda$-accessible, for $\lambda$ big-enough, does it have an object of size $\lambda$? (see Theorem \[ls-acc-thm\]) 4. Is any large accessible category with directed colimits LS-accessible? (This is asked already in [@ct-accessible-jsl]). 5. What other types of compactness can be expressed using accessible functors (Section \[set-func-sec\])? 6. Are there other local methods than Theorem \[ap-thm\] that allow us to prove amalgamation in ZFC? Regarding the second question, we can give a version that does not mention category-theoretic sizes: does there exist a $\mu$-AEC ${\mathbf{K}}$ such that, for a proper class of cardinals $\lambda$ with $\lambda < \lambda^{<\mu}$, ${\mathbf{K}}$ is categorical in $\lambda^{<\mu}$, and ${\mathbf{K}}$ has no objects of cardinality in $[\lambda, \lambda^{<\mu})$? See [@internal-sizes-jpaa 4.16] for why such a $\mu$-AEC is not LS-accessible. Questions on accessible categories vs AECs ------------------------------------------ 1. What is the role of the coherence axiom of AECs? 2. Is there a short characterization of AECs that is completely category-theoretic, in the sense that it does not refer to concrete functors (as in [@ct-accessible-jsl]), or embeddings into (variations on) category of structures (as in [@beke-rosicky 5.7])? 3. Is there a natural logic axiomatizing AECs (see [@logic-intersection-bpas §4])? Questions on stable independence -------------------------------- 1. What are more occurrences of stable independence in mainstream mathematics? 2. How does stable independence interact with accessible functors? 3. Can one characterize when the uniqueness of limit objects holds in terms of properties of stable independence (say in accessible categories with directed colimits)? 4. Does stable independence tell us anything interesting about metric classes (Banach spaces, Hilbert spaces, etc.)? See [@byuscontlog] on first-order stability theory for metric classes. 5. Is there a theory of independence in accessible categories mirroring that of independence in simple unstable first-order theories (see [@kp97])? What about other model-theoretic classes of unstable theories? Questions on the model theory of AECs ------------------------------------- The questions below are more technical, and cannot be understood just from the material of this paper. I chose to collect them here for the convenience of the expert reader: 1. If ${\mathbf{K}}$ is a $\lambda$-superstable AEC, does it have $\lambda$-symmetry (see [@vandieren-symmetry-apal])? More generally, what are the exact relationships between uniqueness of limit objects in $\lambda$, $\lambda$-symmetry, and $\lambda$-superstability? 2. If ${\mathbf{K}}$ is a $\lambda$-stable AEC and ${\mathbf{K}}_\lambda$ has amalgamation and no maximal objects, can we prove uniqueness of long-enough limit objects in $\lambda$, as in [@limit-strictly-stable-v4], but without using a continuity property for splitting? 3. Is the following stability spectrum theorem true? In a large stable ${\text{LS}}({\mathbf{K}})$-tame AEC with amalgamation, there exists $\chi < {\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}$ such that for all $\lambda \ge {\beth_{\left(2^{{\text{LS}}({\mathbf{K}})}\right)^+}}$, ${\mathbf{K}}$ is stable in $\lambda$ if and only if $\lambda = \lambda^{<\chi}$. Approximations are in [@stab-spec-jml]. Further reading {#reading-sec} =============== In addition to all the references cited already, we mention some resources that may help newcomers become acquainted with the field. We repeat again that Makkai-Paré [@makkai-pare] and especially Adámek-Rosický [@adamek-rosicky] are the standard textbooks on accessible categories. The category-theoretic singular compactness theorem (Theorem \[sing-compact\]) appears in [@cellular-singular-jpaa], which has numerous examples and explanations. The relationship between accessible categories and abstract elementary classes is investigated in, for example, [@lieberman-categ; @beke-rosicky; @ct-accessible-jsl]. The Beke-Rosický paper, specifically, started the abstract study of category-theoretic sizes continued in [@internal-sizes-jpaa; @internal-improved-v3-toappear]. The category-theoretic notion of stable independence is introduced in [@indep-categ-advances], and a follow-up (establishing the connection with cofibrant generation) is [@more-indep-v2]. To become acquainted with abstract elementary classes specifically, two introductions are Grossberg’s survey [@grossberg2002] and Baldwin’s book [@baldwinbook09]. Recently, classes about AECs were given at Harvard University by both Will Boney and myself, and both classes had lecture notes [@wb-aec-notes; @sv-aec-notes] that give an updated take on the subject. The survey about tame AECs [@bv-survey-bfo] may also be helpful to get acquainted with the literature. When one starts studying independence for types, Shelah’s good frames, a “pointed” and localized version of stable independence, become an unavoidable concept. Currently, the best introduction to good frames is the paper of Jarden and Shelah [@jrsh875]. Finally, it is impossible not to mention Shelah’s two volume book [@shelahaecbook; @shelahaecbook2] which has a very interesting and readable introduction, and is a gold mine of deep (but not always easily readable) results on good frames and AECs generally. Forcing and construction categories =================================== I give here a general category-theoretic framework for point by point “exhaustion arguments” such as building algebraic closure of fields (or more generally saturated models), or proving a given extension realizing all types many times is universal as in Theorem \[univ-ext\]. It is also a natural framework in which to understand set-theoretic forcing. To the best of my knowledge, this is new. A construction category is a triple ${\mathbf{K}}= ({\mathcal {K}}, U, U_0)$, where: 1. ${\mathcal {K}}$ is a category. 2. $U: {\mathcal {K}}\to {\operatorname{\bf Set}}$ is a faithful functor. 3. $U_0: {\mathcal {K}}\to {\operatorname{\bf Set}}$ is a faithful[^28] subfunctor of $U$: a faithful functor such that for all morphisms $A \xrightarrow{f} B$, $U_0 A \subseteq U A$ and $U_0 f = (U f) {\upharpoonright}U_0 A$. The idea is that, for an object $A$, $U A$ gives the elements that could “conceivably” be constructed at some point (e.g. ${\mathcal {K}}$ could be the category of fields and $U A$ give the polynomials with coefficients from $A$, see Example \[forcing-ex\]), while $U_0 A$ gives the elements that have been constructed already (e.g. in the example of the category of fields, $U_0 A$ could give the polynomials with coefficients from $A$ that have a root in $A$). We will be trying to find an object (called full below) where everything that can be constructed in some extension has been constructed already. It may help the reader to think of the category ${\mathcal {K}}$ as a partially ordered set. Let ${\mathbb{P}}$ be a partially ordered set (we think of it also as a category). A notion of forcing for ${\mathbb{P}}$ (e.g. in the sense of [@forcing-omitting]) associates to each $p \in {\mathbb{P}}$ a set of sentences that it “forces” in such a way that if $p \le q$ then $q$ forces more sentences than $p$. Setting $U_0 p$ to be the formulas that $p$ forces, and $U p$ to be the set of all formulas, we obtain a construction category. Let ${\mathbf{K}}$ be a construction category. 1. Given an object $A$ and an element $x \in U A$, we say that $x$ is constructed by stage $A$ if $x \in U_0 A$. We say that $x$ is constructible from $A$ if there exists a morphism $A \xrightarrow{f} B$ so that $f (x)$ is constructed by stage $B$. 2. A directed diagram $D: I \to {\mathbf{K}}$ with maps $D_i \xrightarrow{d_{ij}} D_j$ is full whenever the following is true: for any $i \in I$ and any $x \in U D_i$, if for all $j \ge i$, $d_{ij} (x)$ is constructible from $D_j$, then there exists $j \ge i$ such that $d_{ij} (x)$ is constructed by stage $D_j$. 3. For an object $A$ and a set $X \subseteq U A$, we say that $A$ is full for $X$ if any $x \in X$ that is constructible from $A$ is constructed by stage $A$. We say that $A$ is full if it is full for $U A$. The following are basic properties of the definitions: Let ${\mathbf{K}}$ be a construction category, $A \xrightarrow{f} B$ be a morphism, and $x \in U A$. 1. If $x$ is constructed by stage $A$, then $x$ is constructible from $A$. 2. If $x$ is constructed by stage $A$ then $f (x)$ is constructed by stage $B$. 3. If $f(x)$ is constructible from $B$, then $x$ is constructible from $A$. 4. If $A$ is an amalgamation base (Definition \[ap-def\]) and $x$ is constructible from $A$, then $f (x)$ is constructible from $B$. 5. If $A$ is full for $X$, then $B$ is full for $f[X]$. 6. An object $A$ is full if and only if the corresponding directed diagram with one object is full. \[full-lem\] Let ${\mathbf{K}}$ be a construction category and let $D: I \to {\mathcal {K}}$ be a directed diagram with maps $D_i \xrightarrow{d_{ij}} D_j$. 1. If $D$ is full, then for any cocone $(D_i \xrightarrow{f_i} A)_{i \in I}$ for $D$ and any $i \in I$, $f_i^{-1}[U_0 A] \cap U D_i \subseteq \bigcup_{j \ge i} d_{ij}^{-1}[U_0 D_j]$. 2. An object $A$ is full if and only if for any morphism $A \xrightarrow{f} B$, $U A \cap f^{-1}[U_0 B] = U_0 A$. 3. \[full-lem-3\] If $D$ is full and $(D_i \xrightarrow{f_i} A)_{i \in I}$ is a cocone for $D$, then $A$ is full for $\bigcup_{i \in I} f_i[U D_i]$. In particular, if $U A = \bigcup_{i \in I} f_i[U D_i]$ then $A$ is full. <!-- --> 1. Let $(D_i \xrightarrow{f_i} A)_{i \in I}$ be a cocone for $D$ and let $i \in I$. Let $x \in f_i^{-1}[U_0 A] \cap U D_i$. Let $y := f_i (x)$ (so $y \in U_0 A$). For any $j \ge i$, $f_j$ (and the fact that $y \in U_0 A$) witnesses that $d_{ij} (x)$ is constructible from $D_j$. By definition of a full diagram, there exists $j \ge i$ such that $d_{ij} (x)$ is constructed by stage $D_j$, i.e. $d_{ij} (x) \in U_0 D_j$. Thus $x \in d_{ij}^{-1} (U_0 D_j)$. 2. If $A$ is full, then the previous part gives that for any morphism $A \xrightarrow{f} B$, $U A \cap f^{-1}[U_0 B] \subseteq U_0 A$. The reverse inclusion is immediate because $U_0$ is a subfunctor of $U$. Conversely, assume that for any morphism $A \xrightarrow{f} B$, $U A \cap f^{-1}[U_0 B] = U_0 A$. Let $x \in U A$ be constructible from $A$. This means there exists $A \xrightarrow{f} B$ such that $f (x) \in U_0 B$, i.e. $x \in f^{-1}[U_0 B]$. By assumption, $x \in U_0 A$, so $x$ is constructed by stage $A$. 3. Assume that $D$ is full. Let $y \in \bigcup_{i \in I} f_i[U D_i]$ be constructible from $A$. There exists $i \in I$ such that $y = f_i (x)$ for some $x \in U D_i$. Since $y$ is constructible from $A$, $x$ is constructible from $D_i$. In fact, for any $j \ge i$, $f_j$ witnesses that $d_{ij} (x)$ is constructible from $D_j$. Since $D$ is full, there exists $j \ge i$ so that $d_{ij} (x) \in U_0 D_j$. Thus $y = f_j d_{ij} (x) \in f_j[U_0 D_j] \subseteq U_0 A$, so $y$ is constructed by stage $A$. Although we will in the end mostly be interested in full objects, it is often helpful (and easier) to first verify that a full diagram exists. Its colimit will then usually be the desired full object. In order for full diagrams to exist, objects should not be too big. We will measure size using cofinality of a certain “order of construction”. Let ${\mathbf{K}}$ be a construction category and let $A$ be an object. 1. Define a relation $\le$ on $U A$ as follows: $x \le y$ if for all $A \xrightarrow{f} B$, if $f (y)$ is constructed by stage $B$, then $f (x)$ is constructed by stage $B$. 2. Let $\\|A\\|$ denote the cofinality of $(U A, \le)$. Note that $\le$ is a preorder on $U A$, and $\\|A\\| \le \|U A\|$. The following will be our main tool to verify existence of full diagrams. \[full-existence\] Let ${\mathbf{K}}$ be a construction category and let $\lambda$ be an infinite cardinal. If $\\|A\\| \le \lambda$ for all objects $A$, and (for $\alpha < \lambda$) any $\alpha$-indexed diagram has a cocone, then ${\mathbf{K}}$ has a $\lambda$-indexed full diagram. This will be a consequence of the following more general version (used in the proof of Theorem \[univ-ext-pf\]), where we only require diagrams with “small” objects (according to a certain rank) to have an upper bound. \[full-existence-lem\] Let ${\mathbf{K}}$ be a construction category, let $\lambda$ be an infinite cardinal, and let $R : {\mathbf{K}}\to \lambda$ be given. If: 1. $\\|A\\| \le \lambda$ for all objects $A$. 2. For any object $A$ and any $x \in U A$ that is constructible from $A$, there exists $A \xrightarrow{f} B$ such that $R B \le R A + 1$ and $f(x)$ is constructed by stage $B$. 3. For every $\alpha < \lambda$ and every diagram $D: \alpha \to {\mathbf{K}}$, if $R D_i < \alpha$ for all $i < \alpha$, then $D$ has a cocone $(D_i \to A)_{i < \alpha}$ with $R A \le \alpha$. Then there is a $\lambda$-indexed full diagram $D: \lambda \to {\mathbf{K}}$ with $R D_i \le i$ for all $i < \lambda$. Apply Lemma \[full-existence-lem\], with $R A = 0$ for all objects $A$. First, we fix a function $F: \lambda \to \lambda \times \lambda$ such that for each pair $(\alpha, \beta)$ in $\lambda \times \lambda$, there exists unboundedly-many $i < \lambda$ so that $F (i) = (\alpha, \beta)$. This can be done by first partitioning $\lambda$ into $\lambda$-many disjoint pieces of cardinality $\lambda$, then bijecting each of these pieces with $\lambda \times \lambda$. Write $(\alpha_i, \beta_i)$ for $F (i)$. We inductively build objects ${\langle D_i : i < \lambda \rangle}$, morphisms ${\langle D_i \xrightarrow{d_{ij}} D_j : i \le j < \lambda \rangle}$, and sequences ${\langle x_{i, j} : i, j < \lambda \rangle}$ such that: 1. $R D_i \le i$ for all $i < \lambda$. 2. $d_{ii} = {\operatorname{id}}_{D_i}$, $d_{jk} d_{ij} = d_{ik}$ for all $i \le j \le k < \lambda$. 3. For each $i < \lambda$, ${\langle x_{i, j} ; j < \lambda \rangle}$ enumerates the elements of a cofinal set of $U D_i$ (perhaps with repetitions). 4. \[forcing-3\] For each $i < \lambda$, if $\alpha_i \le i$ and $d_{\alpha_i, i} (x_{\alpha_i, \beta_i})$ is constructible from $D_i$, then $d_{\alpha_i, i + 1} (x_{\alpha_i, \beta_i})$ is constructed by stage $D_{i + 1}$. Moreover, if there exists an object that is full over $D_i$, then $D_{i + 1}$ is full over $D_i$. : Let $D: \lambda \to {\mathcal {K}}$ be the diagram with maps $d_{ij}$. Then $D$ is the desired full diagram. Indeed, let $i < \lambda$, and let $x \in U D_i$ be such that $d_{ij} (x)$ is constructible from $D_j$ for all $j \ge i$. Let $\alpha := i$. Without loss of generality (using cofinality), $x = x_{\alpha, \beta}$ for some $\beta < \lambda$. By construction, there exists $j \ge i$ such that $(\alpha_j, \beta_j) = (\alpha, \beta)$. By assumption, $d_{i, j} (x)$ is constructible from $D_j$, so by (\[forcing-3\]), $d_{i, j + 1} (x)$ is constructed by stage $D_{j + 1}$, as desired. : We proceed by induction on $j < \lambda$. Assume inductively that ${\langle D_i : i < j \rangle}$, ${\langle d_{ii'} : i \le i' < j \rangle}$, and ${\langle x_{i, i'} : i < j, i' < \lambda \rangle}$ have been constructed. We will build $D_j$, ${\langle d_{i j} : i \le j \rangle}$, and ${\langle x_{j, j'} : j' < \lambda \rangle}$. First assume that $j$ is limit or zero. By assumption, the diagram $D: j \to {\mathcal {K}}$ with maps ${\langle d_{ii'} : i \le i' < j \rangle}$ has a cocone $(D_i \xrightarrow{d_{ij}} D_j)_{i < j}$, with $R D_j \le j$. Set $d_{jj} = {\operatorname{id}}_{D_j}$, and let ${\langle x_{j, j'} : j' < \lambda \rangle}$ be any enumeration of $U D_j$. Now assume that $j$ is a successor: $j = i + 1$. If $\alpha_i > i$, or $\alpha_i \le i$ but $d_{\alpha_i, i} (x_{\alpha_i, \beta_i})$ is not constructible from $D_i$, then set $B = D_i$, $f = {\operatorname{id}}_{D_j}$. If $\alpha_i \le i$ and $d_{\alpha_i, i} (x_{\alpha_i, \beta_i})$ is constructible from $D_i$, then let $D_i \xrightarrow{f} B$ witness this, with $R B \le i + 1$. Set $D_i = B$, $d_{i, j} = f$, and let $d_{j, j} = {\operatorname{id}}_{D_j}$, $d_{i_0, j} = d_{i, j} d_{i_0, i}$ for all $i_0 < i$, and ${\langle x_{j, j'} : j' < \lambda \rangle}$ be any enumeration of $U D_j$. Even if we are unable to directly build full objects, we can also find a lot of them in continuous-enough chains: \[refl-thm\] Let $\lambda$ be a regular uncountable cardinal and let ${\mathbf{K}}= ({\mathcal {K}}, U, U_0)$ be a construction category where ${\mathcal {K}}$ is just the ordered set $\lambda$ and $U$-images of morphisms are inclusions. If $\\|j\\| < \lambda$ for all $j < \lambda$, then the set $\{j < \lambda \mid j \text{ is full for } \bigcup_{i < j} U i \}$ is closed unbounded. Let $C := \{j < \lambda \mid j \text{ is full for } \bigcup_{i < j} U i \}$. - : let $j < \lambda$ be a limit ordinal such that unboundedly-many $i < j$ are in $C$. Let $x \in \bigcup_{i < j} U i$ be constructible from $j$. Pick $i' \in C \cap j$ such that $x \in \bigcup_{i < i'} U i$. Then $x$ is constructible from $i'$ hence, by definition of $C$, $x$ is constructed by stage $i'$, hence by stage $j$. - : let $\alpha < \lambda$. We build ${\langle \alpha_n : n < \omega \rangle}$ an increasing sequence of ordinals below $\lambda$ such that for all $n < \omega$: 1. $\alpha_0 = \alpha$. 2. Any $x \in U \alpha_n$ that is constructible in $\alpha_n$ is constructed by stage $\alpha_{n + 1}$. This is possible: given $\alpha_n$, for each $x \in U \alpha_n$ that is constructible in $\alpha_n$, there exists a least $i_x < \lambda$ such that $x$ is constructed by stage $i_x$. Since $\\|\alpha_n\\| < \lambda$ and $\lambda$ is regular, there exists $\alpha_{n + 1} > \alpha_n$ such that $\alpha_{n + 1} \ge i_x$ for all $x \in U \alpha_n$ constructible in $\alpha_n$. This is enough: let $\beta := \sup_{n < \omega} \alpha_n$. Since $\lambda$ is regular uncountable, $\beta < \lambda$. Moreover, $\beta$ is full for $\bigcup_{i < \beta} U i$. Indeed, if $x \in \bigcup_{i < \beta} U_i$ is constructible from $\beta$, then there exists $n < \omega$ such that $x \in U \alpha_n$ and $x$ is constructible from $\alpha_n$, hence constructed by stage $\alpha_{n + 1}$, hence by stage $\beta$. Many well known constructions can easily be seen as special cases of the theorems just stated: \[forcing-ex\] 1. (Zorn’s lemma) If ${\mathbb{P}}$ is a partially ordered set where each chain has an upper bound, then ${\mathbb{P}}$ has a maximal element. This can be obtained by applying Theorem \[full-existence\] to $\lambda = \|{\mathbb{P}}\| + \aleph_0$, ${\mathbf{K}}= ({\mathbb{P}}, U, U_0)$, where $U p = {\mathbb{P}}$ for all $p \in {\mathbb{P}}$ and $U_0 p = \{q \in {\mathbb{P}}\mid q \le p\}$. Any upper bound to the full diagram gives the desired maximal element. 2. (Existence of generics) Let ${\mathbb{P}}$ be the poset of all finite partial functions from $\omega$ to $\{0, 1\}$. Let ${\mathbf{K}}= ({\mathbb{P}}, U, U_0)$, where $U s = \omega$, $U_0 s = {\operatorname{dom}}(s)$. Then a full diagram $D: \omega \to {\mathbf{K}}$ corresponds to a (total) function $f: \omega \to \{0,1\}$. This generalizes to the existence of generics, in the sense of set-theoretic forcing [@jechbook 14.4]. 3. (Existence of algebraic closure) Every field $F$ has an algebraic closure: take $\lambda = \|F\| + \aleph_0$, ${\mathbf{K}}= ({\mathcal {K}}, U, U_0)$ where ${\mathcal {K}}$ is the category of field extensions of $F$ of cardinality at most $\lambda$, $U A$ is the set of all polynomials with coefficients from $A$, and $U_0 A$ is the set of all such polynomials with a root in $A$. The colimit of the full diagram given by Theorem \[full-existence\] is full (Lemma \[full-lem\](\[full-lem-3\])), hence algebraically closed. 4. (Existence of saturated models) Let ${\mathbf{K}}^\ast$ be an AEC with amalgamation and $\lambda > {\text{LS}}({\mathbf{K}}^\ast)$ be a regular cardinal such that ${\mathbf{K}}^\ast$ is stable in $\lambda$ and ${\mathbf{K}}_\lambda^\ast \neq \emptyset$. Let ${\mathbf{K}}= ({\mathcal {K}}, U, U_0)$, where ${\mathcal {K}}$ is the full subcategory of ${\mathbf{K}}^\ast$ with objects of cardinality $\lambda$, $U A$ is the set of all types over a substructure of $A$, and $U_0 A$ is the set of all such types that are realized in $A$. Note that we may well have $\|U A\| > \lambda$, but by stability we still have that $\\|A\\| \le \lambda$ (the types over $A$ form a cofinal set of size $\lambda$). The colimit of the full diagram given by Theorem \[full-existence\] is full, hence is a saturated object of ${\mathbf{K}}_\lambda^\ast$ (i.e. all types over substructure of cardinality strictly less than $\lambda$ are realized). 5. (Disjointness of filtrations on a club) If $\lambda$ is a regular uncountable cardinal, $A \subseteq B$ are sets of cardinality $\lambda$, and ${\langle A_i : i < \lambda \rangle}$, ${\langle B_i : i < \lambda \rangle}$ are increasing continuous chains of subsets of cardinality strictly less than $\lambda$ such that $A = \bigcup_{i < \lambda} A_i$ and $B = \bigcup_{i < \lambda} B_i$, then the set of $i < \lambda$ such that $A \cap B_i = A_i$ is closed unbounded (in particular, if $A = B$ then $A_i = B_i$ on a closed unbounded set). Indeed, let ${\mathbf{K}}= (\lambda, U, U_0)$, where $U i = A_i \cup B_i$, $U_0 i = A_i \cap B_i$. By Theorem \[refl-thm\], the set of $i$ such that $i$ is full is closed unbounded. Now, if $i$ is full and $x \in A \cap B_i$, then $x \in A_j \cap B_j$ for some $j$, so $x$ is constructible from $i$, so is in $A_i \cap B_i \subseteq A_i$. Conversely, if $x \in A_i$ then it is constructible from $i$, hence in $A_i \cap B_i \subseteq A \cap B_i$. Thus $A \cap B_i = A_i$. 6. Similarly to the previous example, density of reduced towers (in the study of uniqueness models, see e.g. [@gvv-mlq 5.5]) can be seen as describing the existence of a full object in an appropriate construction category. Let’s now give more details on the proof of Theorem \[univ-ext\]: \[univ-ext-pf\] Let ${\mathbf{K}}$ be an AEC, let $\lambda \ge {\text{LS}}({\mathbf{K}})$ be such that ${\mathbf{K}}_\lambda$ has amalgamation, and let ${\langle M_i : i \le \lambda \rangle}$ be increasing continuous in ${\mathbf{K}}_\lambda$ such that $M_{i + 1}$ realizes all types over $M_i$. Then $M_\lambda$ is universal over $M_0$. Let ${\mathbf{K}}^\ast = ({\mathcal {K}}^\ast, V, V_0)$ be defined as follows: - The objects of ${\mathcal {K}}^\ast$ are morphisms $M_i \xrightarrow{f} M$ for $M \in {\mathbf{K}}_\lambda$, $i < \lambda$, such that $f {\upharpoonright}M_0 = {\operatorname{id}}_{M_0}$. - A morphism from $M_i \xrightarrow{f} M$ to $M_j \xrightarrow{g} N$, with $i \le j < \lambda$, is a map $h: M \to N$ such that the following diagram commutes: $$\xymatrix@=3pc{ M \ar[r]^h & N \\ M_i \ar[u]_f \ar[r] & M_j \ar[u]_g \\ }$$ - $V (M_i \to M) = U M$, and the $V$-image of a morphism $h$ from $M_i \to M$ to $M_j \to N$ is $U h$ (where $U$ is the universe functor from ${\mathbf{K}}$ to ${\operatorname{\bf Set}}$) - $V_0 (M_i \xrightarrow{f} M) = U f[M_i]$, and the $V_0$-image of a morphism $h$ from $M_i \to M$ to $M_j \to N$ is $U h {\upharpoonright}f[M_i]$. This is easily checked to be a construction category. Now let $M_i \xrightarrow{f} N$ be given, with $i < \lambda$. We show that any $x \in U N$ is constructible from $f$. Indeed, $M_{i + 1}$ realizes all types over $M_i$, so in particular it realizes the type of $(x, M_i \xrightarrow{f} N)$. Thus there is $x' \in U M_{i + 1}$ and a commutative diagram: $$\xymatrix@=3pc{ M \ar[r]^h & N \\ M_i \ar[u]_f \ar[r] & M_{i + 1} \ar[u]_g \\ }$$ with $g (x') = h (x)$. In particular, $h (x) \in V_0 (g)$, so is constructed by stage $g$. Assume now that a diagram $D : \lambda \to {\mathcal {K}}^\ast$ is full. Let $M_j \xrightarrow{f} N$ be a colimit (in ${\mathbf{K}}$) of $D$ (so $j \le \lambda$). From the previous discussion, it is easy to see that $f$ is surjective, hence an isomorphism. This then gives the result: for any $N_0 \in {\mathbf{K}}_\lambda$, $M_0 \xrightarrow{f_0} N_0$, the subcategory ${\mathbf{K}}^\ast_{f_0}$ of objects of ${\mathbf{K}}^\ast$ above $f_0$ satisfies the hypotheses of Lemma \[full-existence-lem\] (with $R (M_i \to M) = i$), hence has a full diagram, whose colimit must therefore induce an embedding of $N_0$ into $M_\lambda$. We can similarly prove that model-homogeneous is equivalent to saturated (Remark \[mh-sat-rmk\]): \[mh-sat-pf\] Let ${\mathbf{K}}$ be an AEC, let $\lambda \ge {\text{LS}}({\mathbf{K}})$, and assume that ${\mathbf{K}}_{\lambda}$ has amalgamation. If $M \in {\mathbf{K}}_{\ge \lambda}$ realizes all types over every substructure of size $\lambda$, then $M$ is universal over every substructure of size $\lambda$. Similar to the proof of Theorem \[univ-ext-pf\]: this time the objects of the construction categories are maps $M_0 \to N_0$ with $M_0 {{\le_{{\mathbf{K}}}}}M$ and $M_0, N_0$ both of size $\lambda$, and the rest of the definition is analogous. A very short introduction to first-order stability {#fo-sec} ================================================== For the convenience of the unacquainted reader, I give a quick and self-contained construction of stable independence in the first-order case, and derive two consequences: the equivalence of stability (in terms of counting types) with no order property, and the ability to extract indiscernibles from long-enough sequences. All the material in this appendix is well known but I am not aware of a place where it appears in such compressed form. I assume a very basic knowledge of model theory, but not previous knowledge of stability theory. Throughout, we fix a complete first-order theory $T$ with only infinite models in a vocabulary $\tau$. For notational simplicity, we assume that $\|T\| = \|\tau\| + \aleph_0$. We fix a proper class sized “monster model” ${\mathfrak{C}}$ for $T$. This means that ${\mathfrak{C}}$ is universal and homogeneous, so for convenience we work inside ${\mathfrak{C}}$. We use the letters ${\bar{a}}, {\bar{b}}, {\bar{c}}$ for (possibly infinite) sequences of elements from ${\mathfrak{C}}$, ${\bar{x}}, {\bar{y}}, {\bar{z}}$ for (possibly infinite) sequences of variables, $A, B, C$ for subsets of ${\mathfrak{C}}$, and $M, N$ for elementary substructures of ${\mathfrak{C}}$. For a sequence ${\bar{a}}$, ${\operatorname{ran}}({\bar{a}})$ denotes its set of elements (i.e. its range when thought of as a function). We may write $A \cup B$ instead of $AB$, $A {\bar{b}}$ instead of $A \cup {\operatorname{ran}}({\bar{b}})$, etc. Formulas are denoted by $\phi ({\bar{x}}) , \psi ({\bar{x}})$, where ${\bar{x}}$ is a sequence of variables that contains all free variables from $\phi$ (but may contain more – $\phi$ always has finitely-many free variables of course). We write $\models \phi[{\bar{a}}]$ instead of ${\mathfrak{C}}\models \phi[{\bar{a}}]$, which means that $\phi$ holds in ${\mathfrak{C}}$ when ${\bar{a}}$ replaces ${\bar{x}}$ (we are very casual with arities). As usual, we often do not distinguish between $M$ and its universe $U M$. We also abuse notation by writing ${\bar{a}}\in A$ instead of the more proper ${\bar{a}}\in {{}^{<\infty}A}$. We write ${\mathbf{tp}}({\bar{b}}/ A)$ (the type of ${\bar{b}}$ over $A$) for the set of formulas $\phi ({\bar{x}}, {\bar{a}})$, where ${\bar{a}}\in A$ and $\models \phi ({\bar{b}}, {\bar{a}})$. We write ${\bar{b}}_1 \equiv_A {\bar{b}}_2$ to mean that ${\mathbf{tp}}({\bar{b}}_1 / A) = {\mathbf{tp}}({\bar{b}}_2 / A)$. Recall that, by the compactness theorem, this holds if and only if there exists an automorphism $f$ of ${\mathfrak{C}}$ that fixes $A$ pointwise and sends ${\bar{b}}_1$ to ${\bar{b}}_2$. Thus this corresponds to the notion defined in Definition \[type-def\]. We let ${{\textbf{S}}}(A) := \{{\mathbf{tp}}(b / A) \mid b \in {\mathfrak{C}}\}$, and more generally ${{\textbf{S}}}^{\alpha} (A) := \{{\mathbf{tp}}({\bar{b}}/ A) \mid {\bar{b}}\in {{}^{\alpha}{\mathfrak{C}}}\}$, ${{\textbf{S}}}^{<\infty} (A) = \bigcup_{\alpha} {{\textbf{S}}}^\alpha (A)$. Note that $\|{{\textbf{S}}}(A)\| \le 2^{\|T\| + \|A\|}$, and that if $A \subseteq B$ then there is a natural surjection of ${{\textbf{S}}}(B)$ into ${{\textbf{S}}}(A)$ (so $\|{{\textbf{S}}}(A)\| \le \|{{\textbf{S}}}(B)\|$). For $p \in {{\textbf{S}}}(B)$ and $A \subseteq B$, we write $p {\upharpoonright}A$ for the restriction of $p$ to ${{\textbf{S}}}(A)$: the set of formulas from $p$ with parameters in $A$. It is time to define the classes “well-behaved” of theories we will work with. As in Definition \[stable-def\], we will say that $T$ is $\lambda$-stable (or stable in $\lambda$) if for any $A$ of cardinality $\lambda$, $\|{{\textbf{S}}}(A)\| = \lambda$ (of course, this is exactly the same as stability in $\lambda$ in the sense of \[stable-def\], in the AEC of models of $T$ ordered by elementary substructure). We say that $T$ is stable if it is stable in some cardinal $\lambda \ge \|T\|$. The following closely related property will play a key role: \[op-def\] $T$ has the order property if there exists a sequence ${\langle {\bar{a}}_i : i < \omega \rangle}$ and a formula $\phi ({\bar{x}}, {\bar{y}})$ such that $\models \phi[{\bar{a}}_i, {\bar{a}}_j]$ if and only if $i < j$. It turns out that $T$ is stable if and only if it does not have the order property. We prove one direction now. The other will be dealt with after we have constructed stable independence. \[stab-nop\] If $T$ has the order property, then $T$ is unstable. Fix a cardinal $\lambda \ge \|T\|$, and fix a linear order $I$ of cardinality $\lambda$ with strictly more than $\lambda$ Dedekind cuts (if $\lambda = \aleph_0$, the rationals are such an order; in general take $\sigma$ minimal such that $\lambda < \lambda^{\sigma}$ and the set ${{}^{<\sigma}\lambda}$ ordered lexicographically will do the trick). Fix a formula $\phi ({\bar{x}}, {\bar{y}})$ witnessing the order property. Using the compactness theorem, there exists a ${\langle {\bar{a}}_i : i \in I \rangle}$ such that for all $i, j \in I$, $\models \phi[{\bar{a}}_i, {\bar{a}}_j]$ if and only if $i < j$. Since $\phi$ has finitely-many free variables we can of course assume wihout loss of generality that the ${\bar{a}}_i$’s are of finite length. Now each Dedekind cut of $I$ induces a distinct type over $\bigcup_{i \in I} {\operatorname{ran}}({\bar{a}}_i)$, a set of size $\lambda$. Thus $T$ is not stable in $\lambda$. We now define what it means for two sets to be “as independent as possible” over a base. For simplicity, the base is required to be a model. More advanced introductions investigate what happens when the base is an arbitrary set. We write ${\bar{a}}{\unionstick}_M {\bar{b}}$, and say that ${\bar{a}}$ and ${\bar{b}}$ are independent over $M$, if whenever ${\bar{c}}\in M$ and $\models \phi[{\bar{a}}, {\bar{b}}, {\bar{c}}]$, there exists ${\bar{a}}' \in M$ such that $\models \phi[{\bar{a}}', {\bar{b}}, {\bar{c}}]$. Note that if ${\bar{a}}'$, ${\bar{b}}'$ have the same range as ${\bar{a}}$, ${\bar{a}}'$ respectively, then ${\bar{a}}{\unionstick}_M {\bar{b}}$ if and only if ${\bar{a}}' {\unionstick}_M {\bar{b}}'$. Thus we will also write for example $A {\unionstick}_M B$ to mean that ${\bar{a}}{\unionstick}_M {\bar{b}}$ for some (equivalently any) enumerations ${\bar{a}}$, ${\bar{b}}$ of $A$ and $B$ respectively. Another way of saying the same thing: if $M \subseteq B$ and $p \in {{\textbf{S}}}^{<\infty} (B)$, we say that $p$ is free over $M$ if it is finitely satisfiable over $M$: any formula $\phi ({\bar{x}}, {\bar{b}})$ in $p$ is realized in $M$. Note that ${\bar{a}}{\unionstick}_M B$ if and only if ${\mathbf{tp}}({\bar{a}}/ MB)$ is free over $M$. We will freely go back and forth between these two point of views (types and independence notion ${\unionstick}$). Which one is easier to work with depends on the specific concepts we are studying. \[indep-constr\] Assume that $T$ does not have the order property. 1. (Invariance) If $A {\unionstick}_M B$ and $f$ is an automorphism of ${\mathfrak{C}}$, then $f[A] {\unionstick}_{f[M]} f[B]$. 2. (Normality) If $A {\unionstick}_M B$, then $A M {\unionstick}_M BM$. 3. (Left and right monotonicity) If $A {\unionstick}_M B$ and $A_0 \subseteq A$, $B_0 \subseteq B$, then $A_0 {\unionstick}_M B_0$. 4. (Base monotonicity) If $A {\unionstick}_M B$ and $M \preceq N \subseteq B$, then $A {\unionstick}_N B$. 5. (Finite character) $A {\unionstick}_M B$ if and only if $A_0 {\unionstick}_M B_0$ for all finite $A_0 \subseteq A$, $B_0 \subseteq B$. 6. (Disjointness) If $A {\unionstick}_M B$, then $A \cap B \subseteq M$. 7. (Symmetry) $A {\unionstick}_M B$ if and only if $B {\unionstick}_M A$. 8. (Transitivity) If $M_0 \preceq M_1 \preceq M_2$, $A {\unionstick}_{M_0} M_1$, and $A {\unionstick}_{M_1} M_2$, then $A {\unionstick}_{M_0} M_2$. 9. (Local character) For any $A$ and $N$, there exists $M \preceq N$ of cardinality at most $\|A\| + \|T\|$ such that $A {\unionstick}_M N$. 10. (Uniqueness) If $M \subseteq B$, $p, q \in {{\textbf{S}}}^{<\infty} (B)$ are both free over $M$ and $p {\upharpoonright}M = q {\upharpoonright}M$, then $p = q$. 11. (Extension) If $p \in {{\textbf{S}}}^{<\infty} (M)$ and $M \subseteq B$ is a set, there exists $q \in {{\textbf{S}}}^{<\infty} (B)$ that extends $p$ and is free over $B$. Invariance, normality, the monotonicity properties, and finite character are immediate from the definition. To see disjointness, it is enough to see that if $a {\unionstick}_M a$ then $a \in M$. This follows from the definition applied with the formula $x = y$. Let us prove the other properties: - : Suppose not. Fix ${\bar{a}}$, ${\bar{b}}$, and $M$ so that ${\bar{a}}{\unionstick}_M {\bar{b}}$ but ${\bar{b}}{\nunionstick}_M {\bar{a}}$. Without loss of generality, ${\bar{a}}$ and ${\bar{b}}$ are finite and we can pick a formula $\phi ({\bar{x}}, {\bar{y}})$ witnessing ${\bar{b}}{\nunionstick}_M {\bar{a}}$ that has all parameters from $M$ already incorporated in ${\bar{a}}$: $\models \phi[{\bar{a}}, {\bar{b}}]$ but for all ${\bar{b}}' \in M$, $\models \neg \phi[{\bar{a}}, {\bar{b}}']$ (we have swapped the role of ${\bar{x}}$ and ${\bar{y}}$ for convenience in the proof that follows). We inductively build two sequences ${\langle {\bar{a}}_i : i < \omega \rangle}, {\langle {\bar{b}}_i : i < \omega \rangle}$ of tuples in $M$ such that $\models \phi[{\bar{a}}_i, {\bar{b}}_j]$ if and only if $i \le j$, and $\models \phi[{\bar{a}}_i, {\bar{b}}]$ for all $i < \omega$. This is enough: set $\psi ({\bar{x}}_1, {\bar{y}}_1, {\bar{x}}_2, {\bar{y}}_2)$ to be $\phi ({\bar{x}}_1, {\bar{y}}_2) \land {\bar{x}}_1{\bar{y}}_1 \neq {\bar{x}}_2{\bar{y}}_2$. Then $\psi$ and the sequence ${\langle {\bar{a}}_i{\bar{b}}_i : i < \omega \rangle}$ witness the order property, contradiction. This is possible: Fix $j < \omega$ and assume we are given ${\bar{a}}_i$ and ${\bar{b}}_i$ for all $i < j$. By the induction hypothesis, we know that: $$\models \phi ({\bar{a}}, {\bar{b}}) \land \bigwedge_{i < j} \phi ({\bar{a}}_i, {\bar{b}}) \land \bigwedge_{i < j} \neg \phi ({\bar{a}}, {\bar{b}}_i)$$ (the last part is from the hypothesis that $\models \neg \phi ({\bar{a}}, {\bar{b}}')$ for any ${\bar{b}}' \in M$). Because ${\bar{a}}{\unionstick}_M {\bar{b}}$, there exists ${\bar{a}}' \in M$ such that: $$\models \phi ({\bar{a}}', {\bar{b}}) \land \bigwedge_{i < j} \phi ({\bar{a}}_i, {\bar{b}}) \land \bigwedge_{i < j} \neg \phi ({\bar{a}}', {\bar{b}}_i)$$ Since $M \preceq {\mathfrak{C}}$, there exists ${\bar{b}}' \in M$ such that: $$\models \phi ({\bar{a}}', {\bar{b}}') \land \bigwedge_{i < j} \phi ({\bar{a}}_i, {\bar{b}}') \land \bigwedge_{i < j} \neg \phi ({\bar{a}}', {\bar{b}}_i)$$ Set ${\bar{a}}_j := {\bar{a}}'$, ${\bar{b}}_j := {\bar{b}}'$. - : Using the definition of independence, it is easy to check the “left” version of transitivity: if $M_0 \preceq M_1 \preceq M_2$, $M_2 {\unionstick}_{M_1} A$, and $M_1 {\unionstick}_{M_0} A$, then $M_2 {\unionstick}_{M_0} A$. The “right” version of transitivity then follows from symmetry. - : This is a downward Löwenheim-Skolem closure argument, that we could do explicitly. Instead, fix $A$ and $N$, and pick a pair of models $M \preceq M'$ such that $A \subseteq M'$, $\|U M'\| \le \|A\| + \|T\|$, and $(M, M') \preceq (N, {\mathfrak{C}})$ (in the vocabulary with an additional predicate for $M$). From the definition of independence, it follows that $N {\unionstick}_M M'$, so by monotonicity $N {\unionstick}_M A$, hence by symmetry $A {\unionstick}_M N$. - : This is similar to symmetry: let ${\bar{b}}$ be an enumeration of $B - M$. Suppose $p = {\mathbf{tp}}({\bar{a}}/ M {\bar{b}})$, $q = {\mathbf{tp}}({\bar{a}}' / M {\bar{b}})$ both are free over $M$ (so ${\bar{a}}{\unionstick}_M {\bar{b}}$, ${\bar{a}}' {\unionstick}_M {\bar{b}}$), and $p {\upharpoonright}M = q {\upharpoonright}M$. We have to see that $p = q$. Without loss of generality, ${\bar{b}}$, ${\bar{a}}$, and ${\bar{a}}'$ are finite. Suppose $p \neq q$, and let $\phi ({\bar{x}}, {\bar{y}})$ be such that $\models \phi[{\bar{a}}, {\bar{b}}] \land \neg \phi[{\bar{a}}', {\bar{b}}]$. Define sequences ${\langle {\bar{a}}_i : i < \omega \rangle}$, ${\langle {\bar{b}}_i : i < \omega \rangle}$ in $M$ such that for all $i, j < \omega$: 1. $\models \phi[{\bar{a}}_i, {\bar{b}}]$. 2. $\models \phi[{\bar{a}}_i, {\bar{b}}_j]$ if and only if $i \le j$. 3. \[cond3\] $\models \neg \phi[{\bar{a}}, {\bar{b}}_i]$. This is enough: Then $\psi ({\bar{x}}_1, {\bar{y}}_1, {\bar{x}}_2, {\bar{y}}_2) := \phi ({\bar{x}}_1, {\bar{y}}_2) \land {\bar{x}}_1 {\bar{y}}_1 \neq {\bar{x}}_2 {\bar{y}}_2$ together with the sequence ${\langle {\bar{a}}_i{\bar{b}}_i : i < \omega \rangle}$ witness the order property. This is possible: Suppose that ${\bar{a}}_i, {\bar{b}}_i$ have been defined for all $i < j$. By the induction hypothesis, we have: $$\models \bigwedge_{i < j} \phi [{\bar{a}}_i, {\bar{b}}] \land \bigwedge_{i < j} \neg \phi [{\bar{a}}, {\bar{b}}_i] \land \phi [{\bar{a}}, {\bar{b}}]$$ Since ${\bar{a}}{\unionstick}_M {\bar{b}}$, there is ${\bar{a}}'' \in M$ such that: $$\models \bigwedge_{i < j} \phi [{\bar{a}}_i, {\bar{b}}] \land \bigwedge_{i < j} \neg \phi [{\bar{a}}'', {\bar{b}}_i] \land \phi [{\bar{a}}'', {\bar{b}}]$$ We also know that $\models \neg \phi[{\bar{a}}', {\bar{b}}]$. Combining this with the above and the fact that ${\bar{a}}' {\unionstick}_M {\bar{b}}$, hence by symmetry ${\bar{b}}{\unionstick}_M {\bar{a}}'$, we obtain a ${\bar{b}}'' \in M$ such that: $$\models \bigwedge_{i < j} \phi [{\bar{a}}_i, {\bar{b}}''] \land \bigwedge_{i < j} \neg \phi [{\bar{a}}'', {\bar{b}}_i] \land \phi [{\bar{a}}'', {\bar{b}}''] \land \neg \phi[{\bar{a}}', {\bar{b}}'']$$ Let ${\bar{a}}_j := {\bar{a}}''$, ${\bar{b}}_j := {\bar{b}}''$. It is easy to check that this works (for condition (\[cond3\]), we use that $p {\upharpoonright}M = q {\upharpoonright}M$ so $\models \neg \phi[{\bar{a}}', {\bar{b}}'']$ implies $\models \neg \phi[{\bar{a}}, {\bar{b}}'']$). - : This is the only place where we use the compactness theorem. Consider the set $q$ of formulas $\phi ({\bar{x}}, {\bar{b}})$ where ${\bar{b}}\in B$, $p \cup \{\phi ({\bar{x}}, {\bar{b}})\}$ is consistent, and there exists ${\bar{a}}' \in M$ such that $\models \phi [{\bar{a}}', {\bar{b}}]$. Note that $q$ is closed under conjunctions, hence by construction and compactness is consistent. It remains to check that $q$ is complete. Indeed, assume that $\neg \phi ({\bar{x}}, {\bar{b}}) \notin q$. There are two cases. If $p \cup \{\neg \phi ({\bar{x}}, {\bar{b}})\}$ is inconsistent, then $p \models \phi ({\bar{x}}, {\bar{b}})$, so there is $\psi ({\bar{x}}, {\bar{c}}) \in p$ such that $\psi ({\bar{x}}, {\bar{c}}) \models \phi ({\bar{x}}, {\bar{b}})$, with ${\bar{c}}\in M$. We know that ${\mathfrak{C}}\models \exists {\bar{x}}\psi ({\bar{x}}, {\bar{c}})$, so $M \models \exists {\bar{x}}\psi ({\bar{x}}, {\bar{c}})$, hence $\phi ({\bar{x}}, {\bar{b}})$ is also realized in $M$. In the second case, $p \cup \{\neg \phi ({\bar{x}}, {\bar{b}})\}$ is consistent but $\neg \phi ({\bar{x}}, {\bar{b}})$ is not realized in $M$. In particular, $\models \phi[{\bar{a}}', {\bar{b}}]$ for all ${\bar{a}}' \in M$. As before, for any $\psi ({\bar{x}}, {\bar{c}}) \in p$, there exists ${\bar{a}}' \in M$ so that $M \models \psi [{\bar{a}}', {\bar{c}}]$, so $M \models \psi[{\bar{a}}', {\bar{c}}] \land \phi[{\bar{a}}', {\bar{b}}]$, hence $p \cup \{\phi ({\bar{x}}, {\bar{b}})\}$ is finitely consistent, hence consistent. This shows that $\phi ({\bar{x}}, {\bar{b}}) \in q$, as desired. It is straightforward to check that ${\unionstick}$ as defined here yields a stable independence notion (in the sense of Definition \[indep-def\]). The existence of an independence notion as in the theorem implies stability, thus we get: \[nop-stab\] If $T$ does not have the order property (or just the conclusion of Theorem \[indep-constr\] holds), then $T$ is stable in every infinite cardinal $\lambda$ with $\lambda = \lambda^{\|T\|}$. In particular, $T$ is stable if and only if $T$ does not have the order property. The “in particular” part will follow from Theorem \[stab-nop\]. Now assume that $T$ does not have the order property, or just that the conclusion of Theorem \[indep-constr\] holds. Fix an infinite cardinal $\lambda$ such that $\lambda = \lambda^{\|T\|}$. Since any set of cardinality $\lambda$ is contained in a model of cardinality $\lambda$, it is enough to count types over models. Fix $M$ of cardinality $\lambda$ and a sequence ${\langle p_i : i < \lambda^+ \rangle}$ of types over $M$. We will show that there exists $i < j$ so that $p_i = p_j$. First, for each $i < \lambda^+$, local character tells us there exists $M_i \preceq M$ of cardinality at most $\|T\|$ such that $p$ is free over $M_i$. Note that $\|\{M_i : i < \lambda^+\}\| \le \lambda^{\|T\|} = \lambda$, so by the pigeonhole principle there exists $S \subseteq \lambda^+$ of cardinality $\lambda^+$ and $i_0 < \lambda^+$ such that for any $i \in S$, $p_i$ is free over $M_{i_0}$. Now $\|{{\textbf{S}}}(M_{i_0})\| \le 2^{\|T\|} \le \lambda^{\|T\|} = \lambda$, so by the pigeonhole principle again, there exists $i < j$ in $S$ such that $p_i {\upharpoonright}M_{i_0} = p_j {\upharpoonright}M_{i_0}$. Since both $p_i$ and $p_j$ are free over $M_{i_0}$, uniqueness implies that $p_i = p_j$. From now on, we will freely use the equivalence between stability and no order property, as well as the conclusion of Theorem \[indep-constr\] (but not the exact definition of independence). \[split-lem\] Assume that $T$ is stable. If ${\bar{a}}{\unionstick}_M B$ and ${\bar{b}}_1, {\bar{b}}_2 \in B$ are such that ${\bar{b}}_1 \equiv_M {\bar{b}}_2$, then ${\bar{b}}_1 {\bar{a}}\equiv_M {\bar{b}}_2 {\bar{a}}$. Fix an automorphism $f$ of ${\mathfrak{C}}$ fixing $M$ and sending ${\bar{b}}_1$ to ${\bar{b}}_2$. Let ${\bar{a}}' := f ({\bar{a}})$. Then ${\bar{b}}_1 {\bar{a}}\equiv_M {\bar{b}}_2 {\bar{a}}'$. On the other hand, ${\bar{a}}{\unionstick}_M {\bar{b}}_1$, so ${\bar{a}}' {\unionstick}_M {\bar{b}}_2$. By monotonicity, also ${\bar{a}}{\unionstick}_M {\bar{b}}_2$, so by uniqueness, ${\bar{b}}_2 {\bar{a}}' \equiv_{M} {\bar{b}}_2 {\bar{a}}$. Combining the two equality of types shows that ${\bar{b}}_1 {\bar{a}}\equiv_M {\bar{b}}_2 {\bar{a}}$. As a final application, we will show how to extract indiscernibles in stable theories. \[indisc-extraction\] Assume that $\lambda > \|T\|$ and $T$ is stable in $\lambda$. If ${\langle a_i : i < \lambda^+ \rangle}$ is a sequence, there exists $I \subseteq \lambda^+$ of cardinality $\lambda^+$ such that ${\langle a_i : i \in I \rangle}$ is indiscernible. First, build ${\langle M_i : i \le \lambda^+ \rangle}$ an increasing continuous sequence of models of size $\lambda$ such that $a_i \in M_{i + 1}$ for all $i < \lambda^+$. Let $S := \{i < \lambda^+ \mid {\text{cf} (i)} \ge \|T\|^+\}$. This is a stationary set. By local character, for each $i \in S$ there exists $j_i < i$ such that $a_i {\unionstick}_{M_{j_i}} M_i$. By Fodor’s lemma, there exists $S_0 \subseteq S$ stationary and $j^\ast < \lambda^+$ such that $j_i = j^\ast$ for all $i \in S_0$. Let’s prune a bit more: by stability $\|{{\textbf{S}}}(M_{j^\ast})\| = \lambda$. Thus by the pigeonhole principle we can find an unbounded $I \subseteq S$ such that for $i < i'$ in $I$, $a_{i} \equiv_{M_{j^\ast}} a_{i'}$. We prove that ${\langle a_i : i \in I \rangle}$ is indiscernible over $M := M_{j^\ast}$. For this, we show by induction on $n < \omega$ that for any $i_0 < \ldots < i_{n}$, $i_0' < \ldots < i_{n}'$ in $I$, $a_{i_0} \ldots a_{i_n} \equiv_M a_{i_0'} \ldots a_{i_n'}$. The base case has just been observed. Assume now that $n = m + 1$, set ${\bar{b}}_1 := a_{i_0} \ldots a_{i_m}$, ${\bar{b}}_2 := a_{i_0'} \ldots a_{i_m'}$, and assume we know ${\bar{b}}_1 \equiv_M {\bar{b}}_2$. Without loss of generality, $i_n' \le i_n$. By monotonicity, $a_{i_n} {\unionstick}_M M_{i_n'}$, so by uniqueness $a_{i_n} \equiv_{M_{i_n'}} a_{i_n'}$. In particular, ${\bar{b}}_2 a_{i_n} \equiv_M {\bar{b}}_2 a_{i_n'}$. By Lemma \[split-lem\], we also have that ${\bar{b}}_1 a_{i_n} \equiv_M {\bar{b}}_2 a_{i_n}$. Combining these two type equalities gives that ${\bar{b}}_1 a_{i_n} \equiv_M {\bar{b}}_2 a_{i_n'}$, as desired. [^1]: Roughly, ${\mathbb{L}}_{\infty, \infty}$ is the logic where we allow infinitary quantifications, conjunctions, and disjunctions. See Section \[struct-sec\]. [^2]: In fact I have seen categorical model theory described as “model theory without logic”. [^3]: Shelah does not give a precise meaning to “external”: the interpretation as “invariant under equivalence of category” is my own suggestion. [^4]: In fact, the condition in the definition of an accessible category that every object is a directed colimit of a fixed set of small subobjects can itself be thought of as a weak type of compactness. [^5]: Since any $\infty$-AEC has all morphisms monomorphisms, the precise correspondence should say that accessible categories with all morphisms monos are equivalent to $\infty$-AECs. [^6]: In this paper, we will be more interested in colimits than limits, so we also define cocones rather than cones. [^7]: This example was pointed out to me by Ivan Di Liberti. [^8]: We will work with a single sort for simplicity. [^9]: For example, in the category of divisible abelian groups, the quotient map $\mathbb{Q} \to \mathbb{Q} / \mathbb{Z}$ turns out to be a monomorphism [@joy-of-cats 7.33(5)]. [^10]: When $\lambda$ is regular, this is written $\mu \ll \lambda$ in [@htt-lurie A.2.6.3]. The notation can be a misleading though, because it is not true that $\mu \ll \lambda < \lambda'$ implies $\mu \ll \lambda'$ (take for example $\mu = \aleph_1$, $\lambda = \aleph_2$, $\lambda' = \aleph_{\omega + 1}$). [^11]: For example, the categories of finite dimensional vector spaces over $\mathbb{R}$ and of real matrices (where the objects are natural numbers and the morphisms matrices with the right dimension, with composition defined by matrix multiplication) are equivalent but not isomorphic, see [@joy-of-cats 3.35(2)]. [^12]: That is, a functor which is faithful and injective on objects. [^13]: Shelah stated his result in terms of a class of models omitting types, but the proof of Theorem \[tarski\] shows that any universal class is a type-omitting class: omit the types of $\mu$-generated models outside of $K$. [^14]: A cardinal is weakly inaccessible if it is regular and limit. [^15]: The singular cardinal hypothesis is the statement that $\lambda^{{\text{cf} (\lambda)}} = \lambda^+ + 2^{{\text{cf} (\lambda)}}$ for every infinite cardinal $\lambda$. [^16]: A regular cardinal $\kappa$ is strongly compact if every theory in ${\mathbb{L}}_{\kappa, \kappa}$ with all its subsets of size strictly less than $\kappa$ consistent (i.e. with a model) is consistent. See for example [@jechbook 20.2]. [^17]: A subset of a regular uncountable cardinal $\lambda$ is called stationary if it intersects every closed unbounded subset of $\lambda$. If we think of closed unbounded subsets as having full measure, being stationary means having positive measure. [^18]: The statement that $2^\lambda = \lambda^+$ for any infinite cardinal $\lambda$. [^19]: This is in fact an instance of Theorem \[refl-thm\] in the appendix, see Example \[forcing-ex\]. [^20]: Model-theorists call this property categoricity, see \[categ-def\]. [^21]: The terminology comes from posets. [^22]: For example, the proofs of existence and uniqueness of differential closure in differentially closed fields (Example \[indep-ex-0\](\[diff-field-ex\])) rely on properties of the independence notion. [^23]: This is defined by generalizing the definition given in Example \[indep-ex-0\](\[indep-stable-ex-0\]) and in \[op-def\]: there is no long sequence that can be ordered by a type in the sense to be given in Section \[aec-sec\]. See [@indep-categ-advances 9.7]. [^24]: In the AEC literature, the terms “Galois types” or “orbital types” are used. This is to avoid confusion with the model-theoretic syntactic types. However, there is not really a Galois theory and types are not necessarily orbits. Moreover in the first-order case the syntactic types are the same as the orbital types, and we will never refer to syntactic type. Thus we prefer the simpler terminology. [^25]: We shift the notation to use $M$ and $N$ instead of $A$ and $B$ – this is to emphasize that the objects are structures (models). [^26]: Recall that this means that $2^\lambda < 2^{\lambda^+}$ for all infinite cardinals $\lambda$. [^27]: Such an example would yield to a failure of eventual categoricity: take the coproduct with the category of sets [@beke-rosicky 6.3]. [^28]: It seems the faithfulness of $U$ and $U_0$ is never used.
--- abstract: 'Let $X$ be an Alexandrov space (with curvature bounded below). We determine the maximal dimension of the isometry group $\operatorname{Isom}(X)$ of $X$ and show that $X$ is isometric to a Riemannian manifold, provided the dimension of $\operatorname{Isom}(X)$ is maximal. We determine gaps in the possible dimensions of $\operatorname{Isom}(X)$. We determine the maximal dimension of $\operatorname{Isom}(X)$ when the boundary $\partial X$ is non-empty and classify up to homeomorphism Alexandrov spaces with boundary and isometry group of maximal dimension. We also show that a symmetric Alexandrov space is isometric to a Riemannian manifold.' address: - 'Mathematisches Institut, WWU Münster, Germany.' - ' Department of Mathematics, Universidad Autónoma de Madrid, and ICMAT CSIC-UAM-UCM-UC3M, Spain.' author: - 'Fernando Galaz-Garcia\' - 'Luis Guijarro\\' title: Isometry groups of Alexandrov spaces --- [^1] [^2] Introduction ============ Alexandrov spaces (with a lower curvature bound) are synthetic generalizations of Riemannian manifolds with a lower (sectional) curvature bound. These spaces arise naturally as Gromov-Hausdorff limits of sequences of Riemannian $n$-manifolds with a uniform lower curvature bound or as orbit spaces of isometric group actions on Riemannian manifolds with curvature bounded below. Alexandrov spaces have provided a useful tool in the study of smooth and Riemannian manifolds (see, for example, [@Gr1; @Gr2; @Pe2; @Pe3; @Pe4; @SY; @Wi]). Since the class of Alexandrov spaces properly contains the class of Riemannian manifolds, it is natural to ask to what extent Riemannian geometry can be generalized to the Alexandrov setting. This problem has received significant attention, especially from an analytic point of view (see, for example, [@KMS] and references therein). In this paper we adopt the viewpoint of transformation groups and focus our attention on isometries of Alexandrov spaces, inspired by the fact that in the Riemannian case the isometry group has been extensively studied (see, for example, [@Ko] and related bibliography). In [@FY], Fukaya and Yamaguchi showed that, as in the case of Riemannian manifolds (see [@MS]), the isometry group of an Alexandrov space is a Lie group. By a theorem of van Dantzig and van der Waerden [@vDvW] (cf. Theorem 1.1 in [@Ko Chapter II]), the isometry group of a compact Alexandrov space is compact. The presence of an isometric action has been used to obtain further information on the structure of Alexandrov spaces. In [@Ber], Berestovski[ĭ]{} considered finite dimensional homogeneous metric spaces and showed that when they have a lower curvature bound, they are also smooth manifolds (see also [@BePl]). The structure of Alexandrov spaces of cohomogeneity one and their classification in low dimensions appears in [@GaSe]. The main difference between isometries of Alexandrov spaces and isometries of Riemannian manifolds is that the former must preserve the metric singularities of the space, i.e., they must map extremal sets to extremal sets, points with a given type of space of directions to others of the same type, etc. This not only imposes severe restrictions on the possible isometries but, reciprocally, the existence of isometries usually results in the appearance of additional structure in the singular sets of an Alexandrov space. The present paper explores this theme in several directions, extending various results on isometry groups of Riemannian manifolds to the Alexandrov case. We start with section \[S:Prelims\], where we have collected some background material, along with Alexandrov versions of the Isotropy Lemma and the Principal Isotropy Theorem. In section \[S:Bounds\] we derive an upper bound on the dimension of the isometry group of an Alexandrov space of dimension $n$; this bound is the same as the one for Riemannian $n$-manifolds (cf. [@Ko]), which is natural, since the existence of singular points in an Alexandrov space should mean fewer isometries than in the Riemannian case, as pointed out above. We make this more explicit in section \[S:Rigidity\], where we prove that the maximal dimension for the isometry group of an Alexandrov space is only attained in the Riemannian case. The results in sections \[S:Bounds\] and \[S:Rigidity\] were proved for Riemannian orbifolds, a special class of Alexandrov spaces, in [@BZ]. In section \[S:bounds\_extremal\] we bound the dimension of the isometry group of an Alexandrov space when there are extremal subsets. In section \[S:Gap\_thms\] we show that the gaps in the possible dimensions of the isometry group of a Riemannian manifold derived in [@Ma] also occur for Alexandrov spaces. In section \[S:Bdry\] we bound the dimension of the isometry group of an Alexandrov space with boundary and classify these spaces up to homeomorphism when the isometry group has maximal dimension. This extends to the Alexandrov setting the work carried out in [@CSX] for Riemannian manifolds with boundary. In our case, there appears a non-manifold with boundary and isometry group of maximal dimension, namely, the cone over a real projective space. Finally, in section \[S:Sym\_spaces\], we study symmetric Alexandrov spaces following previous work of Berestovski[ĭ]{} [@Be]. The first named author thanks the Department of Mathematics of the Universidad Autónoma de Madrid for its hospitality while part of the work presented in this paper was carried out. Both authors would also like to thank Martin Weilandt and John Harvey for helpful comments on a first version of this paper. Preliminaries {#S:Prelims} ============= In this section we fix notation and collect some preliminary material that we will use in the rest of the paper. For the sake of completeness, in subsection \[SS:PIThm\] we prove the Isotropy Lemma and the Principal Orbit Theorem for isometric Lie actions on Alexandrov spaces. These results are well known in the smooth case (cf. [@Gr2]). Background and notation ----------------------- Recall that a length space $(X,\mathrm{dist})$ of finite (Hausdorff) dimension is an Alexandrov space* (with curvature bounded below) if it has curvature bounded from below in the triangle comparison sense. We refer the reader to [@BBI; @BGP] for the main definitions and theorems pertaining to these spaces. Further developments can be found in [@Pet2]. The primary source of technical difficulties in the study of Alexandrov spaces is usually due to the presence of singularities. However, given an Alexandrov space $X$, work of Otsu and Shioya [@OS], and of Perelman [@Pe], allows one to introduce a $C^0$ structure on a large open subset of $X$. More precisely, call a point $p\in X$ regular if its space of directions $\Sigma_p$ is isometric to the unit round sphere ${\mathrm{\mathbb{S}}}^{n-1}$; otherwise, $p$ is called singular. We will denote the set of regular points of $X$ by $R_X$, and the set of singular points by $S_X$, so that $R_X=X\setminus S_X$. The set $R_X$ is the intersection of countably many open and dense subsets of $X$ and is dense in $X$; any regular point has an open neighborhood $U$ and a map $\phi:U\to{\mathbb{R}}^n$ that is a homeomorphism onto its image (cf. [@OS; @Pe]). In fact, there have been improvements on the smoothness of the structure (cf. [@Ot]), although we shall omit them here, since they are not necessary in what follows. Proving some of our statements in the Riemannian case requires using the differential of a map. Fortunately, this can also be defined for Alexandrov spaces (see, for example, [@Ly] for the more general statements). Given an Alexandrov space $X$, we will denote its isometry group by $\operatorname{Isom}(X)$. The distance between two points $p,q\in X$ will be denoted by $\|pq\|$. Given an isometric action $G\times X\to X$ of a group $G\leq \operatorname{Isom}(X)$, we will denote the orbit of a point $p\in X$ by $G p$. The isotropy group at $p$ will be denoted by $G_p$. Given a subset $A\subset X$, we will denote its image in $X/G$ under the orbit projection map $\pi:X\rightarrow X/G$ by $A^$. Following this notation, we will denote the orbit space of the action $G\times X\rightarrow X$ by $X^$, and a point in $X^$ will be denoted by $p^$, corresponding to the orbit $G p$. A useful fact about $\pi$ is that it is a submetry, and as such admits horizontal lifts of geodesics from $X^$ to $X$ (see, for instance, [@BeG]). We will also use the fact that the natural action of $\operatorname{Isom}(X)$ on $X$ has closed orbits (see, for instance, [@KN], section I.4). We will assume all actions to be effective and all Alexandrov spaces to be connected. We refer the reader to [@Br] for further background on the theory of transformation groups. Isotropy Lemma and Principal Orbit Theorem {#SS:PIThm} ------------------------------------------ We now enunciate and prove Alexandrov versions of the Isotropy Lemma and the Principal Orbit Theorem. The main ideas follow closely Lemma 1.3 and Theorem 1.4 in [@Gr2], which are originally stated for the smooth case. We start with the Isotropy Lemma, whose smooth version is due to Kleiner [@Kl]. Let $G$ be a Lie group acting isometrically on an Alexandrov space $X$. If $c:[0,d]\rightarrow X$ is a minimal geodesic between the orbits $Gc(0)$ and $Gc(d)$, then, for any $t\in (0,d)$, the isotropy group $G_{c(t)}=G_c$ is a subgroup of $G_{c(0)}$ and of $G_{c(d)}$. Fix $t\in (0,d)$ and let $g\in G_{c(t)}$. Proceeding by contradiction, suppose that $g\notin G_{c(d)}$. The image of the minimal geodesic segment $c([t,d])$ under $g$ is a horizontal minimal geodesic segment from $gc(t)=c(t)$ to $gc(d)\neq c(d)$; it follows that the curve $c[0,t]$ followed by $gc[t,d]$ still realizes the distance between the orbits $Gc(0)$ and $Gc(d)$, hence it is a geodesic. This is impossible, since we would have bifurcation of geodesics at $c(t)$ (see Figure \[F:isot\_lemma\]). Let $G$ be a compact Lie group acting isometrically on an $n$-dimensional Alexandrov space $X$. Then there is a unique maximal orbit type and the orbits with maximal orbit type, the so-called principal orbits, form an open and dense subset of $X$. ![Proof of the Isotropy Lemma[]{data-label="F:isot_lemma"}](isot_lemma.eps) Among the isotropy groups of least dimension, let $H_0$ be one with the least number of connected components. Such an $H_0$ exists because isotropy groups are compact. Let $p_0\in X$ be a point with isotropy group $H_0$. We claim that $H_0$ corresponds to a maximal orbit type. To see this, fix $p\in X$ distinct from $p_0$ and let $c:[0,1]\to X$ be a minimal geodesic between the orbits $G p_0$ and $G p$, with $c(0)=p_0$ and $c(1)=p$. Observe that the Isotropy Lemma and the choice of $H_0$ imply that $G_{c(t)}=H_0$ for $t\in [0,1)$. Thus, $H_0$ (or one of its conjugates) is a subgroup of the isotropy group of any other point in $X$, and therefore the orbit through $p_0$ is a maximal orbit Denote by $X_{(H_0)}$ the set of maximal orbits. Since any point $q\in X$ is the endpoint of a minimal geodesic between orbits starting at the orbit through $p_0$, it is clear that $X_{(H_0)}$ is dense in $X$. It remains to show that $X_{(H_0)}$ is open. This follows from the fact that, given an orbit $P\in X_{(H_0)}$, there exists a neighborhood of $P$ such that for any other orbit $Q$ in this neighborhood $\mathrm{type}(Q)\geq \mathrm{type}(P)$ (cf. [@Br Cor. II.5.5]). Since the orbit type of $P$ is maximal, the conclusion follows. Dimension bound {#S:Bounds} =============== We start by obtaining upper bounds for the dimension of the isometry group of an Alexandrov space. \[dimension\_bound\] Let $X$ be an Alexandrov space of dimension $n$. Then the dimension of its isometry group $G$ is at most $n(n+1)/2$. We will use induction on the dimension of $X$. There is clearly nothing to prove in the case $n=0$, so we will assume that the theorem holds for all positive integers less than $n$. \[L:identity\] Let $X$ be a connected Alexandrov space and $f:X\rightarrow X$ an isometry. If there is some $p_0\in X$ such that $f(p_0)=p_0$ and $df_{p_{0}}:\Sigma_{p_0}\rightarrow \Sigma_{p_0}$ is the identity, then $f(p)=p$ for all $p$ in $X$. It suffices to prove that the set of fixed points of $f$ is open in $X$. This follows from observing that $f(\exp_p(t v))=\exp_{f(p)}(t df_p(v))$, whenever both sides are defined. Since any point of $X$ can be connected to $p$ by a shortest geodesic, the set $$\{\, q\in X : q=\exp_p(tv)\text{ for some $v\in \Sigma_p$, $t\geq 0$}\,\}$$ contains an open neighborhood of $p$. Therefore, since $f(p)=p$ and $df_p={\mathsf{1}}$, the isometry $f$ must fix that same open neighborhood. The main consequence of the lemma is that for any $p\in X$, the isotropy group $G_{p}$ acts faithfully by isometries on $\Sigma_{p}$. It follows from the induction hypothesis that $\dim G_{p}\leq \dim \operatorname{Isom}(\Sigma_{p})\leq n(n-1)/2$. Choose now a regular point $p_0\in R_X$. We define the Myers-Steenrod map* $F:G\rightarrow X$ as $F(g)=g\cdot p_0$. Clearly, $F$ is continuous, because the topology of $G$ agrees with the compact-open topology; also, $F(G)\subset R_X$, since each isometry of $X$ maps regular points to regular points and singular points to singular points. Let $X^{ob}\supseteq R_X$ be the set of points in $X$ for which there are at least $n$ directions making obtuse angles among them (cf. [@Pe]). Fix some $g_0\in G$ and take a chart $(U,\varphi)$ of $X^{ob}$ around $g_0\cdot p_0$. Observe that $\varphi:U\rightarrow \mathbb{R}^n$ is a homeomorphism on $R_X\cap U$ for the structure introduced in [@OS] or [@Pe]. Now choose open neighborhoods $V$, $W$ around $g_0$ in $G$ such that $\overline{V}\subset W$ is compact and with $\varphi\circ F$ defined on $W$. Since the quotient map $\pi:G\to G/G_{p_0}$ is a fiber bundle, we can assume, reducing the size of $W$ if necessary, that there is a section $s: \widetilde{W}\subseteq G/G_{p_0}\longrightarrow W$ with $\pi(W)=\widetilde{W}$ and $s([g_0])=g_0$. Composing the maps $s$, $F$ and $\varphi$ results in a map $\tilde{F}:\widetilde{W}\longrightarrow \mathbb{R}^n$, given explicitly by $$\tilde{F}([g])=\varphi(s[g]\cdot p_0),$$ which is clearly continuous and injective. Finally, observe that $\pi(\overline{V})$ is compact, $\tilde{F}\lvert_{\pi(\overline{V})}$ is continuous and injective, and $\mathbb{R}^n$ is Hausdorff. It follows that the map $$\tilde{F}\lvert_{\pi(\overline{V})}:\pi(\overline{V})\rightarrow \tilde{F}(\pi(\overline{V}))\subseteq \mathbb{R}^n$$ is a homeomorphism onto a subspace of $\mathbb{R}^n$. Hence $\dim \pi(\overline{V})\leq \dim\mathbb{R}^n=n$, and $$\label{dimension_sum} \dim G=\dim \pi(\overline{V})+\dim G_{p_0} \leq n+\frac{n(n-1)}{2}=\frac{n(n+1)}{2}.$$ Rigidity {#S:Rigidity} ======== In this section we consider the case in which the dimension of the isometry group attains its maximal possible value. \[T:Rigidity\] Let $X$ be an $n$-dimensional Alexandrov space. If the dimension of the isometry group of $X$ is $n(n+1)/2$, then $X$ is isometric to one of the following space forms: 1. An $n$-dimensional Euclidean space $\mathbb{R}^n$. 2. An $n$-dimensional sphere $\mathbb{S}^n$. 3. An $n$-dimensional projective space $\mathbb{R}P^n$. 4. An $n$-dimensional simply connected hyperbolic space $\mathbb{H}^n$. It is enough to show that $X$ is isometric to a Riemannian manifold, since the conclusion of the theorem holds in the Riemannian case (see, for instance, Theorem 3.1 in [@Ko], p. 46). As in the the previous section, we proceed by induction, and we will continue to use the set up and notation introduced in the proof of Theorem \[dimension\_bound\]. Clearly, the starting case is trivial and hence we assume that the theorem holds for every positive integer less than $n$. Assume now that equality holds in equation \[dimension\_sum\], so that $\dim G_{p_0}=n(n-1)/2$ and $\dim \pi(\overline{V})=n$. Since $\Sigma_{p_0}$ is isometric to the unit round sphere $\mathbb{S}^{n-1}$, then $\dim G_{p_0}=n(n-1)/2$ implies that, up to finite factors, $G_{p_0}=\mathrm{SO}(n)$. We need now the following two lemmas: \[L:empty\] The set of singular points $S_X$ is empty. Let $q$ be a point in $S_X$. Since $G_{p_0}$ acts transitively on $\Sigma_{p_0}$, the metric sphere $S_{p_0}(\|p_0q\|)$ is entirely composed of singular points. Let $p_1\not\in\overline{B_{p_0}(\|p_0q\|)}$ be a regular point. A geodesic from $p_0$ to $p_1$ must intersect $S_{p_0}(\|p_0q\|)$ at some singular point. This contradicts the fact that regular points in an Alexandrov space form a convex set (see [@Pet]); hence $X=\overline{B_{p_0}(\|p_0q\|)}$. In fact, we could assume $q$ to be the singular point closest to $p_0$ (and therefore we have shown that in our situation such singular points can not be dense). Since $p_0$ was an arbitrary regular point in $X$ and these form a dense subset of $X$, we conclude that $S_X$ is empty. \[L:onto\] In the above situation, the Myers-Steenrod map $F:G\longrightarrow X$ is surjective. We will first show that $F:G\longrightarrow R_X$ is an open map. Consider the following commutative diagram: $$\xymatrix{ W\ar[r]^F & U \ar[d]^{\phi} \\ \pi(\overline{V}) \ar[r]^{\tilde{F}} \ar[u]^{s} & {\mathbb{R}}^n }$$ where $U$, $V$ and $W$ are defined as in the proof of Theorem \[dimension\_bound\]. We have $\dim \pi(\overline{V})=n$ and $\pi(\overline{V})$ has nonempty interior. Since $\tilde{F}$ is a homeomorphism onto its image, $\tilde{F}(\operatorname{int}\pi(\overline{V}))$ is an open subset of dimension $n$ in $\tilde{F}(\pi(\overline{V}))\subseteq {\mathbb{R}}^n$. Hence $\operatorname{Image}(\tilde{F})$ is open in ${\mathbb{R}}^n$ and $\operatorname{Image}(F) =G p_0$ is open in $X$. Since $X$ is connected, and the orbits of $G$ give a partition of $X$, it follows that there can exist only one orbit. Therefore, $X=G p_0$. Lemmas \[L:empty\] and \[L:onto\] imply that $X$ is a homogeneous metric space with a lower curvature bound. It then follows from a result of Berestovski[ĭ]{} [@Ber Theorem 7] that $X$ is isometric to a Riemannian manifold, as claimed. This ends the proof of the main theorem in this section. To prove Theorem \[T:Rigidity\] one could also show first that $\operatorname{Isom}(X)$ acts transitively on the regular set $R_X$ and then use that the orbits of the action of the isometry group are closed sets. Nevertheless, the authors did not see a significant advantage in proceeding in this way. Dimension bound in the presence of extremal sets {#S:bounds_extremal} ================================================ If an Alexandrov space contains extremal sets, then any isometry must preserve these, thus having restricted its possible image. As pointed out in the introduction, this has consequences for the dimension of the isometry group. We make this explicit in the next theorem. \[T:Bound\_ext\_set\] Let $X$ be an $n$-dimensional Alexandrov space. If $X$ contains a $k$-dimensional connected extremal subset $E$, then the dimension of the isometry group of $X$ cannot exceed $${k+1 \choose 2} + {n-k \choose 2}.$$ We will use reverse induction on the dimension $k$. Naturally, if $k=0$, then $E$ is an isolated point $p$. Any element in the connected component of the identity of $\operatorname{Isom}(X)$ must preserve $p$, and hence its differential induces an isometry on $\Sigma_p$. Since the action of $\operatorname{Isom}(X)_p$ is faithful on $\Sigma_p$, it follows from Theorem \[dimension\_bound\] that $\dim\operatorname{Isom}(X)\leq n(n-1)/2$, as desired. Assume now that the theorem is true for any Alexandrov space of dimension $n$ or less, and for all extremal subsets of dimension up to $k-1$. Any element in the connected component of the identity of $\operatorname{Isom}(X)$ must map $E$ to points of $E$, and thus the orbit of some $p\in E$ has dimension less than or equal to $k$. On the other hand, elements in $\operatorname{Isom}(X)_p$ act faithfully on $\Sigma_p$ and it is well known that the set of tangent directions to $E$ forms an extremal subset of $\Sigma_p$ of dimension $k-1$ (cf. [@PP]). Hence, by the induction hypothesis, $$\dim \operatorname{Isom}_p\leq {k \choose 2} + {n-k \choose 2}$$ and $$\dim\operatorname{Isom}(X)\leq k+\dim\operatorname{Isom}_p(X)\leq {k+1 \choose 2} + {n-k \choose 2}.$$ The bounds given in the preceding theorem are optimal, as the following example shows. Let $X$ be the Alexandrov space obtained from gluing in ${\mathbb{R}}^{n+1}$ a cone with vertex $p_0$ with a hemisphere along their boundaries. Both spaces have nonnegative sectional curvature, thus $X$ has the same lower curvature bound. If the cone is taken with angle less than $\pi/2$, then its vertex $p_0$ is an extremal set in $X$. Since $X$ has spherical symmetry about its axis, its isometry group is $O(n)$, thus proving that the bound can not be improved when $k=0$. For higher $k$ take the product of this example with ${\mathbb{R}}^k$, and let $E$ be $\{p_0\}\times {\mathbb{R}}^k$. Clearly the isometry group of the whole space is $O(k)\times O(n)$, which corresponds to the bound given in Theorem \[T:Bound\_ext\_set\] once dimensions are readjusted. Dimension gaps {#S:Gap_thms} ============== We generalize to Alexandrov spaces Theorems 3.2 and 3.3 in [@Ko section II.3]. The first theorem below shows that, as in the Riemannian case, there exists a gap in the possible dimensions of the isometry group of an Alexandrov space $X$ of dimension $n\neq 4$. The second theorem shows that $X$ is isometric to a Riemannian manifold when $\operatorname{Isom}(X)$ has the next possible largest dimension. As pointed out in [@Ko], the techniques used in the proof of the Riemannian versions of these theorems do not work when $n=4$, due to the fact that $\mathrm{SO}(4)$ is not simple, and thus this case must be considered separately (cf. [@Is]). Mann [@Ma] showed that, for Riemannian manifolds, there is a more general gap phenomenon which includes as a particular case Theorem 3.2 in [@Ko]. The third theorem in this section shows that this phenomenon also occurs for Alexandrov spaces. \[T:gaps\] Let $X^n$ be an Alexandrov space of dimension $n\neq 4$. Then $\operatorname{Isom}(X)$ contains no closed subgroups of dimension $m$ for $$\frac{1}{2}n(n-1)+1<m<\frac{1}{2}n(n+1).$$ Let $G\leq \operatorname{Isom}(X)$ be a subgroup of dimension $m$, and let $G_p$ be the isotropy group of $G$ at a regular point $p\in R_X$. Observe first that the action of $G$ on $X$ induces an embedding $F:G/G_p \hookrightarrow X$, so that $$\label{EQ:isotropy_ineq} \dim G_p\geq \dim G -\dim X.$$ Let $m>\frac{1}{2}n(n-1)+1$. Then, using inequality \[EQ:isotropy\_ineq\], we get $$\begin{aligned} \dim G_p & > & \frac{1}{2}n(n-1)+1-n\\[7pt] & = & \frac{1}{2}(n-1)(n-2)+1.\end{aligned}$$ Since $\Sigma_p$ is isometric to the unit round sphere $\mathbb{S}^{n-1}$, the isotropy group $G_p$ is a closed subgroup of $\mathrm{O}(n)$. It follows from the lemma on page 48 of [@Ko] that $G_p=\mathrm{SO}(n)$ or $\mathrm{O}(n)$, and therefore it acts transitively on $\Sigma_p$. Recall that the set $R_X$ of regular points of $X$ is convex (see [@Pet]). Thus, given points $p,q\in R_X$ and a geodesic $\gamma$ between them, its midpoint $r$ is in $R_X$, and thus $G_r$ acts transitively on $\Sigma_r$. There is then an isometry fixing $r$ and mapping $\gamma$ to $\gamma^{\mathrm{op}}$ $(=\gamma(a-t))$, hence mapping $p$ to $q$ and vice versa. This implies that $\operatorname{Isom}(X)$ acts transitively on $R_X$. Since $R_X$ is dense in $X$ and the orbits of $\operatorname{Isom}(X)$ are closed in $X$, it follows that $X=R_X$ and $\operatorname{Isom}(X)$ acts transitively on $X$. Hence $$\begin{aligned} m=\dim G & = & \dim X + \dim G_p \\ & = & n+ \dim \mathrm{O}(n)\\ & = & \frac{1}{2}n(n+1).\end{aligned}$$ We can also prove the following rigidity result. \[rigidity\] Let $X$ be an Alexandrov space of dimension $n\neq 4$. If $\operatorname{Isom}(X)$ has dimension $\frac{1}{2}n(n-1)+1$, then $X$ is isometric to a Riemannian manifold, and thus is one of those listed in Theorem 3.3, p. 54 in [@Ko]. Since a homogeneous Alexandrov space must be isometric to a Riemannian manifold (cf. [@Ber theorem 7]), it suffices to show that $\operatorname{Isom}(X)$ acts transitively on $X$. If $\operatorname{Isom}(X)$ does not act transitively on $X$, then $$\dim \operatorname{Isom}(X)_p \geq \dim \operatorname{Isom}(X)-(n-1)=\frac{1}{2}(n-1)(n-2)+1.$$ As in the proof of the previous theorem, $\operatorname{Isom}(X)_p=\mathrm{O}(n)$ or $\mathrm{SO}(n)$, $R_X=X$ and $\operatorname{Isom}(X)$ acts transitively on $X$. The case $n=4$ is slightly different from the cases considered in Theorems \[T:gaps\] and \[rigidity\] above, and we treat it separately. As pointed out at the beginning of this section, its peculiarity arises from the fact that $\mathrm{SO}(4)$ is not simple. The main tool used is Ishihara’s paper [@Is], where the Riemannian case was analyzed. For $n=4$, the statement of Theorem \[T:gaps\] changes to allow for the possibility of $8$-dimensional subgroups in $\operatorname{Isom}(X)$; its proof follows along the same lines as for general $n$, although one needs to use the lemma in [@Is p. 347], ruling out the existence of $5$-dimensional subgroups of $\mathrm{SO}(4)$. Thus, the dimension of $G_p$ must be $6$, and therefore it acts transitively on $\mathbb{S}^3$. To obtain the corresponding rigidity results one proceeds as in the proof of Theorem \[rigidity\]. Using the fact that $\mathrm{SO}(4)$ has no $5$-dimensional subgroups, it is easy to see that a $4$-dimensional Alexandrov space $X$ with a group of isometries $G$ of dimension $7$ or $8$ must be a homogeneous space. Therefore, $X$ must be isometric to a homogeneous Riemannian manifold, and hence one of those considered by Ishihara in [@Is]. When $G$ is $7$-dimensional, this yields the analog of Theorem \[rigidity\] in dimension $4$. In the exceptional case, where $G=8$, the space $X$ must be isometric to a Kähler manifold of constant holomorphic sectional curvature (cf. [@Is section 4]. These Kähler manifolds do not have higher dimensional analogues in the list of manifolds that occur for general $n$ in Theorem \[rigidity\], in contrast to the $4$-dimensional spaces with a $7$-dimensional group of isometries. We conclude this section with an extension to Alexandrov spaces of Mann’s gap theorem in [@Ma]. Observe that we recover Theorem \[T:gaps\] by setting $k=1$ in the theorem below. Let $X$ be an $n$-dimensional Alexandrov space with $n\neq 4, 6, 10$. Then the isometry group $\operatorname{Isom}(X)$ contains no closed subgroup $G$ such that the dimension of $G$ falls into any of the ranges:\ $${n-k+1 \choose 2}+{k+1 \choose 2} < \dim G < {n-k+2 \choose 2}, \quad k\geq 1.$$ Mann’s arguments in [@Ma] carry over verbatim to our case, except for one detail: In Cases A and B in the proof of Theorem 1 in [@Ma], one should choose the point $x\in X$ to be regular and lying in a principal orbit; this can be done, since the principal stratum is open and dense in $X$ and the set of regular points is dense in $X$ (cf. section \[S:Prelims\]). Isometry groups of Alexandrov spaces with boundary {#S:Bdry} =================================================== We now extend to Alexandrov spaces results by Chen, Shi and Xu [@CSX] for Riemannian manifolds with boundary. We follow their proof closely, noting that one must make appropriate modifications for it to work in the Alexandrov case. We start with the following result. \[T:Dim\_bound\_bdry\] If $X$ is an $n$-dimensional Alexandrov space with non-empty boundary $\partial X$, then $$\dim \operatorname{Isom}(X) \leq \frac{1}{2}(n-1)(n-2).$$ Since it is not known whether the boundary of an Alexandrov space is also an Alexandrov space, we must proceed with some care. We know that $\partial X$ is a metric space of Hausdorff dimension $n-1$. Given a point $p\in \partial X$, the orbit of $p$ under the action of $G=\operatorname{Isom}(X)$ is entirely contained in $\partial X$, and hence is of dimension at most $n-1$. Now, we proceed by induction, the case $n=1$ being trivial. Let $p\in \partial X$ and observe that the space of directions $\Sigma_p$ is an $(n-1)$-dimensional Alexandrov space with boundary. The isotropy $G_p$ acts on $\Sigma_p$ by isometries and, by the induction hypothesis, $\dim G_p\leq \frac{1}{2}(n-2)(n-3)$. The conclusion is immediate As pointed out in the proof of the previous theorem, it is not known, in general, whether the boundary of an Alexandrov space is an Alexandrov space (with its induced intrinsic metric). The following proposition shows that, if the isometry group of an Alexandrov space with boundary has maximal dimension, then the boundary is, in fact, a Riemannian manifold. Let $X$ be an $n$-dimensional Alexandrov space with non-empty boundary $\partial X$. If $\dim \operatorname{Isom}(X)=\frac{1}{2}(n-1)(n-2)$, then each connected component of $\partial X$, with its induced intrinsic metric, is isometric to one of the space forms in Theorem \[T:Rigidity\]. Let $G=\operatorname{Isom}(X)$ and fix $p\in \partial X$. It follows from the proof of Theorem \[T:Dim\_bound\_bdry\] that, since the dimension of $G$ is maximal, the dimension of the orbit $Gp\subset \partial X$ is $n-1$ and the dimension of the isotropy group $G_p$ is maximal. By the Theorem of Invariance of Domain, the orbit $G p$ is open and, since it is also closed, it must correspond to a connected component $B$ of $\partial X$. It follows from Theorem 7 in [@Ber] that $B$ is a Finsler homogeneous manifold. Since $G_p$ is of maximal dimension, the unit vectors at each point tangent to the boundary must be a round sphere. Thus, the Finsler structure is Riemannian and the conclusion follows from Kobayashi’s rigidity result [@Ko Theorem 3.1]. We conclude this section with the topological classification of Alexandrov spaces with boundary and isometry group of maximal dimension. \[T:Rigidity\_bdry\] Let $X$ be an $n$-dimensional Alexandrov space with non-empty boundary $\partial X$. If $\dim \operatorname{Isom}(X)=\frac{1}{2}(n-1)(n-2)$, then the following hold: 1. If $X$ is compact, then it is homeomorphic to one of the following spaces: 1. A closed $n$-dimensional unit ball in ${\mathbb{R}}^n$. 2. A cylinder $\mathbb{S}^{n-1}\times [0,1]$ or ${\mathbb{R}}P^{n-1}\times [0,1]$. 3. ${\mathbb{R}}P^{n}$ with a small open disc removed. 4. A cone over ${\mathbb{R}}P^{n-1}$.\ 2. If $X$ is non-compact and $\partial X$ has a compact component, then $X$ is homeomorphic to a half open cylinder $\mathbb{S}^{n-1}\times [0,1)$ or ${\mathbb{R}}P^{n-1}\times [0,1)$.\ 3. If $X$ is non-compact and every component of $\partial X$ is non-compact, then $X$ is homeomorphic to ${\mathbb{R}}^{n-1}\times [0,1)$ or to ${\mathbb{R}}^{n-1}\times [0,1]$. The only new space in Theorem \[T:Rigidity\_bdry\] not appearing in the Riemannian case is the cone over ${\mathbb{R}}P^{n-1}$. By looking at level sets of the distance function to a component of $\partial X$, we observe, as in [@CSX], that the action of $\operatorname{Isom}(X)$ on $X$ is of cohomogeneity one. Thus the orbit space $X^=X/\operatorname{Isom}(X)$ is homeomorphic to either $[0,1]$ or $[0,1)$. In case (1), the orbit space $X^$ is homeomorphic to $[0,1]$. The isotropy at the endpoint of $X^=[0,1]$ corresponding to a component of $\partial X$ is ${\mathrm{SO}}(n-1)$ or $\mathrm{O}(n-1)$, while the isotropy at the orbits corresponding to points in the interior of $[0,1]$ is ${\mathrm{SO}}(n-1)$. The isotropy at the other endpoint must contain ${\mathrm{SO}}(n-1)$ and thus can only be $\mathrm{O}(n-1)$ or ${\mathrm{SO}}(n)$, yielding the desired spaces. The only case not appearing in [@CSX] is the one in which the points $0$ and $1$ in $X^\simeq [0,1]$ have, respectively, isotropy $\mathrm{O}(n-1)$ and ${\mathrm{SO}}(n)$. This corresponds to the cone over ${\mathbb{R}}P^{n-1}$. Cases (2) and (3) follow as in [@CSX]. Since the action of $\operatorname{Isom}(X)$ on $X$ is of cohomogeneity one, the metric of $X$ is quite restricted. This idea was examined for smooth manifolds in [@CSX], where the authors proved that the metrics were warped over an interval. It is natural to expect that similar versions can be given for Alexandrov spaces, although with certain modification on the warping function to reflect the possible lack of smoothness. The results in [@AB] can be used in this context to show that the metric is warped over an interval with a semiconcave function as warping factor. We have omitted the details for concision’s sake, since they can be easily completed. Symmetric and locally symmetric Alexandrov spaces {#S:Sym_spaces} ================================================= Following [@Be], we say that an Alexandrov space $X$ is a locally (uniformly) symmetric space if for every $p\in X$ there is a neighborhood $U(p)$ of $p$ and a number $r>0$ such that for all $q\in U(p)$, the ball $B_r(q)$ admits an isometric involution with $q$ as its only fixed point. The space is symmetric if the involutions extend to all the space. In the above reference, Berestovski[ĭ]{} shows that for simply connected $G$-spaces, locally symmetric and symmetric are equivalent conditions. However, the situation is different for Alexandrov spaces, as we show in the following example. \[E:LSYM\_NOTSYM\] Consider the double disk $X$. It is locally symmetric, simply connected and it is not symmetric. In fact, geodesic involutions do not extend to isometries of the whole space for points different of the centers of the disks, or of the glued boundary. More generally, if $\Omega\subset M^n$ is a geodesic ball in a symmetric space, then $\mathrm{Double}(\Omega)$ is locally symmetric but not symmetric. Any regular polyhedral surface with an even number of edges meeting at each vertex, for example, the surface of an icosahedron, is locally symmetric but not symmetric, and is not the double of a geodesic ball in a symmetric space. The same holds for the double of a right cone over a circle of length less than $2\pi$. Thus, Example \[E:LSYM\_NOTSYM\] does not exhaust all the possibilities for locally symmetric Alexandrov spaces that are not symmetric. The abundance of examples makes it difficult to give a general structure result for these spaces. The double of the half-disc $D^2\cap \{\,(x,y)\in \mathbb{R}^2: y\geq 0\,\}$ in $\mathbb{R}^2$ is a non-negatively curved Alexandrov space that is neither symmetric nor locally symmetric, as can be seen by considering its vertex points. This shows that not every double of a convex domain in a symmetric space is locally symmetric. We conclude our paper with the main result of this section. A symmetric Alexandrov space is isometric to a Riemannian manifold. Let $X^n$ be an $n$-dimensional Alexandrov space. Let $p\in X^n$ be a regular point, so that $\Sigma_p$ is isometric to the unit round sphere $\mathbb{S}^{n-1}$. Let $\sigma_p$ be the isometric involution at $p$, i.e., $\sigma_p^2=\mathrm{Id}$. Since $X^n$ is symmetric, $\sigma_p\in \mathrm{Iso}(X)$. We will now show that $\mathrm{d}\sigma_p:\Sigma_p \rightarrow \Sigma_p$ exists and agrees with $-\mathrm{Id}\|_{\mathbb{S}^{n-1}}$. Consider the sequence of isometries $$\sigma^a: (aX^n,p)\rightarrow (aX^n,p).$$ Letting $a\rightarrow \infty$ we get an isometry $$\sigma^{\infty} : C_{o} (\mathbb{S}^{n-1})\simeq \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and $\mathrm{d}\sigma_p = \sigma^\infty \|_{\mathbb{S}^{n-1}}$. Observe that $\mathrm{d}\sigma_p$ is $-\mathrm{Id}\|_{\mathbb{S}^{n-1}}$, since $\sigma^\infty$ has no fixed points away from the origin and it is an orthogonal involution. Let $x,y \in R_X$ and let $\gamma:[-a,a]\rightarrow X$ be a geodesic joining $x$ to $y$. Observe that the image of $\gamma$ is contained in $R_X$. Let $m=\gamma(0)$. Since $\mathrm{d}\sigma_{m}=-\mathrm{Id}$, $\sigma_m$ sends $\exp_m(tv)$ to $\exp_m(-tv)$, we have that $\gamma(t)$ goes to $\gamma(-t)$. It follows that $\sigma_m$ preserves such geodesics, so $\mathrm{Iso}(X)$ acts transitively on $R_X$. 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--- abstract: 'We theoretically study the finite-size effects in the dynamical response of a quantum anomalous Hall insulator in the disk geometry. Semi-analytic and numerical results are obtained for the wavefunctions and energies of the disk within a continuum Dirac Hamiltonian description subject to a topological infinite mass boundary condition. Using the Kubo formula, we obtain the frequency-dependent longitudinal and Hall conductivities and find that optical transitions between edge states contribute dominantly to the real part of the dynamic Hall conductivity for frequency values both within and beyond the bulk band gap. We also find that the topological infinite mass boundary condition changes the low-frequency Hall conductivity to $ e^2/h $ in a finite-size system from the well-known value $ e^2/2h $ in an extended system. The magneto-optical Faraday rotation is then studied as a function of frequency for the setup of a quantum anomalous Hall insulator mounted on a dielectric substrate, showing both finite-size effects of the disk and Fabry-Pérot resonances due to the substrate. Our work demonstrates the important role played by the boundary condition in the topological properties of finite-size systems through its effects on the electronic wavefunctions.' author: - Junjie Zeng - Tao Hou - Zhenhua Qiao - 'Wang-Kong Tse' title: \| Finite-Size Effects in the Dynamic Conductivity and Faraday Effect\ of Quantum Anomalous Hall Insulators --- Introduction ============ Topological properties are usually studied in extended systems without a confining boundary condition. As the example of integer quantum Hall effect illustrates, edge states existing in a realistic finite-size geometry are indispensable to the explanation of the underlying quantization phenomenon. In this connection, models with exactly solvable edge state wavefunctions are particularly valuable in delineating the role played by edge states in transport and optical phenomena [@Wunsch2008; @Lu2010; @Shan2010; @Grujic2011; @Christensen2014PRB]. The quantum anomalous Hall insulator [@Liu2016ARoCMP] is a two-dimensional topological state of matter characterized under electrical transport conditions by a quantized value of Hall conductivity and a zero longitudinal conductivity due to spin splitting under broken time-reversal symmetry. For the particular case when the quantized value is an integer multiple of the conductance quantum, the system has an integer Chern number and is also called a Chern insulator [@Haldane1988PRL]. A half-quantized Hall conductivity is also possible [@Qi_PRB2008], as is the case for the surface states of a three-dimensional topological insulator. Quantum anomalous Hall insulator has been experimentally realized in magnetically doped three-dimensional topological insulator thin films [@Chang2013S]. Two-dimensional atomically thin materials doped with heavy magnetic adatoms [@Ren2016RoPiP] have also been proposed as platforms for realizing quantum anomalous Hall insulator [@Qiao2010PRB; @Tse2011PRBa; @Deng2017PRB]. This class of system has the advantage that they provide an additional tunability of topological phases due to interplay between the magnetization and an applied out-of-plane electric field [@Qiao_Tse_PRL2011; @Qiao_Tse_PRB2013; @Zeng2017PRB]. Frequency-dependent conductivity provides a useful probe for charge carriers’ dynamical response and elementary excitations. In systems where the electron’s momentum is coupled to the pseudospin or spin degrees of freedom, it reveals unusual interaction renormalization [@Tse_MD_PRB2009; @Li_Tse_PRB2017] and strong-field [@Lee_Tse_PRB2017] effects. The complex dynamic Hall conductivity can yield valuable information in topological materials beyond the direct current limit, with its real part providing the dynamics of the reactive carrier response and imaginary part the dissipative response. Dynamic Hall response can be probed optically by the magneto-optical Faraday and Kerr rotations. Three-dimensional topological insulator thin films under broken time-reversal symmetry exhibit dramatic Faraday and Kerr effects in the low-frequency regime that signifies the underlying topological quantization of the Hall conductivity [@Tse2010PRL; @Maciejko_PRL2010; @wu2016quantized; @okada2016terahertz; @dziom2017observation]. The dynamic Hall response and magneto-optical effects of topological materials are often theoretically studied assuming an infinitely extended system, and so far there has been few study on the finite-size effects of these properties due to the finite planar dimensions of the system. It is the purpose of this work to perform a semi-analytical and numerical study of the dynamic conductivities and magneto-optical Faraday effect of quantum anomalous Hall insulator in a finite circular disk geometry. Our theory is based on the low-energy continuum description of quantum anomalous Hall insulator subject to a vanishing radial current boundary condition (‘no-spill’ boundary condition). The system is described by a massive Dirac Hamiltonian, which would give a half-quantized Hall conductivity when the system is infinitely extended. We compute the exact energies and wavefunctions of the finite disk as a function of the orbital angular momentum quantum number, and use the Kubo formula to evaluate the dynamic longitudinal and Hall conductivities. Contrary to the extended system case, we find that imposing a change of the topological character of the system across the boundary through the no-spill boundary condition causes the finite-sized massive Dirac model to carry a Chern number of $ 1 $ instead of $ 1/2 $. Our calculations also show that optical dipole transitions between edge states contributes to an almost constant value of $ e^2/h $ in the dynamic Hall conductivity that remains constant even for frequencies exceeding the band gap. Combining all three types of optical transitions among the edge and bulk states, the total dynamic conductivities of the finite disk are found to agree with the main features calculated from the massive Dirac model with an additional parabolic dispersion term that breaks electron-hole symmetry in an extended system. The finite-size effects in the conductivities are also seen in the magneto-optical Faraday rotations. Here, we also study the ‘finite-size effects’ along the out-of-plane direction by considering a substrate interfacing the quantum anomalous Hall insulator. The finite thickness of the substrate gives rise to Fabry-Pérot oscillations of the Faraday effect, which are found to exert a stronger influence on the Faraday rotation spectrum than the effect of the finite disk size. This article is organized as follows. In Sec. \[sec:model\] we describe the model Hamiltonian and boundary condition of the quantum anomalous Hall insulator disk and derive semi-analytic expressions for the eigenfunctions and energies. Sec. \[sec:conduc\] describes our calculations and results for the dynamic longitudinal and Hall conductivities using the Kubo formula. A flow diagram plotting the real parts of the longitudinal and Hall conductivities is discussed in Sec. \[sec:flow\], which approaches the behavior of a Chern insulator with increasing disk radius. In Sec. \[sec:F\], we provide results for the magneto-optical Faraday rotation of the quantum anomalous Hall insulator disk and study the effects of finite disk radius and the presence of an underlying substrate. Sec. \[sec:disc\_conc\] summarizes our work. Theoretical Model {#sec:model} ================= As depicted in Fig. \[fig:schematic\], we consider a quantum anomalous Hall insulator in a disk geometry with a radius $ R $. The low-energy physics of a quantum anomalous Hall insulator is described by the massive Dirac Hamiltonian, $$\begin{aligned} \label{eq:Hamiltonian} h_\text{D} &= v\ (\bm{\sigma}\times\bm{p})\cdot\hat{\bm{z}} + M\sigma_z, \end{aligned}$$ where $ v $ is the band velocity of the Dirac model, $ M > 0 $ is the Zeeman interaction that breaks time-reversal symmetry and provides a bulk band gap, and $ \bm{\sigma}=(\sigma_x, \sigma_y, \sigma_z) $ is the 3-vector formed by the Pauli matrices corresponding to the spin degree of freedom. Under periodic boundary conditions, the massive Dirac Hamiltonian gives a Chern number $ 1/2 $. To incorporate finite-size effects, we need to specify the appropriate boundary condition at the edge of the disk. While a vanishing wavefunction boundary condition is suitable for graphene with zigzag edge termination, there is no reason to assume that such a condition also applies in our case without appealing to atomistic details at the boundary. In particular, since we aim to provide a treatment of the finite-size effects of quantum anomalous Hall insulator as generally as possible, we use the no-spill current condition at the boundary, which is more suitable to be used with a continuum Hamiltonian model. We will discuss this boundary condition in more details. ![\[fig:schematic\] (Color online) Schematic of the system under investigation. The cyan (dark) area is a quantum anomalous Hall insulator nanodisk with a radius $ R $ and a positive finite mass in the Dirac model Eq. (\[eq:Hamiltonian\]), surrounded by a yellow (light) domain with a mass going to negative infinity, to ensure a topologically nontrivial domain wall. The outer region is infinitely large. The linear response dynamics of this system is studied in Sec. \[sec:conduc\] by initializing a weak perpendicular incident light with energy $ \omega $.](fig1_system_schematic.eps){width=".45\textwidth"} The wavefunctions and energy eigenvalues are obtained by solving the eigenvalue problem $ h_\text{D}\psi=\epsilon\psi $, where $ \psi $ is a two-component spinor. In view of the rotational symmetry of the system, the eigenvalue problem can be facilitated by using the following wavefunction ansatz in polar coordinates [@Christensen2014PRB], $$\begin{aligned} \label{eq:ansatz} \psi(\rho,\phi)&= \begin{pmatrix} \psi_\uparrow\\ \psi_\downarrow \end{pmatrix} =\mathrm{e}^{\mathrm{i} l\phi} \begin{pmatrix}[r] A_l(\rho)\\ \mathrm{e}^{\mathrm{i}\phi}B_l(\rho) \end{pmatrix}, \end{aligned}$$ where $ \rho $ is the radial position, $ \phi $ the azimuthal angle, and $ l $ is an integer corresponding to the orbital angular momentum quantum number. Combining Eq. (\[eq:Hamiltonian\]) and Eq. (\[eq:ansatz\]), one obtains the wavefunctions for states lying both outside and inside of the bulk energy gap $$\begin{aligned} \label{eq:wave_function} \begin{dcases} \Psi_{ln}(\rho,\phi)=\dfrac{\mathrm{e}^{\mathrm{i} l\phi}}{\sqrt{N_{ln}}} \begin{pmatrix}[r] (\epsilon+M)\ \mathrm{J}_{l}(\beta_{ln}\tilde\rho)\\ +\mathrm{e}^{\mathrm{i}\phi}\sqrt{\epsilon^2-M^2}\ \mathrm{J}_{l+1}(\beta_{ln}\tilde\rho) \end{pmatrix}, & \|\epsilon\| > M\\ \\ \Phi_{ln}(\rho,\phi)=\dfrac{\mathrm{e}^{\mathrm{i} l\phi}}{\sqrt{\mathcal{N}_{ln}}} \begin{pmatrix}[r] (M+\epsilon)\ \mathrm{I}_{l}(b_{ln}\tilde\rho)\\ -\mathrm{e}^{\mathrm{i}\phi}\sqrt{M^2-\epsilon^2}\ \mathrm{I}_{l+1}(b_{ln}\tilde\rho) \end{pmatrix}, & \|\epsilon\| < M \end{dcases} \end{aligned}$$ where we have defined the system’s characteristic frequency $ \omega_R\equiv v/R $, the dimensionless wavenumbers $ \beta=\sqrt{\epsilon^2-M^2}/(\hbar\omega_R) $ and $ b=\sqrt{M^2-\epsilon^2}/(\hbar\omega_R) $, as well as the scaled dimensionless radial distance $ \tilde{\rho}\equiv\rho/R\in[0,1] $. Here $ \text{J}_l(x) $ is the Bessel function and $ \text{I}_l(x) $ the modified Bessel function, both of the first kind. The normalization coefficients $ N_{ln} $ and $ \mathcal{N}_{ln} $ follow from the normalization condition of the wavefunctions $ 1=\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{R}\mathrm{d}\rho\rho\|\psi(\rho,\phi)\|^2=2\pi R^2\int_{0}^{1}\mathrm{d}\tilde{\rho}\tilde{\rho}\|\psi(\rho,\phi)\|^2 $, and are consequently given by $$\begin{aligned} \label{eq:normalization_coeff} \begin{dcases} N_{ln}=2\pi R^2\epsilon_{ln}(\epsilon_{ln}+M)\left[\mathrm{J}_l^2(\beta_{ln})+\mathrm{J}_{l+1}^2(\beta_{ln})-\dfrac{2l+1-M/\epsilon_{ln}}{\beta_{ln}}\mathrm{J}_l(\beta_{ln})\mathrm{J}_{l+1}(\beta_{ln})\right], & \|\epsilon\| > M,\\ \mathcal{N}_{ln}=2\pi R^2(M+\epsilon_{ln})\epsilon_{ln}\left[\mathrm{I}_l^2(b_{ln})-\mathrm{I}_{l+1}^2(b_{ln})-\dfrac{2l+1-M/\epsilon_{ln}}{b_{ln}}\mathrm{I}_l(b_{ln})\mathrm{I}_{l+1}(b_{ln})\right], & \|\epsilon\| < M. \end{dcases} \end{aligned}$$ In a finite-sized geometry, a clear delineation between bulk and edge states does not exist because there is always some degree of mixing between bulk and edge states. However, for the sake of terminology, we shall label $ \Psi_{ln} $ (having energies beyond the bulk band gap) and $ \Phi_{ln} $ (having energies within the bulk band gap) as the bulk and edge states, respectively. This terminology is supported by examining the radial probability density profile as a function of the scaled dimensionless radius $ \tilde{\rho} $ in Fig. \[fig:edge&bulk\]. It is seen that the in-gap states $ \Phi_{ln} $ displays a monotonically increasing profile towards the edge of the disk and indeed behaves like edge states. On the other hand, the out-of-gap states $ \Psi_{ln} $ exhibits oscillations as a function of the radial position $ \tilde{\rho} $ and can therefore be associated with bulk states. ![\[fig:edge&bulk\] (Color online) Radial probability density as a function of the scaled dimensionless radius $ \tilde{\rho}$ (with $ R=\SI{30}{\nano\meter} $) for several typical values of angular momentum $ l $ and energies $ \epsilon_{ln} $. Hollow-shape-dotted curves indicate the edge states $ \Phi_{ln} $ and solid-shape-dotted curves the bulk states $ \Psi_{ln} $.](fig2_radial_PDF.eps){width=".47\textwidth"} To determine the energy eigenvalues, we impose the no-spill boundary condition that the outgoing radial component of the current vanishes at the edge of the disk. Within the massive Dirac model, this condition corresponds to taking the mass term $ M $ outside the disk region to infinity and is sometimes called the infinite mass boundary condition [@Christensen2014PRB]. We note that in principle there can be two choices for the sign of the mass term, with $ M(\rho>R)\to\pm\infty $. Since we assume $ M(\rho\leq R) > 0 $, the choice $ M(\rho >R) \to -\infty $ is appropriate here for a topologically nontrivial domain wall, ensuring a change in the topological character across the disk boundary [@Trushin2016PRB; @FZhang_PRB2012]. This results in the following constraint between the two components of the wavefunction [@Berry1987PotRSoLA; @Peres2009JoPCM] $ \psi_\downarrow/\psi_\uparrow=\alpha\mathrm{e}^{\mathrm{i}\phi} $, with $ \alpha=-1 $, (cf. Appendix \[appsec:IMBC\] for more information). This choice of $ M( \rho > R ) $ differs from that in the original Berry’s paper [@Berry1987PotRSoLA], which addressed a situation without a topological domain wall. For this reason, we refer to the boundary condition we use here as the topological infinite mass boundary condition. Armed with the above, one has the following equations for the eigenenergies $ \epsilon_{ln} $ from Eq. (\[eq:wave\_function\]) $$\begin{aligned} \label{eq:massive_IM_spec} \begin{dcases} (\epsilon_{ln}+M)\,\mathrm{J}_{l}(\beta_{ln})=-\sqrt{\epsilon_{ln}^2-M^2}\,\mathrm{J}_{l+1}(\beta_{ln}), & \|\epsilon\| > M,\\ (M+\epsilon_{ln})\,\mathrm{I}_{l}(b_{ln})=+\sqrt{M^2-\epsilon_{ln}^2}\,\mathrm{I}_{l+1}(b_{ln}), & \|\epsilon\| < M. \end{dcases} \end{aligned}$$ Equation (\[eq:massive\_IM\_spec\]) is transcendental equations with multiple roots $ \epsilon_{ln} $ for each given $ l $, where $ n = 1, 2, \dots $ indicates the multiplicity. While a number of candidates for Chern insulators have been proposed in the literature [@Deng2019a; @Jin2018PRB; @He2018ARoCMP; @Liu2016ARoCMP; @Garrity2013PRL; @Yu2010S], we choose the band parameters corresponding to the quantum anomalous Hall insulator of a thin film with $v = \SI{5.e5}{\meter\cdot\second^{-1}}$ and a Zeeman energy $ 2M = \SI{0.1}{\electronvolt} $. As a low-energy effective theory, the massive Dirac model is valid up to a certain energy cutoff $ \epsilon_{\mathrm{c}} $. For concreteness we use $ \epsilon_{\mathrm{c}} = \SI{.3}{\electronvolt} $ as the energy cutoff, noting that the main findings of our work are not dependent on the precise value of $ \epsilon_{\mathrm{c}}$ with $ \epsilon_{\mathrm{c}} \ll M $. We therefore only seek the numerical roots of Eq. (\[eq:massive\_IM\_spec\]) within the range $ [-\epsilon_{\mathrm{c}}, \epsilon_{\mathrm{c}}] $. ![\[fig:massive\_spec&dos\](Color online) Energy spectrum (energy $ \epsilon $ versus angular momentum quantum number $ l $) and the corresponding density of states (DOS) for $ v=\SI{5.e5}{\meter\cdot\second^{-1}} $ and $ M=\SI{.05}{\electronvolt} $. The region shaded in light blue (gray) indicates the bulk energy gap, within which the red (darker) line shows the analytic dispersion Eq. (\[eq:linear\_disp\]) of the chiral edge state.](fig3_spectrum_dos.eps){width=".47\textwidth"} Figures \[fig:massive\_spec&dos\] (a), (c) and (e) show the obtained energy spectrum for $ \epsilon $ as a function of $ l $ for different values of $ R $. One can identify features that correspond predominantly to bulk states and edge states. The bulk state spectrum contains a gap in which a one-way chiral edge dispersion runs across. As $ R $ increases from , the delineation between the bulk and edge spectra becomes more evident. In the limit $ R\to\infty $ that can be achieved physically when $ R \gg \hbar v/M $, the chiral edge state dispersion can be obtained by expanding the second equation in Eq. (\[eq:massive\_IM\_spec\]) using a large $ R $ expansion: $$\begin{aligned} \label{eq:linear_disp} \epsilon_l &\simeq -\epsilon_0 \left(l+1/2\right), \end{aligned}$$ with $ \epsilon_0 \equiv \hbar\omega_R[1+\hbar\omega_R/(2M)] $. We note that the $ 1/2 $ on the right hand side originates from the spin angular momentum of the electron. Gratifyingly, this approximate analytic dispersion is in excellent agreement with the exact numerical results as shown by the red solid line in Fig. \[fig:massive\_spec&dos\]. Using Eq. (\[eq:linear\_disp\]), the number of the in-gap states can be estimated as $ \lfloor2M/(\hbar\omega_R)\rfloor $, where $ \lfloor\dots\rfloor $ denotes the floor function. Figures \[fig:massive\_spec&dos\] (b), (d) and (f) show the density of states of the calculated spectrum from the expression [@Datta2005] $$\begin{aligned} \label{eq:dos} D(\epsilon)&=\dfrac{1}{\pi\mathcal{A}}\sum_{\nu}\mathrm{Im}\dfrac{1}{\epsilon_\nu-\epsilon-\mathrm{i}\eta}, \end{aligned}$$ where ‘Im’ stands for imaginary part, $ \mathcal{A} $ is the area of the disk and the broadening parameter is set as $ \eta=\SI{2.4e-3}{\electronvolt} $ throughout this work. The nonvanishing peaks of the density of states in the gap indicate the existence of the in-gap chiral edge states. When the radius increases, the DOS profile expectedly becomes smoother as more states are introduced into the system. On the other hand, a small radius ($ R \lesssim \hbar v/M $) enhances the quantum confinement effect as seen from the more prominent resonances from the individual quantum states. Dynamic Conductivity {#sec:conduc} ==================== We now introduce into the system a weak, linearly polarized alternating current probe field that is normally incident on the quantum anomalous Hall insulator disk. With the obtained energy spectrum and wavefunctions, we proceed to calculate the longitudinal and Hall optical conductivities using the Kubo formula [@Mahan2000] in the real space representation (derivation is provided in Appendix \[appsec:Kubo\]): $$\begin{aligned} \label{eq:opt_cond_final} \sigma_{ij}(\omega)&=2\mathrm{i}\omega\frac{e^2}{h}\sum_{mm'}\dfrac{f_0'-f_0}{\Delta\epsilon-\omega-\mathrm{i}\eta}\dfrac{\braket{m\|x_i\|m'}\braket{m'\|x_j\|m}}{R^2}, \end{aligned}$$ where $ i,j \in \set{x,y} $, $ m (m') $ is a collective label for the relevant quantum numbers, $ f_0 $ is the Fermi-Dirac distribution function, and $ \Delta\epsilon=\epsilon'-\epsilon $ is the energy difference between the final (primed) and the initial (unprimed) states in a transition, and $ \omega $ is the photon energy of the incident light. The matrix elements in Eq. (\[eq:opt\_cond\_final\]) capture the transition processes among the bulk states $ \Psi_{ln} $ and edge states $ \Phi_{ln} $ and there are three types of transitions, i.e., edge-to-edge (E-E), edge-to-bulk (B-E), and bulk-to-bulk (B-B). Using the expressions of the wavefunctions Eq. (\[eq:wave\_function\]) together with their normalization coefficients Eq. (\[eq:normalization\_coeff\]), we obtain the following matrix elements for the three types of transitions $$\begin{aligned} \label{eq:matrix_element} \Braket{\psi_{l'n'}^{S'}\|\begin{pmatrix} x\\ y \end{pmatrix}\|\psi_{ln}^{S}} &=R\ \mathcal{I}_{l'n',ln}^{S'S} \begin{pmatrix}[r] 1 & 1\\ \mathrm{i} & -\mathrm{i} \end{pmatrix} \begin{pmatrix} \delta_{l,l'+1}\\ \delta_{l,l'-1} \end{pmatrix}, \end{aligned}$$ where $ S', S\in\set{\text{B}, \text{E}} $ stand for bulk and edge states, $ \delta_{l,l'} $ is the Kronecker delta symbol, and $ \mathcal{I}_{l'n',ln}^{S'S} $ is a dimensionless radial integral defined by $$\begin{aligned} \label{eq:matrix_element_integrals} \mathcal{I}_{l'n',ln}^{S'S}=\int_{0}^{1}\mathrm{d}\tilde{\rho}\tilde{\rho}^2 \begin{dcases} \dfrac{(M+\epsilon')(M+\epsilon)\ \mathrm{I}_{l'}(b_{l'n'}\tilde{\rho})\ \mathrm{I}_{l}(b_{ln}\tilde{\rho}) + \sqrt{(M^2-{\epsilon'}^2)(M^2-\epsilon^2)}\ \mathrm{I}_{l'+1}(b_{l'n'}\tilde{\rho})\ \mathrm{I}_{l+1}(b_{ln}\tilde{\rho})}{\sqrt{\mathcal{N}_{l'n'}\mathcal{N}_{ln}}/(\pi R^2)}, & S'=S=\text{E},\\ \dfrac{(\epsilon'+M)(M+\epsilon)\ \mathrm{J}_{l'}(\beta_{l'n'}\tilde{\rho})\ \mathrm{I}_{l}(b_{ln}\tilde{\rho}) - \sqrt{({\epsilon'}^2-M^2)(M^2-\epsilon^2)}\ \mathrm{J}_{l'+1}(\beta_{l'n'}\tilde{\rho})\ \mathrm{I}_{l+1}(b_{ln}\tilde{\rho})}{\sqrt{N_{l'n'}\mathcal{N}_{ln}}/(\pi R^2)}, & S'=\text{B}, S=\text{E},\\ \dfrac{(\epsilon'+M)(\epsilon+M)\ \mathrm{J}_{l'}(\beta_{l'n'}\tilde{\rho})\ \mathrm{J}_{l}(\beta_{ln}\tilde{\rho}) + \sqrt{({\epsilon'}^2-M^2)(\epsilon^2-M^2)}\ \mathrm{J}_{l'+1}(\beta_{l'n'}\tilde{\rho})\ \mathrm{J}_{l+1}(\beta_{ln}\tilde{\rho})}{\sqrt{N_{l'n'} N_{ln}}/(\pi R^2)}, & S'=S=\text{B}. \end{dcases} \end{aligned}$$ Equation (\[eq:matrix\_element\]) expresses an angular momentum selection rule: transitions are allowed only between states with a change in angular momenta $ \Delta l=\pm 1 $. The remaining radial integrations ($ \mathcal{I}_{l'n',ln}^{S'S} $) in Eq. (\[eq:matrix\_element\_integrals\]) are computed numerically. Figure \[fig:sigma\_xx\] shows our results for the real (blue) and imaginary (red) parts of the longitudinal conductivity $ \sigma_{xx}(\omega) $ when the Fermi level $ \epsilon_\text{F} = 0 $ for different values of $ R $ separated into the three contributions: E-E (first row), B-E (second row) and B-B (third row). First, we note that the finite size of the disk has a different effect on the E-E conductivity contribution compared to the other two types of contributions involving the bulk. As $ R $ is increased, the E-E conductivity \[Figs. \[fig:sigma\_xx\] (a1)-(a3)\] remains approximately the same while the B-E and B-B contributions \[Figs. \[fig:sigma\_xx\] (b1)-(b3) and (c1)-(c3)\] display considerable changes. According to the selection rule in Eq. (\[eq:matrix\_element\]), there can only be one E-E transition along the chiral edge state dispersion from below to above the Fermi level, therefore there is always only one peak in the E-E conductivity regardless of the size of the disk. The peak’s position is seen to shift toward $ \omega = 0 $ with increasing $ R $, because the edge states become more closely spaced with their energy separation $ \approx \epsilon_0 \propto 1/R $ \[Eq. (\[eq:linear\_disp\]), to first order\]. In contrast, for the B-E \[Figs. \[fig:sigma\_xx\] (b1)-(b3)\] and B-B contributions \[Figs. \[fig:sigma\_xx\] (c1)-(c3)\], since the number of possible transitions is directly proportional to the number of bulk states, the number of peaks increases and the conductivity approaches a smooth continuous curve as $ R $ increases. The threshold beyond which the B-B contribution becomes finite corresponds to the bulk energy gap $ 2M $. ![image](fig4_sigma_xx.eps){width=".85\textwidth"} ![image](fig5_sigma_xy.eps){width=".87\textwidth"} The corresponding dynamic Hall conductivity $ \sigma_{xy}(\omega) $ is shown in Fig. \[fig:sigma\_xy\]. The above description for the longitudinal conductivity is also applicable here, if we note that the roles of the reactive and dissipative components are played by the real and imaginary parts of $ \sigma_{xy}(\omega) $ respectively. For both the longitudinal and Hall conductivities, the E-E contribution displays a smooth profile across all values of frequency, which is the result of only one possible edge-to-edge transition. The B-E contribution is smaller than both E-E and B-B contributions by an order of magnitude for the smallest radius $R = 30\,\mathrm{nm}$ and is further suppressed with increasing radius. For frequency within the bulk gap $2M$, its profile exhibits many closely spaced sharp peaks corresponding to the many possible edge-to-bulk transitions, and is smooth for frequency beyond the gap. The opposite behavior is seen in the B-B contribution. Its profile exhibits wild fluctuations due to even more possible bulk-to-bulk transitions for frequency beyond the bulk gap, which are suppressed when the disk radius is increased. There is an important distinction between Fig. \[fig:sigma\_xx\] and Fig. \[fig:sigma\_xy\]. As shown in Fig. \[fig:sigma\_xx\], the E-E contribution of $ \text{Im}[\sigma_{xx}(\omega)] $ quickly drops to zero with increasing $ \omega $. However, the corresponding reactive contribution in the Hall conductivity, $ \mathrm{Re}[\sigma_{xy}(\omega)] $, becomes a constant $ e^2/h $ when $ \omega $ exceeds the value of the bulk gap $ 2M $. We find that the flatness of this plateau as a function of $ \omega $ is not sensitive to the system size, as seen by comparing Figs. \[fig:sigma\_xy\] (a1)-(a3). Adding the three E-E, B-E and B-B contributions, we conclude that the total Hall conductivity in the direct current limit within our finite-size model is $ e^2/h $, rather than $ e^2/2h $ as would have been expected for massive Dirac electrons in an extended system. Therefore, in the direct current limit, our system behaves as a quantum anomalous Hall insulator with an (integer) Chern number, i.e., a Chern insulator. This motivates the question whether our system behaves as a Chern insulator in response to an a.c. field as well. To address this question, we compare the calculated conductivities of our finite-size system for large $ R $ with the conductivities from the low-energy Dirac model $ H=\bm{d}(\bm{k})\cdot\bm{\sigma} $, defined on an extended system. Here we consider two cases and take the spin vector $ \bm{d}(\bm{k}) $ to be linear in momentum with $ \bm{d}_{1}(\bm{k}) = \left(A k_y, -A k_x, M\right) $, and quadratic in momentum with $ \bm{d}_{2}(\bm{k}) = \left(A k_y, -A k_x, M - B(k_x^2+k_y^2)\right) $, where $ A, B(>0),\text{ and } M $ are band parameters independent of momentum. The winding number spanned by the vector $ \bm{d}(\bm{k}) $ is evaluated by this integral $$\begin{aligned} \label{eq:winding_number} \mathcal{C} &= \dfrac{1}{4\pi}\int\mathrm{d}^2k\ \dfrac{\bm{d}\cdot(\partial_{k_x}\bm{d}\times\partial_{k_y}\bm{d}\ )}{d^3}. \end{aligned}$$ Accordingly, the quantized Hall response of the linear model, as applicable for the surface states of three-dimensional topological insulators, is $ \sigma_{xy} = \mathrm{sgn}(M)e^2/2h $ [@Qi_RMP], and that of the second model, as applicable for Chern insulators, is $ \sigma_{xy} = 0 $ when $ M < 0 $ and $ \sigma_{xy} = e^2/h$ when $ M > 0 $. For extended systems, momentum is a good quantum number and the dynamic conductivity for the above models can be calculated from the Kubo formula [@Allen2006] in the momentum representation as usual $$\begin{aligned} \label{eq:Kubo_2nd} \sigma_{\alpha\beta}(\omega) &= \mathrm{i}\hbar\int\dfrac{\mathrm{d}^2k}{(2\pi)^2}\sum_{mn}\dfrac{f_m-f_n}{\epsilon_m-\epsilon_n}\dfrac{\braket{n\|j_\alpha\|m}\braket{m\|j_\beta\|n}}{\epsilon_m-\epsilon_n-(\omega+\mathrm{i}\eta)}\\ &= \dfrac{\mathrm{i} G_0}{4\pi}\int\dfrac{\mathrm{d}^2k}{d}\left[\dfrac{\tilde{j}_\alpha^{-+}\tilde{j}_\beta^{+-}}{\omega-2d+\mathrm{i}\eta}+\dfrac{\left(\tilde{j}_\alpha^{-+}\tilde{j}_\beta^{+-}\right)^}{\omega+2d+\mathrm{i}\eta}\right]\notag, \end{aligned}$$ where $ \tilde{j}_\alpha=\partial_{k_\alpha}H $. Analytic results of the dynamic conductivity tensor for the linear Dirac model can be obtained and is available [@Tse2011PRB]. For the quadratic Dirac model, we compute the dynamic conductivity numerically from Eq. (\[eq:Kubo\_2nd\]) with parameters $ A = \hbar v \simeq \SI{0.3291}{\electronvolt\cdot\nano\meter}, B = \hbar^2/(\SI{2}{\electronmass}) \simeq \SI{0.0381}{\electronvolt\cdot{\nano\meter}^2} $ and $ M = \SI{0.0500}{\electronvolt} $. Interestingly, our results show that not only electronic wavefunctions and dispersions, but boundary conditions can also change the topological property of a system. Figure \[fig:cond\_comp\] shows the dynamic longitudinal and Hall conductivities for the three cases of finite-sized disk with $ R=\SI{150}{\nano\meter} $, linear and quadratic Dirac models defined on an extended system. For the longitudinal conductivity $ \sigma_{xx} $, we see from panel (a) that the qualitative behavior of all three sets of results resemble each other closely for frequencies beyond the band gap. Within the gap, there is a peak in $ \mathrm{Re}(\sigma_{xx}) $ and a corresponding zero-crossing in $ \mathrm{Im}(\sigma_{xx}) $ near $ \omega = 0 $ in the case of finite-sized disk, which are absent in the extended systems. These features originate from the E-E transition \[Fig. \[fig:sigma\_xx\] (a3)\]. For the Hall conductivity $ \sigma_{xy} $, we see from panel (b) that the qualitative behavior the finite-size result matches closely with the extended quadratic Dirac model result, while the extended linear Dirac model result displays a similar trend with increasing frequency but the overall profile is shifted upward. This shift is consistent with the direct current limit of the Hall conductivity for the three cases. Close to $ \omega = 0 $, the finite-size and extended quadratic Dirac models both give a Hall conductivity $ \mathrm{Re}(\sigma_{xy}) $ of $ e^2/h $ whereas the linear Dirac model gives $ e^2/2h $. In addition, a strong peak is apparent near $ \omega = 0 $ in $ \mathrm{Im}(\sigma_{xy}) $ for finite-sized disk due to E-E transition \[Fig. \[fig:sigma\_xy\] (a3)\]. ![image](fig6_tri-sigma_comparison.eps){width=".8\textwidth"} Flow Diagram {#sec:flow} ============ ![\[fig:flow\_diag\] (Color online) Scaling behavior \[$ \text{Re}(\sigma_{xx}) $ versus $ \text{Re}(\sigma_{xy}) $\] for the E-E contribution. The black dot-dashed line shows the behavior of the topological insulator surface states and the black dashed line the integer quantum Hall state. The various shape-dotted lines show the behavior for the quantum anomalous Hall insulator disk with different radii, which approaches the behavior of the integer quantum Hall state as $ R $ increases. Conductivities are expressed in unit of $ G_0 = e^2/h $.](fig7_Re_flow.eps){width=".45\textwidth"} Further insights can be obtained by mapping $ \sigma_{xx} $ and $ \sigma_{xy} $ onto a parametric plot using $ \omega $ as the parameter. In the direct current regime, such a $ \sigma_{xx} $-$ \sigma_{xy} $ plot generates a flow diagram and was studied for quantum anomalous Hall insulators theoretically [@Nomura2011PRL] and experimentally [@Checkelsky2014NP; @Grauer2017PRL] using the system’s size, temperature and gate voltage as parameters. The flow diagram of a quantum Hall insulator consists of two semicircles with a unit diameter located at $ (\pm e^2/2h, 0 ) $ [@Ruzin; @Hilke_1999] \[dashed black semicircles in Fig. \[fig:flow\_diag\]\], while the flow diagram of the topological insulator surface states forms a semicircle centered at the origin \[dot-dashed black semicircle in Fig. \[fig:flow\_diag\]\]. To study the flow diagram for our finite quantum anomalous Hall insulator disk, we focus on only the E-E contribution of $ \mathrm{Re}(\sigma_{xx}) $ and $ \mathrm{Re}(\sigma_{xy}) $ for frequency $ \omega < 2M $. In this frequency range, contributions involving bulk states are almost suppressed and thus will not show in the flow diagram. The various shape-dotted lines in Fig. \[fig:flow\_diag\] show the flow diagram for different values of disk radius from . The diagram resembles large circular arcs for small $ R $s, and gradually approaches a semicircle centered at $ (-e^2/2h, 0) $ as $ R $ is increased (contrast the purple curve for and the blue curve for ). This is consistent with our finding that the finite disk subjected to the topological infinite mass boundary condition becomes a Chern insulator. Faraday Effect {#sec:F} ============== ![image](fig8_F_theta_vs_omega.eps){width=".85\textwidth"} The dynamic Hall conductivity and its quantized value near the direct current limit can be probed with magneto-optical Faraday effect in topological systems [@Tse2010PRL; @Tse2010PRB; @Tse2011PRB]. We consider a setup consisting of a quantum anomalous Hall insulator nanodisk on top of a dielectric substrate, surrounded by vacuum. For concreteness, we choose silicon [@Chang2013AM] as the substrate, which has a dielectric constant $ \varepsilon_{\ce{Si}} = 11.68 $. Using the scattering matrix formalism, the transmitted electric field can be expressed as $$\begin{aligned} \label{eq:trans} {E}^{\mathrm{t}} &= \bar{t}_{\mathrm{B}}\left(\mathbb{I}-\bar{r}_{\mathrm{T}}'\bar{r}_{\mathrm{B}}\right)^{-1}\bar{t}_{\mathrm{T}} \, {E}^{0}, \end{aligned}$$ where $ E^0 $ is the incident field, $ \bar{r}, \bar{r}' $ and $ \bar{t}, \bar{t}' $ are $ 2\times 2 $ matrices accounting for single-interface reflection and transmission, respectively. The subscript ‘T’ labels the top interface at the quantum anomalous Hall insulator between vacuum and the substrate while ‘B’ labels the bottom interface between the substrate and vacuum. Equation (\[eq:trans\]) expresses the transmitted field in the form of geometric series resulting from Fabry-Pérot–like repeated scattering at the top and bottom interfaces. The incident light is normally illuminated onto the setup and is linearly polarized. Resolving the transmitted electric field into its positive and negative helicity components $ E_{\pm}^{\mathrm{t}} = E_x^{\mathrm{t}}\pm \mathrm{i} E_y^{\mathrm{t}}$, the Faraday rotation angle is given by $$\begin{aligned} \theta_{\mathrm{F}} &= \dfrac{1}{2}\left(\arg E_+^{\mathrm{t}} - \arg E_-^{\mathrm{t}} \right). \end{aligned}$$ Our results for the Faraday angle are shown in Fig. \[fig:F\_rotations\], computed with different values of disk radius $ R $ and substrate thickness $ a $. $ \theta_{\mathrm{F}} $ shows both finite-size effects due to a finite $ R $ and Fabry-Pérot resonances [@Tse2016] due to the presence of an underlying substrate, which serves as an optical cavity. Let us first focus on the first row (a1)-(a4) with a small radius $ R = \SI{30}{\nano\meter} $. The effect of discrete energy eigenstates in a finite disk is reflected in $ \theta_{\mathrm{F}} $ as fluctuations with frequency \[panel (a1)\], similar to the behavior seen in the dynamic conductivities. As the substrate thickness is increased \[panels (a2)-(a4)\], these fluctuations become overwhelmed by the periodic oscillations resulting from the Fabry-Pérot resonance. For larger radius values across the rows, the fluctuations due to finite disk effect become smoothened and only Fabry-Pérot oscillations remain. Since the lowest Fabry-Pérot resonance frequency is $ c/(2a\sqrt{\varepsilon_{\ce{Si}}}) $, more resonances appear for a thicker substrate. When the frequency value is equal to the band gap, we notice a kink in $ \theta_{\mathrm{F}} $ that becomes more apparent for thinner substrates, resulting from the onset and peak features in $ \sigma_{xx} $ and $ \sigma_{xy} $. Another interesting feature is the non-monotonic increase in the envelope of the Fabry-Pérot oscillations, which become apparent when both $ a $ and $ R $ are large \[panels (b4), (c4), and (d4)\]. The envelope of oscillations is seen to increase with frequency when the frequency is increased towards the gap and then decreases when the frequency is further increased outside the gap. For large disks, the low-frequency Faraday rotation is closest to the universal value $ -\arctan\alpha\approx\SI{-7.3}{\milli\radian} $ \[indicated by the lower dashed line in panels (c1) and (d1)\] as predicted for extended systems [@Tse2010PRL] when the substrate thickness $ a $ is small, where $ \alpha = 1/137$ is the fine structure constant. Here the value $ \theta_{\mathrm{F}} \simeq -\arctan\alpha $ is consistent with the low-frequency limit of the dynamic Hall conductivity $ \mathrm{Re}(\sigma_{xy}) \simeq e^2/h $. For small radius \[panel (a1)\], $ \theta_{\mathrm{F}} $ deviates from this universal value due to finite size effects. Conclusion {#sec:disc_conc} ========== In conclusion, we have studied the finite-size effects in the dynamical conductivities and magneto-optical Faraday rotation of a quantum anomalous Hall insulator disk. We find that the continuum massive Dirac model subjected to the topological infinite mass boundary condition becomes a Chern insulator with a unit Chern number. In addition, the overall frequency dependence of both the longitudinal and Hall conductivities matches those calculated from an infinitely extended Chern insulator model described by the massive Dirac Hamiltonian with an additional parabolic term. A flow diagram plotting the edge state contribution to $ \mathrm{Re}(\sigma_{xx}) $ versus $ \mathrm{Re}(\sigma_{xy}) $ shows a semicircle in agreement with that expected for a integer quantum Hall insulator. Our numerical results further show that the edge-to-edge transitions constitute the dominant contribution in the Hall conductivity for frequencies not only within but also beyond the bulk band gap. Studies on the magneto-optical Faraday rotation of a small disk show fluctuational features as a function of frequency. These features arise due to optical transitions between discrete bulk states and is present only for frequencies beyond the band gap. The direct current limit of the Faraday angle shows noticeable deviations with decreasing disk radius from its theoretical value $ \arctan\alpha $ (where $\alpha = 1/137$) based on extended model. In the presence of an underlying substrate, we find an interplay between the Fabry-Pérot resonances and the finite-size effects due to the size quantization in the Faraday rotation spectrum. Our findings highlight the importance of finite-size effects in optical measurements of dynamic Hall conductivity and Faraday effect in which the laser spot coverage exceeds the sample size. Acknowledgment ============== Work at USTC was financially supported by the National Key Research and Development Program (Grant No.s: 2017YFB0405703 and 2016YFA0301700), the National Natural Science Foundation of China (Grant No.s: 11474265 and 11704366), and the China Government Youth 1000-Plan Talent Program. Work at Alabama was supported by startup funds from the University of Alabama and the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Early Career Award [\#]{}[DE]{}-[SC]{}0019326. We are grateful to the supercomputing service of AM-HPC and the Supercomputing Center of USTC for providing the high-performance computing resources. Infinite mass boundary condition {#appsec:IMBC} ================================ For completeness, here we include the derivation for the infinite mass boundary condition following Ref. . Rather than vanishing wavefunction, this boundary condition requires that the normal component of the current at each point on the boundary vanishes, and for our case this means $ \bm{e}_\rho\cdot\bm{j}(R)=0 $ along the radial direction. For a general two-dimensional massive Dirac model defined on a domain $ \mathcal{D} $ in the real space, $$\begin{aligned} H= C\mathbb{I} + A\bm{\sigma}\cdot(-\mathrm{i}\nabla) + M\sigma_z, \end{aligned}$$ we consider the total energy $ E=\int_\mathcal{D}\mathrm{d}^2x\ \Psi^\dagger H\Psi$, which can be written as $$\begin{aligned} E=&\int_\mathcal{D}\mathrm{d}^2x\left[\Psi^\dagger(C\mathbb{I}+M\sigma_z)\Psi\right] \nonumber \\ &-\mathrm{i} A\int_\mathcal{D}\mathrm{d}^2x\left[\nabla\cdot(\Psi^\dagger\bm{\sigma}\Psi)-\nabla\Psi^\dagger\cdot\bm{\sigma}\Psi\right] \nonumber \\ =&\left(\int_\mathcal{D}\mathrm{d}^2x\left[\Psi^\dagger(C\mathbb{I}+M\sigma_z)\Psi\right]-\mathrm{i} A\int_\mathcal{D}\mathrm{d}^2x\left[\Psi^\dagger\bm{\sigma}\cdot\nabla\Psi\right]\right)^ \nonumber \\ &-\mathrm{i} A\oint_{\partial\mathcal{D}}\mathrm{d} l\ \bm{e}_\text{n}\cdot\bm{j}. \label{eq:EJ} \end{aligned}$$ In the last equality, Gauss’ theorem in two dimensions is used to rewrite the last term into a line integral. Equation (\[eq:EJ\]) implies that $ E = E^-\mathrm{i} A\oint_{\partial\mathcal{D}}\mathrm{d} l\ \bm{e}_\text{n}\cdot\bm{j} $, where $ \bm{j}=\Psi^\dagger\bm{\sigma}\Psi $ is the current operator. Since $ E $ is real, we have $ \bm{e}_\text{n}\cdot\bm{j}=0 $. Next we show how this condition leads to the constraint between the two components of the wavefunction $\Psi=(\Psi_\uparrow,\Psi_\downarrow)^\top $: $$\begin{aligned} 0&=\bm{e}_\rho\cdot\bm{j}(R)=\Psi^\dagger\bm{e}_\rho\cdot\bm{\sigma}\Psi=\Psi^\dagger(\sigma_x\cos\phi+\sigma_y\sin\phi)\Psi\\ &= \begin{pmatrix} \Psi_\uparrow^ & \Psi_\downarrow^* \end{pmatrix} \begin{pmatrix} 0 & \mathrm{e}^{-\mathrm{i}\phi}\\ \mathrm{e}^{\mathrm{i}\phi} & 0 \end{pmatrix} \begin{pmatrix} \Psi_\uparrow\\ \Psi_\downarrow \end{pmatrix}=2\mathrm{Re}\left\{ \Psi_\downarrow^\Psi_\uparrow\mathrm{e}^{\mathrm{i}\phi} \right\}. \end{aligned}$$ Therefore, on the boundary, the two components of the wavefunction satisfy $ \Psi_\downarrow/\Psi_\uparrow=\mathrm{i}\alpha\mathrm{e}^{\mathrm{i}\phi} $, with $ \alpha=\pm 1 $ for $ \mathrm{sgn}M_\text{in}\ \mathrm{sgn}M_\text{out}=\pm 1 \,$[@Berry1987PotRSoLA], where $ M_\text{in/out} $ represents the mass term for the region of interest (the nanodisk) or the region surrounding it. Vanishing wavefunction boundary condition {#appsec:ZZBC} ========================================= The zigzag boundary condition is used in graphene along a zigzag edge where one of the pseudospin components is taken to vanish. In addition to the fact that there is no microscopic justification for applying the same boundary condition to two-dimensional quantum anomalous Hall insulators, here we also show explicitly that in the massive Dirac model with a finite $ M $ such a boundary condition cannot be applied. If we try to find edge states by setting one component, say the first one, of the spinor wavefunction in Eq. (\[eq:wave\_function\]) to vanish on the boundary ($ \tilde{\rho}=1 $) just as what was done in Ref. [@Christensen2014PRB], that means for $ l\neq 0 $, we have $ \epsilon=\pm M $. However, this makes the second component also vanish, so states with $ \epsilon=\pm M $ do not exist. Then how about the states with $ \epsilon=0 $? To that end, we need to first set $ \epsilon=0 $ and then solve the Schrödinger equation which results in a modified Bessel equation as the radial equation. The wavefunction takes the form below \[up to a normalization coefficient and $ \kappa=M/(\hbar v)>0 $\] $$\begin{aligned} \Phi_{ln}(\rho,\phi)&\sim\mathrm{e}^{\mathrm{i} l\phi} \begin{pmatrix}[r] \mathrm{I}_l(\kappa\rho)\\ -\mathrm{i}\mathrm{e}^{\mathrm{i}\phi}\mathrm{I}_{l+1}(\kappa\rho) \end{pmatrix}. \end{aligned}$$ For this wavefunction, neither of the component can vanish on the boundary with a finite $ M $. Derivation of the Kubo Formula in the Real-Space Representation {#appsec:Kubo} =============================================================== Here we provide a derivation of the real-space Kubo formula used in this paper based on density matrix. Let us start from the Liouville-von Neumann equation $ \mathrm{i}\hbar\partial_t\rho=[H,\rho] $ and consider a system subjected to a time-dependent perturbation $ H=H_0+H'(t)$. In linear response, the density matrix is expanded to first order $ \rho=\rho_0+\delta\rho $, with $\delta\rho $ satisfying $ \mathrm{i}\hbar\partial_t\delta\rho=[H_0,\delta\rho]+[H',\rho_0] $. Assuming a sinusoidal time dependence of $H'(t) \propto \mathrm{e}^{-\mathrm{i}\omega t}$, $\delta\rho $ therefore satisfies $ \hbar\omega\delta\rho = [H_0,\delta\rho]+[H',\rho_0] $. Solving for the matrix element of $\delta\rho $ from above and using the relations $ H_0\ket{m}=\epsilon\ket{m}, \rho_0\ket{m}=f_0(\epsilon)\ket{m} $ gives $$\begin{aligned} \braket{m'\|\delta\rho\|m} &= \dfrac{f_0(\epsilon')-f_0(\epsilon)}{\epsilon'-\epsilon-\hbar(\omega+\mathrm{i}\eta)}\braket{m'\|H'\|m},\label{eq:drho_el} \end{aligned}$$ where $ f_0 $ is the Fermi-Dirac distribution. Using the summation convention and denoting the elementary charge as $ e=\|e\|>0 $, the perturbation is $ H' = {e}A_bp_b/m = ev_bE_b/(\mathrm{i}\omega) $. Using the Heisenberg equation of motion, $ \braket{m'\|v_a\|m} = \braket{m'\|\dot{x}_a\|m} = {\braket{m'\|[x_a,H]\|m}}/(\mathrm{i}\hbar) \approx {\braket{m'\|x_a\|m}(\epsilon-\epsilon')}/(\mathrm{i}\hbar) $. For a two-dimensional system of area $ \mathcal{A} $, the average paramagnetic current density is $ J_a^{\mathrm{p}} = \text{Tr}\{\delta\rho j_a\} $ where $ j_a=(-e)v_a/\mathcal{A} $ is the single-particle current density operator. This gives $$\begin{aligned} J_a^{\mathrm{p}} & =\sum_{mm'}\braket{m'\|\delta\rho\|m}\braket{m\|j_a\|m'} \label{Jpara} \\ %\nonumber \\ &=-\frac{2\pi}{\hbar} G_0\sum_{mm'}\dfrac{(f_0'-f_0)\Delta\epsilon^2}{\Delta\epsilon-\hbar(\omega+\mathrm{i}\eta)}\dfrac{\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}}{\mathcal{A}}A_b, \nonumber \end{aligned}$$ where $ G_0\equiv e^2/h $ is the conductance quantum and $ \Delta\epsilon\equiv\epsilon'-\epsilon $ is the energy difference between two transition states. The paramagnetic current-current correlation function $ \Pi_{ab}(\omega) $, defined through $ J_a^{\mathrm{p}} = \Pi_{ab}(\omega)A_b $, is therefore $$\begin{aligned} \label{eq:cond_ab} \Pi_{ab}(\omega)&=-\frac{2\pi}{\hbar} G_0\sum_{mm'}\dfrac{(f_0'-f_0)\Delta\epsilon^2}{\Delta\epsilon-\hbar(\omega+\mathrm{i}\eta)}\dfrac{\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}}{\mathcal{A}}. \end{aligned}$$ The conductivity, consisting of both paramagnetic and diamagnetic contributions [@Allen2006], can now be obtained as $$\begin{aligned} \label{eq:opt_cond} \sigma_{ab}(\omega)= & \frac{\Pi_{ab}^{\mathrm{p}}(\omega)-\Pi_{ab}^{\mathrm{p}}(0)}{\mathrm{i}\omega} \nonumber \\ =& \dfrac{2\pi\mathrm{i}}{\mathcal{A}} G_0\sum_{mm'}\dfrac{(f_0'-f_0)\Delta\omega}{\Delta\omega-\omega-\mathrm{i}\eta}{\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}}\nonumber \\ =& \dfrac{2\pi\mathrm{i}}{\mathcal{A}}G_0\sum_{mm'}(f_0'-f_0)\left(\dfrac{\omega}{\Delta\omega-\omega-\mathrm{i}\eta}+1\right) \nonumber \\ & \times{\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}}\nonumber \\ =& \dfrac{2\pi\mathrm{i}}{\mathcal{A}}G_0\sum_{mm'}\dfrac{(f_0'-f_0)\omega}{\Delta\omega-\omega-\mathrm{i}\eta}{\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}}\nonumber \\ & +\dfrac{2\pi\mathrm{i}}{\mathcal{A}}G_0\sum_{mm'}(f_0'-f_0)\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}, \end{aligned}$$ where $ \Delta\omega\equiv(\epsilon'-\epsilon)/\hbar $. The second term vanishes identically because of the commutativity among the components of the coordinate operator, $$\begin{aligned} \label{eq:opt_cond_ide} & \sum_{mm'}(f_0'-f_0)\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}\nonumber \\ = & \sum_{m'} f_0' \braket{m'\|x_b x_a\|m'} - \sum_m f_0 \braket{m\|x_a x_b\|m}\nonumber \\ = & \sum_m f_0 \braket{m\|[x_b, x_a]\|m} = 0, \end{aligned}$$ where in deriving the second line we have used the completeness of states. Therefore substituting Eq. (\[eq:opt\_cond\_ide\]) back into Eq. (\[eq:opt\_cond\]), we obtain the final expression of the conductivity Eq. (\[eq:opt\_cond\_final\]) in the main text with area $ \mathcal{A}=\pi R^2 $ for a disk of radius $ R $, $$\begin{aligned} \label{eq:opt_cond_final2} \tilde\sigma_{ab}(\omega) & =2\mathrm{i}\sum_{mm'}\dfrac{(f_0'-f_0)\omega}{\Delta\omega-\omega-\mathrm{i}\eta}\dfrac{\braket{m\|x_a\|m'}\braket{m'\|x_b\|m}}{R^2}. \end{aligned}$$ [99]{} B. Wunsch, T. Stauber, and F. Guinea, Phys. Rev. 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--- abstract: 'Solving sequential decision prediction problems, including those in imitation learning settings, requires mitigating the problem of covariate shift. The standard approach, DAgger, relies on capturing expert behaviour in all states that the agent reaches. In real-world settings, querying an expert is costly. We propose a new active learning algorithm that selectively queries the expert, based on both a prediction of agent error and a proxy for agent risk, that maintains the performance of unrestrained expert querying systems while substantially reducing the number of expert queries made. We show that our approach, `RadGrad`, has the potential to improve upon existing safety-aware algorithms, and matches or exceeds the performance of DAgger and variants (i.e., SafeDAgger) in one simulated environment. However, we also find that a more complex environment poses challenges not only to our proposed method, but also to existing safety-aware algorithms, which do not match the performance of DAgger in our experiments.' bibliography: - 'output.bib' title: 'RadGrad: Active learning with loss gradients' --- Introduction ============ Sequential decision prediction problems, including imitation learning, differ from typical supervised learning tasks in that the actions of the agent affect the distribution of future observed states. The violation of the distributional stationarity assumption inherent in standard machine learning practice results in error compounding. As the agent drifts into states an expert would not have, error increases due to a lack of relevant training data. Consider, for instance, the task of teaching an autonomous car to stay within road boundaries. A facile approach would be to simply train a supervised learning system where environment states (e.g., road markings) are mapped to expert actions (e.g., the angle and velocity of a human driver) from a dataset of expert driving. During testing, if the car begins to drift off-course (inevitable for any algorithm that does not achieve perfect accuracy), the observed states would begin to differ from the training states. Compounding errors may cause the car to veer completely off-track. To mitigate this issue, algorithms have been designed (i.e., DAgger [@ross2011reduction] and its derivatives) to iteratively aggregate training data on expert behaviour in states that the agent visits. An underlying goal of these works is to maximize accuracy while minimizing the number of expert queries. Yet current approaches still require a high amount of expert input, making them infeasible for many real-world tasks. Consider teaching a robot surgeon: Having a surgeon demonstrate tens of thousands of surgeries is impractical. We propose a new algorithm that requires less expert input than DAgger while performing similarly. Our approach outperforms current state-of-the-art DAgger alternatives (i.e., SafeDAgger [@zhang2016query]) in query efficiency at a similar computational cost, increasing the breadth of real-world problems that can be solved with an imitation learning approach. Related Work ============ We begin by introducing some notation to facilitate comparison of approaches and guide the rest of this paper. In the most general setting, we are given a set of expert demonstrations consisting of states $s$ and the corresponding expert actions (determined by the expert policy $\pi^{}$): $\mathcal{D} = \{s_i, \pi^{}(s_i)\}$. We seek to find a policy $\pi \in \Pi$ that closely mimics the expert policy $\pi^{}$. We define a surrogate loss function that captures how “close" the two policies are $\ell(\pi^{}, \pi)$ and seek to minimize it: $$\hat{\pi} = arg\,min_{\pi \in \Pi}\; \mathbb{E}_s [\ell(\pi^{}, \pi)]$$ Unfortunately, we do not have knowledge of the underlying expert policy and instead have access only to its manifestation as a map from observed states to actions. Training only on states observed by the expert, as in supervised learning, is known to generally lead to poor performance due to covariate shift [@ross2011reduction]. We may improve our estimate of $\pi^{}$, and thus potentially improve $\ell(\pi^{}, \pi)$, by collecting additional expert demonstrations at cost $C(s_i)$ during agent-observed state $s_i$. Accordingly, we minimize subject to a maximum cost $\mathcal{C}$ $$min_{\pi \in \Pi}\; \mathbb{E}_s [\ell(\pi^{}, \pi)] \; s.t. \; \sum_{i=1}^N C(s_i) < \mathcal{C}$$ and must make a choice at each agent-observed $s_i$ whether we wish to query the expert. In the original DAgger algorithm [@ross2011reduction], the cost $C(s_i)$ is implicitly assumed to be zero and the expert is always queried. A number of works have enhanced standard DAgger with probabilistic active learning machinery to determine when querying the expert is optimal under non-zero expert cost. SafeDAgger [@zhang2016query] uses an initial set of demonstrations to train a binary risk classifier that predicts whether the agent will make a mistake in a given state, and then uses this classifier to choose when to query the expert. DropoutDAgger [@menda2017dropoutdagger] uses the Bayesian interpretation of neural networks with dropout to measure the epistemic uncertainty associated with a state. However, it uses this estimate only to guide action selection, while still querying the expert every time. BAgger [@cronrathbagger] incorporates these two ideas by directly modelling the agent’s error with respect to the expert as a Gaussian Process or Bayesian Neural Network. Then, it obtains an empirical estimate of a percentile-based worst-case loss to decide whether to query the expert. In pure reinforcement learning, Bayesian Q-learning [@dearden1998bayesian] and Bayesian Deep Q-learning [@azizzadenesheli2018efficient] learn a probabilistic $Q$ function that incorporates the agent’s uncertainty about future rewards. However, the goal of these works is to achieve an optimal exploration-exploitation trade-off, and they do not address how agents could benefit from access to expert demonstrations. Unlike other approaches, `RadGrad` introduces the concept of a loss gradient (Figure \[fig:flowchart\]). SafeDAgger estimates whether a proposed agent action will exceed the unknown expert action beyond a safety threshold $\tau$ using what we term a loss network. We assume that the loss network is differentiable and query the expert both when the threshold is exceeded but also when the norm of the gradient of the prediction with respect to the concatenated vector of state and proposed action exceeds a separate threshold $\epsilon$. This gradient is a crude proxy for risk. As we describe later, this differs from the concept of uncertainty that Bayesian approaches and ensemble methods are well-suited for. ![The `RadGrad` algorithm queries the expert both when the proposed agent action is predicted to be far from the expert’s action, or if the gradient of this error is high with respect to the state and proposed action.[]{data-label="fig:flowchart"}](flowchart.pdf){height="2in"} Method ====== RadGrad Algorithm ----------------- The four parts of our `RadGrad` approach (the primary network, the loss network, the loss gradient, and data aggregation) are summarized in Algorithm \[alg:radgrad\] and detailed below. #### Primary Network To find a policy $\pi \in \Pi$ that minimizes $\ell(\pi^{}, \pi)$, we require a way to express $\pi$. Accordingly, we learn a function that maps from states in $\mathbb{R}^m$ to actions in $\mathbb{R}^n$. We employ a feed-forward neural network with hidden layer sizes $[128, 128, 32, 8]$ and dropout rate of $0.2$, trained using our set of expert demonstrations $\mathcal{D}$. At test time, an observed state $s_i$ is inputted into this function to generate a proposed agent action $\hat{a}_i$. #### Loss Network We additionally learn a differentiable function, which we term a loss network, that maps state and proposed agent action pairs in $\mathbb{R}^{m+n}$ into an estimate of the difference between the proposed agent action and the unknown expert action in $\mathbb{R}^n$. We employ a feed-forward neural network with hidden layer sizes $[128, 128, 64, 64, 32, 32, 16, 16, 8]$ and dropout rate of $0.2$. When the norm of the loss network output exceeds a safety threshold $\tau$, the expert is queried for expert action $a^{}_i$. This is the strategy specified in Algorithm \[alg:radgrad\]. An alternative implementation of the loss network, more similar to SafeDAgger, maps state and proposed agent action pairs in $\mathbb{R}^{m+n}$ to the probability that the norm of the loss network output exceeds the safety threshold $\tau$. In the latter case, the expert is queried for $a^{}_i$ if the predicted probability exceeds $\frac{1}{2}$. #### Loss Gradient Additionally, we calculate the norm of the gradient of the output of the loss network with respect to its input. If the norm exceeds a threshold $\epsilon$, then we query the expert for $a^{}_i$. This norm is a proxy for risk. A large norm implies that a small change in either the state or proposed action would have a large impact on the probability of exceeding the threshold $\tau$, and thus the agent is in as, we define, a risky state with high potential for error. Without the computationally-costly endeavour of building an ensemble or Bayesian neural network to measure uncertainty proper, we have built a proxy for measuring risk (which we treat as distinct from uncertainty). #### Data Aggregation Whenever the expert is queried, we append the state and expert action pair $(s_i, a^{}_i)$ to $\mathcal{D}$. In this work, we choose to take the expert action whenever we query the expert, although more fine-grained rules could be explored in future work. Appending these state-action pairs serves to shift the distribution of training states from those an expert would see to those the agent sees. We retrain the primary network on this new, aggregated dataset to improve agent performance, separating $20\%$ of the dataset for validation so as to reduce the risk of overfitting. Initialize $\mathcal{D} \gets \emptyset$ Initialize $\pi_{agent, \: 1}$ Initialize loss network $l_{agent, \:1}$ Initialize environment and agent Observe state $s_i$ $\hat{a}_i \gets \pi_{agent,\:k}(s_i)$ $\hat l \gets l_{agent,\:k}(s_i, \hat{a}_i)$ Query the expert to obtain $a^_i \gets \pi^(s_i)$ Execute $a^_i$ $\mathcal{D} \gets \mathcal{D} \cup \{(s_i, \hat{a}_i, \hat{a}^_i)\}$ Execute $\hat{a}_i$ Train $\pi_{agent,\:k+1}$ and $l_{agent,\:k+1}$ on $\mathcal{D}$ return* best $\pi_{agent,\:k}$ on validation set \[alg:radgrad\] Experimental Setup ------------------ #### Algorithms We compare the performance of five primary algorithms in our analysis. These five include three non-gradient algorithms (DAgger, SafeDAgger, and Loss Network) and two gradient algorithms (SafeDAgger Gradient and Loss Network Gradient, or `RadGrad`). The non-gradient algorithms query the expert and execute the returned action if the loss network threshold is surpassed; the gradient algorithms query the expert in this case as well, but also if the gradient threshold is exceeded. SafeDAgger and SafeDAgger Gradient refer to a loss network that outputs the probability of surpassing $\tau$, as opposed to an output in $\mathbb{R}^n$. Additionally, we consider the performance of three baseline methods: expert actions, supervised learning, and random selection. The expert action baseline is the reward achieved by the expert on the task. We consider the expert baseline only implicitly; we present loss measures as the difference in reward between the agent algorithm and the expert. We present the results of a simple supervised learning algorithm (which trains on expert demonstrations in expert-observed states only) to display the issue of covariate shift we wish to resolve. Finally, we present the random selection baseline. Random selection queries and follows the expert at random agent-observed states. A random selection baseline is necessary to establish the complexity of active learning approaches is warranted. Finally, because we observe that random selection performs quite well in practice and hypothesize that an unbiased sampling strategy can be beneficial for convergence and stability, we test two hybrid algorithms: Loss Gradient Random (`RadGrad` Random) and SafeDAgger Gradient Random. At each timestep a fair coin is flipped to determine whether to use the loss-based versus random strategy. #### Environment We test our approach in the Reacher-v2 and Hopper-v2 OpenAI gym environments [@brockman2016openai]. In Reacher-v2, a robotic arm with two degrees of freedom rotates to reach a randomly-positioned target. This environment maps from $s_i \in \mathbb{R}^{11}$ to $a_i \in \mathbb{R}^2$. In Hopper-v2, a two-dimensional one-legged robot hops as quickly as possible towards a target. The Hopper-v2 environment maps from $s_i \in \mathbb{R}^{11}$ to $a_i \in \mathbb{R}^{3}$. We selected open source environments for easy reproducibility. #### Hyperparameters Table \[tab:hyper\] summarizes the hyperparameters we used in our evaluations. These were chosen so as to optimize expert query efficiency while maintaining convergence to the algorithm’s best policy. The random baseline hyperparameter was chosen so as to make that strategy competitive with active learning strategies in query efficiency (Equation \[eq:effic\]). Algorithm Hyperparameter Reacher-v2 Hopper-v2 --------------------------- ------------------- ------------ ----------- -- Loss $\tau$ 0.02 0.3 Loss Gradient (`RadGrad`) $\epsilon$ 0.002 0.2 SafeDAgger $\tau$ 0.04 0.3 SafeDAgger Gradient $\epsilon$ 1 200 Random $P(\text{Query})$ $30\%$ $30\%$ : Hyperparameters for proposed algorithms and baselines. $\tau$ is the threshold on the norm of predicted loss, and $\epsilon$ is the threshold on the norm of the gradient of predicted loss with respect to input and action space. Note the two values of $\tau$ for Reacher-v2 differ since displayed values were individually-optimal.[]{data-label="tab:hyper"} Results ======= Our results show that gradient-based methods can outperform their non-gradient-based counterparts in that they may yield higher rewards with only a modest increase in the number of expert queries required. To compare algorithm performance, we define the query efficiency of estimated policy $\hat{\pi}$: $$\text{Efficiency}(\hat{\pi}) \propto \frac{\text{reward}_{\hat{\pi}} - \text{reward}_{supervised}}{\sum_{i \in \mathcal{D}_{\hat{\pi}}} C(s_i)} \label{eq:effic}$$ This is the difference in loss between a supervised learning policy (trained only on expert actions in expert-observed states) and the active learning policy in question, divided by the total cost of querying the expert at states $s_i$ during the estimation of $\hat{\pi}$. For our purposes, we let $C(s_i) = 1 \: \forall \: s_i$, and thus $\sum_{i \in \mathcal{D}_{\hat{\pi}}} C(s_i)$ is simply the number of times the expert was queried in the estimation of $\hat{\pi}$ (i.e. $\#\mathcal{D}_{\hat{\pi}}$). Accurate policies that require few queries to estimate are query efficient. Table \[tab:performance\] shows the test-time performance, that is, average reward when expert demonstrations are not available, of all algorithms in each environment. It also shows, for each setting, the number of expert queries used to estimate the policy and the resulting efficiency. Figures \[fig:reacher\] and \[fig:hopper\] show test-time performance and training-time number of expert queries used over the course of training iterations for the two environments, Reacher-v2 and Hopper-v2, respectively. At each point in training, the current model is deployed on a batch of random test-time environments to generate the curves of performance over time shown in the graphs. ----------------- ----------- ----------------- -------------- ----------- ---------------- -------------- Algorithm Queries Loss Efficiency Queries Loss Efficiency SafeDAgger (SD) 1424 $1.67 \pm 1.08$ -6.3 60094 $3547 \pm 5$ 22 SD Gradient 1436 $0.94 \pm 0.62$ -1.2 71591 $3626 \pm 203$ 8 SD Gr. Random 1551 $0.57 \pm 0.46$ 1.3 53542 $606 \pm 843$ 574 Loss 556 $3.38 \pm 1.83$ -47 16786 $1547 \pm 631$ 1270 Loss Gradient 3332 $0.70 \pm 0.43$ 0.2 62378 $3567 \pm 4$ 18 Loss Gr. Random 2200 $0.50 \pm 0.30$ 1.2 64053 $2342 \pm 605$ 209 DAgger 3750 $0.41 \pm 0.62$ 0.9 42682 $1890 \pm 148$ 419 Random 1343 $0.56 \pm 0.46$ 1.5 24025 $1892 \pm 122$ 744 Supervised 3750 $0.77 \pm 0.57$ 0 140164 $3679 \pm 19$ 0 ----------------- ----------- ----------------- -------------- ----------- ---------------- -------------- : Comparison of performance and query efficiency of gradient-based and non-gradient approaches. Displayed loss is loss in increment of expert loss, along with intervals of one standard deviation over 100 trials. Expert policies are obtained from Berkeley’s Deep Reinforcement Learning course materials (<https://github.com/berkeleydeeprlcourse/homework/tree/master/hw1/experts>). Results presented are from final iteration of fifteen, with 100 trials at each iteration.[]{data-label="tab:performance"} table \[x=iteration, y=loss, y error=err, col sep=comma\][reacher\_final\_dagger\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][reacher\_final\_loss\_002\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][reacher\_final\_safedagger\_004\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][reacher\_final\_gradient\_loss\_002\_0002\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][reacher\_final\_safedagger\_gradient\_004\_1\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][reacher\_final\_random\_30\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][reacher\_final\_dagger\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][reacher\_final\_loss\_002\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][reacher\_final\_safedagger\_004\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][reacher\_final\_gradient\_loss\_002\_0002\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][reacher\_final\_safedagger\_gradient\_004\_1\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][reacher\_final\_random\_30\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][hopper\_final\_dagger\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][hopper\_final\_loss\_03\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][hopper\_final\_safedagger\_03\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][hopper\_final\_gradient\_loss\_03\_02\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][hopper\_final\_safedagger\_gradient\_03\_200\_for\_plot.csv]{}; table \[x=iteration, y=loss, y error=err, col sep=comma\][hopper\_final\_random\_30\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][hopper\_final\_dagger\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][hopper\_final\_loss\_03\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][hopper\_final\_safedagger\_03\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][hopper\_final\_gradient\_loss\_03\_02\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][hopper\_final\_safedagger\_gradient\_03\_200\_for\_plot.csv]{}; table \[x=iteration, y=total\_obs, col sep=comma\][hopper\_final\_random\_30\_for\_plot.csv]{}; We make three conclusions. First, adding gradient logic to safety-aware baselines (SafeDAgger Gradient and Loss Gradient) improves performance and efficiency on Reacher-v2, and integrating gradient logic with random sampling (SafeDAgger Gradient Random and Loss Gradient Random) further improves average reward as well as efficiency. This result suggests the validity of loss gradients as a proxy for risk, as well as the benefit of unbiased expert sampling. We further note that, although not statistically significant, test-time performance of risk-aware algorithms appears superior to that of DAgger early on. We hypothesize that this occurs because, early in training, risk-aware strategies shift the distribution of training data toward riskier states, causing the trained models to give more importance to those states than DAgger would. Second, although DAgger shows best performance overall, random sampling is a strong baseline, converging to similar performance as DAgger with substantially improved query efficiency in both Reacher-v2 and Hopper-v2. This indicates that an unbiased sampling strategy may be a competitive model against which proposed active learning strategies should be tested. Third, most safety- and risk-aware algorithms fail to converge to DAgger performance in the more complex environment Hopper-2. In this setting, DAgger and random sampling stand out as strong algorithms despite their simplicity. While some proposed algorithms (Loss and SafeDAgger Gradient Random) show promising performance and efficiency numbers, the fact that other safety-aware algorithms, including the established SafeDAgger baseline, fail to converge to DAgger performance makes us cautious to make strong conclusions from these data. While it is possible that these algorithms would converge to DAgger performance under more extensive hyperparameter tuning, this result hints at the challenges posed by richer environments to algorithms that aim at outperforming DAgger and random baselines. A final observation is that in our Reacher-v2 simulations, SafeDAgger Gradient queries the expert fewer times than SafeDAgger proper for much of the training course (Figure \[fig:reacher\]), even though the condition for querying the expert in SafeDAgger Gradient is more relaxed. We hypothesize that this occurs because SafeDAgger Gradient, by using the gradient of loss as a proxy for risk and obtaining expert demonstrations in the face of such risk, is better able to reduce future risk and thus future need for expert queries. We note, however, that this is not the case for Hopper-v2. Limitations =========== While our work suggests the value of gradients as a proxy for risk in active learning, our experiments are hardly conclusive. Most notably, we did not complete an extensive analysis of the value of gradients in all of the major DAgger derivatives and instead focused our efforts on SafeDAgger. Due to computational limitations, even though we deployed each trained agent at each training iteration in multiple randomly sampled test-time environments, we executed this procedure only once per algorithm. In other words, only one agent was trained per algorithm. For a more robust evaluation, we would train a number of agents for each algorithm to produce uncertainty estimates for the number of expert queries made as well. While our Reacher-v2 simulations show that adding gradient logic to active learning decision rules has the potential to improve performance and, thus, should be further investigated, we could not replicate those results in the second, more challenging environment Hopper-v2. Not only did our gradient-based methods not improve performance over DAgger and random selection in Hopper-v2, but even the established SafeDAgger did not converge to a competitive policy in that case. While we do not rule out that more extensive hyperparameter search could improve the performance of those algorithms, we believe this result should be a call for more robust methods that can more easily be transferred to new environments. Similarly, we believe further investigation of the trade-offs of active learning and unbiased random strategies to be necessary. Not only did we find that a random querying strategy is highly competitive to both DAgger and safety-aware strategies, but most importantly, we also found that random selection was the most robust policy when replicating our experiments on a new environment. Thus, unbiased strategies seem to provide stronger practical guarantees of generalization compared to current state-of-the-art active learning strategies. Of course, purely random sampling may not be possible due to safety risk; in these cases observation weighting may offer a compromise. Conclusion ========== DAgger is able to improve over a purely supervised learning approach by mitigating the problem of covariate shift, but does so at a high expert querying cost. Various DAgger derivatives have been created to limit the number of expert queries made while maintaining similar policy quality as DAgger. We have shown that these methods may be possible to improve by incorporating gradients. We experienced difficulty in replicating the performance of both popular active learning strategies and our proposed methods in a more complex environment. Further research on the robustness of active learning algorithms across environments is necessary. Finally, we observed that a random selection algorithm, which obtains unbiased samples of expert demonstrations, is a strongly competitive alternative to both query-intensive and safety-aware methods. Future imitation learning active learning algorithms should compare to a random querying baseline to establish algorithmic complexity is warranted.
--- abstract: 'Motivated by recent work \[D. Cubero [et al.]{}, Phys. Rev. E [82]{}, 041116 (2010)\], we examine the mechanisms which determine current reversals in rocking ratchets as observed by varying the frequency of the drive. We found that a class of these current reversals in the frequency domain are precisely determined by dissipation-induced symmetry breaking. Our experimental and theoretical work thus extends and generalizes the previously identified relationship between dynamical and symmetry-breaking mechanisms in the generation of current reversals.' author: - 'A. Wickenbrock$^{1}$, D. Cubero$^{2}$, N.A. Abdul Wahab$^{1}$, P. Phoonthong$^{1}$, and F. Renzoni$^{1}$' title: 'Current reversals in a rocking ratchet: the frequency domain' --- Introduction ============ In out-of-equilibrium systems, directed transport can be obtained without the application of a net bias force. Such a counter-intuitive phenomenon is usually termed the ratchet effect [@comptes; @magnasco; @adjari; @bartussek; @doering; @cubero06; @reimann; @rmp09]. The occurrence of a directed current in such systems can be precisely related to the breaking of the relevant spatio-temporal symmetries [@flach00; @flach01; @super]. An intriguing feature of ratchets are current reversals, where the sign of the generated current changes following a variation in a system parameter [@bartussek; @jung; @mateos]. As a specific example, consider the case of a rocking ratchet, consisting of Brownian particles in an asymmetric sawtooth potential. A sinusoidal rocking force drives the system out of equilibrium, thus allowing the generation of directed transport. In such a system, current reversals can be observed by varying the noise strength at fixed amplitude of the rocking force, as well as by varying the force amplitude while keeping constant the noise strength [@bartussek]. The key feature of current reversals is that the sign of the current can be reversed by varying a system parameter, although the considered parameter does not change the symmetry of the Hamiltonian. At first sight this suggests that current reversals are a dynamical phenomenon, not traceble back to a symmetry-breaking mechanism. Previous theoretical work aimed to identify the mechanisms underlying current reversals in rocking ratchets. For an underdamped deterministic ratchet, Mateos [@mateos] argued that current reversals induced by a variation of the driving strength correspond to a bifurcation from a chaotic to a periodic regime. However, the proposed mechanisms turned out not to be general, as in the very same system current reversals can be observed also in the absence of such bifurcations, and moreover not all chaos-to-order transitions necessarily lead to current reversals [@barbi; @anatole]. A different approach toward the understanding of current reversals induced by a variation of the rocking force amplitude was recently introduced for a biharmonically driven spatially symmetric rocking ratchet [@cubero]. It was shown that a class of current reversals is precisely determined by symmetry breaking. In this way, a link was established between dynamical and symmetry-breaking mechanisms. The still open issue now is to which extent the above link can be generalized, either to different systems or to current reversals of different type in the same set-up. In the present work, we generalise the link between dynamical and symmetry-breaking mechanisms introduced in the aforementioned work. We consider the same ratchet system and examine the current reversals induced by a variation of the frequency of the rocking force. We notice that a variation of the rocking force frequency cannot be mapped onto a variation of the force amplitude, i.e. the current reversals considered here are not a priori equivalent to the ones observed by varying the force strength. Our experimental and theoretical work shows that also in this case there is a class of current reversals that are determined by symmetry breaking, thus generalizing the argument put forward in Ref. [@cubero]. This work is organized as follows. In Section II we introduce the ratchet set-up used in this work, and recall the elements of the symmetry analysis that will be needed to establish the link between current reversals in the frequency domain and dissipation-induced symmetry breaking. Section III presents experimental results obtained with cold atoms in a driven optical lattice in the regime of weak damping. Section IV analyses theoretically the relationship between dynamical and symmetry-breaking mechanisms for the current reversals induced by a variation in the driving frequency. Conclusions are drawn in Sec. \[sec:conclusions\]. Set-up and symmetries ===================== Ratchet set-up -------------- In this work we consider a ratchet set-up consisting of Brownian particles in a spatially symmetric periodic potential driven by a time-asymmetric force. To capture the main features of the dynamics, in the theoretical analysis we consider the simple case of a linear friction, as usually considered in the ratchet literature. This will also allow us to explore different regimes of damping and noise. The relevant Langevin equation is in this case: $$m\ddot{x}=-\alpha\dot{x}-U'(x)+F(t)+\xi(t)$$ where $U(x)=U_0\cos(2kx)/2$ is a periodic potential, $\alpha$ is the friction coefficient, $\xi(t)$ is a Gaussian white noise: $\langle \xi(t)\rangle = 0$, $\langle \xi(t)\xi(t')\rangle = 2D\delta(t-t')$, and $F(t)$ is a biharmonic drive described by: $$F(t) = F_0\left[ A_1\cos(\omega t)+ A_2\cos(2\omega t+\phi)\right]~. \label{eq:drive}$$ The appearance of a ratchet-effect, i.e. the generation of directed motion, in such a set-up is a very well established fact [@fabio; @chialvo; @dykman; @goychuk; @luchinsky2; @machura]. Symmetry analysis ----------------- We now recall the essential elements of the symmetry analysis that, initially developed to explain the generation of a current [@flach00; @flach01; @super], will allow us to introduce the basic concepts that will be needed to establish the link between current reversals in the frequency domain and dissipation-induced symmetry breaking. In general, the symmetry analysis [@flach00; @flach01; @super] is used to identify the symmetry of the system which prevent directed motion. For the specific case of a spatially symmetric potential, of interest here, there are two of those symmetries: the shift symmetry, which corresponds to invariance under the transformation $(x,p,t)\to (-x,-p,t+T/2)$, with $T$ the period of the drive, and the time-reversal symmetry, which requires invariance under the transformation $(x,p,t)\to (x,-p,-t)$. A bi-harmonic drive of the form of Eq. \[eq:drive\] breaks the shift symmetry for any value of the relative phase $\phi$. For the time-reversal symmetry it is necessary to distinguish different cases, corresponding to different levels of dissipation. For no dissipation (Hamiltonian case), the system is symmetric under time-reversal for $\phi=n\pi$ with $n$ integer, and therefore for these values directed motion cannot be produced. It can be shown [@flach00; @niurka] that the dependence of the average velocity on the phase $\phi$ is, in leading order, $v = A\sin\phi$. Consider now the case of non zero dissipation. For non zero dissipation, the time-reversal symmetry is broken by dissipation also for $\phi=n\pi$ with $n$ integer. Thus, a current can be generated also for these values of the phase $\phi$. For weak dissipation, the dependence of the average velocity on the phase $\phi$ is, in leading order, $v = A\sin(\phi-\phi_0)$, where $\phi_0$ is a dissipation-induced symmetry-breaking phase lag which vanishes in the Hamiltonian limit [@flach01; @gommers; @niurka]. Finally, in the overdamped regime, the system is invariant under the so-called “supersymmetry” [@super] $(x,p,t)\to (x+\lambda/2,-p,-t)$, with $\lambda$ the spatial period of the potential, for $\phi=\pi/2 + n\pi$, with $n$ integer. For these values of the phase $\phi$, direction motion is not allowed. Experimental results ==================== The experimental set-up and procedure are substantially the same as the ones used in our previous work of Ref. [@cubero], and we recall here only the essential elements. Up to $10^8$ $^{87}$Rb atoms are trapped and cooled down to $\sim 50$ $\mu K$ in a magneto-optical trap. The atoms are then loaded in a 1D lin$\perp$lin dissipative optical lattice, in which the atom-light interaction determines both a periodic potential for the atoms and the dissipation mechanism which leads to a friction force and to fluctuations in the atomic dynamics. A driving of the form of Eq. (\[eq:drive\]) is applied by phase modulating one of the lattice beams [@advances]. For all the measurements presented in this work, the ratio between harmonics is kept fixed: $A_1=2$, $A_2=1$. The motion of the atoms in the driven lattice is studied by imaging the atomic cloud with a CCD camera. The velocity of the center of mass of the atomic cloud is then derived by using these images. We first consider the standard configuration for the detection of current reversals: for a fixed Hamiltonian, with broken symmetry, the current is studied as a function of a parameter, whose variation does not change the symmetry of the Hamiltonian. In the present case, we fix the ralative phase between driving harmonics $\phi=\pi/2$, so to break the time reversal symmetry, and study the current as a function of the driving frequency $\omega$. Results of our experiment are shown in Fig. \[fig:fig\_exp1\]. The familiar situation of current reversals in the frequency domain [@luchinsky2; @gommers05] is observed: by varying the frequency of the drive it is possible to revert the direction of the atomic current through the lattice. ![ Experimental results for 1D rocking ratchet for cold atoms. The average atomic velocity, rescaled by the recoil velocity $v_r$ ($v_r=5.88$ mm/s for $^{87}$Rb) is reported as a function of the frequency of the drive, for a fixed value of the relative phase between harmonics of the ac drive. The different data sets correspond to different amplitudes of the driving force. The parameters of the optical lattice are: detuning from resonance $\Delta=-9\Gamma$ and intensity per lattice beam: $I_L=(43.5\pm 0.3)$ mW/cm$^2$. The driving is a biharmonic force of the form of Eq. (\[eq:drive\]), with the relative phase kept fixed at $\phi=\pi/2$ for the presented sets of measurements. The amplitude of the force is $F_0 = - m\lambda g_0$ where $m$ is the atomic mass, $\lambda$ the laser field wavelength, and the values for $g_0$ (in kHz$^2$) are, for the different set of data reported in the figures: (a) $g_0=3.2\cdot 10^3$, (b) $g_0=12.8\cdot 10^3$, (c) $g_0=19.2\cdot 10^3$, (d) $g_0=25.6\cdot 10^3$. The lines are a guide for the eye. []{data-label="fig:fig_exp1"}](fig_exp1.eps){height="4.in"} We now study the atomic current as a function of the phase $\phi$ for different values of the driving frequency, in a range of frequencies around the value at which the current reversal is observed. This will allow us to establish a relationship between the observed current reversal and dissipative effects. ![ Experimental results for 1D rocking ratchet for cold atoms. The driving is a biharmonic force of the form of Eq. (\[eq:drive\]). The average atomic velocity, rescaled by the recoil velocity $v_r$, is plotted as a function of the relative phase $\phi$ between harmonics of the ac drive, for different values of the driving force frequency. The lines are the best fit of the data with the function $v/v_r = A \sin(\phi-\phi_0)$. The lattice parameters are the same as in Fig. \[fig:fig\_exp1\]. The frequency of the drive for the different data sets is: (a) $\omega/(2\pi)= 40$ kHz, (b) $\omega/(2\pi)= 50$ kHz, (c) $\omega/(2\pi)= 60$ kHz, (d) $\omega/(2\pi)= 70$ kHz, (e) $\omega/(2\pi)= 80$ kHz, (f) $\omega/(2\pi)= 100$ kHz, (g) $\omega/(2\pi)= 120$ kHz. The driving force amplitude, for all data sets, is $F_0=-m\lambda g_0$ with $g_0=19.2\cdot 10^3$ kHz$^2$. []{data-label="fig:fig_exp2"}](fig_exp2.eps){height="5.in"} The results of our measurements are shown in Fig. \[fig:fig\_exp2\], for a driving force amplitude corresponding to that of Fig. \[fig:fig\_exp1\](c). For all considered driving frequencies, the dependence of the current on the phase $\phi$ is well described by $v/v_r=A\sin(\phi-\phi_0)$, with $\phi_0$ a dissipation-induced symmetry-breaking phase lag. This is in agreement with previous observations [@gommers; @cubero]. The important, and so far unexplored, fact here is that $\phi_0$ varies significantly when the driving frequency is scanned across the value corresponding to the current reversal in the frequency domain. ![ Left (a-d): dissipation-induced phase lag $\phi_0$, as obtained by fitting data as those in Fig. \[fig:fig\_exp2\], with the function $v/v_r = A \sin(\phi-\phi_0)$, as a function of the frequency of the drive, for different values of the driving strength. Right (e-h): amplitude A, obtained from the same fits. Different rows of plots of the figure correspond to different driving strength, with the amplitudes of the drive the same as in Fig. \[fig:fig\_exp1\], e.g. the data in (a) and (e) correspond to a driving amplitude $g_0=3.2\cdot 10^3$ kHz$^2$ and so on. The lattice parameters are the same as in Fig. \[fig:fig\_exp1\]. The lines are a guide ot the eye. The arrows indicate the frequency at which the current reversal shown in Fig. \[fig:fig\_exp1\] occurs. []{data-label="fig:fig_exp3"}](fig_exp3.eps){height="3.in"} The dependence of the phase-lag $\phi_0$ and the amplitude $A$, as obtained by fitting data as those of Fig. \[fig:fig\_exp2\] with the function $v/v_r = A \sin(\phi-\phi_0)$, on the driving frequency is reported in Fig. \[fig:fig\_exp3\]. The four different data sets correspond to the four different driving strength amplitudes for which the data in Fig. \[fig:fig\_exp1\] were obtained. The phase-lag $\phi_0$ shows a very large variation around the frequency at which the current reversal is observed in Fig. \[fig:fig\_exp1\], with the phase lag varying from $-\pi$ to $0$ around the current reversal, and taking the value $-\pi/2$ at the reversal frequency [@note]. Since in the experiments reported in Fig. \[fig:fig\_exp1\] the relative phase is fixed to $\phi=\pi/2$, this value of the phase-lag, $\phi_0=-\pi/2$, guarantees that the current is suppressed at the reversal frequency, where the directed current inverts its sign. On the other hand, the amplitude of the sin-like curve, see right panels in Fig. \[fig:fig\_exp3\], stays finite around the reversal frequency. Thus, we can conclude that the current reversal in the frequency domain is determined by the large variation, around the reversal frequency, of the dissipation-induced symmetry-breaking phase lag $\phi_0$. This generalizes the link between symmetry-breaking and current reversals established in Ref. [@cubero] in the case of current reversals in the amplitude domain. Numerical analysis ================== The presented experimental results show that the current reversals in the frequency domain are determined by a sharp variation of the dissipative phase lag $\phi_0$ when the driving frequency is varied around the value corresponding to the current reversal. The conditions of our experiment correspond to the weakly damped regime, hence the appearance of a nonzero dissipation-induced phase lag $\phi_0$. The validity of the established link between current reversals and dissipation-induced symmetry breaking can be tested by considering the Hamiltonian and the overdamped limit. In both cases, the phase lag $\phi_0$ is fixed by the symmetries of the system valid in the respective limits: $\phi_0=0$ for the Hamiltonian limit and $\phi_0=-\pi/2$ [@note2] for the overdamped case, as discussed in Sec. II.B. Thus $\phi_0$ cannot vary with the driving frequency, and the current reversals observed in the case of moderate dissipation should disappear. Our experimental set-up is not suitable to explore the two extreme case of no-dissipation, and very large dissipation. Thus, in order to show that these current reversals vanish when the phase lag $\phi_0$ is fixed by the symmetries in the Hamiltonian and overdamped regimes, we resort to numerical simulations of Eq. (1). ![(Color online) Average atomic velocity as a function of the relative phase $\phi$ between harmonics of the ac drive for several values of the driving frequency $\omega$, with $\omega_v=k(2U_0/m)^{1/2}$ and $A_1=A_2=1$. Top panel: simulation data in the Hamiltonian regime ($\alpha=D=0$) for $F_0=0.2 \,U_0k$. Middle panel: simulation data in the weakly damped regime. The driving amplitude, the friction and the noise strength values are fixed to $F_0 = 0.2 \,U_0k$, $\alpha=0.15 \alpha_0$, and $D=1.944 D_0$, respectively, where $\alpha_0=m k v_0$, $D_0=\alpha_0^2 v_0/k$, and $v_0=(U_0/m)^{1/2}/10$. The values about $\phi=\pi/2$ (or $\phi=3\pi/2$) show a current reversal as the driving frequency is increased. Bottom panel: simulation data in the overdamped regime for $F_0= U_0k$, $\alpha=100 \alpha_0$, and $D=1.944\cdot 10^3 D_0$. Lines are a guide to the eye.[]{data-label="fig:fig_theory"}](fig4.eps){height="3in"} Let us examine first the weakly damped regime with this model. The middle panel of Fig. 4 shows the average particles’ velocity as a function of the driving phase for different values of the driving frequency. A current reversal can be clearly observed at about $\phi=\pi/2$ (or $\phi=3\pi/2$) as the driving frequency is increased, being a consequence of a variation of the phase lag $\phi_0$ with the driving frequency. This is the regime explored in our experiment. On the other hand, the phase lag $\phi_0$ is fixed to $0$ in the Hamiltonian regime due to the time-reversal symmetry. Accordingly, the top panel of Fig. 4 shows that no current reversals are observed when the driving frequency is varied. In the limit of strong dissipation, the appearance of the supersymmetry fixes the phase lag to $\phi_0=-\pi/2$, as shown in the bottom panel of Fig. \[fig:fig\_theory\]. This fact prevents the occurrence of the current reversals observed in the presence of moderate dissipation. The numerical simulations thus confirm the validity of the established link between current reversals in the frequency domain and dissipation-induced symmetry breaking. Conclusions {#sec:conclusions} =========== In conclusion, we examined the mechanisms which determine current reversals in rocking ratchets as observed by varying the frequency of the drive. We considered the specific case of a bi-harmonically driven spatially symmetric ratchet. We found that a class of these current reversals in the frequency domain are precisely determined by dissipation-induced symmetry breaking. Correspondingly, these reversals are observed only for moderate dissipation, and disappear in the Hamiltonian and overdamped limit. Our experimental and theoretical work thus extends and generalizes the previously identified relationship [@cubero] between dynamical and symmetry-breaking mechanisms in the generation of current reversals. In the future, it would be interesting to re-examine the issue of the relationship between microscopic dynamics and current reversals in the system studied by Mateos [@mateos]. Whether the current reversals discussed there can be associated to some distinctive features of the microscopic dynamics is still an open question. This research was supported by the Leverhulme Trust. One of us (DC) also thanks the Ministerio de Ciencia e Innovación of Spain for financial support (grant FIS2008-02873). [99]{} A. Ajdari and J. Prost, C.R. Acad. Sci. Paris [315]{}, 1635 (1992). M.O. Magnasco, Phys. Rev. Lett. [71]{}, 1477 (1993). A. Adjari, D. Mukamel, L. Peliti, J. Prost, J. Phys. I (France) [4]{}, 1551 (1994). R. Bartussek, P. Hänggi and J.G. Kissner, Europhys. Lett. [28]{}, 459 (1994). C.R. Doering, W. Horsthemke, and J. Riordan, Phys. Rev. Lett. [72]{}, 2984 (1994). D. Cubero, J. Casado-Pascual, A. Alvarez, M. Morillo, P. Hänggi, Acta Physica Pol. B [37]{}, 1467 (2006) P. Reimann, Phys. Rep. 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A [273]{}, 316 (2000). L. Machura, M. Kostur, J. Luczka, Chem. Phys. [375]{}, 445 (2010). N.R. Quintero, J.A. Cuesta, and R. Alvarez-Nodarse, Phys. Rev. E [81]{}, 030102(R) (2010). R. Gommers, S. Bergamini and F. Renzoni, Phys. Rev. Lett. [95]{}, 073003 (2005). F. Renzoni, Adv. At. Mol. Opt. Phys. [57]{}, 1 (2009). R. Gommers, P. Douglas, S. Bergamini, M. Goonasekera, P.H. Jones and F. Renzoni, Phys. Rev. Lett. 94, 143001 (2005) Not all the plots in the figure show exacly $\phi_0=\pi/2$. This is due to a measurement error, as $\phi_0$ is determined by fitting the entire sin-like curve obtained by varying $\phi$ while the reversal frequency is determined by measuring only the current for $\phi=\pi/2$. Note that the fact that the phase lag is $\phi_0=-\pi/2$ and not $\phi_0=\pi/2$ (assuming a positive $A$) cannot be determined a priori by the symmetry analysis, and here is reported based on the a posteriori observation of the numerical results.
--- title: \| Relativistic Heavy Ion Physics\ [(Accepted for publication in Landolt-Börnstein I/23A)]{} --- Introduction ============ Reinhard Stock\ stock@ikf.uni-frankfurt.de Overview -------- Quantum Chromodynamics (QCD), the gauge theory of strong interaction, is firmly rooted within the Standard Model of elementary interaction. The elementary constituents of QCD, quarks and gluons that carry the color charge field, have all been observed, in a wealth of by now “classical” experiments in particle physics, such as deep inelastic electron-proton scattering or jet production in electron-positron annihilation and proton-antiproton collisions, at up to TeV center of mass energies. Common to all such observations of partons is an extremely high spatial resolution scale (provided by very high momentum transfer), that allows to recognize a universality of parton interaction at vanishingly small distance: irrespective of their attached fields, and large scale environment. The “running” QCD coupling constant becomes small enough to treat partonic interactions in a perturbative QCD framework amenable to exact solutions (analogous to QED).\ On the contrary, QCD at modest resolution remains one of the open sectors of the Standard Model. In particular, the confinement-deconfinement transition between hadrons and partons and, more generally, the transition between hadronic and partonic extended matter have remained un-addressed in the course of particle physics progress. These are of fundamental importance toward the understanding of the primordial cosmological expansion that passed through the QCD color neutralization phase transition to hadrons at about 5 microseconds, giving rise to all ponderable matter in the present universe. One has to address extended continuum QCD matter at extremely high energy density. At hadronization, the energy density of cosmological matter amounts to about 1 GeV per cubic-Fermi ($fm^{3}$), corresponding to about 2$\times$10$^{18}$ kg per cubic meter, and the strong interaction coupling constant is high, almost unity, and thus deep into the non-perturbative sector of QCD.\ Likewise, neutron star interior matter, or matter dynamics in neutron star mergers (that give rise to heavy nuclei in the interstellar medium and in planets) require knowledge of high density hadronic, or perhaps even quark matter. In addition to the QCD confinement-deconfinement phase transformation (believed to result from gluonic screening of the long range part of color forces), a further characteristic QCD phase transition is involved in hadronic matter close to the critical energy density: the restoration of chiral symmetry in QCD matter. This is an invariance of the QCD Lagrangian, at least for the near-massless light quarks that constitute all cold matter in the universe. It is spontaneously broken in the transition from partons to massive hadrons (this breaking being the origin of allmost the entire hadron mass). Hadron mass is the consequence of non-perturbative vacuum condensates of QCD, which are expected to “melt” as matter approaches the critical conditions. Non perturbative QCD can be numerically approached by solutions on the dicretized space-time lattice, and the finite temperature sector of lattice QCD theory is under intense recent development.\ What is required in the research field of matter under the governance of the strong interaction is the PHASE DIAGRAM of extended QCD matter, and the EQUATION OF STATE (EOS) governing the relationship of pressure to density, in each of its characteristic domains in density and temperature. Traditional nuclear physics could only offer insight into the ground state of extended QCD matter, and traditional particle physics has dealt essentially only with near-groundstate hadrons and their intrinsic structure. Both fields have merged in RELATIVISTIC HEAVY ION PHYSICS, the topic of this Volume: the study of collisions of heavy nuclear projectiles at relativistic energy. In such collisions an initial dynamics of compression and heating converts the incident, cold nuclear ground state matter into a “fireball” of hadronic or partonic matter, thus populating the QCD matter phase diagram, and notably the deconfined state of a QUARK-GLUON-PLASMA, predicted by lattice QCD to exist over a wide domain of temperature and density. As it turns out the energy available from current synchrotron (CERN SPS) or collider facilities (RHIC at BNL, LHC at CERN) suffices to reach plasma temperatures of up to 1 GeV, i.e. far beyond the QCD phase transition critical temperature, of about 170 MeV.\ Overall, theoretical studies of QCD in the non-perturbative regimes indicate that QCD matter has a rich phase structure. The phase diagram can be parametrized by the grand canonical variables, temperature T and baryochemical potential $\mu{B}$. Based on the phase diagram, as elucidated by relativistic nuclear collision studies, we obtain perspectives on how the vacuum structure of the early universe evolved in extremely high T states after the Big Bang, as well as what happens in states of extreme baryon density, at the core of neutron stars, and in their merger collisions. Above the deconfinement transition line of the phase diagram we confront a novel partonic continuum state, the quark gluon-plasma(QGP). It turns out to feature strikingly unexpected features, behaving as a strongly coupled liquid state with almost vanishing shear viscosity. In fact this plasma state may turn out to be the first experimentally accessible realization of string theory as it appears that the strongly coupled liquid state lends itself to a calculable framework found in the 5-dimensional AdS/CFT theory. A comprehensive and quantitative understanding of the QCD phase diagram is the most important subject in modern nuclear physics.\ History ------- The research field of Relativistic Heavy Ion Collisions was born in the late 1960’s, from a coincidence of questions arising in astrophysics (neutron star interior matter, supernova dynamics, early stages in the cosmological evolution) and in fundamental nuclear/hadronic physics (extended nuclear matter and its collective properties, excited hadronic matter and its limits of existence). Generalizing such aspects we see that a description was sought of the phase diagram of strongly interacting matter, in the variables of temperature (big bang and hadronic matter limiting temperature) and matter density (notably invited by the extreme baryon density expected in the neutron star interior). In any corner of this phase diagram the macroscopic statics and dynamics would be determined by an appropriate equation of state (EOS), relating pressure to density and temperature of strongly interacting matter.\ The first employ of the EOS concept was made in the hydrostatic equilibrium model for neutron star density profiles and stability, by Oppenheimer and Volkov [@1]. We are dealing with truely macroscopic, if not gigantic nuclear matter extensions here, but it is noteworthy to recall that, by 1960, nuclear physics had arrived at the realization that even the baryonic matter inside heavy nuclei, however small, features a continuous, quasi macroscopic density distribution, with gradients large as compared to the elementary constituent nucleon force range, and featuring collective, quasi macroscopic modes of excitation. From among those, the observation of the collective giant monopole density vibration mode of heavy nuclei [@2] had yielded first information concerning the pressure to density relation (the EOS) of extended nuclear matter, albeit in an extremely narrow density window only, centered at the nuclear matter ground state density, $\rho_{0}$ = 0.15 baryons per $fm^{3}$. In fact, all other nuclear reaction studies performed in the preceding 50 years of nuclear physics had, likewise, never involved bulk nuclear density changes exceeding the percent level, owing to the fact that the employed accelerators yielded projectile energies of below about 20 MeV, commensurable to the first excitation modes of ground state nuclear matter. In marked contrast, neutron star interior densities were then expected to range beyond five times nuclear ground state density.\ Reaching such densities in the laboratory requires an input of about 100 MeV per nucleon into nuclear matter compressional potential energy, and it was the relativistic shock compression model pioneered by Greiner and collaborators [@3] that first promised just that. The idea was to bombard two heavy nuclei head-on at “relativistic” energy, as defined by the requirement that the relative interpenetration velocity of the two nuclear density profiles be well in excess of the nuclear sound velocity, as estimated from the giant monopole resonance energy, such that a relativistic Mach shock flow phenomenon generated “fireballs” of (excited) hadronic matter, compressed to densities exceeding 2$\gamma \rho_{0}$, with gamma the Lorentz-Factor of the nuclear projectile in the overall center of mass frame. This consideration suggested a projectile energy (in a fixed target experiment) in the 1 to 2 GeV per nucleon range. The basic underlying hypothesis was that nuclear matter in collisions of heavy nuclei was both extended and interactive enough to allow for a hydrodynamic description assumptions that we know now to be fully satisfied.\ Concurrently, acceleration of nuclear projectiles to the required energies was successfully accomplished in Synchrotron laboratories (Berkeley, Dubna and Princeton). As acceleration of heavy nuclei implies a substantial effort of creating and maintaining high levels of projectile ionization, the new field became known as “Relativistic Heavy Ion Physics”. Today, relativistic superconducting cyclotrons, and ranges of synchrotrons, provide for any desired nuclear projectile and energy, ranging up to the unfathomable total energy of about 1000 TeV, to be reached in Lead-on-Lead collisions at the CERN LHC facility in 2010.\ The goal, of delineating the hadronic matter equation of state, has indeed been matched in nuclear collision studies in the GeV-domain (see “The Quest for the Nuclear Equation of State” in this Volume), albeit after more than two decades of systematic effort. The obstacle in the path toward the zero temperature EOS, as relevant to neutron star structure, and neutron star mergers: the initial T = 0 matter of nuclear projectiles gets compressed but also heated, to beyond 100 MeV, in such collisions. The collisional reaction dynamics is thus sensitive to isothermes of thermally excited, and compressed matter, and the T = 0 EOS could only be derived in a semi-empirical manner, involving relativistic theory hadron transport occuring at the microscopic level, which required a substantial corresponding theoretical effort and innovation. Quite in general strong interaction theory has received, and reacted to, quite a substantial stimulus, over decades, as emanating from a surprisingly multi-faceted development of highly “provocative” experimental observations. We shall turn to prominent examples below.\ The QCD-revolution of strong interaction physics, occuring in the 1970’s, (the development of the non-abelian gauge field theory, Quantum Chromodynamics, a sector of what concurrently became known as the “Standard Model” of elementary interaction) then provided the field of relativistic nuclear collision studies with an unprecedented uplift of scope, to which it is responding until today. With the realization of partons, i.e. quarks and gluons, as the elementary carriers of the “colour” charge of the strong interaction force field, the study extended strongly interacting matter faced a new goal: to look $beyond$ the limits of hadronic matter stability. Such limits had resulted from Hagedorn [@4] exploring the “statistical bootstrap model” of hadronic/resonance/fireball matter, which had resulted in a limiting energy density of about one GeV per $fm^{3}$, with corresponding temperature in the vicinity of T = 160 - 170 MeV, the famous “Hagedorn limiting temperature”. QCD now implied a partonic matter phase beyond these limits, the “Quark-Gluon Plasma”(QGP), as it was baptized by Shuryak [@5]. This implies the consideration of a phase transition occuring, with energy density transcending the parton-hadron phase boundary (predicted at about 1GeV/$fm^{3}$ by Hagedorn), such that partons confined in colour neutral hadrons become the effective degrees of freedom of colour-conducting QCD-matter. This appeared to be also intuitively plausible because, above the critical density, a single hadronic volume would contain more than one thermal gluon, causing force screening and resulting in deconfinement.\ Immediately, the cosmological temperature/density evolution came into view. Based on considerations of QCD deconfinement by Collins, Perry, Cabibbo and Parisi [@6], supposed to result from the falloff of the QCD coupling constant at extremely high temperatures, in the multi-GeV domain (a phenomenon called asymptotic freedom), Weinberg famously argued in 1976 that “quarks were close enough together in the early universe so that they did not feel these (binding) forces, and could behave like free particles” [@7]. Seen in retrospect, the initial qualitative ideas concerning deconfinement to a Quark-Gluon Plasma did range from dissociation by colour “Debye” screening, to thermal dissociation, and dissolution of partonic bound states by asymptotic freedom. All these pictures are leaning on a perturbative treatment of QCD. However, also expecting the partonic phase to set in just above the Hagedorn limiting hadronic temperature, at T about 160 to 170 MeV, and at the corresponding resolution scale of dimension 1 Fermi, deconfinement clearly was a non perturbative process, and the confinement/deconfinement transition remained a major open problem of QCD, as did the order of the phase transformation.\ It became clear that relativistic nuclear collisions at higher than GeV energies offered the possibility to dive deeply into the QCD matter diagram, transcending the critical QCD energy density. In 1974 T.D. Lee was the first to formulate an appropriate, non perturbative vision of QGP creation in nuclear collisions where “the non-perturbative vacuum condensates could be melted down...by distributing high energy or high nucleon density over a relatively large volume” [@8]. Note that hadrons represent such non-perturbative condensates (chiral symmetry breaking excitations of the vacuum). This vision, of entering the phase transition in a process of chiral symmetry restoration occuring near a critical temperature $T_{c}$, and probably coinciding with the Hagedorn temperature, was substantiated, a decade later, by first calculations employing a lattice discretization scheme for non-perturbative analysis of QCD matter at finite temperature [@9], which indeed exhibited a sharp melting transition of the chiral condensate (the vacuum expectation value of the $<\bar{\psi},\psi>$ term in the QCD Lagrangian) at about 170 MeV [@10].\ The emerging goal of relativistic nuclear collision study was, thus, to locate this transition, elaborate its properties, and gain insight into the detailed nature of the deconfined QCD phase. Required beam energies turned out to be upward of about 10 GeV per nucleon pair in the CM frame, i.e. $\sqrt{s}$ $>$10 GeV, and various experimental programs have been carried out, and are being prepared, at the CERN SPS (up to about 20 GeV), at the BNL RHIC collider (up to 200 GeV), and finally reaching 5.5 TeV at the CERN LHC in 2009. This Volume attempts an overview of the most outstanding results and emerging perspectives.\ The QCD Phase Diagram --------------------- QCD confinement-deconfinement transitions are by no means limited to the domain of the phase diagram that is relevant to cosmological expansion dynamics prior to about 5 microseconds (the time of the hadronization transition), where a vanishingly small excess of baryon over antibaryon density implies near zero baryo-chemical potential $\mu_{B}$. In fact, modern QCD suggests [@11; @12; @13] a detailed phase diagram, with various forms of strongly interacting matter and states, that we sketch in Fig. \[fig:Figure1\]. It is presented in the plane of temperature T and baryochemical potential $\mu_{B}$. We are thus employing the terminology of the grand canonical Gibbs ensemble that describes an extended volume $V$ of partonic or hadronic matter at temperature $T$. In it, total particle number is not conserved at relativistic energy, due to particle production-annihilation processes occurring at the microscopic level. However, the probability distributions (partition functions) describing the relative particle species abundances have to respect the presence of certain, to be conserved net quantum numbers ($i$), notably non-zero net baryon number and zero net strangeness and charm. Their global conservation is achieved by a thermodynamic trick, adding to the system Lagrangian a so-called Lagrange multiplier term, for each of such quantum number conservation tasks. This procedure enters a “chemical potential” $\mu_i$ that modifies the partition function via an extra term $\exp{\left(-\mu_i/T\right)}$ occuring in the phase space integral. It modifies the canonical “punishment factor” ($\exp{\left(-E/T\right)}$), where E is the total particle energy in vacuum, to arrive at an analogous grand canonical factor for the extended medium, of $\exp{\left(-E/T - \mu_i/T\right)}$. This concept is of prime importance for a description of the state of matter created in heavy nuclear collisions, where net-baryon number (valence quarks) carrying objects are considered. The same applies to the matter in the interior of neutron stars. Note that $\mu_B$ is high at low energies of collisions creating a matter fireball. In a head-on collision of two mass 200 nuclei at $\sqrt{s}$ =15 GeV the fireball contains about equal numbers of newly created quark-antiquark pairs (of zero net baryon number), and of initial valence quarks. The accomodation of the latter, into created hadronic species, thus requires a formidable redistribution task of net baryon number, reflecting in a high value of $\mu_B$. Conversely, at LHC energy (5.5TeV for Pb+Pb collisions), the initial valence quarks constitute a mere 5% fraction of the total quark density, correspondingly requiring a small value of $\mu_B$. In the extreme, big bang matter evolves toward hadronization (at $T$=170 MeV) featuring a quark over antiquark density excess of $10^{-9}$ only, resulting in $\mu_B \approx 0$. The limits of existence of the hadronic phase are not only reached by temperature increase, to the so-called Hagedorn value $T_H$ (which coincides with $T_{crit}$ at $\mu_B \rightarrow 0$), but also by density increase to $\varrho > (5-10)\: \varrho_0$: ”cold compression” beyond the nuclear matter ground state baryon density $\varrho_0$ of about 0.16 $B/fm^3$. We are talking about the deep interior sections of neutron stars or about neutron star mergers [@14; @15; @16], at low $T$ but high $\mu_{B}$.\ A sketch of the present view of the QCD phase diagram [@11; @12; @13] is given in Fig. \[fig:Figure1\]. It is dominated by the parton-hadron phase transition line that interpolates smoothly between the extremes of predominant matter heating (high $T$, low $\mu_B$) and predominant matter compression ($T \rightarrow 0, \: \mu_B > 1 \: GeV$). Onward in $T$ from the latter conditions, the transition is expected to be of first order [@17] until a critical point of QCD matter is reached at $200 \le \mu_B \: (E)\: \le 500 \: MeV$. The relatively large position uncertainty reflects the preliminary character of Lattice QCD calculations at finite $\mu_B$ [@11; @12; @13]. Onward from the critical point, E, the phase transformation at lower $\mu_B$ is a cross-over[@13; @18], thus also including the case of primordial cosmological expansion. ![Sketch of the QCD matter phase diagram in the plane of temperature $T$ and baryo-chemical potential $\mu_B$. The parton-hadron phase transition line from lattice QCD [@11; @12; @13] ends in a critical point $E$. A cross-over transition occurs at smaller $\mu_B$. Also shown are the points of hadro-chemical freeze-out from the grand canonical statistical model.[]{data-label="fig:Figure1"}](Phasediagram_Points.eps) This would finally rule out former ideas, based on the picture of a violent first order”explosive” cosmological hadronization phase transition, that might have caused non-homogeneous conditions, prevailing during early nucleo-synthesis [@19], and fluctuations of global matter distribution density that could have served as seedlings of galactic cluster formation [@20]. However, it needs to be stressed that the conjectured order of phase transformation, occuring along the parton - hadron phase boundary line, has not been unambiguously confirmed by experiment, as of now.\ On the other hand, the [position]{} of the QCD phase boundary at low $\mu_B$ has, in fact, been located by the hadronization points in the $T, \: \mu_B$ plane that are also illustrated in Fig. \[fig:Figure1\]. They are obtained from statistical model analysis [@21] of the various hadron multiplicities created in nucleus-nucleus collisions, which results in a \[$T, \: \mu_B$\] determination at each incident energy, which ranges from SIS via AGS and SPS to RHIC energies, i.e. $3\le \sqrt{s} \le 200 \: GeV$. Toward low $\mu_B$ these hadronic freeze-out points merge with the lattice QCD parton-hadron coexistence line: hadron formation coincides with hadronic species freeze-out, at high $\sqrt{s}$. These points also indicate the $\mu_B$ domain of the phase diagram which is accessible to relativistic nuclear collisions. The domain at $\mu_B \ge 1.5 \: GeV$ which is predicted to be in a further new phase of QCD featuring color-flavor locking and color superconductivity [@22] will probably be accessible only to astrophysical observation.\ In Fig. \[fig:Figure1\] we are representing states of QCD matter in thermodynamic equilibrium. What is the relation of such states, e.g. a Quark-Gluon Plasma at some T and $\mu_{B}$, to the dynamics of relativistic nuclear collisions? A detailed answer can be only given based on a microscopic transport description of the dynamical evolution, and many of the articles in this Volume address the occurence, and generation, of local or even global thermal and/or chemical equilibrium conditions. As an example, the points of hadronic freeze-out in Fig. \[fig:Figure1\] refer to the observation of perfect hadro-chemical equilibrium among the created species, the derived (T,$\mu_{B}$) value [@21] thus legitimately appearing in the phase diagram. We may add that, in general, the collisional reaction volume of head-on collisions of heavy nuclei is of dimension 10 fm in space and time whereas the typical microscopic extension, and relaxation time scale of non-perturbative QCD objects is of order 1 fm (the confinemant scale). The A+A collision fireball size thus exceeds, by far, the elementary dimensions of microscopic strong interaction dynamics. Further, one can only get with the help of detailed microscopic models.\ Physics Observables ------------------- One can order the various physics observables, that have been developed in this field and are described in this Volume, in sequence of their origin, from successive stages that characterize the overall dynamical evolution of relativistic collisions at high $\sqrt{s}$. In rough outline this evolution can be seen to proceed in three major steps. An initial period of matter compression and heating occurs in the course of interpenetration of the projectile and target baryon density distributions. Inelastic processes occuring at the microscopic level convert initial beam longitudinal energy to new internal and transverse degrees of freedom, by breaking up the initial baryon structure functions. Their partons thus acquire virtual mass, populating transverse phase space in the course of inelastic perturbative QCD shower multiplication. This stage should be far from thermal equilibrium, initially. However, in step two, inelastic interaction between the two arising parton fields (opposing each other in longitudinal phase space) should lead to a pile-up of partonic energy density centered at mid-rapidity (the longitudinal coordinate of the overall center of mass). Due to this mutual stopping down of the initial target and projectile parton fragmentation showers, and from the concurrent decrease of parton virtuality (with decreasing average square momentum transfer $Q^2$) there results a slowdown of the time scales governing the dynamical evolution. Equilibrium could be approached here, the system ”lands” on the $T, \: \mu$ plane of Fig. \[fig:Figure1\], at temperatures of about 300 and $200 \: MeV$ at top RHIC and top SPS energy, respectively. The third step, system expansion and decay, thus occurs from well above the QCD parton-hadron boundary line. Hadrons and hadronic resonances then form, which decouple swiftly from further inelastic transmutation so that their yield ratios become stationary (”frozen-out”). These freeze-out points are included in Fig \[fig:Figure1\] for various $\sqrt{s}$. A final expansion period dilutes the system to a degree such that strong interaction ceases all together.\ In order to verify in detail this qualitative overall model, and to ascertain the existence (and to study the properties) of the different states of QCD that are populated in sequence, one seeks observable physics quantities that convey information imprinted during distinct stages of the dynamical evolution, and ”freezing-out” without significant obliteration by subsequent stages. Clearly, the dynamical formation of a local or even more extended near-equilibrium quark-gluon plasma medium is of foremost interest, but this process belongs to the very early evolution, right after the termination of the primordial perturbative shower evolution period, at about 0.3 to 0.5 fm/c. The parton density and the microscopic rescattering collision frequency are maximal during this period, but the likelyhood to receive unobliterated signals of such early evolution is evidently minute. Nevertheless this problem has been overcome in a remarkable quest for suitable early-time observables, notably the dissolution of primordially formed charm-anticharm quark pairs that would evolve into charmonium states like $J/\Psi$, in elementary collisions, but get obliterated in the course of their traversal of hot and dense early fireball matter [@23], resulting in a suppression of the eventually observed charmonium production rate. Likewise, the high $p_{t}$ partons emerging in the course of primordial, perturbative first parton shower and jet production experience a dramatic dampening characteristic of early medium opacity conditions, leading to a general, well observable high $p_{t}$ hadron suppression. It thus turned out that the primordial, perturbative mechanisms provide for several “tracer probes” co-travelling with the ongoing early dynamical evolution as diagnostic agents. These observables yield information, almost similar to what would be provided by a fictitious deep inelastic scattering experiment with partonic fireballs as a target. Moreover, it has turned out that early collective partonic flow modes of relativistic hydrodynamic matter get formed in the collisional volume, surviving later expansion stages, including hadronization, and providing for information on e.g. medium viscosity [@24] and equation of state [@25], at times below and at 1 fm/c.\ This focus on primordial time evolution came about in the course of experimental progress, from SPS to RHIC energies. It is, basically, a straight forward consequence of increasing time resolution, brought about by the collider technique that extended the energy range toward $\sqrt{s}$ = 200GeV. Consider the duration of primordial interpenetration of the projectile-target nuclear density distributions, t = $2R/\gamma$, where R is the nuclear radius. At top SPS energy t = 1.6 fm/c whereas, at RHIC top energy, t = 0.14 fm/c. Evidently, the critical early time interval below about 1 fm/c can not be properly resolved at top SPS energy: at time as “late” as about 1.5 fm/c, initial spatial layers of the interaction volume are well past all primordial interaction stages, whereas nucleons at the far end layers of the density distributions still enter their first interaction. Thus, there exists no global, time-synchronized interacting system until later times, of about 2 fm/c and beyond, i.e. just before the hadronization transition sets in: SPS physics captures a partonic system at densities of about 2-3 GeV per $fm^{3}$, in the vicinity of the parton-hadron coexistence line predicted by lattice QCD. Whereas, at top RHIC energy, t = 0.14 fm/c, and synchronization of a global high density QCD matter evolutional trajectory (including a local approach toward partonic equilibrium) may be accomplished at times as low as 0.5 fm/c, thus enabeling the definition of the above-mentioned early-time observables. It is, thus, not surprising that the “elliptic flow” signal, resulting from collective primordial partonic density/pressure gradients, stays relatively small up to top SPS energy but reaches the “hydrodynamic limit” ( i.e. the elliptic flow magnitude predicted by parton hydrodynamics) at top RHIC energy.\ At future LHC energy the interpenetration time is practically zero. Thus even the primordial, perturbative QCD parton shower multiplication phase, occuring at times below about 0.2 fm/c, will now occur in a globally synchronized, extended collisional volume. This gives rise to the expectation [@26] that effects of QCD colour saturation become essential in the primordial evolution: at the relatively low momentum transfers, corresponding to “soft” bulk hadron production, individual colour charges in the projectile-target transverse parton density profiles can not be spatially resoved in the dynamics. A new version of QCD is beeing called for: established QCD “DGLAP” evolution treats only interactions of elementary unit colour charge [@27], unlike classical Maxwell theory that deals with arbitrary charge Z force fields (recall the superposition principle). What one looks for, here, is something like a “classical limit” of QCD interaction, which has been called a “colour glass condensate” theory [@28].\ On the path from top SPS to top RHIC energy, exploiting the physics observables that refer to early time stages in the dynamical evolution has resulted in substantial first contributions to our view of the deconfined Quark-Gluon Plasma state, expected to exist above the parton-hadron coexistence line in Fig. \[fig:Figure1\]. A multiply cross-connected web of theoretical inferrences from the RHIC early time signals (in part also oftentimes called “hard probes” because their initial “tracer” partons emerge from a hard , high $Q^{2}$ process, described in perturbative QCD) leads to the conclusion that the initial evolutionary stages of RHIC Au+Au collisions reach QCD matter at about 10 times higher than the critical density ( $\epsilon_{c}$ = 1GeV/$fm^{3}$), and temperatures of 300-400 MeV. This matter, most remarkably, appears to be $very$ $much$ $unlike$ a free gas of weakly coupled partons, rather behaving like a near-ideal liquid, with a minimal shear viscosity [@29]. As such, however, it is clearly a non-perturbative QCD matter, and a whealth of new focus on the development of npQCD theory has resulted. In particular it was observed [@30] that a “dual” theory might exist for the strongly coupled, non perturbative QCD plasma, which is weakly coupled enabling a quantitative description of viscosity and other transport properties. This dual theory turns out to be a 5-dimensional string theory in anti-de Sitter (AdS) topology. This theory now confronts the alternative QCD lattice theory - both in need, and in evidence, of progress, which results in an unprecedented focus of fundamental interest in this field.\ Concerning the further evolution, the medium conditions governing matter in the direct vicinity of the parton-hadron confinement transition have been illucidated by data gathered at the lower SPS energy , where the dynamical trajectory of A+A collisions can be expected to settle at a “turning point”, occuring inbetween the overall compression-expansion cycles, and situated close to the line of QCD phase transformation. Two fundamental QCD symmetry breaking transitions, with falling energy density, are encountered here. Confinement leads to hadron formation as, at the “critical energy density”, coloured partons acquire lower free energy via pre-hadronic colour singlet formation [@31]. Concurrently, as lattice QCD results suggest, albeit at zero baryo-chemical potential only [@13], chiral symmetry breaking leads to a “dressing” of the partons with non-perturbative vacuum excitation “condensate” mass, eventually leading to the observed hadronic mass. The end product of these two transitions emerges as a multitude of hadron and resonance states, produced with a characteristic multiplicity pattern ranging over several orders of magnitude, while the chiral restoration process to hadrons concurs with electromagnetic interaction decay of hadronic, or pre-hadronic resonance states, to observable dilepton final channels. This decay is active throughout the hadronization process. Two principal physics observables result: the hadron-resonance species yield distribution emerging from hadronization, and the dilepton invariant mass spectra that integrate the yield over the entire hadronization period. The former exhibit a striking resemblance to Gibbs grand-canonical equilibrium distributions [@32], leading to determination of the hadronization temperature and corresponding hadro-chemical potential (recall the entries in Fig. \[fig:Figure1\], of (T,$\mu_{B}$) points for various $\sqrt{s}$). The latter indeed give indications of a gradual broadening and “melting” of the hadronic spectral functions , in the vicinity of the critical temperature [@33].\ The order of this phase transformation, at non-zero baryochemical potential, has not been unambiguously determined, as of yet. State of the art predictions of lattice QCD show initial success in overcoming the corresponding mathematical obstacles [@11; @12; @13], as we have implied by the parton-hadron phase boundary sketched in Fig. \[fig:Figure1\]. This line features, in particular, a critical point, at a still rather uncertain position but expected to occur at a rather high baryochemical potential, corresponding to the low end of SPS energies, covered only rather superficially in the CERN SPS Pb-beam program. The critical point would lead to typical critical fluctuations [@34], and imply an adjacent first order phase transition domain [@35]. Both of these conjectured QCD matter properties should imprint distinct traces onto the dynamical trajectory of A+A collisions- it is just unknown today whether dramatic, or subtle. For appropriate signals one wants to consult observables imprinted during the hadronization stage, e.g. hadronic species equilibrium decoupling freeze-out points, or possible reflections on collective hydrodynamic flow from a “softest point” of the equation of state, reflecting in radial and elliptic flow variables, as well as pion pair Bose-Einstein interferometry study of collective matter flow toward hadronic decoupling. While first potentially significant data exist [@36] it is clear that this region of the phase diagram deserves intense further study. This is, in fact, the goal of a planned low energy running program at RHIC, and one of the major purposes of the planned FAIR-facility of GSI.\ In summary, from among the Standard Model fundamental interactions and force fields, extended matter architecture arises, most prominently, from the electromagnetic, and the strong interactions (gravitational architecture overwhelmingly evident but still not available from a quantized theory). After decades of nuclear, and hadron physics, the 1970’s QCD revolution has, at first, swept aside the non-perturbative sector relevant to QCD as a theory of extended matter, embracing the evidence for a renormalizable gauge field theory resulting from high $Q^{2}$ physics (deep inelastic scattering, jet phenomenology), that triumphantly exploited perturbative QCD, in microscopic processes. 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--- abstract: 'Energetics and conductance in jellium modelled nanowires are investigated using the local-density-functional-based shell correction method. In analogy with studies of other finite-size fermion systems , e.g., simple-metal clusters or $^3$He clusters, we find that the energetics of the wire as a function of its radius (transverse reduced dimension) leads to formation of self-selecting magic wire configurations (MWC’s, i.e., discrete sequence of wire radii with enhanced stability), originating from quantization of the electronic spectrum, namely formation of subbands which are the analogs of electronic shells in clusters. These variations in the energy result in oscillations in the force required to affect a transition from one MWC of the nanowire to another, and are correlated directly with step-wise variations of the quantized conductance of the nanowire in units of $2 e^2/h$.' address: ' School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 ' author: - Constantine Yannouleas and Uzi Landman date: 'LETTER in J. Phys. Chem. B [101]{}, 5780 \[24 July 1997\]' --- [On mesoscopic forces and quantized conductance in model metallic nanowires]{} Introduction ============ Identification and understanding of the physical origins and systematics underlying the variations of materials properties with size, form of aggregation, and dimensionality are some of the main challenges in modern materials research, of ever increasing importance in the face of the accelerated trend toward miniaturization of electronic and mechanical devices. While for over two decades studies of size-evolutionary patterns of materials have focused on atomic and molecular clusters [@haber; @heer; @yann1] in beams or embedded in inert matrices, more recent efforts concentrated on preparation, characterization, and understanding of finite solid-sate structures. These include nanometer-scale metal and semiconductor nanocrystals, [@whet; @lued; @aliv] surface-supported structures and quantum dots, [@avou] and nanoscale junctions or wires.$^{8-27}$ Interestingly, it has emerged that concepts and methodologies developed in the context of isolated gas-phase clusters and atomic nuclei are often most useful for investigations of finite-size solid-state structures. In particular, it has been shown most recently [@land4; @barn] through first-principles molecular dynamics simulations that as metallic (sodium) nanowires are stretched to just a few atoms in diameter, the reduced dimensions, increased surface-to-volume ratio, and impoverished atomic environment, lead to formation of structures, made of the metal atoms in the neck, which can be described in terms of those observed in small gas-phase sodium clusters; hence they were termed as supported [cluster-derived structures (cds)]{}. The above prediction of the occurrence of “magic-number” cds’s in nanowires, due to characteristics of electronic cohesion and atomic bonding in such structures of reduced dimensions, are directly correlated with the energetics of metal clusters, where magic-number sequences of clusters sizes, shapes and structural motifs due to electronic and/or geometric shell effects, have been long predicted and observed. [@heer; @yann1; @mart] Furthermore, these results lead one directly to conclude that other properties of nanowires, derived from their energetics, may be described using methodologies developed previously in the context of clusters. Indeed, in this paper, we show that certain aspects of the mechanical response (i.e., elongation force) and electronic transport (e.g., quantized conductance) in metallic nanowires can be analyzed using the local-density-approximation (LDA) -based shell correction method (SCM), developed and applied previously in studies of metal clusters. [@yann1; @yann2] Specifically, we show that in a jellium-modelled, volume-conserving nanowire, variations of the total energy (particularly terms associated with electronic subband corrections) upon elongation of the wire lead to [self-selection]{} of a sequence of stable “magic” wire configurations (MWC’s, specified in our model by a sequence of the wire’s radii), with the force required to elongate the wire from one configuration to the next exhibiting an oscillatory behavior. Moreover, we show that due to the quantized nature of electronic states in such wires, the electronic conductance varies in a quantized step-wise manner (in units of the conductance quantum $g_0=2e^2/h$), correlated with the transitions between MWC’s and the above-mentioned force oscillations. Prior to introducing the model studied in this paper, it is appropriate to briefly review certain previous theoretical and experimental investigations, which form the background and motivation for this study. Atomistic descriptions, based on realistic interatomic interactions, and/or first-principles modelling and simulations played an essential role in discovering the formation of nanowires, and in predicting and elucidating the microscopic mechanisms underlying their mechanical, spectral, electronic and transport properties. Formation and mechanical properties of interfacial junctions (in the form of crystalline nanowires) have been predicted through early molecular-dynamics simulations, [@land1] where the materials (gold) were modelled using semiempirical embedded-atom potentials. In these studies it has been shown that separation of the contact between materials leads to generation of a connective junction which elongates and narrows through a sequence of structural instabilities; at the early stages, elongation of the junction involves multiple slip events, while at the later stages, when the lateral dimension of the wire necks down to a diameter of about 15 Å, further elongation involves a succession of stress accumulation and fast relief stages associated with a sequence of order-disorder structural transformations localized to the neck region. [@land1; @land2; @land3] These structural evolution patterns have been shown through the simulations to be portrayed in oscillations of the force required to elongate the wire, with a period approximately equal to the interlayer spacing. In addition, the “sawtoothed” character of the predicted force oscillations \[see Fig. 3(b) in Ref. and Fig. 3 in Ref. \] reflects the stress accumulation and relief stages of the elongation mechanism. Moreover, the critical resolved yield stress of gold nanowires has been predicted [@land1; @land2] to be $\sim$ 4GPa, which is over an order of magnitude larger than that of the bulk, and is comparable to the theoretical value for Au (1.5 GPa) in the absence of dislocations. These predictions, as well as anticipated electronic conductance properties, [@land1; @boga1] have been corroborated in a number of experiments using scanning tunneling and force microscopy, [@land1; @pasc1; @oles; @pasc2; @smith; @rubi; @stal] break junctions, [@krans] and pin-plate techniques [@costa; @land2] at ambient environments, as well as under ultrahigh vacuum and/or cryogenic conditions. Particularly, pertinent to our current study are experimental observations of the oscillatory behavior of the elongation forces and the correlations between the changes in the conductance and the force oscillations; see especially the simultaneous measurements of force and conductance in gold nanowires in Ref. , where in addition the predicted “ideal” value of the critical yield stress has also been measured (see also Ref. ). The jellium-based model introduced in this paper, which by construction is devoid of atomic crystallographic structure, does not address issues pertaining to nanowire formation methods, atomistic configurations, and mechnanical response modes \[e.g., plastic deformation mechanisms, interplanar slip, ordering and disordering mechanisms (see detailed descriptions in Refs. and , and a discussion of conductance dips in Ref. ), defects, mechanichal reversibility, [@rubi; @land2] and roughening of the wires’s morphology during elongation [@land3]\], nor does it consider the effects of the above on the electron spectrum, transport properties, and dynamics [@barn] Nevertheless, as shown below, the model offers a useful framework for linking investigations of solid-state structures of reduced dimensions (e.g., nanowires) with methodologies developed in cluster physics, as well as highlighting certain nanowire phenomena of mesoscopic origins and their analogies to clusters. The jellium model for metallic nanowires: Theoretical method and results ======================================================================== Consider a cylindrical jellium wire of length $L$, having a positive background with a circular cross section of radius $R \ll L$. For simplicity, we restrict ourselves here to this symmetry of the wire cross section. Variations in the shape of the nanowire cross section serve to affect the degeneracies of the electronic spectrum [@sche; @boga3] without affecting our general conclusions. We also do not include here variations of the wire’s shape along its axis. Adiabatic variation of the wire’s axial shape introduces a certain amount of smearing of the conductance steps through tunnelling, depending on the axial radius of curvature of the wire. [@sche; @boga2; @boga3] Both the cross-sectional and axial shape of the wire can be included in our model in a rather straightforward manner. The principal idea of the SCM is the separation of the total LDA energy $E_T(R)$ as[@yann1; @yann2; @stru] $$E_T(R) = \widetilde{E}(R) + \Delta E_{\text{sh}} (R)~, \label{etscm}$$ where $\widetilde{E}(R)$ varies smoothly as a function of the system size, and $\Delta E_{\text{sh}} (R)$ is an oscillatory term arising from the discrete quantized nature of the electronic levels. $\Delta E_{\text{sh}} (R)$ is usually called a shell correction in the nuclear [@stru] and cluster [@yann1; @yann2] literature; we continue to use here the same terminology with the understanding that the electronic levels in the nanowire form subbands, which are the analog of electronic shells in clusters, where furthermore the size of the system is usually given by specifying the number of atoms $N$. The SCM method, which has been shown to yield results in excellent agreement with experiments [@yann1; @yann3; @yann5] and self-consistent LDA calculations [@yann1; @yann2] for a number of cluster systems, is equivalent to a Harris functional ($E_{\text{harris}}$) approximation to the Kohn-Sham LDA with the input density obtained through variational minimization of an extended Thomas-Fermi (ETF) energy functional, $E_{\text{ETF}}[\widetilde{\rho}]$ (with the kinetic energy, $T_{\text{ETF}}[\widetilde{\rho}]$, given to 4th order gradients and the potential, $V_{\text{ETF}}$, including the Hartree repulsion and exchange-correlation and positive-background attractions as in LDA). The smooth contribution in Eq. (\[etscm\]) is identified with $E_{\text{ETF}}[\widetilde{\rho}]$. The optimized density [@note1] $\widetilde{\rho}$ at a given radius $R$ is obtained under the normalization condition (charge neutrality) $2 \pi \int \widetilde{\rho} (r) r dr= \rho^{(+)}_L(R)$, where $\rho^{(+)}_L (R) = 3R^2/(4r_s^3)$ is the linear positive background density. Using the optimized $\widetilde{\rho}$, one solves for the eigenvalues $\widetilde{\epsilon}_i$ of the Hamiltonian $H=-(\hbar^2/2m) \nabla^2 + V_{\text{ETF}}[\widetilde{\rho}]$, and the shell correction is given by $$\begin{aligned} \Delta E_{\text{sh}} & \equiv & E_{\text{harris}}[\widetilde{\rho}] - E_{\text{ETF}}[\widetilde{\rho}] \nonumber \\ & = & \sum_{i=1}^{\text{occ}} \widetilde{\epsilon}_i - \int \widetilde{\rho} ({\bf r}) V_{\text{ETF}}[\widetilde{\rho} ({\bf r})] d{\bf r} - T_{\text{ETF}}[\widetilde{\rho}]~, \label{shcor}\end{aligned}$$ where the summation extends over occupied levels. here the dependence of all quantities on the pertinent size variable (i.e., the radius of the wire $R$) is not shown explicitly. Additionally, the index $i$ can be both discrete and continuous, and in the latter case the summation is replaced by an integral. Following the above procedure with a uniform background density of sodium ($r_s=4$ a.u.), a typical potential $V_{\text{ETF}}(r)$ for $R=12.7$ a.u., where $r$ is the radial coordinate in the transverse plane, is shown in Fig. 1, along with the transverse eigenvalues $\widetilde{\epsilon}_{nm}$ and the Fermi level; to simplify the calculations of the electronic spectrum, we have assumed (as noted above) $R \ll L$, which allows us to express the subband electronic spectrum as $$\widetilde{\epsilon}_{nm} (k_z;R) = \widetilde{\epsilon}_{nm}(R) + \frac{\hbar^2 k_z^2}{2m}~, \label{eigval}$$ where $k_z$ is the electron wave number along the axis of the wire ($z$). As indicated earlier, taking the wire to be charge neutral, the electronic linear density, $\rho_L^{\text{(--)}}$($R$), must equal the linear positive background density, $\rho^{(+)}_L (R)$. The chemical potential (at $T=0$ the Fermi energy $\epsilon_F$) for a wire of radius $R$ is determined by setting the expression for the electronic linear density derived from the subband spectra equal to $\rho^{(+)}_L (R)$, i.e., $$\frac{2}{\pi} \sum_{n,m}^{\text{occ}} \sqrt{ \frac{2m}{\hbar^2} [\epsilon_F (R)- \widetilde{\epsilon}_{nm} (R)] } = \rho^{(+)}_L (R)~, \label{linden}$$ where the factor of 2 on the left is due to the spin degeneracy. The summand defines the Fermi wave vector for each subband, $k_{F,nm}$. The resulting variation of $\epsilon_F (R)$ versus $R$ is displayed in Fig. 2(a), showing cusps for values of the radius where a new subband drops below the Fermi level as $R$ increases (or conversely as a subband moves above the Fermi level as $R$ decreases upon elongation of the wire). Using the Landauer expression for the conductance $G$ in the limit of no mode mixing and assuming unit transmission coefficients, $G(R) = g_0 \sum_{n,m} \Theta [\epsilon_F(R)-\widetilde{\epsilon}_{nm}(R)]$, where $\Theta$ is the Heaviside step function. The conductance of the nanowire, shown in Fig. 2(b), exhibits quantized step-wise behavior, with the step-rises coinciding with the locations of the cusps in $\epsilon_F (R)$, and the height sequence of the steps is 1$g_0$, 2$g_0$, 2$g_0$, 1$g_0$, ..., reflecting the circular symmetry of the cylindrical wires’ cross sections, [@boga1] as observed for sodium nanowires. [@krans] Solving for $\epsilon_F (R)$ \[see Eq. (\[linden\])\], the expression for the sum on the right-hand-side of Eq. (\[shcor\]) can be written as $$\begin{aligned} && \sum_i^{\text{occ}} \widetilde{\epsilon}_i = \frac{2}{\pi} \sum_{n,m}^{\text{occ}} \int_0^{k_{F,nm}} dk_z \widetilde{\epsilon}_{nm}(k_z;R) = \nonumber \\ && \frac{2}{3\pi} \sum_{n,m}^{\text{occ}} [\epsilon_F(R) + 2 \widetilde{\epsilon}_{nm}(R)] \sqrt{ \frac{2m}{\hbar^2} [\epsilon_F(R) - \widetilde{\epsilon}_{nm}(R)] }~, \label{sumi}\end{aligned}$$ which allows one to evaluate $\Delta E_{\text{sh}}$ \[Eq. (\[shcor\])\] for each wire radius $R$. Since the expression in Eq. (\[sumi\]) gives the energy per unit length, we also calculate $E_{\text{ETF}}$, $T_{\text{ETF}}$, and the volume integral in the second line of Eq. (\[shcor\]) for cylindrical volumes of unit height. To convert to energies per unit volume \[denoted as $\varepsilon_T (R)$, $\widetilde{\varepsilon}(R)$, and $\Delta \varepsilon_{\text{sh}} (R)$\] all energies are further divided by the wire’s cross-sectional area, $\pi R^2$. The smooth contribution and the shell correction to the wire’s energy are shown respectively in Fig. 3(a) and Fig. 3(b). The smooth contribution decreases slowly towards the bulk value ($-$2.25 eV per atom [@yann2]). On the other hand, the shell corrections are much smaller in magnitude and exhibit an oscillatory behavior. This oscillatory behavior remains visible in the total energy \[Fig. 3(c)\] with the local energy minima occurring for values $R_{\text{min}}$ corresponding to conductance plateaus. The sequence of $R_{\text{min}}$ values defines the MWC’s, that is a sequence of wire configurations of enhanced stability. From the expressions for the total energy of the wire \[i.e., $\Omega \varepsilon_T (R)$, where $\Omega = \pi R^2 L$ is the volume of the wire\] and the smooth and shell (subband) contributions to it, we can calculate the “elongation force” (EF), $$\begin{aligned} F_T (R) & = & -\frac{d[\Omega \varepsilon_T (R)]}{dL} = - \Omega \left\{ \frac{d \widetilde{\varepsilon} (R)}{dL} + \frac{ d [\Delta \varepsilon_{\text{sh}} (R) ]}{dL} \right\} \nonumber\\ & \equiv & \widetilde{F}(R) + \Delta F_{\text{sh}}(R)~. \label{force}\end{aligned}$$ Using the volume conservation, i.e., $d(\pi R^2 L)=0$, these forces can be written as $F_T(R)= (\pi R^3/2) d\varepsilon_T (R)/dR$, $\widetilde{F}(R)= (\pi R^3/2) d\widetilde{\varepsilon} (R)/dR$, and $\Delta F_{\text{sh}} (R)= (\pi R^3/2) d [\Delta \varepsilon_{\text{sh}} (R)]/dR$. $\widetilde{F}(R)$ and $F_T(R)$ are shown in Fig. 3(d,e). The oscillations in the force resulting from the shell-correction contributions dominate. In all cases, the radii corresponding to zeroes of the force situated on the left of the force maxima coincide with the minima in the potential energy curve of the wire, corresponding to the MWC’s. Consequently, these forces may be interpreted as guiding the self-evolution of the wire toward the MWC’s. Also, all the local maxima in the force occur at the locations of step-rises in the conductance \[reproduced in Fig. 3(f)\], signifying the sequential decrease in the number of subbands below the Fermi level (conducting channels) as the wire narrows (i.e., as it is being elongated). Finally the magnitude of the total forces is comparable to the measured ones (i.e., in the nN range). Conclusions and Discussion ========================== We investigated energetics, conductance, and mesoscopic forces in a jellium modelled nanowire (sodium) using the local-density-functional-based shell correction method. The results shown above, particularly, the oscillations in the total energy of the wire as a function of its radius (and consequently the oscillations in the EF), the corresponding discrete sequence of magic wire configurations, and the direct correlation between these oscillations and the step-wise quantized conductance of the nanowires, originate from quantization of the electronic states (i.e., formation of subbands) due to the reduced lateral (transverse) dimension of the nanowires. In fact such oscillatory behavior, as well as the appearance of “magic numbers” and “magic configurations” of enhanced stability, are a general characteristic of finite-size fermionic systems and are in direct analogy with those found in simple-metal clusters (as well as in $^3$He clusters [@yann5] and atomic nuclei[@stru]), where electronic shell effects on the energetics [@heer; @yann1; @yann2; @yann3] (and most recently shape dynamics [@yann4] of jellium modelled clusters driven by forces obtained from shell-corrected energetics) have been studied for over a decade. While these calculations provide a useful and instructive framework, we remark that they are not a substitute for theories where the atomistic nature and specific atomic arrangements are included [@land1; @land2; @land3; @land4; @barn] in evaluation of the energetics (and dynamics) of these systems (see in particular Refs. , where first-principles molecular-dynamics simulations of electronic spectra, geometrical structure, atomic dynamics, electronic transport and fluctuations in sodium nanowires have been discussed). Indeed, the atomistic structural characteristics of nanowires (including the occurrence of cluster-derived structures of particular geometries[@land4; @barn]), which may be observed through the use of high resolution microscopy, [@kizu] influence the electronic spectrum and transport characteristics, as well as the energetics of nanowires and their mechanical properties and response mechanisms. In particular, the mechanical response of materials involves structural changes through displacement and discrete rearrangenent of the atoms. The mechanisms, pathways, and rates of such structural transformations are dependent on the arrangements and coordinations of atoms, the magnitude of structural transformation barriers, and the local shape of the wire, as well as possible dependency on the history of the material and the conditions of the experiment (i.e., fast versus slow extensions). Further evidence for the discrete atomistic nature of the structural transformations is provided by the shape of the force variations (compare the calculated Fig. 3(b) in Ref. and Fig. 3 in Ref. with the measurements shown in Figs. 1 and 2 in Ref. ), and the interlayer spacing period of the force oscillations when the wire narrows. 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--- abstract: 'Ultracold atoms provide clues to an important many-body problem regarding the dependence of Bose-Einstein condensation (BEC) transition temperature $T_c$ on interactions. However, cold atoms are trapped in harmonic potentials and theoretical evaluations of the $T_c$ shift of trapped interacting Bose gases are challenging. While previous predictions of the leading-order shift have been confirmed, more recent experiments exhibit higher-order corrections beyond available mean-field theories. By implementing two large-$\mathcal{N}$ based theories with the local density approximation (LDA), we extract next-order corrections of the $T_c$ shift. The leading-order large-$\mathcal{N}$ theory produces results quantitatively different from the latest experimental data. The leading-order auxiliary field (LOAF) theory containing both normal and anomalous density fields captures the $T_c$ shift accurately in the weak interaction regime. However, the LOAF theory shows incompatible behavior with the LDA and forcing the LDA leads to density discontinuities in the trap profiles. We present a phenomenological model based on the LOAF theory, which repairs the incompatibility and provides a prediction of the $T_c$ shift in stronger interaction regime.' author: - Tom Kim - 'Chih-Chun Chien' title: 'Critical temperature of trapped interacting bosons from large-$\mathcal{N}$ based theories' --- Introduction ============ Bose-Einstein condensation (BEC) occurs when low-energy states are macroscopically occupied by bosons below a critical temperature due to Bose-Einstein statistics, and the advent of ultracold atoms allows detailed analyses of BEC and its related phenomena [@pethick2008bose; @ueda2010fundamentals; @stoof2008ultracold; @RevModPhys.76.599; @RevModPhys.71.463; @RevModPhys.80.885]. The BEC transition temperature, defined as the temperature when BEC starts to form, shifts with self-interactions of bosons and determining the functional form of the shift has been a challenge in many-body physics. The BEC transition temperature of a homogeneous Bose gas has been studied using various analytic and numerical methods (see Refs. [@RevModPhys.76.599; @RevModPhys.80.885] for a review). Experimental determinations of the dependence of the BEC temperature shift on self-interactions, however, have been complicated by the fact that ultracold atoms are usually trapped in optical or magnetic potentials. The trapping potential is usually of the harmonic form and leads to an inhomogeneous density profile of the atomic cloud. In noninteracting bosons, all particles fall to the ground state as the temperature approaches zero but this is no longer the case in interacting bosons since excitations out of the condensate can be finite even at zero temperature [@pethick2008bose; @ueda2010fundamentals; @stoof2008ultracold; @RevModPhys.76.599; @RevModPhys.71.463]. In a harmonic trap, Bose-Einstein condensation starts to form at the trap center where the density is higher. On the other hand, repulsive interactions push particles away from each other and broaden the density profile. Hence, the density of repulsive bosons at the trap center is lower compared to a noninteracting system under similar conditions. Due to the reduction of the density at the trap center, the overall effect was proposed to be a negative $T_c$ shift as the interaction increases [@RevModPhys.71.463; @BF02396737]. There have been experimental works on measuring $T_c$ of trapped interacting Bose gases. Ref. [@PhysRevLett.92.030405] measured the $T_c$ shift by estimating the total atom number after integrating over a cloud image and deducing the condensate fraction using the Thomas-Fermi approximation. The critical temperature is inferred by using a time of flight method and the temperature is inferred by the size of the cloud after an expansion. In experiments, $T_c$ is considered as the temperature at which the deduced condensate fraction vanishes. The $T_c$ shift was estimated as $\delta T_c/T_c^0=\alpha N^{1/6}$, where $\alpha=-0.009$ with the two-body $s$-wave scattering length $a=5.31nm$ and the harmonic-trap length $a_H=1.00\mu m$. Ref. [@PhysRevA.81.053632] prepared atomic clouds at different temperatures above as well as below $T_c$ and derived $T$ and $\mu$ from a fit to the Popov model [@pethick2008bose]. It was reported that $\delta T_c/T_c^0=c^\prime \frac{a}{a_H}N^{1/6}$, where $c^\prime=-1.4$. Ref. [@PhysRevLett.106.250403] prepared two atomic clouds concurrently, one with the targeted interaction and the other with a very small interaction, same trapping frequency, and very similar particle number as a reference point. The $T_c$ shift due to interactions was extracted by the difference of the results from the two clouds, which eliminates finite size effects. The result was $\delta T_c/T_c^0=b_1(a/\lambda_0)+b_2(a/\lambda_0)^2$ with $b_1=-3.5 \pm0.3$ and $b_2=46\pm5$. Here $\lambda_0$ is the thermal length of a noninteracting trapped Bose gas at its critical temperature. On the theoretical side, by using the Popov approximation with the local density approximation (LDA) [@PhysRevA.54.R4633], the $T_c$ shift to the first order in interaction is found to be $\delta T_c/T_c^0=-3.426(a/\lambda_0)$, which agrees well with later experimental results. To capture the higher-order corrections to the $T_c$ shift of trapped interacting bosons, we will implement a large-$\mathcal{N}$ expansion and its generalization called the leading-order auxiliary field (LOAF) theory, both of which are non-perturbative in interaction and temperature, along with the local density approximation (LDA). Here $\mathcal{N}$ counts the number of atomic species and the expansion already works reasonably when modeling ultracold bosons with $\mathcal{N}=1$ [@PhysRevLett.105.240402; @ChienPRA12]. The large-$\mathcal{N}$ based theories, when applied to atomic Bose gases, are derivable from well-defined thermodynamic free energies, show a smooth second-order BEC transition, and are consistent with standard perturbation theory or renormalization analysis in low-temperature and weakly-interacting regimes [@PhysRevLett.105.240402; @PhysRevA.85.023631; @ChienPRA12; @ChienAnnPhys]. The normal state properties from the large-$\mathcal{N}$ expansion also compare favorable with Monte-Carlo simulations [@PhysRevA.91.043631]. Here we will show that while the $T_c$ shift from the LOAF theory with the LDA captures the functional form from the most recent experimental data [@PhysRevLett.106.250403], the LOAF theory exhibits behavior incompatible with the LDA. The deviation from the conventional LDA illustrates another challenge of applying mean-field theories to trapped interacting Bose gases, and we will discuss a phenomenological model based on the LOAF theory allowing us to extract the functional form of the $T_c$ shift in the stronger interaction regime. The paper is organized as follows. We first check the validity of the LDA using trapped noninteracting Bose gases in Sec. \[sec:LDA\] and show that the exact $T_c$ as well as the first-order finite particle number correction can be reproduced in the LDA. Sec. \[sec:largeN\] summarizes the large-N expansion, the leading-order large-N theory, and the LOAF theory. Their integrations with the LDA are summarized in Sec. \[sec:LargeNLDA\]. The numerical results of the trapped profiles and $T_c$ shifts from the two theories are presented in Sec. \[sec:Results\] and compared to the latest experimental results. In the same section, incompatible behavior of the LOAF theory with the LDA is discussed and a phenomenological model based on the LOAF theory is presented, which provides further theoretical predictions. Finally, Sec. \[sec:conclusion\] concludes our work. Local density approximation {#sec:LDA} =========================== To evaluate the $T_c$ shift of a trapped Bose gas, the full density profile needs to be constructed because the emergence of BEC at the trap center depends on the particle density at the center, which has to be determined from a consistent density profile in the whole trap. While most mean-field theories are designed for uniform quantum gases, a powerful tool called the local density approximation (LDA) allows one to construct a full density profile by slicing the system into pieces and treating each piece as a locally uniform system [@RevModPhys.71.463; @pethick2008bose]. By sewing all the pieces with a smooth profile of the chemical potential, an approximated density profile can be obtained. The LDA has been applied to trapped quantum gases and proved to be a versatile treatment [@RevModPhys.71.463; @stoof2008ultracold; @RevModPhys.80.885; @PhysRevLett.98.110404; @Stewart2008]. $T_c$ of noninteracting bosons ------------------------------ At first look the LDA may not accurately describe finite-temperature phenomena like the BEC transition. We begin by checking the validity of LDA for a noninteracting trapped Bose gas at its transition temperature. According to the Bose-Einstein statistics, the number of bosons in excited states is given by $$N_T=\sum_{\epsilon} \frac{1}{\exp(\beta\left(\epsilon-\mu\right))-1}. \label{eq:BCE}$$ Here $\mu$ is the chemical potential, $\beta=1/(k_B T)$ and we set the Boltzmann constant $k_B=1$. For a homogeneous noninteracting Bose gas, the BEC temperature is the lowest $T$ satisfying $N_T/\Omega=\rho$ with $\mu=\epsilon_0$ in the thermodynamic limit, where $\rho$ is the particle density, $\Omega$ is the system volume, and $\epsilon_0$ is the single-particle ground-state energy. For a parabolic energy dispersion $\epsilon_k=\hbar^2 k^2/(2m)$ with wave vector $k$, the Planck constant divided by $2\pi$, $\hbar$, and particle mass $m$, it can be shown that [@fetter2012quantum] $$T_c^{0,homo}=\left(\frac{\rho}{\zeta(3/2)}\right)^{2/3}\frac{2\pi \hbar^2}{ m k_B},$$ where the superscript $0$ denotes quantities of a noninteracting Bose gas and $\zeta(x)$ is the Riemann zeta function. For noninteracting bosons in a harmonic trap, the energy eigenvalues are $E_{n_1,n_2,n_3}=\hbar\omega\left(n_1+n_2+n_3\right)+\frac{3}{2}\hbar\omega$, where $\omega$ denotes the trap frequency. The total particle number is [@Haugerud199718; @Grossmann1995188] $$\label{eq:ExactNonint} N=\sum_{n_1,n_1,n_3} \frac{1}{\exp\left[\beta\hbar\omega\left(n_1+n_2+n_3\right)+\beta\left(E_0-\mu\right)\right]-1}.$$ $T_c^0$ is the lowest temperature when the equation is satisfied with $\mu=E_0=(3/2)\hbar\omega$ in the thermodynamic limit, which can be calculated analytically [@mazenko2000equilibrium; @BF02396737]. Explicitly, $$\label{eq:Tc0trap} T_c^0=\left(\frac{N}{\zeta(3)}\right)^{1/3}\frac{\hbar\omega}{k_B}.$$ One may also apply the LDA to obtain the $T_c$ of a trapped noninteracting Bose gas. By approximating the energy dispersion with a parabolic one $\epsilon=E_0+\frac{\hbar^2 k^2}{2m}+\frac{1}{2} m \omega^2 r^2$ in Eq. and replacing the summation by integrals over space and momentum, one obtains the number of bosons in the thermal cloud as $$N_T=\left(\frac{k_b T}{\hbar\omega}\right)^3 g_{3/2}(e^{(\mu-E_0)/{k_B T}}),$$ where $\operatorname{g}_s(z) = \sum_{k=1}^\infty {z^k \over k^s} $ is the polylogarithm. We will redefine $\mu$ as the chemical potential measured from the zero-point energy of the system, so $\mu-E_0 \rightarrow \mu$ from here on. By expanding the series around $\beta\mu=0$ with $N_T=N$, the leading order gives Eq. and next order reproduces the leading-order finite-size correction presented in Eq. . Therefore, the validity of LDA has been established for noninteracting Bose gases. Higher-order correction from finite $N$ can be found from Eq. by applying the Euler-Maclaurin formula [@Haugerud199718; @Grossmann1995188] or turning the summand into a geometric series and then carrying out the summation by expanding around $\beta \hbar \omega=0$ resulting in polylogarithmic functions [@PhysRevA.54.656; @PhysRevA.54.4188; @PhysRevA.58.1490]. Explicitly, $$\frac{\delta T_c^0}{T_c^0}=-\frac{\zeta(2)}{2(\zeta(3))^{2/3}}N^{-1/3}\approx -0.73 N^{-1/3}. \label{eq:deltaTc0}$$ The correction decays with $N$ and is present regardless of self-interactions. In experiments those finite-size corrections are discarded by taking the difference of the results from two similarly prepared systems at different interaction strengths [@PhysRevLett.106.250403]. Ref. [@PhysRevA.83.023616] derived the next order correction to the $T_c^0$ shift due to finite $N$ by using the LDA with an expansion of $\mu/k_B T$. However, Ref. [@PhysRevA.92.017601] points out an ambiguity in the definition of the critical temperature because of finite-size effects. Thus, the next order correction from finite particle number may not provide a better pointer to the critical regime than the first order term when compared to numerical results. Here we will focus on systems with well-defined thermodynamic limit and will not include corrections from finite particle number already present in the noninteracting system in our later discussions. Interacting bosons ------------------ The shift of $T_c$ from its noninteracting value $T_c^0$ in the presence of interactions has been a great challenge. Even for a uniform Bose gas, it took a long time for results from various studies to converge [@PhysRevLett.106.250403; @RevModPhys.76.599; @RevModPhys.80.885]. The issue has been settled more recently and the leading-order shift is now believed to have a form of $\Delta T_c^{homo}/T_c^{0,homo}= c \rho ^{1/3}a$, where $a$ is the $s$-wave two-body scattering length and $c$ is a positive constant. Different values of $c$ have been reported using various analytic or numerical methods [@RevModPhys.76.599]. In a trapped Bose gas, a repulsive interaction flattens the density profile and lowers the density at the trap center [@RevModPhys.71.463; @PhysRevLett.106.250403]. As a consequence, the leading order of the $T_c$ shift for a trapped interacting Bose gas is believed to have the form $\frac{\delta T_c}{T_c^0}=c^{\prime}\frac{a}{a_{H}}N^{1/6}$, where $c^{\prime}$ should be a negative number. Here $a_{H}=\sqrt{\frac{\hbar}{m\omega}}$ is the harmonic length. An early theoretical analysis using the Popov approximation [@PhysRevA.54.R4633] provided an estimation of $c^{\prime}$ by introducing an approximated dispersion $\epsilon=\frac{\hbar^2 k^2}{2m}+V(r)+2\lambda n_T(r)-\mu$ into Eq. , where $V(r)=\frac{1}{2}m\omega^2r^2$ is the harmonic trap potential, $\lambda$ is the coupling constant, and $n_T(r)$ is the thermal particle density at radius $r$. After expanding the expression to the first order in $\delta T_c$, $\lambda$, and $\mu$, the $T_c$ shift to the first order due to interaction is found to be $\frac{\delta T_c}{T_c^0}\approx-1.33\frac{a}{a_{H}}N^{1/6}$, which agrees well with later experiments [@PhysRevLett.106.250403] An estimation of the second-order $T_c$ shift has been shown in Ref. [@PhysRevA.64.053609] by expanding the distribution function in powers of the fugacity around $\beta\mu=0$, and the following expression was obtained. $$\frac{\delta T_c}{T_c^0}=c_1\frac{a}{\lambda _0}+\left(c_2^{\prime}\ln \left(\frac{a}{\lambda _0}\right)+c_2^{\prime \prime}\right)\left(\frac{a}{\lambda _0}\right)^2,$$ where $c_1 =-3.426$, $c_2^{\prime} =-45.86$, and $c_2^{\prime\prime} =-155.0$. However, corrections from the logarithmic term were not reported in later experiments [@PhysRevLett.106.250403]. Instead of assuming small interaction strength and expanding around the noninteracting limit, we use a path integral formalism to formulate trapped interacting bosons and apply the large-$\mathcal{N}$ expansion to find $T_c$. To handle the background harmonic trap, we adjust the theory to fit the LDA framework. Obtaining a full density profile in a trap is often a difficult task if a theory can only apply to a small range of temperature close to $T=0$ or $T=T_c$. This is because in a trap the local temperature scale $T/T_c^0(r)$ spans a wide range. Here $T_c^0(r)$ is the critical temperature of a noninteracting Bose gas with the same local density. Previous works on weakly interacting bosons have encountered challenges. For instance, the Popov theory exhibits an artificial first order transition at $T_c$ [@PhysRevLett.105.240402; @popov1991functional], and a higher-order large-$\mathcal{N}$ expansion used in Ref. [@0295-5075-49-2-150] mainly focused on a uniform system near its critical temperature. The leading-order large-$\mathcal{N}$ theory and its generalization both exhibit second order transition and is not temperature restrictive [@ChienPRA12; @PhysRevLett.105.240402]. By using the LDA, we calculate the trap density profile and estimate $T_c$ for trapped interacting Bose gases. To compare atomic clouds with the same total particle number, we impose the following condition to fix the total particle number $N$. $$\label{eq:number} N=\int d^3 x \rho(x).$$ large-$\mathcal{N}$ based theories {#sec:largeN} ================================== Leading-order large-$\mathcal{N}$ theory ---------------------------------------- The partition function of a single component Bose gas can be cast in an imaginary-time path-integral formalism [@fetter2012quantum; @ChienPRA12]. Explicitly, $$Z(\mu ,\beta,j)=\int D \phi D \phi ^* e^{-S\left(\phi ,\phi ^,\mu ,\beta \right)+\int[dx](j^\phi+j\phi^)}. \label{eq:partition}$$ Here $\beta=(k_B T)^{-1}$, $\mu$ is the chemical potential, $[dx]\equiv d\tau d^{3}x$, and $j$ (or $j^$) is the source of $\phi^$ (or $\phi$). The action with imaginary time $\tau$ is $S\left(\phi ,\phi ^,\mu ,\beta \right)=\int[dx]\mathcal{L}\left(\phi ,\phi ^,\mu \right)$. For a nonrelativistic dilute Bose gas with contact interactions, the effective Euclidean Lagrangian density is $$\begin{aligned} \mathcal{L}&&=\frac{1}{2} \hbar \left(\phi ^ \frac{\partial \phi }{\partial \tau }-\phi \frac{\partial \phi ^}{\partial \tau }\right) -\frac{1}{2}\left(\phi ^\frac{\hbar ^2\nabla ^2}{2 m}\phi +\phi \frac{\hbar ^2\nabla ^2}{2 m}\phi ^\right) -\nonumber\\ &&\mu \phi ^ \phi + \frac{1}{2} \lambda \left(\phi ^* \phi \right)^2. \label{eq:Lagrangian}\end{aligned}$$ Here $\lambda$ is the bare coupling constant. In what follows we set $\hbar=1$, $k_B=1$, and $2m=1$. By introducing $$\begin{aligned} \Phi &=&\left( \begin{array}{cc} \phi, & \phi ^* \\ \end{array} \right)^{\mathsf{T}},J=\left( \begin{array}{cc} j, & j^* \\ \end{array} \right)^{\mathsf{T}}, \nonumber \\ \bar{G}_0^{-1}&=& \left( \begin{array}{cc} \frac{\partial }{\partial \tau } -\frac{\nabla^2}{2m} & 0 \\ 0 & -\frac{\partial }{\partial \tau }- \frac{\nabla^2}{2m} \\ \end{array} \right),\end{aligned}$$ $S$ can be written as $$S=\int [dx] \left(\frac{1}{2}\Phi ^{\dagger} \bar{G}_0^{-1} \Phi -J^{\dagger}\Phi +\frac{1}{2} \lambda \left(\phi^* \phi \right)^2-\mu \phi ^* \phi\right).$$ The large-$\mathcal{N}$ expansion introduces $\mathcal{N}$ copies of the original systems with $\phi_n$, $n=1,2,\cdots,\mathcal{N}$, rescale the coupling constant as $\lambda/\mathcal{N}$, and sort the Feynman diagrams by powers of $1/\mathcal{N}$. For the single-species Bose gas studied here, we follow Ref. [@ChienPRA12] and introduce an auxiliary field $\alpha$ representing $(\lambda/\mathcal{N})\sum_n\phi_n^\phi_n$ via the following identity: $$\begin{aligned} 1&&=\int D \alpha \delta \left(\alpha -(\lambda/\mathcal{N}) \sum_{n}\phi_n^ \phi_n \right)\nonumber\\ &&=\mathcal{C}\int D\alpha D \chi e^{\frac{\chi \left(\alpha -(\lambda/\mathcal{N}) \sum_{n}\phi_n^* \phi_n \right)}{\lambda }}, \label{eq:dirac_delta}\end{aligned}$$ where $\mathcal{C}$ is a normalization factor and the $\chi$ integration runs parallel to the imaginary axis [@Moshe200369]. After replacing $\sum_n\phi^_n \phi_n $ by $\alpha\mathcal{N}/\lambda$, the Gaussian integration of $\phi_n$ and $\phi^_n$ can be performed and one obtains $Z[J,Y,K]=\int D\alpha D \chi e^{-S_{\text{eff}}}$, where we have introduced the sources $Y$ and $K$ for the auxiliary fields $\chi$ and $\alpha$, respectively. To construct the leading-order theory, we only include up to the leading order of $1/\mathcal{N}$ in the effective action. Higher-order corrections of the $1/\mathcal{N}$ expansion can be constructed following Ref. [@PhysRevA.83.053622]. After obtaining the leading-order $S_{\text{eff}}$, we set $\mathcal{N}=1$ for single-component bosons. Then, $$\begin{aligned} S_{\text{eff}}&=& \int[dx]\left(-\frac{1}{2} J^{\dagger} G_0 J-\frac{\alpha \mu }{\lambda }-\frac{\alpha \chi }{\lambda }+\frac{\alpha^2 }{2 \lambda } -K\alpha - \right. \nonumber \\ & &\left. Y\chi \right)+\frac{1}{2} \text{Tr} \ln G_0^{-1},\end{aligned}$$ $G_0^{-1}$ is defined as $G_0^{-1}\equiv \bar{G}_0^{-1}+diag(\chi,\chi)$. The grand potential, which is also the generator of one-particle irreducible (1PI) graphs, can be obtained from a Legendre transform of the effective action [@nair2006quantum; @zee2010quantum; @PhysRevLett.105.240402] $$\Gamma[\phi_c,\phi_c^,\chi_c,\alpha_c] =\int[dx]\left(J^{\dagger}\Phi _c +K \alpha _c+Y \chi _c\right)+S_{\text{eff}},$$ where the subscript $c$ denotes the expectation values and will be dropped in the following. For static homogeneous fields, the effective potential is $V_{\text{eff}}=\Gamma /(\beta \Omega)$. Using the relation $J=\int[dx]G_0^{-1}\Phi$ we rewrite $V_{\text{eff}}$ in terms of the expectation value of $\Phi$ as $$\begin{aligned} V_{\text{eff}}&&=\frac{1}{2}\Phi ^{\dagger}G_0^{-1} \Phi-\frac{\alpha \mu }{\lambda }-\frac{\alpha \chi }{\lambda }+\frac{\alpha }{2 \lambda }+\nonumber\\ &&\sum _k \left(\frac{\omega _k}{2}+\frac{\ln \left(1-e^{-\beta \omega _k}\right)}{\beta }\right),\end{aligned}$$ where $ \omega _k=\epsilon _k+\chi$ and $\epsilon _k=\frac{k^2}{2 m}$. The last term in $V_{\text{eff}}$ comes from the trace log term, whose derivation is summarized in Appendix \[app:TrLn\]. For static homogeneous fields, only $\chi\phi^\phi$ remains in the first term. The expectation values of the fields can be found as the minimization conditions of the effective potential. From $\frac{\text{$\delta $V}_{\text{eff}}}{\delta \phi ^}=0$, we get $\chi \phi =0$, which imposes the broken-symmetry (BEC) condition that when $\phi =0$ in the normal phase, $\chi$ is finite and $\chi=0$ in the broken symmetry phase when $\phi \neq 0$. The condition $\frac{\text{$\delta $V}_{\text{eff}}}{\delta \alpha }=0$ fixes the relation between $\chi$ and $\alpha$ by $$\chi =\alpha -\mu. \label{eq:1fchi}$$ Since $V_{\text{eff}}$ is ultraviolet divergent, the theory needs to be renormalized. Ref. [@ChienPRA12] detailed the renormalization of the leading-order large-$\mathcal{N}$ theory, and a brief summary is given in Appendix \[app:Renormalization\]. The renormalized effective potential is $$V_{\text{eff}}=-\frac{\alpha ^2}{2 \lambda }+\phi \phi ^ (\alpha -\mu )+\sum _k \frac{\ln \left(1-e^{-\beta \omega _k}\right)}{\beta }. \label{eq:2fVReff}$$ Here renormalized (physical) quantities are used. Importantly, $V_{\text{eff}}$ at the minimum is $-P$ from thermodynamics, where $P$ is the pressure of the system. The equations of state can be derived from $\delta V_{\texttt{eff}}/\delta\phi^=0$, $\frac{\delta V_{\text{eff}}}{\delta \alpha}=0 $ and $-\frac{\delta V_{\text{eff}}}{\delta \mu }=\rho$. Explicitly, $$\begin{aligned} (\alpha-\mu)\phi&=&0, \nonumber \\ \frac{\alpha }{\lambda }&=&\phi \phi ^+\sum _k n(\omega _k), \nonumber \\ \frac{\alpha }{\lambda }&=&\rho. \label{eq:1frho}\end{aligned}$$ Here $n(\omega_k)=[e^{\frac{\omega _k}{k_B T}}-1]^{-1}$ is the Bose distribution function. In the normal phase, $\phi =0$ and $\omega _k=\epsilon _k+\alpha-\mu$. In the broken symmetry phase, $\phi$ is finite, so $\mu =\alpha $ and $\omega _k=\epsilon _k$. By the $U(1)$ symmetry of $\phi$, we can choose $\phi$ to be real in the broken symmetry phase and associate $\phi=\sqrt{\rho_c}$ with the condensate density $\rho_c$. LOAF theory ----------- The leading-order large-$\mathcal{N}$ theory is a conserving theory with a consistent thermodynamic free energy. It also shows a second-order BEC transition. However, one major issue with the leading-order large-$\mathcal{N}$ theory is its inconsistency with the Bogoliubov theory of weakly interacting bosons at zero temperature [@ChienPRA12]. Given theoretical and experimental support of the Bogoliubov dispersion of weakly interacting bosons near zero temperature [@RevModPhys.76.599; @PhysRevLett.88.120407], the leading-order large-$\mathcal{N}$ theory needs further improvement. The main reason of the inconsistency is because the anomalous density representing pairing correlations, $A=\lambda\langle\phi\phi\rangle$, is included in the Bogoliubov theory but not in the leading-order large-$\mathcal{N}$ theory. By including the normal density composite field $\chi=\sqrt{2}\lambda\langle\phi^\phi\rangle$ and $A$, a similar large-$\mathcal{N}$ expansion leads to the LOAF theory [@PhysRevLett.105.240402; @PhysRevA.83.053622; @PhysRevA.84.023603; @PhysRevA.85.023631; @PhysRevLett.105.240402], which is fully consistent with the Bogoliubov theory in the weakly interacting regime. A brief summary of the derivation of the LOAF theory is in Appendix \[app:LOAF\]. The regularized LOAF effective potential is $$\begin{aligned} V_{\text{eff}} &&=\chi ^{\prime} \phi ^ \phi -\frac{1}{2} A^* \left(\phi ^\right)^2-\frac{A \phi ^2}{2}-\frac{\left(\chi ^{\prime}+\mu \right)^2}{4 \lambda }+\frac{A A^}{2 \lambda }+\nonumber\\ &&\sum _k \left(\frac{1}{2} \left(\omega _k-\epsilon _k-\chi ^{\prime}+\frac{A A^}{2 \epsilon _k}\right)+\right.\nonumber\\ &&\left. T \ln \left(1-e^{-\frac{\omega _k}{T}}\right)\right),\end{aligned}$$ where $\chi^{\prime} \equiv \sqrt{2}\chi-\mu$. From the minimization conditions,$-\frac{\text{$\delta $V}_{\text{eff}}}{\delta \mu } =\rho$ and $\frac{\text{$\delta $V}_{\text{eff}}}{\delta \phi }=\frac{\text{$\delta $V}_{\text{eff}}}{\delta \chi ^{\prime}}=\frac{\text{$\delta $V}_{\text{eff}}}{\text{$\delta $A}^}=0$, we arrive at the equations of state $$\begin{aligned} \rho =\frac{\mu +\chi ^{\prime}}{2 \lambda }, \label{eq:2fmu}\end{aligned}$$ $$\begin{aligned} 0 =\phi ^* \chi ^{\prime}-A \phi, \label{eq:2fchi}\end{aligned}$$ $$\begin{aligned} 0 &&=- \frac{\mu +\chi ^{\prime}}{2 \lambda }+\rho_c +\nonumber\\ &&\sum _k \left(\frac{\left(\epsilon _k+\chi ^{\prime}\right) \left(1+2n(\omega_k)\right)}{2 \omega _k}-\frac{1}{2}\right). \label{eq:2frho}\end{aligned}$$ $$\begin{aligned} 0 &&= \frac{A}{\lambda}- \rho_c- A \sum _k \left(\frac{1+2 n(\omega_k)}{2 \omega _k}-\frac{1}{2 \epsilon _k}\right). \label{eq:2fA}\end{aligned}$$ In the BEC phase, we use the $U(1)$ symmetry to choose the expectation value of $\phi$ to be real and equal to $\sqrt{\rho_c}$ with $\rho_c$ being the condensate density. The leading-order large-$\mathcal{N}$ theory has two phases: Above $T>T_c$ it gives a normal phase, where the condensate $\phi=0$ but the composite field $\chi>0$ playing the role of the chemical potential. Below $T_c$ it is a broken symmetry phase corresponding to BEC, where the condensate $\phi>0$ and $\chi=0$ [@ChienPRA12]. The LOAF theory, on the other hand, predicts three possible phases: At high $T$ it is a normal phase, where both the condensate $\phi$ and the anomalous density $A$ vanish. The composite field $\chi^{\prime}>0$ is related to the chemical potential. Below $T_c$ it is a broken symmetry (BEC) phase with $\phi>0$ and $A>0$. The composite field $\chi$ is related to $A$ according to Eq. . Interestingly, there is an intermediate-temperature superfluid phase in the regime $T_c<T<T^$, where the condensate vanishes $\phi=0$ but the anomalous density remains finite, $A>0$. The finite $A$ gives rise to a finite superfluid density as derived in Ref. [@PhysRevA.85.023631]. In the intermediate-temperature superfluid regime, the two composite fields $\chi$ and $A$ are different and need to be determined from a set of coupled equations. Leading-order $T_c$ shift ------------------------- As mentioned before, the leading-order $T_c$ shift of a trapped Bose gas has been evaluated in Refs [@PhysRevA.54.R4633] using the Popov approximation with the LDA. To compare with the experimental results in Ref. [@PhysRevLett.106.250403], we introduce the thermal de Broglie wavelength of a trapped noninteracting Bose gas with the same total particle number $N$ at its critical temperature, $\lambda_0=\sqrt{\frac{2 \pi \hbar ^2}{m k_B T_c^0}}$. The leading order in the Popov approximation is then $$\frac{\delta T_c}{T_c^0}=-3.4260\frac{a}{\lambda_0}.$$ For the leading-order large-$\mathcal{N}$ theory, the dispersion above $T_c$ is $\omega_k(r)=k^2+\chi(r)$, where $\chi(r)=\lambda\rho(r)-\mu(r)$. When compared to the dispersion of the Popov approximation, the dispersion is almost identical if $\lambda$ in the Popov approximation is replaced by $\lambda/2$. This substitution leads to the $T_c$ shift in the leading-order large-$\mathcal{N}$ theory as $$\frac{\delta T_c}{T_c^0}=-1.7130\frac{a}{\lambda_0}.$$ In the LOAF theory, the two critical temperatures $T^$ and $T_c$ merge in the weakly interacting regime [@PhysRevA.85.023631]. Therefore, for analytic calculations we use the shift of $T^$ as a proxy to estimate the $T_c$ shift of a trapped gas in the weak interaction regime. From the discussion of the LOAF theory in Appendix \[app:LOAF\], the local composite fields are $\chi^{\prime}(r)=-\mu(r)+2\lambda\rho(r)$ by Eq. and $A(r)=0$ at $T^$, so Eq. reduces to $\rho(r)=\sum _k n(\omega_k)$, where $\omega_k(r)=k^2+V(r)+2\lambda\rho(r)-\mu_0$. This dispersion is identical to the Popov approximation to the lowest order in the coupling constant [@PhysRevA.54.R4633]. Following a similar calculation, $$\frac{\delta T_c}{T_c^0}=-3.4260\frac{a}{\lambda_0}.$$ Thus the leading order result of the LOAF theory agrees with the Popov theory and experimental data [@PhysRevA.54.R4633; @PhysRevLett.106.250403]. Our numerical results using the LDA agree well with the leading-order estimations presented here. large-$\mathcal{N}$ based theories for trapped Bose gases {#sec:LargeNLDA} ========================================================= The large-$\mathcal{N}$ based theories can be formulated with the LDA, where the trap potential $V(r)$ is grouped with the chemical potential $\mu_0$ and the local chemical potential $\mu(r)=\mu_0-V(r)$ is introduced. Then we search for a solution consistent with the profile of $\mu(r)$ with a given $N$. To find the density profile $\rho(r)$ from large-$\mathcal{N}$ based theories numerically, the following procedures have been implemented. Since the total particle number $N$ and temperature $T$ are given, one needs to find the chemical potential $\mu_0$ satisfying Eq. . This also implies that $\mu_0$ is a function of $T$ and the coupling constant. For harmonically trapped systems, the following units are introduced. $a_H\equiv\sqrt{\frac{\hbar }{m \omega }}$ and $E_0\equiv \frac{\hbar ^2}{2 m a_H^2}$. Then $\frac{\lambda_0 }{a_H}=\sqrt{2 \pi } \left(\frac{\zeta (3)}{N}\right)^{1/6}$. This allows us to use the following dimensionless quantities. $\frac{\epsilon _k}{E_0}=\left(k a_H\right)^2$, $\frac{V(r)}{E_0}=\left(\frac{r}{a_H}\right)^2$, and $\frac{k_B T}{E_0}$. Moreover, the (renormalized) coupling constant is related to the two-body $s$-wave scattering length by $$\frac{\lambda }{E_0 a_H^3}=8\pi\frac{a}{a_H}.$$ Leading-order large-$\mathcal{N}$ theory with LDA ------------------------------------------------- We begin with the leading-order large-$\mathcal{N}$ theory with the LDA. At given $T$ and $a$, $\mu_0$ should satisfy Eq. with a density profile $\rho(r)$ determined from the equations of state on a grid discretizing the geometry. We have chosen the grid size small enough that further reductions of the size do not change our results. Initially, a trial value of $\mu_0$ is guessed and we find the corresponding $\rho(r)$. If BEC is present, we need to locate the size of the condensate. This is equivalent to finding a critical radius $r_c$ where $\rho_c(r_c)=0$. At $r=r_c$ the condition $\chi(r_c)=\rho_c(r_c)=0$ can be used in Eq. and Eq. to obtain $r_c =\sqrt{(\mu _0-\left(\frac{ m k_B T}{2 \pi \hbar ^2}\right)^{3/2} \zeta \left(\frac{3}{2}\right) \lambda)(\frac{2}{m \omega ^2})}$. Once $r_c$ is located, the density profile $\rho(r)$ can be constructed with the information of $T$ and $\mu(r)$. If $T<T_c$, there is a condensate within $r<r_c$, whose condensate density can be found from $\rho_c(r)=\rho(r)-\rho_T(r)$. In this region $\chi(r)=0$ due to the BEC condition, so $\rho(r)=\mu(r)/\lambda$ by Eq. and Eq. gives us $\rho_T(r)=\left(\frac{k_B T}{4 \pi E_0}\right){}^{3/2} \zeta \left(\frac{3}{2}\right)a_H^3$. Outside the condensate region ($r>r_c$), $\rho_c(r)=0$ and one can solve Eq. to obtain $\rho(r)$. After $\rho(r)$ in the whole trap is found, $\mu_0$ can be evaluated by iteratively solving Eq. by treating $\mu_0$ as a function of $N$, $T$, and $a$. Above $T_c$ there is no condensate ($\rho_c(r)=0$), and a similar procedure leads to $\mu_0$ and $\rho(r)$ as well. To find $T_c$, we tune the temperature so that $r_c=0$. This is the temperature when the condensate is about to emerge. The relation between $T_c$ and $\mu_0(T_c)$ are fixed by the expression of $r_c=0$, and $T_c$ can be found by iteratively search for the solution that satisfies Eq. with given $N$ and $a$. LOAF theory with LDA -------------------- As mentioned before, the LOAF theory for a uniform interacting Bose gas exhibits a richer phase diagram with three distinct phases. For a trapped Bose gas below $T_c$, there is a condensate at the center with $\rho_c(r)>0$ and $A(r)=\chi(r)>0$. The condensate vanishes at $r_c$, where $\rho_c(r\ge r_c)=0$ but $A(r_c)$ can still be finite. The anomalous density vanishes at $r=r^$, and outside $r^$ the system is normal with $\rho_c(r)=0$ and $A(r)=0$. To find the density profile and $\mu_0$ with given $N$, $T$, and $a$, we solve Eq. iteratively with the following procedures. The initial value of $\mu_0$ is guessed and we map out the corresponding $\rho(r)$. Next we need to locate $r_c$ where $\rho_c(r_c)=0$. From $A(r_c)=\chi(r_c)$ and $\rho(r_c)=0$, Eq. and Eq. allow us to determine $\chi(r_c)$ and $\mu(r_c)$. Then $r_c$ can be inferred from $\mu_0-\mu(r_c)$. When $r>r_c$, the anomalous density $A(r)$ should decay to zero at $r=r^$. Using $A(r\rightarrow r^)\rightarrow 0$ in Eq. , one can find $\chi(r^)$, which can be used in Eq. to find $\mu(r^)$. Then $r^$ is inferred from $\mu_0-\mu(r^)$. After determining $r_c$ and $r^$, we can map out the whole density profile. To find $\rho(r)$ in the condensate region, we set $A(r)=\chi(r)$ with a finite $\rho_c(r)$. Multiplying Eq. by $\lambda$ and subtracting Eq. lead to an equation for $\chi(r)$. After solving for $\chi(r)$, we can get $\rho(r)$ from Eq with $\mu(r)$ and $\chi(r)$. In the region between $r_c$ and $r^$, $A(r)$ and $\chi(r)$ are found by solving Eq. and Eq. simultaneously. Once $\chi(r)$ is found, $\rho(r)$ is again obtained by Eq. . Outside $r^$, $A(r)=0$ and only Eq. needs to be solved, which will give us $\chi(r)$ and thus $\rho(r)$. After $\rho(r)$ is obtained in all regions, $\mu_0$ is solved iteratively by Eq. , where the integral is split over different regions determined by $r_c$ and $r^$. The critical temperature corresponds to a density profile with $r_c=0$ when the condensate at the center is about to emerge. The condition $r_c=0$ fixes the relation between $T_c$ and $\mu_0$ at the center by Eq. with $\rho_c(r_c)=0$ and $\chi(r_c)=A(r_c)$, and we only need to find $r^$ and the whole density profile. Then by solving Eq. iteratively we obtain $T_c$ from the LOAF theory with the LDA. Results and Discussions {#sec:Results} ======================= Since the thermodynamic limit has been taken in each slice of the LDA, the number-fixing procedure, Eq. , serves to fix the units. In our calculations we set $N=1000$, which corresponds to $\frac{\lambda_0 }{a_H}=0.8173$. By choosing a different value of $N$ and scaling the units accordingly, the coefficients in the expression of $T_c$ shift remain the same. Using the experimental data from Ref. [@PhysRevLett.92.030405] and subtracting finite-size effects, Ref. [@PhysRevLett.96.060404] showed a $T_c$-shift curve as a function of $a/\lambda_0$. Only linear and quadratic terms are used over the range of experimental data and corrections from the logarithmic term suggested in Ref. [@PhysRevA.64.053609] was not found. We follow the clue and did not include the logarithmic term when extracting the functional form of our results. The $T_c$ shift from the leading-order large-$\mathcal{N}$ theory with the LDA is presented in Figure \[fig:1field\]. The curve is almost linear with $a/\lambda_0$ and has a very small curvature. A fitting of the curve gives $$\label{eq:LNTcFit} \frac{\delta T_c}{T_c^0}=-1.71\frac{a}{\lambda_0}+4.55\left(\frac{a}{\lambda_0}\right)^2.$$ When compared to the experimental data from Ref. [@PhysRevLett.106.250403], the coefficient of the leading-order term is only half of the experimental value and the coefficient of the next-order term is even farther away. The $T_c$ shift from the LOAF theory with the LDA is shown in Figure \[fig:2field\]. The density profile exhibit a density discontinuity at the boundary of superfluid and normal phases, and we will comment on this behavior later on. For the weakly interacting regime, a fitting of the $T_c$ shift gives $$\label{eq:LOAFTcFit} \frac{\delta T_c}{T_c^0}=-3.42\frac{a}{\lambda_0}+52.00\left(\frac{a}{\lambda_0}\right)^2.$$ When compared to the expression extracted from the experimental data of Ref. [@PhysRevLett.106.250403], we found excellent agreements for the leading-order as well as the next-order terms. Due to difficulties of formulating mean-field theories of trapped interacting Bose gases, to our knowledge a theoretical evaluation of the quadratic term in the $T_c$ shift has not been available. The LOAF theory with the LDA thus may serve as a manageable mean-field theory for describing the $T_c$ shift of trapped interacting Bose gases. A summary of the coefficients of $T_c$ shift from large-$\mathcal{N}$ based theories and the experimental data of Ref. [@PhysRevLett.106.250403] is given in Table \[tab:comparison\]. One can see that by introducing the anomalous density $A=\lambda\langle\phi\phi\rangle$ originally incorporated in the Bogoliubov theory, the predictions of the $T_c$ shift improve substantially from the leading-order large-$\mathcal{N}$ theory to the LOAF theory. When compared to previous theoretical studies using the Popov theory [@PhysRevA.54.R4633] allowing for an extraction of the leading-term coefficient, the large-$\mathcal{N}$ base theories allow us to fit the functional form of the $T_c$ shift with higher orders and analyze the trap density profiles. Moreover, the agreement between the LOAF theory and experimental data from Ref. [@PhysRevLett.106.250403] does not require the logarithmic term from Ref. [@PhysRevA.64.053609]. $b_1$ $b_2$ -------------------------------------- -------------- ---------- Leading-order large-$\mathcal{N}$ -1.71 4.55 LOAF - 3.42 52.00 Phenomenological model - 3.38 30.35 Experiment [@PhysRevLett.106.250403] $-3.5\pm0.3$ $46\pm5$ : Comparsion of theoretical and experimental results. The $T_c$ shift has the form $\frac{\text{$\delta $T}_c}{T_c^0}=b_1\frac{a}{\lambda_0 }+b_2 \left(\frac{a}{\lambda_0 }\right)^2$. The phenomenological model predicts a cubic term with a coefficient $-152.5$. Choosing different values of $N$ only scales the units, and the coefficients shown here do not change.[]{data-label="tab:comparison"} Incompatible behavior of LOAF theory with LDA --------------------------------------------- As shown in Fig. \[fig:2field\], the LOAF theory with the LDA exhibits an observable density discontinuity between the superfluid and normal phases. The discontinuity increases as $a$ increases. We caution that, although density discontinuities were also found in the Popov theory with the LDA [@BF02396737], the origins of the discontinuities are very different. For the Popov theory, a discontinuous first-order transition already emerges in a homogeneous interacting Bose gas [@RevModPhys.76.599; @BF02396737; @PhysRevLett.105.240402]. In contrast, the LOAF theory predicts a smooth second-order transition for homogeneous interacting Bose gases [@PhysRevLett.106.250403]. The discontinuity in the trap profile of the LOAF theory comes from incompatibility of the theory with the LDA requiring $\mu(r)$ to decrease quadratically in a harmonic trap. The incompatibility will be elaborated, and here we emphasize that the leading-order large-$\mathcal{N}$ theory does not suffer from any density discontinuity even in the stronger interaction regime (illustrated in Figure \[fig:1field\]), but its predictions of the $T_c$ shift do not agree quantitatively with the experimental data of Ref. [@PhysRevLett.106.250403] as summarized in Table \[tab:comparison\]. The density discontinuity of the LOAF theory can be analyzed as follows. For $r\ge r_c$, $\rho_c=0$ in Eq. . Thus, at $r_c$ and $r^$ the following equation has to be satisfied with different energy dispersions. $$1 = \lambda \sum _k \left(\frac{1+2 n(\omega_k)}{2 \omega _k}-\frac{1}{2 \epsilon _k}\right). \label{eq:A_c}$$ The dispersion at $r=r_c$ is $\omega_{kc}=\sqrt{\epsilon_k (\epsilon_k+2\chi_c^{\prime})}$ with $\chi_c^{\prime}$ denoting the value of $\chi^{\prime}$ at $r_c$. At $r=r^$, $\omega_{k}^=\epsilon_k+\chi^{\prime }$ with $\chi^{\prime }$ denoting the value of $\chi^{\prime}$ at $r^$. If $\chi_c^{\prime}$ is less than $\chi^{\prime }$, $\omega_{k}^$ will be greater than $\omega_{kc}$ for any $k$, so Eq. cannot be satisfied by both dispersions. This problem can be circumvented by making $\omega_{k}^$ less than $\omega_{kc}$ at small $k$ so that Eq. can be satisfied by both dispersions. This requires $\chi_c^{\prime}$ to be greater than $\chi_{k}^$. However, if $\lambda\rho(r)$ does not decrease fast enough as $r$ increases, $\chi^{\prime}(r)=-(\mu_0-\frac{1}{2}m\omega^2r^2)+2\lambda\rho(r)$ may be an increasing function. When it happens, $r^>r_c$ does not exist. One solution is to force the LDA form of $\mu(r)$ and connect the superfluid and normal phases similar to the Maxwell construction. Then the solution exhibits a jump of $A(r)$ to zero at $r_c$. Such a discontinuity in $A(r)$ then cause a discontinuity in the density profile according to Eq. , which is observable in Fig. \[fig:2field\] (c). The incompatibility with the LDA is also hinted by the behavior of $\mu$ as a function of $T$. In the LOAF theory of homogeneous Bose gases, $\mu(T)$ can be non-monotonic as $T$ increases [@PhysRevLett.105.240402]. However, for a trapped Bose gas in the LDA, the local chemical potential $\mu(r)=\mu_0-\frac{1}{2}m\omega^2r^2$ should decrease quadratically as $r$ increases. Since the particle density decreases with $r$ and the local temperature scale $T(r)$ is determined by the local density, the temperature ratio $T/T(r)$ increases with $r$. Therefore, when the interaction is too strong and $\mu(T)$ exhibits prominent non-monotonicity, the LOAF theory cannot be pieced together in the LDA. We remark that the leading-order large-$\mathcal{N}$ theory does not have such incompatibility with the LDA. Phenomenological model {#sec:phe} ---------------------- The incompatibility of the LOAF theory with the LDA leads us to contemplate possible alternatives. One may re-derive the whole theory in real space with inhomogeneity. The fields are no longer uniform and the equations of state will be coupled differential equations. Solving the equations is not only numerically demanding, but also loses transparency in explaining the underlying physics. Here we explore a phenomenological alternative by requiring that the local anomalous density $A(r)$, instead of the local chemical potential $\mu(r)$, decays with a quadratic form. Explicitly, the condition $A(r)=A_0-E_0(r/a_H)^2$ is imposed and the anomalous density at the trap center, $A_0$, needs to be solved iteratively. To obtain the density profile and corresponding chemical potential, $A_0$ is guessed initially and we use $A(r)$ to map out $\rho(r)$. Since $A(r)$ has to vanish at $r=r^$, we have $r^/a_H=\sqrt{A_0/E_0}$. To find $r_c$, we first find $A(r_c)$ by solving Eq. at $r_c$ with $\chi(r_c)=A(r_c)$. Then $r_c$ is found by $r_c/a_H=\sqrt{A_0-A(r_c)}$. Moreover, at $r_c$ one has $\chi(r_c)=A(r_c) >0$, so $r_c$ cannot be greater than $r^$. In the region $r<r_c$, $\chi(r)=A(r)$ and $\mu(r)$ is obtained by multiplying Eq. by $\lambda$ and subtracting Eq. . Then the resulting equation can be solved to give $\chi(r)$. In the region $r_c<r<r^$, $\chi(r)$ is obtained by solving Eq. and then Eq. is used to obtain $\mu(r)$. For $r^<r$, $A(r)=0$ and we use the LDA for local chemical potential $\mu(r)=\mu(r^)-E_0\left(\left(r-r^\right)/a_H\right)^2$ to finish the computation. Outside $r^$, $\chi(r)$ is obtained by solving Eq. . After $\chi(r)$ and $\mu(r)$ are found in all regions, $\rho(r)$ can be inferred by Eq. . Then $A_0$ is found by iteratively solving Eq. with given $T$ and $a$. As shown in Figure \[fig:funny\_model\] (a)-(d), both the density profile and local chemical potential of the phenomenological model are continuous in the whole trap. However, the local chemical potential clearly exhibits a deviation from the conventional LDA. To contrast the difference, the red dotted line shows an extrapolation according to the LDA with $\mu(r)=\mu_0^2-E_0(r/a_H)^2$, where $\mu_0$ is calculated to match the chemical potential of this phenomenological model outside $r^$. The non-monotonic local chemical potential as a function of $r$ confirms the incompatibility of the LOAF theory with the LDA, and with a simple reformulation of the local anomalous density $A(r)$ we restore continuity to the trapped system. Figure \[fig:funny\_model\] further illustrates the phenomenological model for different interaction strength close to $T_c$. Apparently, the deviation from the conventional LDA becomes more prominent as the interaction increases. The deviation is understandable because interactions should lead to corrections of the chemical potential in a many-body system, and the simple assumption of grouping $\mu$ and the trapping potential $V(r)$ at the bare level may no longer hold. We caution that the model with an LDA form of $A(r)$ is purely phenomenological, and a full treatment of the LOAF theory with inhomogeneous fields using numerical methods will eventually replace the phenomenological model and the LDA. The $T_c$ shift predicted by the phenomenological model is shown in Fig. \[fig:funny\_model\] (e). In contrast to the LOAF theory with the LDA, we did not find any density jump in the trap profile of the phenomenological model. By fitting the curve in a broader range of interaction strength, we found the functional form of the $T_c$ shift as $$\label{eq:funnyTcFit} \frac{\delta T_c}{T_c^0}=-3.382\frac{a}{\lambda_0}+30.35\left(\frac{a}{\lambda_0}\right)^2-152.5 \left(\frac{a}{\lambda_0}\right)^3.$$ The coefficient of the cubic term serves as a prediction for future experimental measurements in stronger interaction regime. Conclusion {#sec:conclusion} ========== Two large-$\mathcal{N}$ based theories, the leading-order large-$\mathcal{N}$ theory and the LOAF theory, implemented with the LDA capture interesting physics of the $T_c$ shift of harmonically trapped interacting Bose gases. The leading-order large-$\mathcal{N}$ theory produces continuous trap profiles in weak and intermediate interaction regimes, but quantitative comparisons with available experimental data do not agree quantitatively. On the other hand, the LOAF theory with the LDA predicts a functional form with higher-order corrections agreeing well with experimental data. The LOAF theory also signals beyond-LDA behavior in the trap profile. The proposed phenomenological model based on the LOAF theory may serve as a useful patch for studying trapped Bose gases when a full analysis of inhomogeneous fields remains a great challenge. We thank Fred Cooper and Kevin Mitchell for stimulating discussions. Details of large-$\mathcal{N}$ based theories ============================================= Calculation of $Tr \ln \bar{G}_0^{-1}$ {#app:TrLn} -------------------------------------- To evaluate the $Tr \ln \bar{G}_0^{-1}$ term in the effective action, we define $I(s)$ as $$\begin{aligned} I (s) &&\equiv\frac{1}{2}\int d^4x \int \frac{d^3k}{\left(2\pi\right)^3}\nonumber\\ &&T \sum _n \log \left(\omega _n^2+\omega_k^2+s\right),\end{aligned}$$ where $\omega_n=2n\pi/\beta$ is the bosonic Matsubara frequency and $\omega_k$ is the energy dispersion. It can be shown that $I(0)=\frac{1}{2} \text{Tr}\left[\ln \left(\bar{G}_0^{-1}\right)\right]$ [@nair2006quantum; @zee2010quantum]. Then, $$\begin{aligned} \frac{\partial I(s)}{\partial s}&&=\frac{1}{2}\int d^4x \int \frac{d^3k}{\left(2\pi\right)^3}T \sum _n \frac{1}{\omega _n^2+\omega_k^2+s}\nonumber\\ &&=\frac{1}{2}\int d^4x \int \frac{d^3k}{\left(2\pi\right)^3}T\nonumber\\ &&\oint\frac{1}{\left((2 \pi T z)^2+\omega_k^2+s\right) \left(e^{2 \pi i z}-1\right)}\end{aligned}$$ where the contour encircles the real axis counterclockwise. We deform the contour on the upper complex plane to encircle the upper half complex plane and the contour on the lower complex plane to encircle the lower half complex plane, both clockwise. Taking the residues on the imaginary axis, $2\pi T z=\pm i \sqrt{\omega_k^2+s}$, one obtains $$\begin{aligned} \frac{\partial I(s)}{\partial s}&&=\frac{1}{2}\int d^4x \int \frac{d^3k}{\left(2\pi\right)^3}\nonumber\\ &&\left(\frac{1}{\sqrt{\omega_k^2+s}}\left(\frac{1}{e^{\frac{\sqrt{\omega_k^2+s}}{T}}-1}+\frac{1}{2}\right)\right).\end{aligned}$$ Here we have taken care of the minus sign due to the clockwise contours. Integrating it back and setting $s=0$, we arrive at (apart from a constant) $$I(0)=\frac{1}{2}\int d^4x \int \frac{d^3k}{\left(2\pi\right)^3} \left(T \ln \left(1-e^{-\frac{\omega_k }{T}}\right)+\frac{1}{2} \omega_k\right).$$ Renormalization of $V_{\text{eff}}$ {#app:Renormalization} ----------------------------------- Here we follow Ref. [@ChienPRA12] to renormalize $V_{eff}$ of leading-order large-$\mathcal{N}$ theory. The renormalized coupling constant can be defined as $$\frac{1}{\lambda _R}=-\frac{\delta V_{\text{eff}}}{\delta \alpha \delta \alpha}.$$ For the leading-order large-$\mathcal{N}$ theory, $\frac{\delta V_{\text{eff}}}{\delta \alpha \delta \alpha }=-\frac{1}{\lambda }+$ (regular terms). Thus, the renormalized coupling constant can be identified as the bare coupling constant. This lead to $\lambda_R=(4\pi\hbar^2 a/m)$, where $a$ is the two-body $s$-wave scattering length [@pethick2008bose]. By inspecting the classical part $-\frac{\partial V_{\text{eff}}}{\partial \chi }\|_{\chi=0}=\frac{\mu }{\lambda }$, the renormalized chemical potential $\mu_R$ is given by $$-\frac{\partial V_{\text{eff}}}{\partial \chi }=\frac{\mu _R}{\lambda }=\frac{\mu }{\lambda }-\sum _k \frac{1}{2}.$$ The renormalized effective potential is $$V_{R,\text{eff}}=V_{R,0}+\sum _k \frac{\ln \left(1-e^{-\beta \omega _k}\right)}{\beta }-\frac{\left(\mu _R+\chi \right){}^2}{2 \lambda }+\chi \phi ^ \phi,$$ where $V_{R,0}$ is an infinite constant that absorbs the zero-point energy and may be dropped. We define $\alpha_R=\mu_R+\chi$, too. Derivation of the LOAF theory {#app:LOAF} ----------------------------- Here we briefly review the LOAF theory by skipping the derivation with $\mathcal{N}$ copies of the fields and just presenting the leading-order $1/\mathcal{N}$ theory with $\mathcal{N}$ set to $1$. The detailed derivation can be found in Refs. [@PhysRevLett.105.240402; @PhysRevA.83.053622]. In the LOAF theory, we introduce two auxiliary fields $\chi$ and $A$ representing the normal and anomalous densities. They can be introduced by inserting the following identity into the partition function . $$\begin{aligned} 1 &&= \int\mathcal{D}\chi \text{$\mathcal{D}$A} \text{$\mathcal{D}$A}^* \nonumber\\ &&\delta \left(\chi -\sqrt{2} \lambda \phi ^* \phi \right) \delta \left(A-\lambda \phi ^2\right) \delta \left(A^-\lambda \left(\phi ^\right)^2\right)\nonumber\\ &&=\mathcal{C}^{\prime}\int\mathcal{D}\chi \mathcal{D} \tilde{\chi } \text{$\mathcal{D}$A} \mathcal{D} \tilde{A} \text{$\mathcal{D}$A}^* \mathcal{D} \tilde{A^}\nonumber\\ && e^{\frac{\tilde{\chi } \left(\chi -\sqrt{2} \lambda \phi ^ \phi \right)}{\lambda }} e^{\frac{\tilde{A^} \left(A-\lambda \phi ^2\right)}{\lambda }} e^{\frac{\tilde{A} \left(A^-\lambda \left(\phi ^\right)^2\right)}{\lambda }}.\end{aligned}$$ Here $\mathcal{C}^{\prime}$ is a normalization factor, and the contour of integration follows the description below Eq. . Then the quartic term in $\phi$ can be replaced by $$\begin{aligned} \frac{1}{2} \lambda \left(\phi ^ \phi \right)^2 &&=\frac{1}{2} \left(2 \lambda \left(\phi ^* \phi \right)^2-\lambda \left(\phi ^\right)^2 \phi ^2\right)\nonumber\\ &&=\frac{1}{2} \left(2 \lambda \left(\frac{\chi }{\sqrt{2} \lambda }\right)^2-\frac{\lambda A A^}{\lambda \lambda }\right).\end{aligned}$$ With the quartic terms replaced, the action becomes $$\begin{aligned} &&\int[dx](\frac{1}{2} \Phi \bar{G}_0^{-1} \Phi-\frac{\mu \chi }{\sqrt{2} \lambda }+\frac{\chi ^2}{2 \lambda }-\frac{A A^}{2 \lambda }-J^{\dagger} \Phi\nonumber\\ && -\frac{\tilde{\chi } \chi }{\lambda }-\frac{\tilde{A^} A}{\lambda }-\frac{\tilde{A} A^}{\lambda } -s\chi-\mathcal{S}^A-\mathcal{S} A^),\end{aligned}$$ where $s$, $\mathcal{S}$, and $\mathcal{S}^$ are the source terms for the auxiliary fields, and $\bar{G}_0^{-1}=\tilde{G}_0^{-1}+\left( \begin{array}{cc} \sqrt{2} \tilde{\chi } & 2 \tilde{A^} \\ 2 \tilde{A} & \sqrt{2} \tilde{\chi } \\ \end{array} \right)$. Here $\tilde{G}_0^{-1}=diag(\frac{\partial }{\partial \tau } -\frac{\nabla^2}{2m} ,-\frac{\partial }{\partial \tau }- \frac{\nabla^2}{2m})$. Performing the $\phi$ integral, the effective action becomes $$\begin{aligned} S_{\text{eff}} &&= \frac{1}{2} \text{Tr}\left[\ln \left(\bar{G}_0^{-1}\right)\right] +\int[dx]\left(-\frac{1}{2} J^{\dagger} \bar{G}_0 J-\frac{\mu \chi }{\sqrt{2} \lambda }+\frac{\chi ^2}{2 \lambda }-\right. \nonumber\\ &&\left.\frac{A A^}{2 \lambda }-\frac{\tilde{\chi } \chi }{\lambda }-\frac{\tilde{A^} A}{\lambda }-\frac{\tilde{A} A^}{\lambda }-s\chi-\mathcal{S}^A-\mathcal{S} A^\right).\end{aligned}$$ To obtain the effective potential, we apply a Legendre transform and replace $J$ in terms of $\phi_c$ and $\phi^_c$ by using $J=\int[dx]\bar{G}_0^{-1} \Phi_c$. This leads to the grand potential $$\begin{aligned} \Gamma&&=\int[dx](\Phi_c J^{\dagger}+s\chi_c+\mathcal{S}^A_c+\mathcal{S} A_c^)+S_{\text{eff}}\nonumber\\ =&& \frac{1}{2} \text{Tr}\left[\ln \left(\bar{G}_0^{-1}\right)\right] +\int[dx]\left(\Phi ^{\dagger} \bar{G}_0^{-1} \Phi -\frac{\mu \chi }{\sqrt{2} \lambda }+\frac{\chi ^2}{2 \lambda }-\right.\nonumber\\ &&\left.\frac{A A^}{2 \lambda }-\frac{\tilde{\chi } \chi }{\lambda }-\frac{\tilde{A^} A}{\lambda }-\frac{\tilde{A} A^}{\lambda }\right).\end{aligned}$$ In the following we drop the subscript $c$ denoting the expectation values. The equilibrium state corresponds to the theory at the minimum of $\Gamma$. Thus, $$\begin{aligned} \frac{\delta \Gamma }{\delta \chi } =0\Rightarrow \tilde{\chi } =\chi -\frac{\mu }{\sqrt{2}}.\\ \frac{\delta \Gamma }{\text{$\delta $A}^} =0\Rightarrow \tilde{A}=-\frac{A}{2}.\\ \frac{\delta \Gamma }{\text{$\delta $A}} =0\Rightarrow \tilde{A^}=-\frac{A^}{2}.\end{aligned}$$ At the minimum, $\Gamma$ has the expression $$\Gamma =\int[dx]( \Phi ^{\dagger} \bar{G}_0^{-1} \Phi -\frac{\chi ^2}{2 \lambda }+\frac{A A^}{2 \lambda })+\frac{1}{2} \text{Tr}\left[\ln \left(\bar{G}_0^{-1}\right)\right],$$ where $$\begin{aligned} \bar{G}_0^{-1} &&= \left( \begin{array}{cc} \partial _{\tau }-\frac{\nabla ^2}{2 m}+\chi^{\prime} & -A^* \\ -A & -\partial _{\tau }-\frac{\nabla ^2}{2 m}+\chi^{\prime} \\ \end{array} \right).\end{aligned}$$ Here $\chi^{\prime}=\sqrt{2} \chi -\mu$. Following Appendix \[app:TrLn\], for homogeneous static fields we obtain $$\frac{1}{2} \text{Tr}\left[\ln \left(\bar{G}_0^{-1}\right)\right] =\int[dx]\sum_{k}\left[T \ln \left(1-e^{-\frac{\omega _k}{T}}\right)+\frac{\omega _k}{2}\right],$$ where $\omega _k=\sqrt{\left(\epsilon _k+\chi^{\prime}\right){}^2-A A^}$ and the dispersion is gapless in the presence of BEC. For homogeneous and static fields, the effective potential $V_{\text{eff}}=\Gamma/(\beta\Omega)$ becomes $$\begin{aligned} V_{\text{eff}} &&= \chi^{\prime} \phi ^ \phi -\frac{1}{2} A^* \left(\phi ^\right)^2-\frac{A \phi ^2}{2}-\frac{\left(\chi^{\prime}+\mu\right) ^2}{4 \lambda }+\frac{A A^}{2 \lambda }+\nonumber\\ &&\sum _k \left(\frac{\omega _k}{2}+T \ln \left(1-e^{-\frac{\omega _k}{T}}\right)\right).\end{aligned}$$ The ultraviolet divergence of $V_{eff}$ can be renormalized in the normal and BEC phases, while in the intermediate superfluid phase it can be regularized. The regularization smoothly interpolates the two renormalizations at high and low temperatures. Following the procedures described in Refs. [@PhysRevLett.105.240402; @PhysRevA.83.053622], the effective potential after the renormalization and regularization is shown in Eq. . Then minimizing $V_{eff}$ with respect to the fields and implementing standard thermodynamic relations lead to the equations of state shown in Eqs. -. [37]{} 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 [](https://books.google.com/books?id=G8kgAwAAQBAJ) (, ) [](https://books.google.com/books?id=iix3_pqy6ysC) (, ) [](https://books.google.com/books?id=-9c2Ns2J5P4C), Theoretical and Mathematical Physics(, )[, ()](\doibase 10.1103/RevModPhys.76.599)[, ()](\doibase 10.1103/RevModPhys.71.463)[, ()](\doibase 10.1103/RevModPhys.80.885)[, ()](\doibase 10.1007/BF02396737)[, ()](\doibase 10.1103/PhysRevLett.92.030405)[, ()](\doibase 10.1103/PhysRevA.81.053632)[, ()](\doibase 10.1103/PhysRevLett.106.250403)[, ()](\doibase 10.1103/PhysRevA.54.R4633)[, ()](\doibase 10.1103/PhysRevLett.105.240402)[, ()](\doibase 10.1103/PhysRevA.85.023631)[, ()](\doibase 10.1103/PhysRevA.86.023634)[ ()](http://www.sciencedirect.com/science/article/pii/S0003491614001006)[, ()](http://journals.aps.org/pra/abstract/10.1103/PhysRevA.91.043631)[, ()](\doibase 10.1103/PhysRevLett.98.110404)[, ()](\doibase 10.1038/nature07172)[](https://books.google.com/books?id=t5_DAgAAQBAJ), Dover Books on Physics (, )[**, ()](\doibase 10.1016/S0375-9601(96)08842-1)[, ()](\doibase 10.1016/0375-9601(95)00766-V)[](https://books.google.com/books?id=Vn5GAAAAYAAJ), No. (, )[**, ()](\doibase 10.1103/PhysRevA.54.656)[, ()](\doibase 10.1103/PhysRevA.54.4188)[, ()](\doibase 10.1103/PhysRevA.58.1490)[, ()](\doibase 10.1103/PhysRevA.83.023616)[, ()](\doibase 10.1103/PhysRevA.92.017601)[, ()](\doibase 10.1103/PhysRevA.64.053609)[](https://books.google.com/books?id=Bx59iYMgB84C), Cambridge Monographs on Mathematical Physics (, )[**, ()](http://stacks.iop.org/0295-5075/49/i=2/a=150)[, ()](\doibase http://dx.doi.org/10.1016/S0370-1573(03)00263-1)[, ()](\doibase 10.1103/PhysRevA.83.053622)[, ()](\doibase 10.1103/PhysRevLett.88.120407)[, ()](\doibase 10.1103/PhysRevA.84.023603)[, ()](\doibase 10.1103/PhysRevLett.96.060404)[](https://books.google.com/books?id=rcr0fxDf8jsC), Graduate Texts in Contemporary Physics (,)[**](https://books.google.com/books?id=n8Mmbjtco78C),In a Nutshell (, )
--- abstract: 'We present singlet-majoron couplings to Standard Model particles through two loops at leading order in the seesaw expansion, including couplings to gauge bosons as well as flavor-changing quark interactions. We discuss and compare the relevant phenomenological constraints on majoron production as well as decaying majoron dark matter. A comparison with standard seesaw observables in low-scale settings highlights the importance of searches for lepton-flavor-violating two-body decays $\ell \to \ell'' +$majoron both in the muon and tau sector.' author: - Julian Heeck - 'Hiren H. Patel' bibliography: - 'BIB.bib' title: The majoron at two loops --- Introduction ============ The Standard Model (SM) has emerged as an incredibly accurate description of our world at the particle level. Even its apparently accidental symmetries, baryon number $B$ and lepton number $L$, are seemingly of high quality and have never been observed to be violated. One could however argue that the established observation of non-zero neutrino masses is not only a sign for physics beyond the SM but also for possible lepton number violation by two units. This argument is based on an interpretation of the SM as an effective field theory (EFT) and the observation that the leading non-renormalizable operator is Weinberg’s dimension-five operator $(\bar L H)^2/\Lambda$ [@Weinberg:1979sa]. This operator violates lepton number ($\Delta L =2$) and leads to Majorana neutrino masses of order $\langle H\rangle^2/\Lambda$ after electroweak symmetry breaking, which gives the correct neutrino mass scale for a cutoff $\Lambda\sim\unit[10^{14}]{GeV}$. Besides explaining neutrino oscillation data, an EFT scale this high has little impact on other observables and thus nicely accommodates the absence of non-SM-like signals in our experiments. An ever-increasing number of renormalizable realizations of the Weinberg operator exist in the literature, the simplest of which arguably being the famous type-I seesaw mechanism [@Minkowski:1977sc] which introduces three heavy right-handed neutrinos to the SM field content. While the Weinberg operator explicitly breaks lepton number, underlying renormalizable models could have a dynamical origin for $\Delta L =2$ via spontaneous breaking of the global $U(1)_L$ symmetry. This leads to the same Weinberg operator and thus Majorana neutrino masses, but as a result of the spontaneous breaking of a continuous global symmetry also a Goldstone boson appears in the spectrum. This pseudo-scalar Goldstone boson of the lepton number symmetry has been proposed a long time ago and was dubbed the majoron [@Chikashige:1980ui; @Schechter:1981cv]. The majoron is obviously intimately connected and coupled to Majorana neutrinos, but at loop-level also receives couplings to the other SM particles. This makes it a simple renormalizable example of an axion-like particle (ALP), defined essentially as a light pseudo-scalar with an approximate shift symmetry. Although not our focus here, by coupling the majoron to quarks it is even possible to identify it with the QCD axion [@Mohapatra:1982tc; @Langacker:1986rj; @Ballesteros:2016euj; @Ballesteros:2016xej], thus solving the strong CP problem dynamically. The main appeal of the majoron ALP is that its couplings are not free but rather specified by the seesaw parameters, which opens up the possibility to reconstruct the seesaw Lagrangian by measuring the majoron couplings [@Garcia-Cely:2017oco]. This is aided by the fact that the loop-induced effective operators that couple the majoron to the SM are only suppressed by one power of the lepton-number breaking scale $\Lambda$, whereas right-handed neutrino induced operators without majorons are necessarily suppressed by $1/\Lambda^2$ [@Broncano:2002rw; @Broncano:2003fq; @Broncano:2004tz; @Cirigliano:2005ck; @Abada:2007ux; @Gavela:2009cd; @Coy:2018bxr], rendering it difficult to reconstruct the seesaw parameters in that way. In this article we complete the program that was started in the inaugural majoron article [@Chikashige:1980ui] and derive all majoron couplings to SM particles. The tree-level and one-loop couplings have been obtained a long time ago; here we go to the two-loop level in order to calculate the remaining couplings, which include the phenomenologically important couplings to photons as well as to quarks of different generations. Armed with this complete set of couplings we then discuss various phenomenological consequences and constraints on the parameters. This includes a discussion of majorons as dark matter (DM). The rest of this article is structured as follows: in Sec. \[sec:tree\_and\_one\_loop\] we introduce the singlet majoron model and reproduce the known tree-level and one-loop couplings. In Sec. \[sec:two\_loop\] we present results of our novel two-loop calculations necessary for the majoron couplings to gauge bosons and to quarks of different generations. The phenomenological aspects of all these couplings are discussed in Sec. \[sec:pheno\]. Finally, we conclude in Sec. \[sec:conclusion\]. Majoron couplings at tree level and one loop {#sec:tree_and_one_loop} ============================================ In this article we consider the minimal singlet majoron model [@Chikashige:1980ui], $$\begin{aligned} \L = -\overline{L} y N_R H -\tfrac12\overline{N}_R^c \lambda N_R \sigma +{\ensuremath{\text{h.c.}}}- \text{V}(H,\sigma), \label{eq:lagrangian}\end{aligned}$$ which introduces three right-handed neutrinos $N_R$ coupled to the SM lepton doublets $L$ and Higgs doublet $H$, and one SM singlet complex scalar $\sigma$ carrying lepton number $L = -2$, minimally coupled to the right-handed neutrinos proportional to the Yukawa matrices $y$ and $\lambda$. We do not specify the scalar potential $\text{V}(H,\sigma)$ but simply assume that $\sigma = (f+\sigma^0 + \i J)/\sqrt{2}$ obtains a vacuum expectation value $f$, which then gives rise to the right-handed Majorana mass matrix $M_R = f \lambda/\sqrt2$. $J$ is the majoron, $\sigma^0$ is a massive CP-even scalar with mass around $f$, assumed to be inaccessibly heavy in the following. Both $M_R$ and the charged-lepton mass matrix are chosen to be diagonal without loss of generality, effectively shifting all mixing parameters into $y$. Electroweak symmetry breaking via $\langle H\rangle = (v/\sqrt2,0)^T$ yields the Dirac mass matrix $M_D = y v/\sqrt2$. The full $6\times 6$ neutrino mass matrix in the basis $(\nu_L^c, N_R) = V n_R$ is then $$\begin{aligned} \begin{split} \L &= -\frac12 \bar{n}_R^c V^T {\begin{pmatrix} 0 & M_D \\ M_D^T & M_R \end{pmatrix}} V n_R +{\ensuremath{\text{h.c.}}}\\ &\equiv-\frac12 \bar{n}_R^c M_n n_R +{\ensuremath{\text{h.c.}}}\,, \end{split} \label{eq:neutrino_mass_matrix}\end{aligned}$$ where $V$ is the unitary $6\times 6$ mixing matrix to the states $n_R$, which form the Majorana mass eigenstates $n = n_R + n_R^c$. The diagonal mass matrix $M_n = {\text{diag}}(m_1,\dots, m_6)$ consists of the physical neutrino masses arranged in ascending order. Throughout this article, we denote mass matrices with capital letters $M_x$ and individual mass eigenvalues with small letters $m_i$. In the mass eigenstate basis, the tree-level neutrino couplings to $J$, $Z$, $W^-$, and $h$, take the form [@Pilaftsis:1993af; @Pilaftsis:2008qt] $$\begin{aligned} \begin{split} \L_J &= -\frac{\i J }{2f}\sum_{i,j=1}^6 \overline{n}_i \left[C_{ij} (m_i P_L - m_j P_R) \right.\\ &\qquad\qquad\qquad \left.+ C_{ji}(m_j P_L - m_i P_R) + \delta_{ij} \gamma_5 m_i\right] n_j \,,\nonumber \end{split}\\ \L_Z &= \frac{g_w}{4 \cos\theta_W} \sum_{i,j=1}^6\overline{n}_i \slashed{Z}\left[ C_{ij} P_L - C_{ji} P_R\right] n_j\,, \label{eq:Jnunu}\\ \L_W &= \frac{g_w}{\sqrt2} \sum_{j=1}^6\sum_{\alpha=1}^3 \left( \overline{\ell}_\alpha B_{\alpha j}\slashed{W}^- P_L n_j + \overline{n}_jB_{\alpha j}^* \slashed{W}^+ P_L \ell_\alpha\right) ,\nonumber\\ \begin{split} \L_h &= -\frac{h }{2 v}\sum_{i,j=1}^6 \overline{n}_i \left[ C_{ij} (m_i P_L + m_j P_R) \right.\\ &\qquad\qquad\qquad \left.+ C_{ji}(m_j P_L + m_i P_R) \right] n_j \,,\nonumber \end{split}\end{aligned}$$ where $g_w = e/\sin\theta_W$ with Weinberg angle $\theta_W$ and $$\begin{aligned} C_{ij} \equiv \sum_{k=1}^3 V_{ki} V^_{kj}\,, && B_{\alpha j} &\equiv \sum_{k=1}^{3} V_{\alpha k}^\ell V_{k j}^= V_{\alpha j}^* \,.\end{aligned}$$ In the last equation we used $V^\ell_{\alpha k} ={{1\hspace{-0.87ex}1}}_{\alpha k}$ since we work in the basis where the charged-lepton mass matrix is diagonal. The $6\times 6$ matrix $C$ and the $3\times 6$ matrix $B$ satisfy a number of identities [@Pilaftsis:1991ug; @Pilaftsis:1992st] that are particularly important in order to establish ultraviolet (UV) finiteness of amplitudes involving neutrino loops: $$\begin{gathered} \nonumber C = C^\dagger= C C\,,\\ \begin{aligned} B B^\dagger &= {{1\hspace{-0.87ex}1}}\,,& C M_n C^T &= 0\,,\\ B^\dagger B &= C\,,& B M_n C^T &= 0\,,\\ B C &= B\,, & B M_n B^T &= 0\,.\\ \end{aligned}\label{eq:BandCidentities}\end{gathered}$$ So far we have not made any assumption about the scale of $M_R$. In the following we will work in the seesaw limit $M_D \ll M_R$, resulting in a split neutrino spectrum with three heavy neutrinos with mass matrix $M_R$ and three light neutrinos with seesaw mass matrix $M_\nu \simeq -M_D M_R^{-1} M_D^T$, naturally suppressed compared to the electroweak scale $v$. This hierarchy permits an expansion of all relevant matrices in terms of the small $3\times 3$ matrix $A\equiv U^\dagger M_D M_R^{-1}$, where $U$ is the unitary $3\times 3$ Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix. Parametrically this corresponds to an expansion in the scale hierarchy $v\ll f$ which we refer to as the seesaw expansion. To leading order, the matrices take the form $$\begin{aligned} \begin{split} C &\simeq {\begin{pmatrix} {{1\hspace{-0.87ex}1}}- A A^\dagger & A\\ A^\dagger & A^\dagger A \end{pmatrix}},\\ B &\simeq {\begin{pmatrix} U({{1\hspace{-0.87ex}1}}- \tfrac12 A A^\dagger ) & U A \end{pmatrix}},\\ M_n &\simeq {\begin{pmatrix} -A M_R A^T & 0 \\ 0 & M_R \end{pmatrix}} . \end{split} \label{eq:seesaw_expansion}\end{aligned}$$ Note that $A M_R A^T$ is diagonal, which imposes constraints on $A$ and provides an implicit definition of $U$. These constraints may be automatically satisfied using the Casas–Ibarra parametrization [@Casas:2001sr]; however, more useful for our purpose is the Davidson–Ibarra parametrization [@Davidson:2001zk], which uses $M_\nu = -A M_R A^T$ and $M_D M_D^\dagger$ as the independent matrices containing all seesaw parameters. Since $M_\nu$ is essentially already fixed by neutrino oscillation experiments (modulo the phases, hierarchy, and overall mass scale), the next step is to experimentally determine $M_D M_D^\dagger$. As we will see, this could in principle be achieved by measuring majoron couplings without ever observing the heavy right-handed neutrinos. To this effect let us point out some interesting properties of the hermitian matrix $M_D M_D^\dagger$ [@Garcia-Cely:2017oco]: its determinant is simply $\det M_D M_D^\dagger =\det M_n = \prod_{j=1}^6 m_j$, which is strictly positive in the model at hand even if one of the light neutrinos were massless at tree level [@Davidson:2006tg]. Thus, $M_D M_D^\dagger$ is positive definite, which yields a chain of inequalities for the off-diagonal entries $(M_D M_D^\dagger)_{ij}$, $i\neq j$ (see e.g. Ref. [@Matrix_Analysis]): $$\begin{aligned} \begin{split} \|(M_D M_D^\dagger)_{ij}\| & < \sqrt{(M_D M_D^\dagger)_{ii}(M_D M_D^\dagger)_{jj}}\\ & \leq \frac{(M_D M_D^\dagger)_{ii}+(M_D M_D^\dagger)_{jj}}{2}\\ & \leq \frac{1}{2} \, {\text{tr}}(M_D M_D^\dagger) \,. \end{split} \label{eq:inequality}\end{aligned}$$ This provides a useful way to constrain magnitudes of the elements of $M_D M_D^\dagger$ since its trace appears in many couplings of the majoron. From Eq. all loop-induced majoron couplings are necessarily proportional to $1/f$. But many couplings contain additional powers of $M_R^{-1}\propto 1/f$, which makes them higher order in the seesaw expansion. We will neglect these suppressed couplings and focus on those that are down by only one power of $1/f$. For the sake of generality, we determine the majoron couplings assuming an explicit shift-symmetry-breaking majoron mass term $-\frac{1}{2}m_J^2 J^2$, making $J$ a pseudo-Goldstone boson. This mass could be explicit [@Gu:2010ys; @Frigerio:2011in] or arise from quantum-gravity effects [@Akhmedov:1992hi; @Rothstein:1992rh; @Alonso:2017avz]. Neutrino couplings ------------------ By inserting Eq. into Eq. , the tree-level majoron coupling to the light active Majorana neutrinos in the seesaw limit is $$\begin{aligned} \L_J &= \frac{\i J }{2f}\sum_{j=1}^3 m_j \overline{n}_j\gamma_5 n_j\,.\end{aligned}$$ These diagonal majoron couplings to neutrinos are formally second order in the seesaw expansion since $m_{1,2,3}/f \sim M_D^2/(M_R f)\sim (v/f)^2$. The omitted off-diagonal $Jn_i n_j$ couplings are determined by the matrix $A A^\dagger A M_R A^T/f \sim (v/f)^3$ which are further suppressed, and lead to irrelevantly slow active-neutrino decays $n_i \to n_j J$ [@Schechter:1981cv]. Assuming for simplicity $m_J \gg m_{1,2,3}$, the majoron’s partial decay rate into light neutrinos is $$\begin{aligned} \Gamma (J\to \nu\nu) = \frac{m_J}{16\pi f^2}\sum_{j=1}^3 m_j^2 \,. \label{eq:J_to_nunu}\end{aligned}$$ For sufficiently large $f$ the majoron becomes a long-lived DM candidate [@Rothstein:1992rh; @Berezinsky:1993fm; @Lattanzi:2007ux; @Bazzocchi:2008fh; @Frigerio:2011in; @Lattanzi:2013uza; @Queiroz:2014yna; @Wang:2016vfj], discussed in Sec. \[sec:majoron\_dark\_matter\]. As mentioned earlier, the majoron couplings to all other SM particles are leading order in the seesaw expansion, i.e. proportional to $1/f$, and may easily dominate the phenomenology despite the additional loop suppression [@Garcia-Cely:2017oco]. Therefore, a thorough discussion of the majoron requires knowledge of all loop-induced couplings that are leading order in the seesaw expansion. Using the tree-level couplings of Eq. we calculate the loop-induced majoron couplings to the rest of the SM particles and provide them below. Charged fermion couplings ------------------------- ![ Loop-induced majoron couplings to charged fermions with the Majorana neutrino mass eigenstates $n_i$ running in the loops. []{data-label="fig:Majoron_fermion_coupling"}](Jqq-one-loop){width="0.75\columnwidth"} The leading order couplings to charged fermions are obtained from the one-loop diagrams in Fig. \[fig:Majoron\_fermion\_coupling\]. These were calculated long ago, both in the one-generation case [@Chikashige:1980ui] and in the three-generation case, which leads to off-diagonal majoron couplings to leptons [@Pilaftsis:1993af]. At leading order in the seesaw expansion, these couplings take a simple form [@Garcia-Cely:2017oco], with (diagonal) quark couplings $$\begin{aligned} \hspace{-1ex}\L_{Jqq} = \frac{ \i J }{16\pi^2 v^2 f} {\text{tr}}(M_D M_D^\dagger) \left( \bar{d} M_d \gamma_5 d-\bar{u} M_u \gamma_5 u \right), \label{eq:Jqq}\end{aligned}$$ and charged lepton couplings $$\begin{aligned} \begin{split} \L_{J\ell\ell} &= \frac{ \i J }{16\pi^2 v^2 f} \bar{\ell} \left(M_\ell \, {\text{tr}}(M_D M_D^\dagger) \gamma_5 \right.\\ &\left.\quad +2 M_\ell M_D M_D^\dagger P_L - 2 M_D M_D^\dagger M_\ell P_R \right)\ell \,, \end{split} \label{eq:Jellell}\end{aligned}$$ where $M_{\ell,u,d}$ denote the diagonal mass matrices of the appropriate SM fermions. In addition to exhibiting decoupling in the seesaw limit $M_R\sim f \to\infty$, these couplings vanish in the electroweak symmetric limit $v\to 0$ as expected since $J$ is an electroweak singlet. The quark couplings can be used to derive the majoron couplings to nucleons $N=(p,n)^T$, using the values from Ref. [@diCortona:2015ldu]: $$\begin{aligned} \hspace{-1ex}\L_{JNN} \simeq \frac{ \i J \,{\text{tr}}(M_D M_D^\dagger)}{16\pi^2 v^2 f} \bar{N}{\begin{pmatrix} -1.30 m_p & 0\\0 & 1.24 m_n \end{pmatrix}} \gamma_5 N \,. \label{eq:Jpp}\end{aligned}$$ At this point let us make some remarks about CP violation. Already in the one-loop processes above one encounters loop-induced majoron mixing with the Brout–Englert–Higgs boson $h$, which would result in majoron couplings to the scalar bilinear $\bar{f}f$ as opposed to the pseudo-scalar $\bar{f}\i \gamma_5 f$. It was noted in Ref. [@Pilaftsis:1993af] that the relevant $J$–$h$ mixing diagrams vanish for $m_J =0$. For $m_J\neq 0$ the $J$–$h$ amplitude is of order $(v/f)^2$ in seesaw and hence negligible. This can be understood by noting that CP-violating phases in the Davidson–Ibarra parametrization reside both in the active-neutrino mass matrix $M_\nu = -A M_R A^T$ and in the off-diagonal entries of the hermitian matrix $M_D M_D^\dagger$, each containing three complex phases [@Davidson:2001zk]. CP-violating majoron couplings via $M_\nu$ are unavoidably suppressed by $M_\nu/f\sim (v/f)^2$, leaving only $M_D M_D^\dagger$ as a potential source. However, closing lepton loops implies an amplitude dependence on ${\text{tr}}(M_D M_D^\dagger g(M_\ell))$, with some function $g(M_\ell)$ of the charged-lepton mass matrix. Since the latter is diagonal, the trace depends only on the real diagonal entries of $M_D M_D^\dagger$, resulting in an effectively CP-conserving amplitude. The CP phases of $M_D M_D^\dagger$ thus only appear in the off-diagonal majoron couplings to leptons, at least to lowest order in the seesaw expansion. Couplings to gauge bosons ------------------------- ![ Loop-induced majoron couplings to $WW$ and $ZZ$ at one loop. These diagrams generate couplings that are subdominant in the seesaw expansion.[]{data-label="fig:JVV_1loop"}](JVV-one-loop){width="\columnwidth"} At one loop order, the only non-vanishing couplings to gauge bosons are to $WW$ and $ZZ$, with typical diagrams shown in Fig. \[fig:JVV\_1loop\]. However, they are higher order in the seesaw expansion, which can be understood as follows: the amplitudes come with a factor of $M_D M_D^\dagger/f$ in order to achieve the necessary $N_R$–$\nu_L$ mixing to close the loop; on dimensional grounds there is an additional $M_R^{-2}$ suppression since this is the only high mass scale in the loop. Explicit one-loop formulae can be found in Refs. [@Latosinski:2012qj; @Latosinski:2012ha]. The leading seesaw behavior for coupling to gauge bosons without the $M_R^{-2}$ suppression starts at two-loop order. At this point it is appropriate to discuss the connection to anomalies in the minimal majoron model. There is some confusion in the literature regarding the question of whether the majoron is the Goldstone boson of the anomaly-free $U(1)_{B-L}$ or the anomalous $U(1)_L$. Both choices seem equally valid because baryon number remains unaltered by the Lagrangian in Eq. . Since according to common lore Goldstone couplings to gauge bosons are determined by the anomaly structure of the theory, this leads to a paradox when attempting to guess the form of majoron couplings to $W$ and $Z$. The resolution was recently presented in Ref. [@Quevillon:2019zrd], where it was explained that Goldstone couplings to gauge bosons are driven entirely by non-anomalous processes. Anomalies still serve as a useful bookkeeping device for the couplings to vector-like gauge bosons such as gluons and photons, but fail for chiral gauge bosons. Disregarding anomalies it is then necessary to calculate Goldstone couplings to gauge bosons in perturbation theory, the results of which we present in the next section. Additionally, we emphasize that the non-vanishing majoron coupling to electroweak gauge bosons (Eqs. , and below) does not lead to nonperturbative violation of the shift symmetry beyond $m_J$ due to the absence of electroweak instantons without $B+L$ violation [@Anselm:1992yz; @Anselm:1993uj; @Perez:2014fja]. Therefore, electroweak instantons cannot generate majoron mass. Majoron couplings at two loops {#sec:two_loop} ============================== In this section we present majoron couplings for which the leading seesaw behavior arises at two loops. To automate the evaluation of some $\mathcal{O}(100)$ Feynman diagrams contributing to each effective coupling we implemented this model in Feynman gauge including all Goldstone bosons [@Pilaftsis:2008qt] in `FeynRules` [@Alloul:2013bka], and generated the necessary amplitudes with `FeynArts` [@Hahn:2000kx]. We validated our implementation by reproducing the tree-level and one-loop couplings above. The Feynman diagrams naturally divide into two sets (see Figs. \[fig:JVVdiagrams\] and \[fig:Jds\] below). Set I diagrams contain the one-particle irreducible (1PI) two-loop diagrams. Set II diagrams contain the reducible diagrams which are dominated by $J$–$Z$ mixing. We used an in-house Mathematica implementation of expansion by regions as described in Ref. [@Smirnov:1999bza] to carry out a double asymptotic expansion $m_{4,5,6}\rightarrow\infty$, $m_{1,2,3}\rightarrow0$ of the two-loop vertex integrals, and to algebraically reduce the one-loop [@Passarino:1978jh] and two-loop [@Davydychev:1995nq] tensor integrals in dimensional regularization. We treated $\gamma_5$ naively, such that it anti-commutes with all other Dirac matrices while also preserving cyclic property of traces. Finally, after expanding around four spacetime dimensions, we summed over fermion generations to extract the leading seesaw behavior of each majoron coupling. We found the couplings to be expressible as simple sums of one-loop functions and rational terms. In principle, two-loop self-energy and vacuum integrals may be present, but they cancel away in the course of reduction, leaving behind rational terms. We have checked our results by confirming that all amplitudes are proportional to the expected tensor structures, are UV finite upon using the relations of Eq. , and have the expected limiting low-energy/small-mass behavior. Additionally, we confirmed that our results are insensitive to the treatment of $\gamma_5$ by reevaluating them in several different ways, including projecting the integrals onto form factors and also starting from cyclically reordered Dirac traces, and finding the same answer upon expanding around four spacetime dimensions. We pause to comment on how we quote our results for couplings to gauge bosons $\{VV'\} = \{gg, \gamma\gamma, Z\gamma, ZZ, W^+W^-\}$. We phrase our results in terms of on-shell decay amplitudes $$\begin{gathered} \mathcal{M}(J\rightarrow V(k_1) V'(k_2)) = \\ - g_{JVV'} \epsilon^{\mu\nu\rho\sigma} \epsilon^_{\mu}(k_1)\epsilon^_{\nu}(k_2) k_{1,\rho} k_{2,\sigma} \,.\end{gathered}$$ It is commonplace to see these amplitudes interpreted as effective couplings as they appear to match onto EFT operators of the form [@Brivio:2017ije] $$\label{eq:naiveJVVoperator} \mathcal{L} = -\frac{g_{JVV'}}{4}J V^{\mu\nu}\tilde{V}'_{\mu\nu}\,,$$ where $V^{\mu\nu}$ is the appropriate field-strength tensor and $\tilde{V}^{\mu\nu}$ its dual. However, we caution the reader that the identification with effective couplings in this way is somewhat clumsy for the following reasons. Firstly, matching onto local operators should be carried out for off-shell Green functions which have been expanded in the external momenta. Secondly, our effective couplings $g_{JVV'}$ cannot be viewed in a Wilsonian sense, since degrees of freedom lighter than the majoron contribute in certain mass ranges, nor can it be viewed in the 1PI sense since the couplings include Set II diagrams which are not one-particle irreducible. Therefore, interpreting our results as coefficients of effective operators should be done with care. Coupling to two gluons ---------------------- ![ Representative two-loop diagrams contributing to loop-induced majoron couplings to vector bosons $\{VV'\} = \{gg, \gamma\gamma, Z\gamma, ZZ, W^+W^-\}$. Set I contains the two-loop 1PI diagrams and Set II contains reducible diagrams dominated by $J$–$Z$ mixing. Here, $n_i$ and $n_j$ are Majorana neutrino mass eigenstates and $f$ SM fermions (not necessarily all identical).[]{data-label="fig:JVVdiagrams"}](JVV-two-loop){width="0.80\columnwidth"} Assuming a sufficiently heavy majoron $m_J \gtrsim \Lambda_\text{QCD}$, the coupling to free gluons comes entirely from $J$–$Z$ mixing diagrams of Set II in Fig. \[fig:JVVdiagrams\]. A straightforward evaluation of the decay amplitude $J\to gg$ at leading order in the seesaw expansion yields the simple expression $$\begin{aligned} g_{J g g} = \frac{\alpha_S}{16 \pi ^3 v^2 f} {\text{tr}}(M_D M_D^\dagger)\sum_{q=u,d} T_3^q \ h\Big(\frac{m_J^2}{4 m_q^2}\Big)\,, \end{aligned}$$ with $T_3^u = -T_3^d = 1/2$ and the loop function $$\begin{aligned} \begin{split} h(x) &\equiv -\frac{1}{4 x} \big(\log [1-2 x + 2\sqrt{x (x-1)}]\big)^2 -1\\ & = \begin{cases} \frac{x}{3} + \frac{8x^2}{45} + \frac{4 x^3}{35}+\mathcal{O}(x^4)\,, & x\to0\,,\\ -1 + \frac{(\pi +\i \log (4 x))^2}{4 x}+\mathcal{O}(x^{-2})\,, & x\to\infty\,. \end{cases} \end{split}\end{aligned}$$ For small $m_J$, the amplitude vanishes as $g_{J g g} \sim m_J^2$ and indicates that at leading order in the derivative expansion the amplitude matches onto an operator $(\partial^2J) G^{a \mu\nu}\tilde{G}^a_{\mu\nu}$ instead of $J G^{a \mu\nu} \tilde{G}^{a}_{\mu\nu}$ as in Eq. . This implies that the majoron does not solve the strong CP problem, as this operator is insensitive to a constant shift $J\to J + c$ that could otherwise be used to cancel the strong CP $\theta$ term [@Quevillon:2019zrd]. Furthermore, contrary to the claim in Ref. [@Latosinski:2012qj], majorons without tree-level couplings to quarks cannot solve the strong CP problem even at higher loop order [@Latosinski:2015pba; @Nakawaki:2018cwk]. Coupling to two photons {#sec:diphoton} ----------------------- The majoron coupling to photons at two-loop level receives contributions from both sets of Feynman diagrams in Fig. \[fig:JVVdiagrams\], $$g_{J\gamma\gamma} = g_{J\gamma\gamma}^\text{I} + g_{J\gamma\gamma}^\text{II}\,,$$ and yields the partial decay rate into two photons $$\begin{aligned} \Gamma (J\to\gamma\gamma) = \frac{\|g_{J\gamma\gamma}\|^2 m_J^3}{64\pi} \,.\end{aligned}$$ The contributions from Set II were calculated in Ref. [@Garcia-Cely:2017oco] with the result $$\begin{aligned} \hspace{-1.7ex} g_{J\gamma\gamma}^\text{II} = \frac{\alpha}{8 \pi ^3 v^2 f} {\text{tr}}(M_D M_D^\dagger) \sum_f N_c^f Q_f^2 T_3^f \ h\Big(\frac{m_J^2}{4 m_f^2}\Big) \,, \label{eq:JgammagammaSet1} \end{aligned}$$ already simplified with the help of the electroweak anomaly cancellation condition $\sum_f N_c^f Q_f^2 T_3^f=0$. Here, $ N_c^{u,d} = 3= 3 N_c^\ell$ is the number of colors, $T_3^{u}=1/2 = - T_3^{d,\ell}$ the isospin, and $\{Q_\ell, Q_d, Q_u\}=\{-1,-1/3,+2/3\}$ the electric charge in units of $e=\sqrt{4\pi \alpha}$. We complete the evaluation of $g_{J\gamma\gamma}$ here by computing the additional contributions arising from Set I diagrams, which give $$\begin{aligned} g_{J\gamma\gamma}^\text{I} = \frac{\alpha}{8 \pi ^3 v^2 f} \sum_\ell (M_D M_D^\dagger)_{\ell\ell} \ h\Big(\frac{m_J^2}{4 m_\ell^2}\Big) \,. \end{aligned}$$ Just as for the gluon coupling, the amplitude vanishes as $g_{J\gamma\gamma}\sim m_J^2$ for small majoron masses, implying that the leading effective operator this amplitude matches onto in the derivative expansion is $(\partial^2 J) F_{\mu\nu} \tilde{F}^{\mu\nu}$ rather than the typically occurring $J F_{\mu\nu} \tilde{F}^{\mu\nu}$. For $m_J \ll m_f$ we can relate the total majoron-photon coupling $g_{J\gamma\gamma}$ to the dimensionless diagonal fermion couplings of Eqs. –, $g_{Jff}\,J\bar{f}\i\gamma_5 f$, as $$\begin{aligned} g_{J\gamma\gamma}&\simeq -\frac{\alpha m_J^2}{12 \pi } \sum_f N_c^f Q_f^2 \frac{ g_{Jff}}{m_f^3} \,, \end{aligned}$$ which agrees with the EFT result of Ref. [@Nakayama:2014cza]. Since the $g_{J f f}$ couplings can have different signs and magnitudes, the $J\gamma\gamma$ coupling for $m_J<m_e$ could be heavily suppressed. The key point and crucial result of this full two-loop calculation is that the $J\gamma\gamma$ coupling has richer structure than anticipated in Ref. [@Garcia-Cely:2017oco] based on the evaluation of $g_{J\gamma\gamma}^\text{II}$ alone. This is illustrated in Fig. \[fig:Jgammagamma\_amplitude\] where we show $\|g_{J\gamma\gamma}\|\times f$ for a variety of hierarchies of the diagonal entries $(M_D M_D^\dagger)_{\ell\ell}$. The SM-fermion mass thresholds together with the different signs in $g_{Jff}$ potentially suppress $g_{J\gamma\gamma}$ by orders of magnitude. The typical size of the coupling for $m_J>\unit{MeV}$ is $\|g_{J\gamma\gamma}\|\sim 10^{-5} f^{-1} (M_D M_D^\dagger)/(\unit[100]{GeV})^2$, simply due to the unavoidable suppression factor $\alpha/(8\pi^3)$. In Fig. \[fig:Jgammagamma\_amplitude\] we used the current-quark masses to evaluate $g_{J\gamma\gamma}$; for $m_J \lesssim \Lambda_\text{QCD}$ they should be replaced by hadronic loops. We have not attempted this, but we refer the interested reader to standard axion literature on the topic [@Georgi:1986df; @Bauer:2017ris; @Alonso-Alvarez:2018irt]. ![image](Jgammagamma.pdf){width="88.00000%"} Coupling to Z and photon ------------------------ Next we present the $Z$-photon coupling $g_{J Z\gamma} = g_{J Z\gamma}^\text{I}+g_{J Z\gamma}^\text{II}$, which receives contributions from Set I and II diagrams in Fig. \[fig:JVVdiagrams\]. The results are $$\begin{aligned} g_{J Z\gamma}^\text{I} &= -\frac{\alpha}{16 \pi^3 c_W s_W v^2 f} \left({\text{tr}}(M_D M_D^\dagger) -(1-4 s_W^2) \sum_\ell (M_D M_D^\dagger)_{\ell\ell} \frac{m_J^2 h\left(\frac{m_J^2}{4 m_\ell^2}\right)-m_Z^2 h\left(\frac{m_Z^2}{4 m_\ell^2}\right)}{m_J^2-m_Z^2} \right) ,\\ g_{J Z\gamma}^\text{II} &= -\frac{\alpha}{16 \pi^3 c_W s_W v^2 f} {\text{tr}}(M_D M_D^\dagger) \sum_f 2 N_c^f Q_f T_3^f (2 Q^f s_W^2-T_3^f) \frac{m_J^2 h\left(\frac{m_J^2}{4 m_f^2}\right)-m_Z^2 h\left(\frac{m_Z^2}{4 m_f^2}\right)}{m_J^2-m_Z^2} \,,\end{aligned}$$ with $c_W \equiv \cos\theta_W$ and $s_W \equiv \sin\theta_W$. We have used $\sum_f N_c^f Q_f^2 T_3^f=0=\sum_f N_c^f Q_f (T_3^f)^2$ to simplify the formula. In the limit $m_J, m_Z \to 0$, the amplitude is non-vanishing, $$\label{eq:JZgammaeffCoup} g_{JZ\gamma} \sim -\frac{\alpha\, {\text{tr}}(M_D M_D^\dagger)}{16\pi^3 c_W s_W v^2 f}\,,$$ and matches onto the effective operator $J Z^{\mu\nu}\tilde{F}_{\mu\nu}$, as in Eq. . Coupling to two Z bosons ------------------------ The majoron coupling $g_{JZZ} = g_{JZZ}^\text{I} + g_{JZZ}^\text{II}$ to two $Z$ bosons receives contributions from Sets I and II: $$\begin{gathered} g_{JZZ}^\text{I} = -\frac{\alpha }{32 \pi ^3 c_W^2 s_W^2 v^2 f}\Big[ \big(1-2 s_W^2\big)^2 {\text{tr}}(M_D M_D^\dagger) -\frac{2}{m_J^2-4 m_Z^2} \sum_\ell (M_D M_D^\dagger)_{\ell\ell} m_\ell^2 \Big(g(\textstyle\frac{m_J^2}{4 m_\ell^2})-g(\textstyle\frac{m_Z^2}{4 m_\ell^2})\\ +\Big( 2 s_W^2(1-2 s_W^2)m_J^2 + (1-4s_W^2)^2m_Z^2\Big) C_0(m_J^2,m_Z^2,m_Z^2,m_\ell,m_\ell,m_\ell)\Big)\Big]\,.\end{gathered}$$ $$\begin{gathered} g_{JZZ}^\text{II} = \frac{\alpha}{4 \pi ^3 c_W^2 s_W^2 v^2 f}{\text{tr}}(M_D M_D^\dagger) \frac{1}{m_J^2-4 m_Z^2} \sum_f N_c^f T_3^f m_f^2 \Big[\big(T_3^f\big)^2\big(g(\textstyle\frac{m_J^2}{4 m_f^2})-g(\textstyle\frac{m_Z^2}{4 m_f^2})\big) \\ + \Big(s_W^2 Q_f \big(T_3^f - Q_f s_W^2\big) m_J^2 + \big(T_3^f-2 Q_f s_W^2\big)^2 m_Z^2 \Big)C_0(m_J^2,m_Z^2,m_Z^2,m_f,m_f,m_f) \Big]\,,\end{gathered}$$ where $$\begin{aligned} g(x) &\equiv \sqrt{\textstyle 1-\frac{1}{x}} \log[1-2x+2\sqrt{x(x-1)}]\end{aligned}$$ and $C_0$ is the scalar three-point Passarino–Veltman function. Despite the appearance of $(m_J^2 - 4m_Z^2)^{-1}$, the amplitude is regular at threshold $m_J\to 2 m_Z$. The coupling $g_{JZZ}$ is nonvanishing in the limit $m_J, m_Z \to 0$, $$\label{eq:JZZeffCoup} g_{JZZ} \sim -(1-3 s_W^2) \frac{\alpha\, {\text{tr}}(M_D M_D^\dagger)}{48 \pi ^3 c_W^2 s_W^2 v^2 f} \,,$$ and matches onto $J Z^{\mu\nu}\tilde{Z}_{\mu\nu}$. Coupling to two W bosons ------------------------ Finally, we present results for the two-loop amplitude for $J\to W^+ W^-$ in order to extract the coupling $g_{JWW}$, which receives contributions from diagrams in Set I and Set II $$g_{JWW} = g_{JWW}^\text{I} + g_{JWW}^\text{II,$\ell$} + g_{JWW}^\text{II,$d$} + g_{JWW}^\text{II,$u$}\,,$$ where we have separated the Set II $J$–$Z$ mixing contributions based on the type of SM fermions running in the loop. The Set I diagrams give $$\begin{aligned} \begin{split} g_{JWW}^\text{I} &= \frac{\alpha }{64 \pi^3 s_W^2 v^2 f}\,\frac{1}{m_J^2-4 m_W^2} \sum_\ell \frac{(M_D M_D^\dagger)_{\ell\ell}}{m_W^2} \left\{ 4 m_W^4- m_J^2 m_W^2 + 4 m_\ell^2 m_W^2 g\Big(\frac{m_J^2}{4 m_\ell^2}\Big) \right.\\ &\quad \left. + 4 m_\ell^2 (m_W^2-m_\ell^2) \left[ -\log\big(\frac{m_\ell^2}{m_\ell^2-m_W^2}\big) + m_W^2 C_0(m_J^2,m_W^2,m_W^2,m_\ell,m_\ell,0) \right] \right\} , \end{split}\end{aligned}$$ Set II $J$–$Z$ mixing diagrams with two charged leptons in the loop gives $$\begin{aligned} \begin{split} g_{JWW}^\text{II,$\ell$} &= - \frac{\alpha}{32 \pi ^3 s_W^2 v^2 f} \,\frac{{\text{tr}}(M_D M_D^\dagger) }{m_W^2}\frac{1}{m_J^2-4 m_W^2}\sum_\ell m_\ell^2 \left\{m_W^2 g\Big(\frac{m_J^2}{4 m_\ell^2}\Big) \right.\\ &\quad \left.+ (m_W^2-m_\ell^2) \left[ -\log\big(\frac{m_\ell^2}{m_\ell^2-m_W^2}\big) + m_W^2 C_0(m_J^2,m_W^2,m_W^2,m_\ell,m_\ell,0) \right] \right\} , \end{split}\end{aligned}$$ Set II $J$–$Z$ mixing diagrams with two down quarks in the loop gives $$\begin{aligned} \begin{split} g_{JWW}^\text{II,$d$} &= - \frac{3\, \alpha }{32 \pi ^3 s_W^2 v^2 f} \frac{{\text{tr}}(M_D M_D^\dagger)}{m_W^2} \frac{1}{m_J^2-4 m_W^2}\sum_{i,j} \|(V_q)_{ji}\|^2 m_{d_i}^2 \left\{m_W^2 \left[g\Big(\frac{m_J^2}{4 m_{d_i}^2}\Big) - g\Big(\frac{m_W^2}{m_{d_i}^2},\frac{m_W^2}{m_{u_j}^2}\Big)\right] \right.\\ &\quad \left.+ (m_W^2-m_{d_i}^2+m_{u_j}^2) \left[ \log\big(\frac{m_{u_j}}{m_{d_i}}\big) + m_W^2 C_0(m_J^2,m_W^2,m_W^2,m_{d_i},m_{d_i},m_{u_j}) \right] \right\} , \end{split}\end{aligned}$$ and Set II $J$–$Z$ mixing diagrams with two up quarks in the loop gives $$\begin{aligned} \begin{split} g_{JWW}^\text{II,$u$} &= \frac{3\, \alpha}{32 \pi ^3 s_W^2 v^2 f} \frac{{\text{tr}}(M_D M_D^\dagger)}{ m_W^2} \frac{1}{m_J^2-4 m_W^2}\sum_{i,j} \|(V_q)_{ij}\|^2 m_{u_i}^2 \left\{m_W^2 \left[g\Big(\frac{m_J^2}{4 m_{u_i}^2}\Big) - g\Big(\frac{m_W^2}{m_{u_i}^2},\frac{m_W^2}{m_{d_j}^2}\Big)\right] \right.\\ &\quad \left.+ (m_W^2-m_{u_i}^2+m_{d_j}^2) \left[ \log\big(\frac{m_{d_j}}{m_{u_i}}\big) + m_W^2 C_0(m_J^2,m_W^2,m_W^2,m_{u_i},m_{u_i},m_{d_j}) \right] \right\} . \end{split}\end{aligned}$$ Here, $V_q$ is the unitary Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix. In the last two formulae, the two-argument loop function is $$g(x,y) = \sqrt{(x+y-1)^2-4xy}\,\log\left(\frac{x+y-1}{2\sqrt{xy}}+\sqrt{\frac{(x+y-1)^2}{4xy}-1}\right) .$$ The coupling is regular at threshold $m_J\to 2m_W$, and nonvanishing in the limit $m_J, m_W \rightarrow 0$, $$\label{eq:JWWeffCoup} g_{JWW} \sim -\frac{3\alpha \, {\text{tr}}(M_D M_D^\dagger)}{128 \pi ^3 s_W^2 v^2 f}\left[ 1 - \sum_{i,j} \|(V_q)_{ij}\|^2\left( \frac{m_{u_i}^4-m_{d_j}^4+2 m_{d_j}^2 m_{u_i}^2 \log (m_{d_j}^2/m_{u_i}^2)}{(m_{u_i}^2-m_{d_j}^2)^2} \right) \right] ,$$ and matches onto $JW^{+\,\mu\nu}W^-_{\mu\nu}$. Coupling to photon-Higgs and Z-Higgs ------------------------------------ Besides the usually considered pseudo-scalar couplings to two gauge bosons discussed above, CP-invariance also allows couplings of $J$ to $h Z$ and $h\gamma$. The former arises already at one-loop level but is seesaw suppressed; the dominant contributions to both couplings then arise at two-loop level. Due to the large number of diagrams and low phenomenological relevance of decays such as $h\to \gamma J$ compared to the processes derived above we will however not present the results here. Flavor changing quark couplings ------------------------------- ![ Representative two-loop diagrams for off-diagonal majoron couplings to quarks.[]{data-label="fig:Jds"}](Jqq-two-loop){width="0.80\columnwidth"} At the two-loop level we find off-diagonal majoron couplings to quarks, which can lead for example to $s\to d J$ or $K\to \pi J$ at hadron level. Such flavor-changing couplings have long been advocated to search for light bosons and axions [@Frere:1981cc; @Hall:1981bc] and have enjoyed increased attention in recent years [@Izaguirre:2016dfi; @Dolan:2017osp; @Dobrich:2018jyi; @Gavela:2019wzg], partly because of an improved reach at existing and upcoming experiments such as NA62 and Belle II. In the majoron model the relevant flavor-changing quark-level coupling arise at two loops and involve a large number of diagrams, see Fig. \[fig:Jds\]. The leading logarithmic contribution to coupling to down quarks is $$\begin{gathered} \mathcal{L}_{Jdd'} = -\frac{1}{128\pi^4 v^4 f} {\text{tr}}\Big(M_D \log\big(\textstyle\frac{M_R}{m_W}\big) M_D^\dagger\Big) \\ \times (\i J\, \bar d_R M_d V_q^\dagger M_u^2 V_q \, d_L + {\ensuremath{\text{h.c.}}}) \,,\end{gathered}$$ arising from the Set I diagrams in Fig. \[fig:Jds\]. The couplings are of minimal-flavor-violating (MFV) type [@DAmbrosio:2002vsn], as expected from the fact that $J$ is quark-flavor blind. The subleading contribution from the remaining diagrams is given by $$\L_{Jdd'}^\text{sub} = \frac{- {\text{tr}}(M_D M_D^\dagger)}{512\pi^4 v^4 f} (\i J\, \bar d_R M_d V_q^\dagger F_u V_q \, d_L + {\ensuremath{\text{h.c.}}}) \,,$$ where the diagonal matrix $F_u$ has entries $$\begin{aligned} \begin{split} F_u &= \frac{7 m_u^4 + 3 m_u^2 m_W^2 - 8 m_W^4}{m_u^2-m_W^2} +4 m_u^2 \, g\Big(\frac{m_J^2}{4m_u^2}\Big) \\ &\quad +\frac{2m_u^2 (m_u^4 -2 m_u^2 m_W^2 + 2 m_W^4)}{(m_u^2-m_W^2)^2}\, \log\Big(\frac{m_W^2}{m_u^2}\Big)\\ &\quad -4 m_u^2 m_W^2 C_0(0,0,m_J^2,m_u,m_W,m_u) \,. \end{split}\end{aligned}$$ For sub-TeV majoron mass and large-enough right-handed neutrino masses, $\log\left( M_R/m_W\right) \gtrsim 1$ as is assumed in our seesaw expansion, the leading logarithmic contribution $\L_{Jdd'}$ dominates over $\L_{Jdd'}^\text{sub}$, which we neglect in the following for simplicity. The MFV matrix $ M_d V_q^\dagger M_u^2 V_q$ relevant for $\L_{Jdd'}$ makes it clear that off-diagonal terms would vanish if all up-quarks were degenerate, so these terms are necessarily proportional to up-quark mass differences. Numerically this matrix evaluates to $$\begin{aligned} \hspace{-1ex}\left\| \frac{M_d V_q^\dagger M_u^2 V_q}{v^3}\right\| \simeq {\begin{pmatrix} 0 & 0 & 8\times 10^{-8}\\ 7\times 10^{-8} & 3\times 10^{-7} & 8\times 10^{-6}\\ 7\times 10^{-5} & 3\times 10^{-4} & 8\times 10^{-3} \end{pmatrix}} ,\end{aligned}$$ keeping only the largest entries. The biggest amplitude is therefore $b\to s J$, which is however experimentally less clean than $s\to d J$, further discussed in Sec. \[sec:light\_majorons\]. From the dominant flavor-changing down-quark couplings in $\L_{Jdd'}$ we immediately obtain the corresponding flavor-changing up-quark couplings as $$\begin{gathered} \mathcal{L}_{Juu'} = -\frac{1}{128\pi^4 v^4 f} {\text{tr}}\Big(M_D \log\big(\textstyle\frac{M_R}{m_W}\big) M_D^\dagger\Big) \\ \times (\i J \, \bar u_R M_u V_q M_d^2 V_q^\dagger \, u_L + {\ensuremath{\text{h.c.}}}) \,,\end{gathered}$$ with the markedly smaller MFV coupling matrix $$\begin{aligned} \hspace{-2.5ex}\left\| \frac{M_u V_q M_d^2 V_q^\dagger}{v^3}\right\| \simeq {\begin{pmatrix} 0 & 0 & 0\\ 3\!\times\!10^{-10} & 3\times 10^{-9} & 6\times 10^{-8} \\ 7\times 10^{-7} & 8\times 10^{-6} & 2\times 10^{-4} \end{pmatrix}} .\end{aligned}$$ Taken together with the weaker experimental limits on $u\to u' J$ we can ignore these couplings in practice. From the point of view of the seesaw expansion, the quark-flavor changing couplings are actually the dominant majoron couplings. They only decouple as $\log (M_R)/f$, whereas all other couplings decouple at least as $1/f$. It was noted before that Goldstone bosons with effective diagonal couplings $J m_q \bar{q}\i\gamma_5 q$ yield flavor-changing quark couplings at one loop that depend logarithmically on the UV scale [@Freytsis:2009ct; @Batell:2009jf], whereas an initial coupling $J W_{\mu\nu}\tilde W^{\mu\nu}$ does not have such a dependence [@Izaguirre:2016dfi]. In our case $J W_{\mu\nu}\tilde W^{\mu\nu}$ gives only a seesaw-suppressed contribution to $Jqq'$ and the $\log (M_R)$ terms originate from an effective coupling to Goldstone bosons, $JG^+G^-$. Phenomenology {#sec:pheno} ============= Having obtained all majoron couplings to leading order in the seesaw expansion we can discuss existing constraints and signatures. Light majorons {#sec:light_majorons} -------------- We start with the simplest case of a massless majoron, which most importantly gives a vanishing coupling to photons. It proves convenient to phrase our discussion in terms of the dimensionless parameters $$\begin{aligned} K_{\alpha\beta} \equiv \frac{(M_D M_D^\dagger)_{\alpha\beta}}{v f} \,,\end{aligned}$$ as they capture the majoron couplings in most cases [@Garcia-Cely:2017oco]. The off-diagonal entries of $M_D M_D^\dagger$ are directly constrained by the lepton flavor violating (LFV) decays $\ell \to \ell' J$ [@Pilaftsis:1993af; @Feng:1997tn; @Hirsch:2009ee]. For $m_{\ell'} \ll m_\ell$, the partial widths read $$\begin{aligned} \frac{\Gamma (\ell \to \ell' J)}{\Gamma (\ell \to \ell' \nu_\ell \bar\nu_{\ell'})} \simeq \frac{3}{16 \pi^2}\frac{v^2}{m_\ell^2}\, \|K_{\ell\ell'}\|^2\end{aligned}$$ and involve a left-handed final-state lepton, leading to an anisotropic decay [@Garcia-Cely:2017oco]. The constraints in the tau sector are ${\text{Br}}(\tau \to \ell J)<\mathcal{O}(10^{-3})$ [@Albrecht:1995ht] and lead to [@Garcia-Cely:2017oco] $$\begin{aligned} \|K_{\tau e}\|< 6\times 10^{-3}\,, && \|K_{\tau \mu}\| < 9\times 10^{-3}\,,\end{aligned}$$ which can be improved by Belle and Belle-II [@Heeck:2016xkh; @Heeck:2017xmg; @Yoshinobu:2017jti]. In the muon sector, the best constraints on a majoron with anisotropic emission come from $\mu\to e J$ [@Bayes:2014lxz] (to be improved with Mu3e [@Perrevoort:2018ttp]) and $\mu\to e J \gamma$ [@Goldman:1987hy]. The latter is also sensitive to $m_J=0$ and provides a limit $$\begin{aligned} \|K_{\mu e}\| < 10^{-5}\,.\end{aligned}$$ The diagonal couplings $K_{\ell\ell}$ of a massless majoron are constrained by astrophysics, stellar cooling in particular, and imply $$\begin{aligned} \|K_{ee}-K_{\mu\mu} -K_{\tau\tau}\| &< 2\times 10^{-5} \,, \\ {\text{tr}}(K) &< 5\times 10^{-6}\,,\end{aligned}$$ from the electron [@Raffelt:1994ry] and nucleon coupling [@Keil:1996ju], respectively. The bound on the trace ${\text{tr}}(K)$ is particularly powerful since it provides upper bounds on all entries of the positive-definite $K$ [@Garcia-Cely:2017oco] by means of the inequality of Eq. . This then puts an upper bound on the majoron coupling to muons and taus which is far better than any direct bound on these couplings. It also ensures that rare decays such as $K\to \pi J$ and $Z\to \gamma J$ [@Bauer:2017ris; @Craig:2018kne; @Bauer:2018uxu; @Alonso-Alvarez:2018irt] are unobservably suppressed for a massless majoron. Two-loop couplings are hence irrelevant for massless majorons. Overall we see that a massless majoron gives seesaw-parameter constraints of order $M_D M_D^\dagger/(v f) \lesssim 10^{-5}$–$10^{-6}$. While this is far off the “natural” value $M_D M_D^\dagger/(v f)\sim M_\nu/v \sim 10^{-13}$, it can be realized by assuming certain matrix structures in $M_D$ that suppress $M_\nu \simeq -M_D M_R^{-1} M_D^T$ but not $M_D M_D^\dagger$, to be discussed in more detail in Sec. \[sec:comparison\_to\_seesaw\]. As we have seen, the relevant couplings of a massless majoron are those to nucleons and electrons, but even $\mu\to e J$ could be observable. ![ Upper limits on combinations of $K_{\alpha\beta}= (M_D M_D^\dagger)_{\alpha\beta}/(v f) $ for majoron masses above MeV. The shaded regions exclude $\|K_{\alpha\beta}\|$ or ${\text{tr}}(K)$ by non-observed rare decays, the dashed lines show the potential future reach, see text for details. $K\to \pi J$ and $B\to KJ$ further scale with $\log (M_R/m_W)$, which has been set to ${{1\hspace{-0.87ex}1}}$ here. The black region is a very naive estimate of SN1987 constraints on the di-photon coupling, setting for simplicity $K_{\ell\ell}={\text{tr}}(K)/3$. The yellow region is the SN1987 constraint on the $JNN$ coupling [@Lee:2018lcj]. The off-diagonal entries have to satisfy $\|K_{\alpha\beta} \| < {\text{tr}}(K)/2$, see Eq. .[]{data-label="fig:light_majoron_constraints"}](light_majoron_constraints.pdf){width="48.00000%"} The phenomenology becomes more interesting for non-zero majoron mass, specifically values above $\sim\unit[10]{keV}$ in order to kinematically evade the stellar cooling constraints. This is shown in Fig. \[fig:light\_majoron\_constraints\] for majoron masses above . In addition to the one-loop lepton-flavor-violating decays that probe $K_{\alpha\beta}$ we now also have relevant constraints from the two-loop quark-flavor-violating decays, esp. $K\to \pi J$ and $B\to K J$ (limits and future sensitivity taken from Ref. [@Gavela:2019wzg], based on Refs. [@Artamonov:2008qb; @Adler:2004hp; @Izaguirre:2016dfi]). These decays probe the quantity ${\text{tr}}(M_D \log (M_R/m_W) M_D^\dagger)$, but to simplify comparison with the other limits we set $\log (M_R/m_W) = {{1\hspace{-0.87ex}1}}$ to obtain a limit on ${\text{tr}}(K)$. It should be kept in mind, however, that a larger log-enhancement can make these rare processes even more relevant. Also potentially relevant are the two-loop majoron couplings to photons via the effective coupling $g_{J\gamma\gamma}$ from Sec. \[sec:diphoton\]. Astrophysical limits on this coupling are extremely strong for $m_J \lesssim \unit[100]{MeV}$ [@Jaeckel:2017tud]; since $g_{J\gamma\gamma}$ is $m_J^2$ suppressed for $m_J \lesssim \unit{MeV} $, the region where the photon coupling is important is between MeV and $\unit[100]{MeV}$. This is unfortunately precisely the mass region where the light quarks that run in the $J$–$\gamma$–$\gamma$ loops should be replaced by hadrons; as a very naive way to incorporate this we simply set $m_u = m_d = m_\pi$ and $m_s = m_K$ in Eq. . The $g_{J\gamma\gamma}$ limit in Fig. \[fig:light\_majoron\_constraints\] should therefore not be taken too seriously. In light of these uncertainties we do not discuss how the $g_{J\gamma\gamma}$ coupling depends on the various diagonal $K_{\ell\ell}$, but rather set them all equal to ${\text{tr}}(K)/3$ to allow a comparison to the other limits. It is clearly possible to suppress $g_{J\gamma\gamma}$ significantly in the region of interest by choosing hierarchical $K_{\ell\ell}$, as shown in Fig. \[fig:Jgammagamma\_amplitude\]. Also illustrated in Fig. \[fig:light\_majoron\_constraints\] are SN1987 constraints on the majoron–nucleon coupling, Eq. , adopted from Ref. [@Lee:2018lcj], which reach up to $m_J\sim \unit[250]{MeV}$ and constrain ${\text{tr}}(K)$ between $5\times 10^{-5}$ and $0.06$. As can be appreciated from Fig. \[fig:light\_majoron\_constraints\], even the strong astrophysical constraints on majorons do not rule out flavor-violating rare decays, with significant experimental progress expected in the near future. Even the two-loop suppressed $d\to d' J$ decays provide meaningful constraints. $\mu\to e J$ is well constrained already and we expect it to eventually become the most sensitive probe of the $K$ matrix entries for $m_J < m_\mu$, even beating out stellar cooling limits. $\tau\to \ell J$ on the other hand is mainly relevant for $m_J$ between $\sim\unit[100]{MeV}$ and $m_\tau$. For smaller $m_J$ the flavor-conserving constraints on $K$ from $g_{J\gamma\gamma}$ and $g_{JNN}$ become stronger, which suppresses the LFV modes via the inequality of Eq. . For $m_J > m_\tau$, the main rare decays are $B\to K J$ and $Z\to J\gamma$ [@Brivio:2017ije], the former is shown in Fig. \[fig:light\_majoron\_constraints\] and the latter gives irrelevant constraints on the $K$ entries of order $10^4$. Comparison with seesaw observables {#sec:comparison_to_seesaw} ---------------------------------- So far we have discussed the interactions of the majoron, but of course the right-handed neutrinos $N_R$ also mediate non-majoron processes, discussed at length in the literature [@Broncano:2002rw; @Broncano:2003fq; @Broncano:2004tz; @Cirigliano:2005ck; @Abada:2007ux; @Gavela:2009cd; @Coy:2018bxr]. Assuming again that the $N_R$ are heavy enough to be integrated out, the relevant dimension-six operators involving SM fields all depend on the matrices $$\begin{aligned} M_D M_R^{-2} M_D^\dagger \,\, \text{ and }\,\, M_D M_R^{-2}\log\left( M_R/m_W\right) M_D^\dagger \,, \label{eq:seesaw_matrices}\end{aligned}$$ which drive LFV processes such as $\ell \to \ell' \gamma$ as well as lepton-universality violating effects such as $\Gamma(Z\to \ell\bar{\ell})/\Gamma(Z\to \ell'\bar{\ell}')$, recently discussed thoroughly in Ref. [@Coy:2018bxr]. In comparison, we have seen above that all majoron operators depend on the matrices $$\begin{aligned} \frac{M_D M_D^\dagger}{f} \,\, \text{ and }\,\, \frac{M_D \log\left( M_R/m_W\right) M_D^\dagger}{f} \,. \label{eq:majoron_matrices}\end{aligned}$$ For $f\sim M_R$, this makes the majoron operators potentially dominant, while $f\gg M_R$ suppresses them to an arbitrary degree [@Garcia-Cely:2017oco]. To properly compare majoron and non-majoron processes it is necessary to pick a structure for $M_D$, which is guided by our experimental reach. ![image](seesaw_limits_etau.pdf){width="48.00000%"} ![image](seesaw_limits_emu.pdf){width="48.00000%"} As we have seen above, even future limits on majoron production cannot reach the natural seesaw scale $M_R\sim\unit[10^{14}]{GeV}$, and the same is true for other $N_R$-mediated processes [@Coy:2018bxr]. By no means does this preclude observable effects, since it is possible to use the matrix structure of $M_D$ to suppress $M_\nu \simeq -M_D M_R^{-1} M_D^T$ while keeping $M_D M_D^\dagger$ large [@Buchmuller:1990du; @Buchmuller:1991tu], potentially realizing a lepton-number symmetry [@Kersten:2007vk] as in the inverse seesaw [@Wyler:1982dd; @Mohapatra:1986bd; @GonzalezGarcia:1988rw]. Following Refs. [@Ingelman:1993ve; @Coy:2018bxr] we can solve $M_\nu =0$ for $M_D$, which then requires only tiny perturbations $\delta M_D$ to produce the observable neutrino masses via $$\begin{aligned} M_\nu \simeq - \delta M_D M_R^{-1} M_D^T- M_D M_R^{-1} (\delta M_D)^T \,.\end{aligned}$$ The key observation is that $\delta M_D$ is negligible in $M_D M_D^\dagger$. $M_\nu =0$ requires the low-scale seesaw structure $$\begin{aligned} M_D = v {\begin{pmatrix} \xi_e \\ \xi_\mu \\ \xi_\tau \end{pmatrix}} {\begin{pmatrix} 1 && z \sqrt{\frac{m_5}{m_4}} && \pm \i \sqrt{1+z^2}\sqrt{\frac{m_6}{m_4}} \end{pmatrix}} , \label{eq:low_scale_texture}\end{aligned}$$ with complex $z$ and real $\xi_{\alpha}$ without loss of generality [@Coy:2018bxr].[^1] This product structure of $M_D$ implies $$\begin{aligned} (M_D \,b(M_R) M_D^\dagger)_{\alpha\beta} \propto \xi_\alpha \xi_\beta \,,\end{aligned}$$ for any function $b$. The off-diagonal entries are then real and entirely determined by the diagonal ones: $$\begin{aligned} &(M_D \,b(M_R) M_D^\dagger)_{\alpha\beta} \label{eq:equality}\\ &\qquad = \sqrt{(M_D \,b(M_R) M_D^\dagger)_{\alpha\alpha} (M_D \,b(M_R) M_D^\dagger)_{\beta\beta}} \,.\nonumber\end{aligned}$$ Notice that this violates the strict inequality derived earlier in Eq. , which assumed non-vanishing neutrino masses. Eq. drastically simplifies the discussion of the seesaw parameter space, seeing as all observables now only depend on the three real diagonal entries of $M_D\, b(M_R) M_D^\dagger$ instead of the nine parameters it could contain in general. Furthermore, all majoron and non-majoron parameter matrices have the same flavor structure and only differ in their absolute magnitude. Eq. also shows that any low-scale seesaw texture automatically maximizes the off-diagonal flavor-violating entries of the relevant coupling matrices (Eq. or Eq. ). We compare majoron and non-majoron limits in Fig. \[fig:seesaw\_limits\], the latter adopted from Ref. [@Coy:2018bxr]. We set $M_R/\unit{TeV} ={{1\hspace{-0.87ex}1}}$ as well as $f=\unit[1]{TeV}$ and stress again that the majoron limits can be suppressed arbitrarily by increasing $f$. Fig. \[fig:seesaw\_limits\] (left) shows the $(M_D M_D^\dagger)_{ee}$ vs. $(M_D M_D^\dagger)_{\tau\tau}$ parameter space, setting $(M_D M_D^\dagger)_{\mu\mu}=0$. As already noted in Ref. [@Coy:2018bxr], the non-majoron limits are completely dominated by flavor-conserving observables such as $Z\to e^+e^-$ and other electroweak precision data. LFV such as $\tau\to e\gamma$ and $\tau\to eee$ resides far in the already excluded region and even future improvements, e.g. in Belle-II, do not reach the allowed parameter space. In comparison, majorons with masses between $\sim\unit[100]{MeV}$ and $m_\tau$ do give relevant constraints from $\tau\to e J$ and will probe significantly more parameter space with upcoming Belle and Belle-II analyses [@Yoshinobu:2017jti]. Standard LFV in the $\tau e$ (and $\tau\mu$ in complete analogy) sector is hence doomed to be unobservable in the seesaw model, but the majoron LFV channels $\tau \to \ell J$ could be observable and deserve more experimental attention. Fig. \[fig:seesaw\_limits\] (right) shows the $(M_D M_D^\dagger)_{ee}$ vs. $(M_D M_D^\dagger)_{\mu\mu}$ parameter space, setting $(M_D M_D^\dagger)_{\tau\tau}=0$. Standard LFV, currently dominated by $\mu$ conversion in nuclei [@Bertl:2006up], provides important constraints on the parameter space and all future $\mu e$ LFV will probe uncharted terrain. For $m_J < m_\mu$, the majoron channel $\mu \to e J$ sets already better limits than $\mu N \to e N$ and can continue to dominate over $\mu\to e \gamma$ and $\mu\to 3e$ in the future. Ultimately, $\mu N \to e N$ conversion in Mu2e [@Abrams:2012er] and COMET [@Adamov:2018vin] has the best future reach. Majoron dark matter {#sec:majoron_dark_matter} ------------------- Returning to the “standard” high-scale seesaw scenario with huge hierarchy $v\ll f$ it is clear that a massive majoron can be long-lived even on cosmological scales, e.g. from Eq. , $$\begin{aligned} \Gamma (J\to \nu\nu) \sim \frac{1}{\unit[400]{Gyr}}\left(\frac{m_J}{\unit{MeV}}\right)\left(\frac{\unit[10^9]{GeV}}{f}\right)^2 .\end{aligned}$$ In this region of parameter space majorons can form DM [@Rothstein:1992rh; @Berezinsky:1993fm; @Lattanzi:2007ux; @Bazzocchi:2008fh; @Frigerio:2011in; @Lattanzi:2013uza; @Queiroz:2014yna; @Wang:2016vfj], with a production mechanism that can be unrelated to the small decay couplings [@Frigerio:2011in; @Garcia-Cely:2017oco; @Heeck:2017xbu; @Boulebnane:2017fxw]. The defining signature of majoron DM is a flux of neutrinos from DM decay with $E_\nu \simeq m_J/2$ and a known flavor composition [@Garcia-Cely:2017oco]. For $m_J \gtrsim \unit{MeV}$ these neutrino lines could be potentially observable via charged-current processes in detectors such as Borexino or Super-Kamiokande [@Garcia-Cely:2017oco], while lower masses are more difficult to probe [@McKeen:2018xyz]. The loop-level couplings generate the much more constrained decays $J \to f\bar f', \gamma\gamma$, which however depend on the matrix $K$ and are hence complementary to the neutrino signature, as discussed in detail in Ref. [@Garcia-Cely:2017oco]. This analysis used only the Set II of diagrams to calculate $J\to\gamma\gamma$, namely the expression proportional to ${\text{tr}}(K)$, Eq. . The new full expression presented in Sec. \[sec:diphoton\] leads in general to a suppression of the diphoton rate and a more involved dependence on the $K$ matrix entries (Fig. \[fig:Jgammagamma\_amplitude\]). We omit a full recasting of existing DM$\to\gamma\gamma$ limits onto our $J\to\gamma\gamma$ expression since it is not very illuminating, but stress that this diphoton suppression makes the neutrino modes even more dominant. Conclusion {#sec:conclusion} ========== Majorons, the Goldstone bosons of spontaneously broken lepton number, were proposed in the early 1980s in models for Majorana neutrino masses. Since then experiments have indeed found evidence for non-zero neutrino masses, although it is not clear yet whether they are of Majorana type. With the motivation for majorons as strong as ever, we have set out in this article to complete the program that was started almost 40 years ago and calculate all majoron couplings to SM particles. The couplings to neutrinos (tree level) as well as charged leptons and diagonal quarks (one loop) were known previously. Here we presented the two-loop couplings to gauge bosons ($J\gamma\gamma$, $J\gamma Z$, $JZZ$, $JWW$, $Jgg$) and flavor-changing quarks ($Jdd'$, $Juu'$). Phenomenologically relevant of these are currently only the majoron coupling to photons as well as the $Jdd'$ couplings behind the rare decays $K\to \pi J$ and $B\to K J$. Standard seesaw effects in an EFT approach are encoded in the matrix $M_D M_R^{-2} M_D^\dagger$, which drives for example $\ell\to \ell' \gamma$. Majoron couplings on the other hand depend on the matrix $M_D M_D^\dagger/f$, which is parametrically larger in the seesaw limit and can indeed give better constraints in parts of the parameter space. For example, while $\tau \to \ell \gamma$ and other $\tau$ LFV is unlikely to be observable in the seesaw model, $\tau\to \ell J$ can be observably large and deserves more experimental attention. The singlet majoron model together with the coupling texture of Eq. implied by low-scale seesaw is a very minimal UV-complete realization of an axion-like particle and thus a well-defined benchmark model. We expect future studies to elucidate additional aspects of this model, in particular when the majoron is used as a portal to dark matter. Acknowledgements {#acknowledgements .unnumbered} ================ JH would like to thank Arvind Rajaraman, Raghuveer Garani, and C[é]{}dric Weiland for useful discussions. HHP would like to thank Wolfgang Altmannshofer, Michael Dine, and Stefano Profumo for useful discussions. JH is supported, in part, by the National Science Foundation under Grant No. PHY-1620638, and by a Feodor Lynen Research Fellowship of the Alexander von Humboldt Foundation. The work of JH was performed in part at the Aspen Center for Physics, which is supported by the National Science Foundation under Grant No. PHY-1607611. HHP is partly supported by U.S. Department of Energy grant number de-sc0010107. [^1]: To ensure $M_\nu =0$ to all orders in the seesaw expansion and at loop level one has to further impose either $m_5=m_4$ and $z=\pm \i$ or $m_6=m_4$ and $z=0$, both of which correspond to a conserved lepton number [@Kersten:2007vk; @Moffat:2017feq].
--- author: - 'Christopher S. Wang' - 'Jacob C. Curtis' - 'Brian J. Lester' - Yaxing Zhang - 'Yvonne Y. Gao' - Jessica Freeze - 'Victor S. Batista' - 'Patrick H. Vaccaro' - 'Isaac L. Chuang' - Luigi Frunzio - Liang Jiang - 'S. M. Girvin' - 'Robert J. Schoelkopf' title: 'Supplemental Material: Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor' --- Obtaining Doktorov parameters ============================= The Doktorov parameters originate from the physical properties of a given molecule in the two electronic states of interest. Specifically, it is the structural information of the molecular configurations and the relationship between the two that fully parametrize the problem. Description of Quantum-Chemical Analyses ---------------------------------------- Theoretical predictions of optimized equilibrium geometries (with imposed $C_{2v}$ symmetry constraints), harmonic (normal-mode) vibrational displacements, and Franck-Condon parameters (Duschinsky rotation matrices and associated shift vectors) exploited the commercial (G16 rev. A.03) version of the Gaussian quantum-chemical suite (TABLE I), [@Gaussian2016] with canonical Franck-Condon matrix elements for specific vibronic bands being evaluated through use of the open-source ezSpectrum (ver. 3.0) package. [@ezSpectrum2014] All analyses relied on the CCSD(T) coupled-cluster paradigm, which includes single and double excitations along with non-iterative correction for triples. Dunning’s correlation-consistent basis sets [@Dunning1989; @Kendall1992; @Woon1993] of triple-$\zeta$ quality augmented by supplementary diffuse functions (aug-cc-pVTZ $\equiv$ apVTZ) were deployed for all targeted molecules except water, where a larger doubly augmented, quadruple-$\zeta$ basis was employed (daug-cc-pVQZ $\equiv$ dapVQZ). The Duschinsky rotation matrices and associated shift vectors provided by the commercial package Gaussian are defined via: $$\textbf{Q}\boldsymbol{'} = \textbf{JQ}\boldsymbol{''} + \textbf{K} \tag{S1}$$ where $\textbf{Q}\boldsymbol{'}$ and $\textbf{Q}\boldsymbol{''}$ are mass-weighted normal coordinates of the pre- and post-transition molecular configurations, respectively. Because our simulation considers the transformation from a vibrational state in the pre-transition configuration to the post-transition configuration, we must redefine the Duschinsky rotation matrices and associated shift vectors accordingly: $$\begin{gathered} U = \left(\begin{array}{cc} \textrm{cos}\theta & -\textrm{sin}\theta \\ \textrm{sin}\theta & \textrm{cos}\theta \end{array}\right) = \textbf{J}^T \tag{S2} \\ \textbf{d} = -\textbf{J}^T\textbf{K} \tag{S3}\end{gathered}$$ -- --------- --------- --------- --------- ------------ --------------------- 3830.91 1649.27 2619.09 1602.85 $-0.16598$ (5.05, 49.47) 1031.10 582.58 1147.04 713.39 $-0.0417$ (27.36, 14.33) 1297.27 783.55 2633.34 796.94 2.40146 (35.67, $-38.01$) 1136.38 506.27 1056.79 396.11 0.19012 ($-8.86$, $-58.34$) -- --------- --------- --------- --------- ------------ --------------------- Conversion from molecular parameters to Doktorov parameters ----------------------------------------------------------- The Doktorov transformation as given in Eq. (2) of the main text is: $$\hat{U}_\textrm{Dok} = \hat{\boldsymbol{D}}(\boldsymbol{\alpha})\hat{\boldsymbol{S}}^\dagger(\boldsymbol{\zeta'})\hat{\boldsymbol{R}}(U)\hat{\boldsymbol{S}}(\boldsymbol{\zeta}) \tag{S4}$$ where for $N$ = 2 modes, the squeezing and displacement operations are defined as: $$\begin{aligned} \hat{\textbf{S}}^{(\dagger)}(\boldsymbol{\zeta}^{(')}) & = \hat{S}^{(\dagger)}_A(\zeta^{(')}_1) \otimes \hat{S}^{(\dagger)}_B(\zeta^{(')}_2) \nonumber \\ & = \textrm{exp}\big(\frac{1}{2}(\zeta^{(')}_1\hat{c}^2_A - \zeta^{(')}_1\hat{c}^{\dagger 2}_A)\big) \otimes \textrm{exp}\big(\frac{1}{2}(\zeta^{(')}_2\hat{c}^2_B - \zeta^{(')}_2\hat{c}^{\dagger 2}_B)\big) \tag{S5} \\ \hat{\textbf{D}}(\boldsymbol{\alpha}) & = \hat{D}_A(\alpha_1) \otimes \hat{D}_B(\alpha_2) \nonumber \\ & = \textrm{exp}(\alpha_1\hat{c}^\dagger_A - \alpha^_1\hat{c}_A) \otimes \textrm{exp}(\alpha_2\hat{c}^\dagger_B - \alpha^_2\hat{c}_B) \tag{S6}\end{aligned}$$ where $\zeta^{(')}_i = \textrm{ln}\bigg(\sqrt{\tilde{\nu}^{(')}_i}\bigg)$ and $\tilde{\nu}^{(')}_i$ is the vibrational frequency of mode $i$ in the pre- (post-) transition configuration and $\alpha_i = \sqrt{\frac{\omega^{(')}_i}{2\hbar}}d_i$ where $\{d_i\}$ are the vector elements of $\textbf{d}$ in Eq. (S3). The Duschinsky rotation matrix $U$ generates the $N$-mode rotation operator $\hat{\textbf{R}}(U)$. The multi-mode mixing elements implemented in this experiment are two-mode beamsplitters, necessitating a decomposition of $U$ into nearest-neighbor rotations, and thus $\hat{\textbf{R}}$ into nearest-neighbor beamsplitters. $\hat{\textbf{R}}(U)$ becomes a product of two mode beamsplitters parametrized by $\{\theta_k\}$ and $\{i_k,j_k\}$, a sequence of angles and rotation axes derived from the decomposition of $U = \prod_k R_{i_k,j_k}(\theta_k)$. We can then write: $$\hat{\textbf{R}}(U) = \prod_k \textrm{exp}\big(\theta_k (\hat{c}_{i_k}\hat{c}^\dagger_{j_k} - \hat{c}^\dagger_{i_k}\hat{c}_{j_k})\big) \tag{S7}$$ The decomposition of $U$ is analogous to generalizing Euler angles to SO($N$); any rotation in $\mathbb{R}^N$ can be written as a product of rotations in a plane $R_{i_k,j_k}(\theta_k)$, known as Givens rotations. Following an algorithm similar to that in [@Reck1994; @Cybenko2001], but simplified to real orthogonal matrices, produces a decomposition of $U$ as a product of nearest-neighbor rotation matrices. The Duschinsky matrix for $N$ = 2 is a single Givens rotation parametrized by an angle $\theta$ which is enacted with one beamsplitter: $$\hat{\textbf{R}}(\hat{U}(\theta)) = \textrm{exp}\big(\theta(\hat{c}^\dagger_A\hat{c}_B - \hat{c}_A\hat{c}^\dagger_B)\big) \tag{S8}$$ Optimization of squeezing parameters ------------------------------------ The modification of the creation and annihilation operators under the mode transformation is given in [@Huh2015]: $$\hat{\textbf{a}}^{'\dagger} = \frac{1}{2}(L-(L^T)^{-1})\hat{\textbf{a}} + \frac{1}{2}(L+(L^T)^{-1})\hat{\textbf{a}}^\dagger +\vec{\boldsymbol{\alpha}} \tag{S9}$$ where $$\begin{gathered} L = \Omega'U\Omega^{-1} \nonumber \\ \Omega = \left( \begin{array}{ccc} \sqrt{\tilde{\nu}_1} & & 0 \\ & \ddots & \\ 0 & & \sqrt{\tilde{\nu}_N} \end{array} \right) \quad \Omega' = \left( \begin{array}{ccc} \sqrt{\tilde{\nu}'_1} & & 0 \\ & \ddots & \\ 0 & & \sqrt{\tilde{\nu}'_N} \end{array} \right) \tag{S10}\end{gathered}$$ The structure of $L$ allows for a free scaling parameter $\eta$ which leaves $L$ invarant, namely: $$\begin{aligned} \tilde{\Omega}^{(')} & = \Omega^{(')} / \eta \nonumber \\ L(\Omega, \Omega') & = L(\tilde{\Omega}, \tilde{\Omega}') \tag{S11}\end{aligned}$$ Given that the squeezing operations of the Doktorov transformation take $\boldsymbol{\zeta} = \textrm{ln}(\vec{\Omega})$ as inputs, an optimization may be performed, as done in [@Shen2018], that minimizes the total amount of squeezing while leaving the unitary invariant. This is desirable as less squeezing corresponds to shorter gate times in the simulation, which reduces the overall error rate. TABLE II lists the final set of dimensionless Doktorov parameters used in the experiment. [\|c\|c\|c\|c\|c\|]{} & --------------------------------------- H$_2$O $\xrightarrow{h\nu}$ H$_2$O$^+ (\tilde{B}^2$B$_2)$ + e$^-$ --------------------------------------- & ---------------------------- O$^-_3 \xrightarrow{h\nu}$ O$_3$ + e$^-$ ---------------------------- & ----------------------------- NO$^-_2 \xrightarrow{h\nu}$ NO$_2$ + e$^-$ ----------------------------- & --------------------------- SO$_2 \xrightarrow{h\nu}$ SO$^+_2$ + e$^-$ --------------------------- \ $\zeta_1$ & 0.262 & 0.104 & 0.035 & 0.242\ $\zeta_2$ & $-0.160$ & $-0.181$ & $-0.217$ & $-0.162$\ $\theta$ & $-0.166$ & $-0.042$ & 2.402 & 0.19\ $\zeta'_1$ & 0.072 & 0.157 & 0.389 & 0.206\ $\zeta'_2$ & $-0.174$ & $-0.080$ & $-0.208$ & $-0.285$\ $\alpha_1$ & $-1.0162$ & $-1.4278$ & 0.0546 & $-0.1140$\ $\alpha_2$ & $-2.8977$ & $-0.5311$ & $-2.2207$ & 1.7713\ $\eta$ & 47.6381 & 28.9364 & 34.7639 & 26.4676\ Theoretically predicted Hamiltonian terms ========================================= Derivation of ancilla-mediated operations ----------------------------------------- In this section, we derive the ancilla-mediated beamsplitter and single-mode squeezing interactions as shown in the main text as well as the associated ancilla-induced cavity frequency shifts. Derivations based on the perturbative four wave frequency mixing enabled by a weak ancilla nonlinearity have been presented previously in [@Gao2018]. Here, we follow the formalism used in [@Zhang2019] and sketch the general results without assuming weak ancilla nonlinearity or weak pumps. We also give explicit expressions for the strength of the engineered interactions in the case of weak pumps. We start from the Hamiltonian of the two bare cavity modes A and B coupled to the coupler transmon in module C: $$\hat{H}/\hbar = \omega_A\hat{c}^\dagger_A\hat{c}_A + \omega_B\hat{c}^\dagger_B\hat{c}_B + \hat{H}_C + \hat{H}_I + \hat{H}_\textrm{pump}(t) \tag{S12}$$ We emphasize that here the operators $\hat{c}_A$ and $\hat{c}_B$ are the annihilation operators for the bare cavity modes whereas in the main text the operators correspond to the dressed cavity modes that are weakly hybridized with the ancilla transmons. $\hat{H}_C$ is the Hamiltonian of the bare coupler transmon in module C. After expanding the transmon potential energy to fourth order in the phase across the Josephson junction and neglecting counter rotating terms, we obtain [@Koch2007]: $$\hat{H}_C / \hbar = \omega_C\hat{t}^\dagger_C\hat{t}_C - \frac{K_C}{2}\hat{t}^{\dagger 2}_C\hat{t}^2_C \tag{S13}$$ where $\hat{t}^{(\dagger)}_C$ again is the bare annihilation (creation) operator for the coupler transmon with frequency $\omega_C$ and anharmonicity $K_C$. $\hat{H}_I$ is the interaction energy between the coupler transmon and the two cavity modes. Neglecting counter rotating terms, it may be written as: $$\hat{H}_I / \hbar = (g_A\hat{c}_A + g_B\hat{c}_B)\hat{t}^\dagger_C + \textrm{h.c.} \tag{S14}$$ $\hat{H}_\textrm{pump}(t)$ represents two pumps on the coupler transmon: $$\hat{H}_\textrm{pump}(t) / \hbar = (\Omega_1 e^{-i\omega_1 t} + \Omega_2 e^{-i\omega_2 t})\hat{t}^\dagger_C + \textrm{h.c.} \tag{S15}$$ Of primary interest to us is the dispersive regime where the cavity-transmon coupling strengths are much smaller than their detuning: $\|g_{A,B}\| \ll \|\omega_{A,B} - \omega_C\|$[^1]. In this regime, we can treat the cavity-transmon interaction as a perturbation (while treating the remaining parts of the Hamiltonian exactly), and to second order in the interaction strength, we obtain an effective Hamiltonian: $$\hat{H}_\textrm{eff} / \hbar = \sum_m (\delta\omega_{A,m}\hat{c}^\dagger_A\hat{c}_A + \delta\omega_{B,m}\hat{c}^\dagger_B\hat{c}_B) \otimes \ket{\Psi_m}\bra{\Psi_m} + \hat{V} \tag{S16}$$ where $\ket{\Psi_m}$ is the $m^\textrm{th}$ Floquet state that quasi-adiabatically connects to the $m^\textrm{th}$ Fock state $\ket{m}$ of the bare transmon as the pumps are ramped up or down [@Zhang2019]. At zero pump amplitudes, $\ket{\Psi_m} = \ket{m}$. The first term in $\hat{H}_\textrm{eff}$ thus represents transmon-induced cavity frequency shifts $\delta\omega_{A,m}$ and $\delta\omega_{B,m}$ when the transmon is in $\ket{\Psi_m}$. The difference between $\delta\omega_{A,m}$ or $\delta\omega_{B,m}$ with different $m$ leads to cavity-photon-number dependent transmon transition frequencies. In particular, at zero pump amplitudes, the transmon’s transition frequency from the ground to the first excited state decreases linearly with the cavity photon number with a proportionality constant: $$\chi_{iC} = \delta\omega_{i,0} - \delta\omega_{i,1} = 2K_C\abs{\frac{g_i}{\delta_i}}^2\frac{\delta_i}{\delta_i + K_C}, \quad i \in \{A, B\} \tag{S17}$$ where $\delta_i = \omega_i - \omega_C$. Physically, the factor $\abs{g_i/\delta_i}^2$ quantifies the participation ratio of cavity A or B in the coupler transmon. In the experiment, this factor is 0.3$\%$ for cavity A and 0.2$\%$ for cavity B. Pumps on the coupler transmon can induce effective inter- or intra-cavity interactions (single-mode squeezing or beamsplitter) denoted as $\hat{V}$ in $\hat{H}_\textrm{eff}$. For the case of the beamsplitter interaction $(\omega_2 - \omega_1 = \omega_B - \omega_A)$, we have: $$\hat{V} / \hbar = \hat{H}_\textrm{BS} / \hbar = \sum_m g_{\textrm{BS},m}(e^{i\varphi^{(m)}}\hat{c}_A\hat{c}^\dagger_B + e^{-i\varphi^{(m)}}\hat{c}^\dagger_A\hat{c}_B) \otimes \ket{\Psi_m}\bra{\Psi_m} \tag{S18}$$ For the case of single-mode squeezing ($\omega_1 + \omega_2 = 2\omega_A$ or $2\omega_B$), we have: $$\hat{V} / \hbar = \hat{H}_{\textrm{sq},i} / \hbar = \sum_m g_{\textrm{sq},i,m}(e^{i\phi^{(m)}_{\textrm{sq},i}}\hat{c}^2_i +e^{-i\phi^{(m)}_{\textrm{sq},i}}\hat{c}^{\dagger 2}_i) \otimes \ket{\Psi_m}\bra{\Psi_m}, \quad i \in \{A, B\} \tag{S19}$$ Similar to the transmon-induced frequency shifts on the cavities, here both the strength and phase of the transmon-mediated interactions depend on the state of the transmon. Of primary interest to us is the strengths of the transmon-mediated interactions when the transmon is in the Floquet state $\ket{\Psi_0}$. For weak drives, these strengths are: $$\begin{gathered} g_{\textrm{BS},0} \approx 2K_C \abs{\frac{g_A}{\delta_A}\frac{g_B}{\delta_B}\frac{\Omega_1}{\delta_1}\frac{\Omega_2}{\delta_2}\frac{\delta_A + \delta_2}{\delta_A + \delta_2 + K_C}} = \sqrt{\chi_{AC}\chi_{BC}}\sqrt{\abs{\frac{(\delta_A + K_C)(\delta_B + K_C)}{\delta_A \delta_B}}}\abs{\frac{\Omega_1}{\delta_1}\frac{\Omega_2}{\delta_2}\frac{\delta_A + \delta_2}{\delta_A + \delta_2 + K_C}} \tag{S20} \\ g_{\textrm{sq},i,0} \approx 2K_C\abs{(\frac{g_i}{\delta_A})^2\frac{\Omega_1}{\delta_1}\frac{\Omega_2}{\delta_2}\frac{\delta_i}{2\delta_i + K_C}} = \chi_{iC}\abs{\frac{\Omega_1}{\delta_1}\frac{\Omega_2}{\delta_2}\frac{\delta_i + K_C}{2\delta_i + K_C}}, \quad i \in \{A, B\} \tag{S21}\end{gathered}$$ where $\delta_{1,2} = \omega_{1,2} - \omega_C$. In the case where the transmon anharmonicity $K_C$ is small compared to the detunings $\abs{\delta_{1,2,A,B}}$, the expressions above reduce to those obtained based on perturbative multiwave frequency mixing [@Gao2018]. Note that the interactions strengths presented in the main text and the rest of the supplementary material refer to the values of $g_{\textrm{BS},0}$ and $g_{\textrm{sq},i,0}$. For weak drives, Eqs. (S20, S21) show that the strengths of the engineered beamsplitter and single-mode squeezing increase linearly with both drive amplitudes $\Omega_{1,2}$. For strong drives, this dependence becomes nonlinear in $\Omega_{1,2}$ and can be accurately captured using Floquet theory for the driven transmon [@Zhang2019]. We have verified that the experimentally measured beamsplitter and single-mode squeezing rates match the expressions (S20, S21) for weak drives and the full Floquet analysis at strong drives. Transmon-induced cavity Kerr ---------------------------- Another important effect and a source of infidelity is the cavity nonlinearity induced by the transmons. To fourth order in the cavity-transmon coupling, this nonlinearity is a Kerr nonlinearity and has the following form: $$\hat{H}_\textrm{Kerr} / \hbar = \sum_m (-\frac{K_{A,m}}{2}\hat{c}^{\dagger 2}_A\hat{c}^2_A - \frac{K_{B,m}}{2}\hat{c}^{\dagger 2}_B\hat{c}^2_B - K_{AB,m}\hat{c}^\dagger_A\hat{c}_A\hat{c}^\dagger_B\hat{c}_B) \otimes \ket{\Psi_m}\bra{\Psi_m} \tag{S22}$$ where $K_{A,m}$ and $K_{B,m}$ are the self-Kerr of cavities A and B and $K_{AB,m}$ is the cross-Kerr between cavities A and B when the transmon is in the state $\ket{\Psi_m}$. First, we consider the case in the absence of pumps. Of interest to us is the cavity Kerr when the transmon is in the ground state $\ket{0}$: $$\begin{gathered} K_{i,0} = 2K_C\abs{\frac{g_i}{\delta_i}}^4\frac{\delta_i}{2\delta_i + K_C} = \frac{\chi^2_{iC}}{2K_C}\frac{(\delta_i + K_C)^2}{\delta_i(2\delta_i + K_C)}, \quad i \in \{A, B\} \tag{S23} \\ K_{AB,0} = 2\abs{\frac{g_A}{\delta_A}\frac{g_B}{\delta_B}}^2\frac{K_C(\delta_A + \delta_B)}{\delta_A + \delta_B + K_C} = \frac{\chi_{AC}\chi_{BC}}{2K_C}\frac{(\delta_A + K_C)(\delta_B + K_C)}{\delta_A \delta_B}\frac{\delta_A + \delta+B}{\delta_A + \delta_B + K_C} \tag{S24}\end{gathered}$$ Also of interest to us is the difference between and $K_{i,0}$ and $K_{i,1}$. This difference leads to a nonlinear dependence of the transmon transition frequency on the cavity photon number. This difference is usually denoted as: $$\chi'_{iC} = \frac{K_{i,0}-K_{i,1}}{2} = \frac{\chi^2_{iC}}{\delta_i}f(\delta_1/K_C), \quad i \in \{A, B\} \tag{S25}$$ where $f(x) = (18x^3 + 30x^2 + 22x + 6)/(4(x+1)(4x^2+8x+3))$. We note that there is also a contribution to $\chi'_{iC}$ from a term in the sixth order expansion of the transmon cosine potential, but for $\omega_C \gg \abs{\delta_i}$ this correction is negligible. Here we have only considered the cavity Kerr induced by the coupler transmon. In general, the transmon ancillas in modules A and B also induce Kerr in their respective cavities. The total Kerr of each cavity will then be the sum of all contributions. In the presence of pumps on the coupler transmon, the cavity Kerr can be strongly modified due to a relatively strong hybridization between cavity photons and excitations of the coupler transmon. To illustrate this effect, we consider as an example the pumps used in generating the beamsplitter interaction between the two cavities. For the choice of pumps used in the experiment, the sum of the frequency of cavity A and the higher-frequency pump is close to the frequency of transition from transmon ground to the second excited state: $\omega_A + \omega_2 \approx \omega_{02}$. As a result, the cavity photons become relatively strongly hybridized with the second excited state of the transmon, thus modifying their nonlinearity. Using a sixth-order perturbation theory (fourth-order in $g_A$ and second order in $\Omega_2$), we find that the modification to the cavity Kerr is: $$\delta K_{A,0} \approx 2K_C\abs{\frac{\Omega_2}{\delta_2}}^2\frac{\chi^2_{AC}}{\Delta^2}\frac{(2\delta_2 + K_C)\delta_2}{(\delta_2 + K_C)(\delta_2 - K_C)} \tag{S26}$$ where $\Delta = \omega_A + \omega_2 - \omega_{02}$, and $\omega_{02}$ is the Stark shifted transmon transition frequency from the ground to the second excited state. The above expression, which applies for small $\abs{\Delta}$, qualitatively captures the observed enhanced self Kerr of cavity A in the experiment during the beamsplitter operation. Comparing this expression with that of the bare cavity Kerr $K_{A,0}$ without pumps, we see that $\delta K_{A,0}$ becomes comparable to $K_{A,0}$ when $K_C\abs{\Omega_2/\delta_2} \sim \abs{\Delta}$. We note that such dependence of the cavity Kerr on the drive parameters also potentially provides a knob to control the cavity Kerr for the purpose of simulating nonlinear bosonic modes. System characterization ======================= Calibration of Gaussian operations ---------------------------------- In the dispersive regime, the transition frequency $\omega^l_{t_i}$ of ancilla $\hat{t}_i$ depends on the photon number $l$ in the respective cavity: $$\omega^l_{t_i} = \omega^0_{t_i}-l\chi_i+(l^2-l)\frac{\chi'_i}{2} \tag{S27}$$ where $\omega^0_{t_i}$ is the ancilla frequency when there are no photons in its respective cavity and $\chi_i$ and $\chi'_i$ are the dispersive shifts originating from fourth and sixth order Hamiltonian terms, respectively, as introduced in the main text. Using this, the photon number population of each cavity can be extracted via $\pi$ pulses selective on each photon number after applying various strengths of each operation (FIG. 1). These populations are then fit to the corresponding expected models, including an overall offset and scaling factor to take into account errors due to ancilla relaxation and readout imperfections (TABLE III). For the beamsplitter, we assume an effective detuning between cavities A and B in a frame where $\delta_\textrm{BS} = 0$ if the beamsplitter resonance condition is satisfied. Transmon heating leads to fluctuations in $\delta_\textrm{BS}$, which dephases the beamsplitter operation with a dephasing rate: $\kappa^\textrm{BS}_{ph} = \int_0^\infty \langle\delta_\textrm{BS}(t)\delta_\textrm{BS}(0)\rangle dt$. Thus, the oscillating populations of a single photon in each cavity $P_{10/01}$ is given to leading order in $\kappa_{A,B}/g_\textrm{BS}$ and $\kappa^\textrm{BS}_{ph} / g_\textrm{BS}$ by the expression in TABLE III, where $\bar{\kappa} = (\kappa^\textrm{BS}_A + \kappa^\textrm{BS}_B)/2$ and $\kappa^\textrm{BS}_{A,B}$ are the effective linewidths of cavities A and B during the beamsplitter operation. Measurement of system parameters -------------------------------- Static and pump-induced self-Kerr Hamiltonian terms, $-\frac{K_A}{2}\hat{c}^{\dagger 2}_A\hat{c}^2_A$ and $-\frac{K_B}{2}\hat{c}^{\dagger 2}_B\hat{c}^2_B$ , are estimated using the protocol detailed in [@ChouThesis] (FIG. 2). For the pump-induced cases, the one of the two pumps are detuned by $\delta$ = 20 kHz and 50 kHz for squeezing and beamsplitter operations, respectively, to make the engineered interaction off-resonant. We assume that the induced self-Kerr is not a strong function of this detuning. Static and pump-induced cavity decay rates are measured via $T_1$ experiments (FIG. 3). A single photon is prepared in each cavity, followed by either a delay or an off-resonant pumped operation (with the same detunings as above). Again, we assume that the pumped-induced decay rates are not a strong function of the pump detuning. The ancillas are then flipped via selective $\pi$ rotations conditioned on $n$ = 1 photon. In both cases, the data is post-selected on the ancilla being in the ground state before the selective $\pi$ rotation. We attribute the higher decay rate to the hybridization of the cavities with the shorter-lived coupler transmon. Measured cavity Kerr and $T_1$ values are given in TABLE IV. ![Calibrations for Gaussian operations. A) Displacement calibrations for cavities A (left) and B (right) starting in vacuum. Here, the amplitude of a resonant pulse is varied. B) Squeezing calibration for cavities A (left) and B (right) starting in vacuum. Here, the length of two squeezing pump tones is varied. Legends indicate population in photon number $n$. C) Beamsplitter calibration for beginning with a single photon in cavities A (left, $\ket{\psi_0} = \ket{1,0}$) and B (right, $\ket{\psi_0} = \ket{0,1}$). The length of two beamsplitter pump tones is varied, and the probability that the photon remains in the cavity that it started in is plotted over time.](fig_s1.png) ![Estimation of intrinsic $\&$ pump-induced self-Kerr. In all plots, the y-coordinate corresponds to the effective frequency of a coherent state with average photon number $\bar{n}$. The slope determines the self-Kerr, and the offsets reflect pump-induced stark shifts. Experiments for estimating the self-Kerr in the A) absence of pumps, B) presence of off-resonant squeezing pumps, and C) presence of off-resonant beamsplitter pumps for cavity A (left panels) and B (right panels).](fig_s2.png) ![Measurement of intrinsic $\&$ pump-induced decay rates. Cavity $T_1$ experiments with either a varying delay (red) or the application of an off-resonant squeezing operation (purple) during the delay for cavities A (left) and B (right).](fig_s3.png) Operation Model Cavity Calibrated rate --------------- ------------------------------------------------------------------------------------------------------- ------------ -------------------------------------------- A $\tau_{\alpha = 1} = 72 \, \textrm{ns}$ B $\tau_{\alpha = 1} = 72 \, \textrm{ns}$ A $g_\textrm{sq} \approx 60 \, \textrm{kHz}$ B $g_\textrm{sq} \approx 60 \, \textrm{kHz}$ $P_{10/01} = \frac{1}{2}\textrm{exp}\big(-\bar{\kappa}(t-t_0)\big)\times$ A $\big(1 + \textrm{exp}(-\kappa^\textrm{BS}_{ph} (t - t_0)/2)\textrm{cos}(2g_\textrm{BS}(t-t_0))\big)$ B Cavity Operation $K/2\pi$ (kHz) $T_1$ ($\mu$s) ------------ --------------- ---------------- ---------------- Native 1.8 280 Squeezing 2 200 Beamsplitter 30 170 Native 3.2 320 Squeezing 1.9 280 Beamsplitter 5 170 Circuit Implementation ====================== ![Circuit implementation of the Franck-Condon simulation. A) Overview of the quantum simulation algorithm, consisting of state preparation, unitary Doktorov transformation, and measurement. A set of verification measurements is performed after the unitary Doktorov transformation for the purpose of post-selecting the final data on measuring the transmons in their ground state. B) The two-mode circuit decomposition of the Doktorov transformation used in this experiment. The nonlinearity of the coupler transmon is primarily utilized to perform all three pumped operations, though in principle that of the ancilla transmons could have been used as well. C) Single-bit extraction. Selective $\pi$-pulses $(R_\pi)$ flip each ancilla transmon conditioned on having $n'$ and $m'$ photons in cavities A and B, respectively, for a given run of the experiment. The ancillas are then simultaneously read out using standard dispersive techniques. Subsequent runs of the experiment thus need to scan $n'$ and $m'$ over the photon number range of interest up to the desired $n_\textrm{max}$. D) Sampling. Optimal control pulses are designed to excite each ancilla transmon from $\ket{g}$ to $\ket{e}$ conditioned on the value of the binary bits $b_i$ of each cavity state, followed by dispersive readouts. Here, we measure the first 4 bits on a given run of the experiment, thus resolving the first 16 Fock states for each cavity. Real-time feedforward control is used to dynamically reset the state of the ancilla in between bit measurements to minimize errors due to ancilla relaxation.](fig_s4.png) The full quantum circuit implemented in our experiment is shown in FIG. 4. State preparation in our experiment (FIG. 5) consists of first performing measurement-based feedback cooling of all modes to their ground state (this protocol is described in full detail in the supplement of [@Elder2019]). For preparing Fock states, optimal control pulses are then played that perform the following state transfers: $$\begin{aligned} \ket{0}_A \otimes \ket{g}_{t_A} & \rightarrow \ket{n}_A \otimes \ket{g}_{t_A} \nonumber \\ \ket{0}_B \otimes \ket{g}_{t_B} & \rightarrow \ket{m}_B \otimes \ket{g}_{t_B} \tag{S28}\end{aligned}$$ These state transfers, however, suffer a finite error probability on the order of a few percent due to decoherence during the operation. This error is suppressed by performing a series of QND measurements of each cavity photon number and post-selecting on outcomes that verify that the correct state was prepared. This is done via $k$ selective $\pi$ rotations on the ancilla transmons conditioned on the desired photon numbers in the cavities following by measurements, even if the desired state is joint vacuum. The final data is post-selected on the ancilla measurement outcomes being $(``e",``g")^{\otimes k/2}$ for both modules, where $k$ is chosen to be even. In our experiment, we choose $k$ = 2 for the “single-bit extraction" measurement scheme and $k$ = 6 for the “sampling" measurement scheme. ![Circuit implementation of heralded state preparation. Measurement-based feedback cooling techniques prepare the full system in its ground state (i.e. both cavities in $\ket{0}$ and all transmons in $\ket{g}$). Optimal control pulses of the form listed in Table 1 of the main text are played simultaneously on each module to prepare a desired photon number state, followed by a set of $k$ check measurements.](fig_s5.png) Correcting systematic errors due to transmon decoherence during single-bit extraction ===================================================================================== Errors due to ancilla decoherence during the “single-bit extraction" measurement scheme may be systematically calibrated out. Specifically, decay and heating events during selective $\pi$ rotations and readout errors result in a systematic bias in the final estimate of the photon number population. For the case of a single ancilla qubit coupled to a single cavity, these effects result in a reduction of contrast for a Rabi experiment when both the ancilla and the cavity are prepared in their ground state (FIG. 6). ![Calibration of systematic measurement errors using selective ancilla pulses. A standard Rabi calibration experiment of a selective pulse used for measurement (here shown for ancilla B). The maximum probability $t$ is limited by decoherence of the transmon during both the pulse and the readout. The floor $f$ is set by the probability of heating out of the ground state during both the pulse and the readout.](fig_s6.png) When using this pulse to infer cavity photon number populations, we assume that there is no photon number dependence to either the Rabi or decoherence rates of the ancilla. Under this model, we can relate the measured probabilities $\vec{Q}$ to the true probabilities $\vec{P}$ via: $$\vec{P} = \frac{\vec{Q} - f}{t-f} \tag{S29}$$ where $f$ and $t$ are the probabilities of assigning the ancilla measurement to the excited state when it is prepared in the ground and excited states, respectively. Thus, inferring the true probabilities from the measured probabilities is a relatively straightforward task. For two modes, however, the problem becomes more complicated as a measurement of a joint probability relies on shot-by-shot correlations of the individual ancilla outcomes. Thus, false positive counts due to heating and readout errors lead to misassignment in a nonlinear fashion. We can again write what a given joint measured probability $Q_{nm}$ is in terms of the true distribution $P_{nm}$: $$Q_{nm} = t_A t_B P_{nm} + t_A f_B P_{n\bar{m}} + f_A t_B P_{\bar{n}m} + f_A f_B P_{\overline{nm}} \tag{S30}$$ Eq. (S30) may be solved for $P_{nm}$ by noting that: $$\begin{aligned} P_{\bar{n}m} & = \sum_k(1-\delta_{nk})P_{km} \nonumber \\ P_{n\bar{m}} & = \sum_l(1-\delta_{lm})P_{nl} \nonumber \\ P_{\overline{nm}} & = 1 - P_{nm} \tag{S31}\end{aligned}$$ It is worth noting that this requires $Q_{nm}$ to be a square matrix, which translates to measuring both $n'$ and $m'$ up to a pre-specified $n_\textrm{max}$. Numerical Franck-Condon data ============================ Additional experimental data is provided in this section. TABLE V provides an overview of the different molecular processes that are simulated and corresponding information regarding post-selection, systematic offsets (see supplementary text V), and distance metrics. The data for each molecular process in the following tables is calculated as follows. For the “single-bit extraction" scheme, the probability and standard error for a given joint photon number of interest is: $$\begin{aligned} q^\textrm{meas}_{n',m'} & = \frac{n^{ee}_{n',m'}}{N^\textrm{runs}_{n',m'}} \tag{S33} \\ \sigma_{n',m'} & = \sqrt{\frac{q^\textrm{meas}_{n',m'}(1 - q^\textrm{meas}_{n',m'})}{N^\textrm{runs}_{n',m'}}} \tag{S34}\end{aligned}$$ where $n^{ee}_{n',m'}$ is the number of counts where both ancillas are measured in their excited state, indicating a measure for population in $\ket{n',m'}$ (see Eq. 4 in the main text), and $N^\textrm{runs}_{n',m'}$ is the total number of runs of the experiment for probing $\ket{n',m'}$. The number of runs varies slightly among different final states due to varying post-selection probabilities. The correction protocol outlined in the supplementary text V is then applied to $q^\textrm{meas}_{n',m'}$ to retrieve a new probability distribution $p^\textrm{meas}_{n',m'}$. The standard error $\sigma_{n',m'}$ is truncated to one significant digit and $p^\textrm{meas}_{n',m'}$ is then rounded to the precision set by $\sigma_{n',m'}$. The data reported is $p^\textrm{meas}_{n',m'} \pm \sigma_{n',m'}$ only for probabilities with significant support relative to the precision of the experiment $(p^\textrm{ideal}_{n',m'} \gtrsim 10^{-4})$. The same method (sans the correction protocol) is applied to the data for the “sampling" scheme, except there the probabilities and standard error are given by: $$\begin{aligned} q^\textrm{meas}_{n',m'} & = \frac{n_{n',m'}}{N_\textrm{runs}} \tag{S35} \\ \sigma_{n',m'} & = \sqrt{\frac{q^\textrm{meas}_{n',m'}(1 - q^\textrm{meas}_{n',m'})}{N_\textrm{runs}}} \tag{S36}\end{aligned}$$ where $n_{n',m'}$ is the number of times the joint photon number $\ket{n',m'}$ is sampled from the total number of runs of the experiment $N_\textrm{runs}$. Sometimes, there will be no counts reported for a given $\ket{n',m'}$ (i.e., $n^{ee}_{n',m'}$ or $n_{n',m'}$ = 0). In this case, we simply report a probability of zero. Furthermore, sometimes the correction protocol will return negative elements in the probability distribution due to statistical noise; these unphysical cases are also nulled and a zero is reported. All distances $D = \frac{1}{2}\sum_{i=0}^{n_\textrm{max}}\sum_{j=0}^{n_\textrm{max}}\|p^\textrm{meas}_{ij} - p^\textrm{ideal}_{ij}\|$ are calculated after this correction process, with the corresponding values for $n_\textrm{max}$ specified in TABLE V. Full time-domain master equation simulations are performed using QuTiP and consider only the Hilbert space of the two cavities with $n_\textrm{max} = 30$. Each Gaussian operation is simulated by evolving the associated Hamiltonian term, while also including the corresponding self-Kerr terms and photon loss for each operation. While squeezing cavity A, for instance, the native self-Kerr and photon loss rates for cavity B are used, assuming that the pumped process for squeezing cavity A does not change the participation of cavity B in any nonlinear lossy modes (and vice-versa). The simulation also takes into account photon loss during the verification measurement, which takes 2.5 $\mu$s. The simulation does not consider imperfect state preparation and systematic errors in calibrations, which we believe to account for the remaining difference between the measured distances for the “single-bit extraction" scheme and predicted distances from the master equation simulations. [\|c\|c\|c\|c\|c\|c\|c\|c\|c\|]{} & & & & & &\ & & & & & Single-bit & & Master &\ & & & & & extraction & & Equation &\ --------------------------------------- H$_2$O $\xrightarrow{h\nu}$ H$_2$O$^+ (\tilde{B}^2$B$_2)$ + e$^-$ --------------------------------------- & $(0, 0)$ & $\sim 95\%$ & 16 & -------------- 0.937, 0.946 0.005, 0.002 -------------- & 0.049(1) & 0.151(9) & 0.0123 & FIG. 7\ & $(0, 0)$ & $\sim 96\%$ & 12 & -------------- 0.937, 0.948 0.005, 0.002 -------------- & 0.039(9) & 0.075(2) & 0.0052 & FIG. 8\ & $(1, 0)$ & $\sim 95\%$ & 10 & -------------- 0.937, 0.950 0.004, 0.002 -------------- & 0.057(5) & 0.085(5) & 0.0131 & FIG. 9\ & $(1, 2)$ & $\sim 93\%$ & 12 & -------------- 0.938, 0.950 0.004, 0.001 -------------- & 0.105(3) & 0.148(4) & 0.0217 & FIG. 10\ & $(0, 0)$ & $\sim 94\%$ & 12 & -------------- 0.935, 0.943 0.005, 0.003 -------------- & 0.034(0) & 0.110(9) & 0.0331 & FIG. 11\ & $(1, 0)$ & $\sim 92\%$ & 14 & -------------- 0.934, 0.951 0.004, 0.002 -------------- & 0.202(2) & 0.209(7) & 0.1269 & FIG. 12\ & $(0, 0)$ & $\sim 96\%$ & 12 & -------------- 0.938, 0.950 0.004, 0.002 -------------- & 0.019(6) & 0.095(3) & 0.0065 & FIG. 13\ & $(0, 1)$ & $\sim 94\%$ & 12 & -------------- 0.931, 0.951 0.004, 0.001 -------------- & 0.063(7) & 0.136(6) & 0.0213 & FIG. 14\ ![Photoionization of neutral water to the $(\tilde{B}^2\textrm{B}_2)$ excited state of the cation starting in the vibrationless state $n$ = 0, $m$ = 0.](h2onp_b2_spectra_00.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 7.92E-05 & 8.26E-05 & 0 & 0.006 $\pm$ 0.0001\ (0,1) & 6.67E-04 & 7.01E-04 & 0.0005 $\pm$ 0.0002 & 0.00396 $\pm$ 0.0001\ (0,2) & 2.80E-03 & 2.94E-03 & 0.0024 $\pm$ 0.0003 & 0.0084 $\pm$ 0.0001\ (0,3) & 7.76E-03 & 8.15E-03 & 0.0077 $\pm$ 0.0005 & 0.0122 $\pm$ 0.0002\ (0,4) & 1.60E-02 & 1.68E-02 & 0.0162 $\pm$ 0.0007 & 0.0229 $\pm$ 0.0002\ (0,5) & 2.64E-02 & 2.74E-02 & 0.0255 $\pm$ 0.0008 & 0.0288 $\pm$ 0.0003\ (0,6) & 3.59E-02 & 3.70E-02 & 0.0357 $\pm$ 0.001 & 0.0336 $\pm$ 0.0003\ (0,7) & 4.16E-02 & 4.26E-02 & 0.043 $\pm$ 0.001 & 0.0332 $\pm$ 0.0003\ (0,8) & 4.19E-02 & 4.27E-02 & 0.044 $\pm$ 0.001 & 0.0391 $\pm$ 0.0003\ (0,9) & 3.73E-02 & 3.77E-02 & 0.039 $\pm$ 0.001 & 0.0355 $\pm$ 0.0003\ (0,10) & 2.96E-02 & 2.98E-02 & 0.0316 $\pm$ 0.0009 & 0.026 $\pm$ 0.0003\ (0,11) & 2.13E-02 & 2.12E-02 & 0.0228 $\pm$ 0.0008 & 0.0202 $\pm$ 0.0002\ (0,12) & 1.39E-02 & 1.38E-02 & 0.0154 $\pm$ 0.0006 & 0.0118 $\pm$ 0.0002\ (0,13) & 8.33E-03 & 8.20E-03 & 0.0098 $\pm$ 0.0005 & 0.0079 $\pm$ 0.0001\ (0,14) & 4.60E-03 & 4.50E-03 & 0.0051 $\pm$ 0.0004 & 0.009 $\pm$ 0.0001\ (0,15) & 2.36E-03 & 2.29E-03 & 0.0029 $\pm$ 0.0003 & 0.0084 $\pm$ 0.0001\ (1,0) & 7.82E-05 & 8.11E-05 & 0.0002 $\pm$ 0.0002 & 0.0072 $\pm$ 0.0001\ (1,1) & 6.91E-04 & 7.19E-04 & 0.0005 $\pm$ 0.0002 & 0.0046 $\pm$ 0.0001\ (1,2) & 3.03E-03 & 3.16E-03 & 0.0029 $\pm$ 0.0003 & 0.009 $\pm$ 0.0001\ (1,3) & 8.81E-03 & 9.14E-03 & 0.0078 $\pm$ 0.0005 & 0.0131 $\pm$ 0.0002\ (1,4) & 1.91E-02 & 1.97E-02 & 0.0192 $\pm$ 0.0007 & 0.0254 $\pm$ 0.0002\ (1,5) & 3.28E-02 & 3.36E-02 & 0.0321 $\pm$ 0.0009 & 0.031 $\pm$ 0.0003\ (1,6) & 4.66E-02 & 4.75E-02 & 0.046 $\pm$ 0.001 & 0.0383 $\pm$ 0.0003\ (1,7) & 5.63E-02 & 5.71E-02 & 0.053 $\pm$ 0.001 & 0.0402 $\pm$ 0.0003\ (1,8) & 5.92E-02 & 5.97E-02 & 0.056 $\pm$ 0.001 & 0.0475 $\pm$ 0.0003\ (1,9) & 5.49E-02 & 5.50E-02 & 0.053 $\pm$ 0.001 & 0.0439 $\pm$ 0.0003\ (1,10) & 4.55E-02 & 4.53E-02 & 0.042 $\pm$ 0.001 & 0.033 $\pm$ 0.0003\ (1,11) & 3.40E-02 & 3.37E-02 & 0.0321 $\pm$ 0.0009 & 0.0247 $\pm$ 0.0002\ (1,12) & 2.32E-02 & 2.28E-02 & 0.0214 $\pm$ 0.0008 & 0.0149 $\pm$ 0.0002\ (1,13) & 1.44E-02 & 1.41E-02 & 0.0143 $\pm$ 0.0006 & 0.0106 $\pm$ 0.0002\ (1,14) & 8.30E-03 & 8.04E-03 & 0.0073 $\pm$ 0.0005 & 0.0111 $\pm$ 0.0002\ (1,15) & 4.42E-03 & 4.25E-03 & 0.004 $\pm$ 0.0004 & 0.0101 $\pm$ 0.0002\ (2,0) & 2.61E-05 & 2.68E-05 & 0.0004 $\pm$ 0.0001 & 0.0044 $\pm$ 0.0001\ (2,1) & 2.47E-04 & 2.52E-04 & 0.0001 $\pm$ 0.0001 & 0.00261 $\pm$ 8E-05\ (2,2) & 1.15E-03 & 1.17E-03 & 0.0013 $\pm$ 0.0002 & 0.0041 $\pm$ 0.0001\ (2,3) & 3.57E-03 & 3.60E-03 & 0.0031 $\pm$ 0.0003 & 0.0057 $\pm$ 0.0001\ (2,4) & 8.19E-03 & 8.21E-03 & 0.0082 $\pm$ 0.0005 & 0.0127 $\pm$ 0.0002\ (2,5) & 1.49E-02 & 1.48E-02 & 0.016 $\pm$ 0.0007 & 0.0165 $\pm$ 0.0002\ (2,6) & 2.25E-02 & 2.22E-02 & 0.0211 $\pm$ 0.0008 & 0.0203 $\pm$ 0.0002\ (2,7) & 2.87E-02 & 2.81E-02 & 0.0307 $\pm$ 0.0009 & 0.0214 $\pm$ 0.0002\ (2,8) & 3.19E-02 & 3.10E-02 & 0.0318 $\pm$ 0.0009 & 0.027 $\pm$ 0.0003\ (2,9) & 3.12E-02 & 3.01E-02 & 0.0312 $\pm$ 0.0009 & 0.0256 $\pm$ 0.0002\ (2,10) & 2.72E-02 & 2.61E-02 & 0.0276 $\pm$ 0.0009 & 0.0205 $\pm$ 0.0002\ (2,11) & 2.14E-02 & 2.04E-02 & 0.0211 $\pm$ 0.0008 & 0.0167 $\pm$ 0.0002\ (2,12) & 1.53E-02 & 1.45E-02 & 0.0162 $\pm$ 0.0007 & 0.0097 $\pm$ 0.0002\ (2,13) & 1.00E-02 & 9.46E-03 & 0.0111 $\pm$ 0.0006 & 0.0069 $\pm$ 0.0001\ (2,14) & 6.02E-03 & 5.67E-03 & 0.0064 $\pm$ 0.0004 & 0.0069 $\pm$ 0.0001\ (2,15) & 3.36E-03 & 3.14E-03 & 0.0038 $\pm$ 0.0003 & 0.0067 $\pm$ 0.0001\ (3,0) & 2.78E-06 & 3.35E-06 & 0 & 0.00174 $\pm$ 7E-05\ (3,1) & 2.97E-05 & 3.43E-05 & 8E-05 $\pm$ 8E-05 & 0.00091 $\pm$ 5E-05\ (3,2) & 1.56E-04 & 1.74E-04 & 9E-05 $\pm$ 8E-05 & 0.00142 $\pm$ 6E-05\ (3,3) & 5.37E-04 & 5.79E-04 & 0.0006 $\pm$ 0.0002 & 0.00189 $\pm$ 7E-05\ (3,4) & 1.36E-03 & 1.43E-03 & 0.0021 $\pm$ 0.0003 & 0.0044 $\pm$ 0.0001\ (3,5) & 2.73E-03 & 2.79E-03 & 0.0034 $\pm$ 0.0003 & 0.005 $\pm$ 0.0001\ (3,6) & 4.50E-03 & 4.49E-03 & 0.0053 $\pm$ 0.0004 & 0.0069 $\pm$ 0.0001\ (3,7) & 6.26E-03 & 6.12E-03 & 0.0078 $\pm$ 0.0005 & 0.0073 $\pm$ 0.0001\ (3,8) & 7.52E-03 & 7.22E-03 & 0.0093 $\pm$ 0.0005 & 0.0092 $\pm$ 0.0002\ (3,9) & 7.94E-03 & 7.50E-03 & 0.0097 $\pm$ 0.0005 & 0.0095 $\pm$ 0.0002\ (3,10) & 7.46E-03 & 6.94E-03 & 0.0082 $\pm$ 0.0005 & 0.0078 $\pm$ 0.0001\ (3,11) & 6.29E-03 & 5.78E-03 & 0.0076 $\pm$ 0.0005 & 0.0063 $\pm$ 0.0001\ (3,12) & 4.81E-03 & 4.37E-03 & 0.0062 $\pm$ 0.0004 & 0.0041 $\pm$ 0.0001\ (3,13) & 3.36E-03 & 3.01E-03 & 0.0033 $\pm$ 0.0003 & 0.00279 $\pm$ 8E-05\ (3,14) & 2.16E-03 & 1.91E-03 & 0.0026 $\pm$ 0.0003 & 0.00277 $\pm$ 8E-05\ (3,15) & 1.28E-03 & 1.12E-03 & 0.0012 $\pm$ 0.0002 & 0.00253 $\pm$ 8E-05\ (4,0) & 2.66E-09 & 1.53E-07 & 0 & 0.00029 $\pm$ 3E-05\ (4,1) & 1.45E-07 & 1.74E-06 & 6E-05 $\pm$ 5E-05 & 0.00017 $\pm$ 2E-05\ (4,2) & 1.73E-06 & 9.99E-06 & 6E-05 $\pm$ 6E-05 & 0.0002 $\pm$ 2E-05\ (4,3) & 1.02E-05 & 3.79E-05 & 0.0001 $\pm$ 8E-05 & 0.00033 $\pm$ 3E-05\ (4,4) & 3.86E-05 & 1.06E-04 & 0.0004 $\pm$ 0.0001 & 0.00065 $\pm$ 4E-05\ (4,5) & 1.06E-04 & 2.33E-04 & 0.0005 $\pm$ 0.0002 & 0.00073 $\pm$ 4E-05\ (4,6) & 2.25E-04 & 4.19E-04 & 0.001 $\pm$ 0.0002 & 0.0009 $\pm$ 5E-05\ (4,7) & 3.88E-04 & 6.34E-04 & 0.0014 $\pm$ 0.0002 & 0.00092 $\pm$ 5E-05\ (4,8) & 5.63E-04 & 8.24E-04 & 0.0014 $\pm$ 0.0002 & 0.00144 $\pm$ 6E-05\ (4,9) & 7.01E-04 & 9.36E-04 & 0.0019 $\pm$ 0.0003 & 0.00139 $\pm$ 6E-05\ (4,10) & 7.63E-04 & 9.43E-04 & 0.0014 $\pm$ 0.0002 & 0.00127 $\pm$ 6E-05\ (4,11) & 7.36E-04 & 8.50E-04 & 0.0014 $\pm$ 0.0002 & 0.00094 $\pm$ 5E-05\ (4,12) & 6.36E-04 & 6.92E-04 & 0.0011 $\pm$ 0.0002 & 0.00067 $\pm$ 4E-05\ (4,13) & 4.97E-04 & 5.13E-04 & 0.0008 $\pm$ 0.0002 & 0.00051 $\pm$ 4E-05\ (4,14) & 3.54E-04 & 3.49E-04 & 0.0005 $\pm$ 0.0001 & 0.00049 $\pm$ 3E-05\ (4,15) & 2.31E-04 & 2.18E-04 & 0.0004 $\pm$ 0.0001 & 0.00046 $\pm$ 3E-05\ ![Photoionization of the ozone anion to neutral ozone starting in the vibrationless state $n$ = 0, $m$ = 0.](o3mn_spectra_00.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 1.14E-01 & 1.16E-01 & 0.127 $\pm$ 0.002 & 0.1372 $\pm$ 0.0005\ (0,1) & 2.42E-02 & 2.47E-02 & 0.0271 $\pm$ 0.0008 & 0.0286 $\pm$ 0.0003\ (0,2) & 5.57E-03 & 5.60E-03 & 0.0063 $\pm$ 0.0004 & 0.0071 $\pm$ 0.0001\ (0,3) & 1.06E-03 & 1.06E-03 & 0.0013 $\pm$ 0.0002 & 0.0024 $\pm$ 8E-05\ (1,0) & 2.06E-01 & 2.08E-01 & 0.215 $\pm$ 0.002 & 0.1998 $\pm$ 0.0006\ (1,1) & 4.55E-02 & 4.60E-02 & 0.046 $\pm$ 0.001 & 0.0439 $\pm$ 0.0003\ (1,2) & 1.06E-02 & 1.06E-02 & 0.01 $\pm$ 0.0005 & 0.0109 $\pm$ 0.0002\ (1,3) & 2.06E-03 & 2.04E-03 & 0.0022 $\pm$ 0.0003 & 0.0036 $\pm$ 0.0001\ (2,0) & 1.98E-01 & 1.98E-01 & 0.201 $\pm$ 0.002 & 0.1766 $\pm$ 0.0006\ (2,1) & 4.52E-02 & 4.52E-02 & 0.045 $\pm$ 0.001 & 0.0403 $\pm$ 0.0003\ (2,2) & 1.07E-02 & 1.05E-02 & 0.0104 $\pm$ 0.0005 & 0.0101 $\pm$ 0.0002\ (2,3) & 2.11E-03 & 2.06E-03 & 0.0025 $\pm$ 0.0003 & 0.00349 $\pm$ 9E-05\ (3,0) & 1.33E-01 & 1.32E-01 & 0.128 $\pm$ 0.002 & 0.1177 $\pm$ 0.0005\ (3,1) & 3.15E-02 & 3.12E-02 & 0.031 $\pm$ 0.0009 & 0.0274 $\pm$ 0.0003\ (3,2) & 7.54E-03 & 7.37E-03 & 0.0073 $\pm$ 0.0005 & 0.0071 $\pm$ 0.0001\ (3,3) & 1.51E-03 & 1.47E-03 & 0.0012 $\pm$ 0.0002 & 0.00248 $\pm$ 8E-05\ (4,0) & 7.08E-02 & 6.97E-02 & 0.066 $\pm$ 0.001 & 0.0546 $\pm$ 0.0004\ (4,1) & 1.73E-02 & 1.70E-02 & 0.0173 $\pm$ 0.0007 & 0.0133 $\pm$ 0.0002\ (4,2) & 4.19E-03 & 4.06E-03 & 0.0047 $\pm$ 0.0004 & 0.00342 $\pm$ 9E-05\ (4,3) & 8.51E-04 & 8.19E-04 & 0.0008 $\pm$ 0.0002 & 0.0012 $\pm$ 6E-05\ (5,0) & 3.15E-02 & 3.09E-02 & 0.0307 $\pm$ 0.0009 & 0.0253 $\pm$ 0.0003\ (5,1) & 7.91E-03 & 7.74E-03 & 0.0092 $\pm$ 0.0005 & 0.0065 $\pm$ 0.0001\ (5,2) & 1.94E-03 & 1.87E-03 & 0.0022 $\pm$ 0.0003 & 0.00173 $\pm$ 7E-05\ (5,3) & 4.01E-04 & 3.83E-04 & 0.0004 $\pm$ 0.0001 & 0.00064 $\pm$ 4E-05\ (6,0) & 1.22E-02 & 1.19E-02 & 0.0132 $\pm$ 0.0007 & 0.0165 $\pm$ 0.0002\ (6,1) & 3.15E-03 & 3.07E-03 & 0.0039 $\pm$ 0.0004 & 0.0041 $\pm$ 0.0001\ (6,2) & 7.83E-04 & 7.52E-04 & 0.0007 $\pm$ 0.0002 & 0.00105 $\pm$ 5E-05\ (6,3) & 1.64E-04 & 1.56E-04 & 0.00024 $\pm$ 9E-05 & 0.00036 $\pm$ 3E-05\ (7,0) & 4.23E-03 & 4.10E-03 & 0.0066 $\pm$ 0.0005 & 0.0102 $\pm$ 0.0002\ (7,1) & 1.12E-03 & 1.09E-03 & 0.0021 $\pm$ 0.0003 & 0.00252 $\pm$ 8E-05\ (7,2) & 2.81E-04 & 2.69E-04 & 0.0005 $\pm$ 0.0001 & 0.00068 $\pm$ 4E-05\ (7,3) & 5.96E-05 & 5.64E-05 & 0.00019 $\pm$ 8E-05 & 0.00017 $\pm$ 2E-05\ (8,0) & 1.33E-03 & 1.28E-03 & 0.0049 $\pm$ 0.0005 & 0.00132 $\pm$ 6E-05\ (8,1) & 3.61E-04 & 3.48E-04 & 0.001 $\pm$ 0.0002 & 0.00035 $\pm$ 3E-05\ (8,2) & 9.18E-05 & 8.73E-05 & 0.0002 $\pm$ 0.0001 & 9E-05 $\pm$ 2E-05\ (8,3) & 1.97E-05 & 1.85E-05 & 0.00013 $\pm$ 7E-05 & 2.8E-05 $\pm$ 8E-06\ (9,0) & 3.86E-04 & 3.68E-04 & 0.0047 $\pm$ 0.0005 & 0.00156 $\pm$ 6E-05\ (9,1) & 1.07E-04 & 1.03E-04 & 0.0012 $\pm$ 0.0002 & 0.00038 $\pm$ 3E-05\ (9,2) & 2.76E-05 & 2.60E-05 & 0.0004 $\pm$ 0.0001 & 8E-05 $\pm$ 1E-05\ (9,3) & 5.98E-06 & 5.59E-06 & 0.00013 $\pm$ 7E-05 & 3.6E-05 $\pm$ 1E-05\ ![Photoionization of the ozone anion to neutral ozone starting with one quanta in the symmetric-stretching mode and zero in the bending mode $n$ = 1, $m$ = 0.](o3mn_spectra_10.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 2.24E-01 & 2.23E-01 & 0.226 $\pm$ 0.002 & 0.2226 $\pm$ 0.0007\ (0,1) & 5.42E-02 & 5.40E-02 & 0.054 $\pm$ 0.001 & 0.0537 $\pm$ 0.0004\ (0,2) & 1.31E-02 & 1.28E-02 & 0.0121 $\pm$ 0.0006 & 0.0139 $\pm$ 0.0002\ (0,3) & 2.64E-03 & 2.56E-03 & 0.0023 $\pm$ 0.0003 & 0.0044 $\pm$ 0.0001\ (1,0) & 8.96E-02 & 9.09E-02 & 0.095 $\pm$ 0.002 & 0.0923 $\pm$ 0.0005\ (1,1) & 2.67E-02 & 2.69E-02 & 0.0304 $\pm$ 0.0009 & 0.0288 $\pm$ 0.0003\ (1,2) & 6.90E-03 & 6.83E-03 & 0.0075 $\pm$ 0.0005 & 0.0075 $\pm$ 0.0001\ (1,3) & 1.51E-03 & 1.48E-03 & 0.002 $\pm$ 0.0003 & 0.00263 $\pm$ 9E-05\ (2,0) & 3.14E-04 & 5.76E-03 & 0.0265 $\pm$ 0.0009 & 0.0299 $\pm$ 0.0003\ (2,1) & 2.69E-04 & 1.39E-03 & 0.006 $\pm$ 0.0004 & 0.0064 $\pm$ 0.0001\ (2,2) & 1.59E-04 & 4.07E-04 & 0.0013 $\pm$ 0.0002 & 0.00174 $\pm$ 7E-05\ (2,3) & 6.61E-05 & 1.11E-04 & 0.0004 $\pm$ 0.0001 & 0.00059 $\pm$ 4E-05\ (3,0) & 6.59E-02 & 6.96E-02 & 0.088 $\pm$ 0.002 & 0.074 $\pm$ 0.0004\ (3,1) & 1.05E-02 & 1.14E-02 & 0.0129 $\pm$ 0.0006 & 0.0124 $\pm$ 0.0002\ (3,2) & 2.11E-03 & 2.27E-03 & 0.0028 $\pm$ 0.0003 & 0.00267 $\pm$ 9E-05\ (3,3) & 3.34E-04 & 3.67E-04 & 0.0005 $\pm$ 0.0001 & 0.00078 $\pm$ 5E-05\ (4,0) & 1.26E-01 & 1.25E-01 & 0.132 $\pm$ 0.002 & 0.1097 $\pm$ 0.0005\ (4,1) & 2.51E-02 & 2.49E-02 & 0.0264 $\pm$ 0.0009 & 0.0216 $\pm$ 0.0002\ (4,2) & 5.55E-03 & 5.42E-03 & 0.0054 $\pm$ 0.0004 & 0.0049 $\pm$ 0.0001\ (4,3) & 1.01E-03 & 9.81E-04 & 0.0009 $\pm$ 0.0002 & 0.00165 $\pm$ 7E-05\ (5,0) & 1.19E-01 & 1.16E-01 & 0.111 $\pm$ 0.002 & 0.0925 $\pm$ 0.0005\ (5,1) & 2.58E-02 & 2.51E-02 & 0.0261 $\pm$ 0.0009 & 0.0198 $\pm$ 0.0002\ (5,2) & 5.94E-03 & 5.68E-03 & 0.0054 $\pm$ 0.0004 & 0.0048 $\pm$ 0.0001\ (5,3) & 1.14E-03 & 1.08E-03 & 0.0014 $\pm$ 0.0002 & 0.00153 $\pm$ 7E-05\ (6,0) & 7.79E-02 & 7.54E-02 & 0.071 $\pm$ 0.001 & 0.056 $\pm$ 0.0004\ (6,1) & 1.79E-02 & 1.73E-02 & 0.0171 $\pm$ 0.0007 & 0.013 $\pm$ 0.0002\ (6,2) & 4.23E-03 & 4.01E-03 & 0.0041 $\pm$ 0.0004 & 0.00313 $\pm$ 9E-05\ (6,3) & 8.33E-04 & 7.82E-04 & 0.0012 $\pm$ 0.0002 & 0.00091 $\pm$ 5E-05\ (7,0) & 4.03E-02 & 3.89E-02 & 0.038 $\pm$ 0.001 & 0.0344 $\pm$ 0.0003\ (7,1) & 9.69E-03 & 9.34E-03 & 0.0098 $\pm$ 0.0005 & 0.008 $\pm$ 0.0001\ (7,2) & 2.33E-03 & 2.21E-03 & 0.0024 $\pm$ 0.0003 & 0.00207 $\pm$ 8E-05\ (7,3) & 4.70E-04 & 4.40E-04 & 0.0004 $\pm$ 0.0001 & 0.00065 $\pm$ 4E-05\ (8,0) & 1.75E-02 & 1.69E-02 & 0.0186 $\pm$ 0.0008 & 0.0089 $\pm$ 0.0002\ (8,1) & 4.38E-03 & 4.22E-03 & 0.0045 $\pm$ 0.0004 & 0.00239 $\pm$ 8E-05\ (8,2) & 1.07E-03 & 1.02E-03 & 0.0013 $\pm$ 0.0002 & 0.00059 $\pm$ 4E-05\ (8,3) & 2.20E-04 & 2.06E-04 & 0.00016 $\pm$ 8E-05 & 0.0002 $\pm$ 2E-05\ (9,0) & 6.66E-03 & 6.43E-03 & 0.0105 $\pm$ 0.0006 & 0.0046 $\pm$ 0.0001\ (9,1) & 1.72E-03 & 1.66E-03 & 0.0033 $\pm$ 0.0003 & 0.00132 $\pm$ 6E-05\ (9,2) & 4.28E-04 & 4.05E-04 & 0.0006 $\pm$ 0.0002 & 0.00034 $\pm$ 3E-05\ (9,3) & 8.93E-05 & 8.37E-05 & 3E-05 $\pm$ 6E-05 & 0.00013 $\pm$ 2E-05\ ![Photoionization of the ozone anion to neutral ozone starting with one quanta in the symmetric-stretching mode and two in the bending mode $n$ = 1, $m$ = 2.](o3mn_spectra_12.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 6.24E-03 & 9.10E-03 & 0.0208 $\pm$ 0.0008 & 0.0255 $\pm$ 0.0003\ (0,1) & 1.21E-01 & 1.20E-01 & 0.122 $\pm$ 0.002 & 0.1215 $\pm$ 0.0006\ (0,2) & 4.73E-02 & 4.75E-02 & 0.039 $\pm$ 0.001 & 0.0403 $\pm$ 0.0004\ (0,3) & 6.01E-02 & 6.02E-02 & 0.057 $\pm$ 0.001 & 0.0583 $\pm$ 0.0004\ (0,4) & 3.85E-02 & 3.65E-02 & 0.0295 $\pm$ 0.001 & 0.0274 $\pm$ 0.0003\ (0,5) & 1.46E-02 & 1.39E-02 & 0.0137 $\pm$ 0.0007 & 0.0124 $\pm$ 0.0002\ (0,6) & 4.57E-03 & 4.18E-03 & 0.0042 $\pm$ 0.0004 & 0.0049 $\pm$ 0.0001\ (0,7) & 1.21E-03 & 1.10E-03 & 0.001 $\pm$ 0.0002 & 0.0042 $\pm$ 0.0001\ (1,0) & 9.99E-04 & 1.99E-03 & 0.0049 $\pm$ 0.0004 & 0.0072 $\pm$ 0.0002\ (1,1) & 3.79E-02 & 3.89E-02 & 0.041 $\pm$ 0.001 & 0.0395 $\pm$ 0.0004\ (1,2) & 1.90E-02 & 1.99E-02 & 0.0198 $\pm$ 0.0008 & 0.0198 $\pm$ 0.0003\ (1,3) & 3.22E-02 & 3.22E-02 & 0.037 $\pm$ 0.001 & 0.0329 $\pm$ 0.0003\ (1,4) & 2.14E-02 & 2.04E-02 & 0.0207 $\pm$ 0.0008 & 0.0174 $\pm$ 0.0002\ (1,5) & 8.78E-03 & 8.31E-03 & 0.0096 $\pm$ 0.0006 & 0.0079 $\pm$ 0.0002\ (1,6) & 2.87E-03 & 2.63E-03 & 0.0033 $\pm$ 0.0003 & 0.0031 $\pm$ 0.0001\ (1,7) & 7.99E-04 & 7.23E-04 & 0.0011 $\pm$ 0.0002 & 0.00258 $\pm$ 0.0001\ (2,0) & 4.14E-04 & 6.10E-04 & 0.0037 $\pm$ 0.0004 & 0.0045 $\pm$ 0.0001\ (2,1) & 1.56E-03 & 4.26E-03 & 0.0145 $\pm$ 0.0007 & 0.0168 $\pm$ 0.0002\ (2,2) & 2.44E-04 & 1.72E-03 & 0.0066 $\pm$ 0.0005 & 0.0085 $\pm$ 0.0002\ (2,3) & 5.64E-04 & 1.90E-03 & 0.0095 $\pm$ 0.0006 & 0.0085 $\pm$ 0.0002\ (2,4) & 6.09E-04 & 1.33E-03 & 0.0047 $\pm$ 0.0004 & 0.0043 $\pm$ 0.0001\ (2,5) & 4.53E-04 & 6.66E-04 & 0.0027 $\pm$ 0.0003 & 0.00229 $\pm$ 9E-05\ (2,6) & 1.95E-04 & 2.51E-04 & 0.0012 $\pm$ 0.0002 & 0.00098 $\pm$ 6E-05\ (2,7) & 6.97E-05 & 7.85E-05 & 0.0003 $\pm$ 0.0001 & 0.0008 $\pm$ 5E-05\ (3,0) & 2.02E-03 & 2.95E-03 & 0.0082 $\pm$ 0.0005 & 0.0079 $\pm$ 0.0002\ (3,1) & 3.84E-02 & 3.97E-02 & 0.049 $\pm$ 0.001 & 0.0422 $\pm$ 0.0004\ (3,2) & 1.87E-02 & 1.94E-02 & 0.0224 $\pm$ 0.0009 & 0.0179 $\pm$ 0.0003\ (3,3) & 1.20E-02 & 1.34E-02 & 0.0155 $\pm$ 0.0007 & 0.0145 $\pm$ 0.0002\ (3,4) & 6.36E-03 & 6.72E-03 & 0.0064 $\pm$ 0.0005 & 0.0051 $\pm$ 0.0001\ (3,5) & 1.83E-03 & 2.05E-03 & 0.0026 $\pm$ 0.0003 & 0.00206 $\pm$ 9E-05\ (3,6) & 4.70E-04 & 5.06E-04 & 0.0006 $\pm$ 0.0002 & 0.00093 $\pm$ 6E-05\ (3,7) & 9.93E-05 & 1.14E-04 & 0.0002 $\pm$ 0.0001 & 0.00086 $\pm$ 6E-05\ (4,0) & 2.05E-03 & 3.41E-03 & 0.0082 $\pm$ 0.0005 & 0.0082 $\pm$ 0.0002\ (4,1) & 6.06E-02 & 6.01E-02 & 0.063 $\pm$ 0.001 & 0.0505 $\pm$ 0.0004\ (4,2) & 3.64E-02 & 3.58E-02 & 0.035 $\pm$ 0.001 & 0.0277 $\pm$ 0.0003\ (4,3) & 3.11E-02 & 3.13E-02 & 0.031 $\pm$ 0.001 & 0.0258 $\pm$ 0.0003\ (4,4) & 1.77E-02 & 1.68E-02 & 0.0149 $\pm$ 0.0007 & 0.01 $\pm$ 0.0002\ (4,5) & 5.91E-03 & 5.69E-03 & 0.0054 $\pm$ 0.0004 & 0.0039 $\pm$ 0.0001\ (4,6) & 1.69E-03 & 1.55E-03 & 0.0014 $\pm$ 0.0002 & 0.00167 $\pm$ 8E-05\ (4,7) & 4.09E-04 & 3.80E-04 & 0.0003 $\pm$ 0.0001 & 0.00151 $\pm$ 7E-05\ (5,0) & 1.18E-03 & 2.29E-03 & 0.0053 $\pm$ 0.0004 & 0.0062 $\pm$ 0.0002\ (5,1) & 5.02E-02 & 4.93E-02 & 0.048 $\pm$ 0.001 & 0.0367 $\pm$ 0.0004\ (5,2) & 3.57E-02 & 3.46E-02 & 0.034 $\pm$ 0.001 & 0.0245 $\pm$ 0.0003\ (5,3) & 3.39E-02 & 3.31E-02 & 0.033 $\pm$ 0.001 & 0.0244 $\pm$ 0.0003\ (5,4) & 1.96E-02 & 1.82E-02 & 0.0156 $\pm$ 0.0007 & 0.0103 $\pm$ 0.0002\ (5,5) & 6.87E-03 & 6.40E-03 & 0.0066 $\pm$ 0.0005 & 0.0043 $\pm$ 0.0001\ (5,6) & 2.03E-03 & 1.81E-03 & 0.0012 $\pm$ 0.0002 & 0.00196 $\pm$ 8E-05\ (5,7) & 5.10E-04 & 4.57E-04 & 0.0005 $\pm$ 0.0002 & 0.00148 $\pm$ 7E-05\ (6,0) & 4.73E-04 & 1.13E-03 & 0.0032 $\pm$ 0.0003 & 0.0038 $\pm$ 0.0001\ (6,1) & 2.94E-02 & 2.89E-02 & 0.0252 $\pm$ 0.0009 & 0.0201 $\pm$ 0.0003\ (6,2) & 2.45E-02 & 2.36E-02 & 0.0225 $\pm$ 0.0009 & 0.0169 $\pm$ 0.0002\ (6,3) & 2.47E-02 & 2.37E-02 & 0.0226 $\pm$ 0.0009 & 0.017 $\pm$ 0.0002\ (6,4) & 1.44E-02 & 1.32E-02 & 0.0113 $\pm$ 0.0006 & 0.0078 $\pm$ 0.0002\ (6,5) & 5.18E-03 & 4.76E-03 & 0.0046 $\pm$ 0.0004 & 0.0033 $\pm$ 0.0001\ (6,6) & 1.56E-03 & 1.38E-03 & 0.0015 $\pm$ 0.0002 & 0.00143 $\pm$ 7E-05\ (6,7) & 4.00E-04 & 3.53E-04 & 0.0005 $\pm$ 0.0001 & 0.00122 $\pm$ 7E-05\ (7,0) & 1.44E-04 & 4.53E-04 & 0.0013 $\pm$ 0.0002 & 0.00221 $\pm$ 9E-05\ (7,1) & 1.37E-02 & 1.35E-02 & 0.013 $\pm$ 0.0007 & 0.0122 $\pm$ 0.0002\ (7,2) & 1.33E-02 & 1.28E-02 & 0.0109 $\pm$ 0.0006 & 0.0112 $\pm$ 0.0002\ (7,3) & 1.39E-02 & 1.33E-02 & 0.0126 $\pm$ 0.0007 & 0.011 $\pm$ 0.0002\ (7,4) & 8.18E-03 & 7.46E-03 & 0.0064 $\pm$ 0.0005 & 0.0054 $\pm$ 0.0001\ (7,5) & 3.00E-03 & 2.73E-03 & 0.0024 $\pm$ 0.0003 & 0.00221 $\pm$ 9E-05\ (7,6) & 9.14E-04 & 8.02E-04 & 0.001 $\pm$ 0.0002 & 0.00102 $\pm$ 6E-05\ (7,7) & 2.38E-04 & 2.08E-04 & 0.0004 $\pm$ 0.0001 & 0.00092 $\pm$ 6E-05\ (8,0) & 3.37E-05 & 1.55E-04 & 0.0005 $\pm$ 0.0002 & 0.00085 $\pm$ 6E-05\ (8,1) & 5.37E-03 & 5.35E-03 & 0.0074 $\pm$ 0.0005 & 0.003 $\pm$ 0.0001\ (8,2) & 6.07E-03 & 5.83E-03 & 0.0058 $\pm$ 0.0005 & 0.0033 $\pm$ 0.0001\ (8,3) & 6.54E-03 & 6.22E-03 & 0.0079 $\pm$ 0.0005 & 0.0034 $\pm$ 0.0001\ (8,4) & 3.86E-03 & 3.52E-03 & 0.0044 $\pm$ 0.0004 & 0.00176 $\pm$ 8E-05\ (8,5) & 1.44E-03 & 1.30E-03 & 0.0018 $\pm$ 0.0003 & 0.00085 $\pm$ 6E-05\ (8,6) & 4.42E-04 & 3.88E-04 & 0.0003 $\pm$ 0.0001 & 0.00033 $\pm$ 3E-05\ (8,7) & 1.17E-04 & 1.02E-04 & 0.0001 $\pm$ 7E-05 & 0.00023 $\pm$ 3E-05\ (9,0) & 5.90E-06 & 4.75E-05 & 0.0005 $\pm$ 0.0002 & 0.00035 $\pm$ 4E-05\ (9,1) & 1.84E-03 & 1.84E-03 & 0.0038 $\pm$ 0.0004 & 0.00149 $\pm$ 7E-05\ (9,2) & 2.42E-03 & 2.32E-03 & 0.0041 $\pm$ 0.0004 & 0.00184 $\pm$ 8E-05\ (9,3) & 2.67E-03 & 2.53E-03 & 0.0044 $\pm$ 0.0004 & 0.00208 $\pm$ 9E-05\ (9,4) & 1.58E-03 & 1.44E-03 & 0.0023 $\pm$ 0.0003 & 0.00105 $\pm$ 6E-05\ (9,5) & 5.95E-04 & 5.39E-04 & 0.0012 $\pm$ 0.0002 & 0.00044 $\pm$ 4E-05\ (9,6) & 1.85E-04 & 1.62E-04 & 0.0003 $\pm$ 0.0001 & 0.00016 $\pm$ 2E-05\ (9,7) & 4.94E-05 & 4.28E-05 & 0.00012 $\pm$ 7E-05 & 0.00015 $\pm$ 2E-05\ ![Photoionization of nitrite to nitrogen dioxide starting in the vibrationless state $n$ = 0, $m$ = 0.](no2mn_spectra_00.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 3.54E-03 & 3.14E-03 & 0.0033 $\pm$ 0.0004 & 0.0222 $\pm$ 0.0002\ (0,1) & 2.17E-02 & 2.07E-02 & 0.0243 $\pm$ 0.0008 & 0.0337 $\pm$ 0.0003\ (0,2) & 6.42E-02 & 6.40E-02 & 0.066 $\pm$ 0.001 & 0.0802 $\pm$ 0.0004\ (0,3) & 1.22E-01 & 1.24E-01 & 0.126 $\pm$ 0.002 & 0.1145 $\pm$ 0.0005\ (0,4) & 1.66E-01 & 1.73E-01 & 0.165 $\pm$ 0.002 & 0.1498 $\pm$ 0.0006\ (0,5) & 1.73E-01 & 1.82E-01 & 0.175 $\pm$ 0.002 & 0.1492 $\pm$ 0.0005\ (0,6) & 1.43E-01 & 1.51E-01 & 0.139 $\pm$ 0.002 & 0.1125 $\pm$ 0.0005\ (0,7) & 9.68E-02 & 9.95E-02 & 0.089 $\pm$ 0.002 & 0.079 $\pm$ 0.0004\ (0,8) & 5.41E-02 & 5.27E-02 & 0.049 $\pm$ 0.001 & 0.0375 $\pm$ 0.0003\ (0,9) & 2.53E-02 & 2.29E-02 & 0.0229 $\pm$ 0.0008 & 0.0179 $\pm$ 0.0002\ (0,10) & 9.93E-03 & 8.49E-03 & 0.0076 $\pm$ 0.0005 & 0.0067 $\pm$ 0.0001\ (0,11) & 3.29E-03 & 2.75E-03 & 0.0014 $\pm$ 0.0003 & 0.0043 $\pm$ 0.0001\ (1,0) & 2.74E-04 & 1.03E-04 & 0.00026 $\pm$ 0.0001 & 0.0008 $\pm$ 4E-05\ (1,1) & 1.18E-03 & 6.52E-04 & 0.0012 $\pm$ 0.0002 & 0.00147 $\pm$ 6E-05\ (1,2) & 2.24E-03 & 1.73E-03 & 0.0027 $\pm$ 0.0003 & 0.00316 $\pm$ 9E-05\ (1,3) & 2.32E-03 & 2.48E-03 & 0.0033 $\pm$ 0.0003 & 0.00384 $\pm$ 0.0001\ (1,4) & 1.26E-03 & 2.04E-03 & 0.0039 $\pm$ 0.0004 & 0.00414 $\pm$ 0.0001\ (1,5) & 2.02E-04 & 1.04E-03 & 0.0022 $\pm$ 0.0003 & 0.00336 $\pm$ 9E-05\ (1,6) & 6.79E-05 & 7.77E-04 & 0.0015 $\pm$ 0.0002 & 0.00245 $\pm$ 8E-05\ (1,7) & 6.34E-04 & 1.27E-03 & 0.0015 $\pm$ 0.0002 & 0.0018 $\pm$ 7E-05\ (1,8) & 1.12E-03 & 1.53E-03 & 0.0012 $\pm$ 0.0002 & 0.00107 $\pm$ 5E-05\ (1,9) & 1.13E-03 & 1.15E-03 & 0.0011 $\pm$ 0.0002 & 0.00067 $\pm$ 4E-05\ (1,10) & 8.04E-04 & 5.44E-04 & 0.0005 $\pm$ 0.0001 & 0.00039 $\pm$ 3E-05\ (1,11) & 4.38E-04 & 1.56E-04 & 0.0004 $\pm$ 0.0001 & 0.00021 $\pm$ 2E-05\ (2,0) & 4.71E-04 & 2.67E-04 & 0.0005 $\pm$ 0.0001 & 0.0026 $\pm$ 8E-05\ (2,1) & 2.62E-03 & 1.71E-03 & 0.0026 $\pm$ 0.0003 & 0.00376 $\pm$ 9E-05\ (2,2) & 7.06E-03 & 5.16E-03 & 0.0077 $\pm$ 0.0005 & 0.0088 $\pm$ 0.0001\ (2,3) & 1.24E-02 & 9.77E-03 & 0.0134 $\pm$ 0.0006 & 0.0128 $\pm$ 0.0002\ (2,4) & 1.58E-02 & 1.31E-02 & 0.017 $\pm$ 0.0007 & 0.0164 $\pm$ 0.0002\ (2,5) & 1.58E-02 & 1.35E-02 & 0.0165 $\pm$ 0.0007 & 0.0154 $\pm$ 0.0002\ (2,6) & 1.28E-02 & 1.09E-02 & 0.0126 $\pm$ 0.0006 & 0.0119 $\pm$ 0.0002\ (2,7) & 8.70E-03 & 7.12E-03 & 0.008 $\pm$ 0.0005 & 0.0084 $\pm$ 0.0001\ (2,8) & 5.07E-03 & 3.79E-03 & 0.0046 $\pm$ 0.0004 & 0.00354 $\pm$ 9E-05\ (2,9) & 2.56E-03 & 1.67E-03 & 0.0017 $\pm$ 0.0002 & 0.00191 $\pm$ 7E-05\ (2,10) & 1.14E-03 & 6.20E-04 & 0.0008 $\pm$ 0.0002 & 0.00064 $\pm$ 4E-05\ (2,11) & 4.45E-04 & 1.96E-04 & 0.0005 $\pm$ 0.0001 & 0.00039 $\pm$ 3E-05\ ![Photoionization of nitrite to nitrogen dioxide starting with one quanta in the symmetric-stretching mode and zero in the bending mode $n$ = 1, $m$ = 0. The more significant errors are primarily due to having a large self-Kerr on cavity A ($\sim$30 kHz) during the beamsplitter operation after starting in a state with higher photon number.](no2mn_spectra_10.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 9.08E-03 & 4.87E-03 & 0.0036 $\pm$ 0.0004 & 0.0166 $\pm$ 0.0002\ (0,1) & 3.77E-02 & 2.40E-02 & 0.0239 $\pm$ 0.0008 & 0.0287 $\pm$ 0.0003\ (0,2) & 6.70E-02 & 5.28E-02 & 0.059 $\pm$ 0.001 & 0.0562 $\pm$ 0.0004\ (0,3) & 6.21E-02 & 6.36E-02 & 0.07 $\pm$ 0.001 & 0.0686 $\pm$ 0.0004\ (0,4) & 2.57E-02 & 4.12E-02 & 0.058 $\pm$ 0.001 & 0.0534 $\pm$ 0.0004\ (0,5) & 5.78E-04 & 1.47E-02 & 0.0306 $\pm$ 0.001 & 0.0288 $\pm$ 0.0003\ (0,6) & 1.25E-02 & 2.20E-02 & 0.034 $\pm$ 0.001 & 0.0321 $\pm$ 0.0003\ (0,7) & 4.34E-02 & 5.43E-02 & 0.065 $\pm$ 0.001 & 0.0474 $\pm$ 0.0004\ (0,8) & 6.17E-02 & 7.15E-02 & 0.078 $\pm$ 0.001 & 0.0625 $\pm$ 0.0004\ (0,9) & 5.65E-02 & 5.71E-02 & 0.06 $\pm$ 0.001 & 0.0507 $\pm$ 0.0004\ (0,10) & 3.80E-02 & 2.95E-02 & 0.0308 $\pm$ 0.001 & 0.0231 $\pm$ 0.0003\ (0,11) & 2.00E-02 & 9.85E-03 & 0.0083 $\pm$ 0.0005 & 0.0124 $\pm$ 0.0002\ (0,12) & 8.41E-03 & 2.11E-03 & 0.002 $\pm$ 0.0003 & 0.004 $\pm$ 0.0001\ (0,13) & 2.87E-03 & 4.62E-04 & 0.0002 $\pm$ 0.0002 & 0.00248 $\pm$ 9E-05\ (0,14) & 8.00E-04 & 2.41E-04 & 0.0004 $\pm$ 0.0002 & 0.0058 $\pm$ 0.0001\ (1,0) & 1.46E-04 & 9.10E-04 & 0.0002 $\pm$ 0.0002 & 0.0071 $\pm$ 0.0001\ (1,1) & 2.73E-03 & 6.17E-03 & 0.0036 $\pm$ 0.0004 & 0.0078 $\pm$ 0.0002\ (1,2) & 1.51E-02 & 2.15E-02 & 0.0125 $\pm$ 0.0006 & 0.0211 $\pm$ 0.0002\ (1,3) & 4.29E-02 & 4.89E-02 & 0.0336 $\pm$ 0.001 & 0.0344 $\pm$ 0.0003\ (1,4) & 7.59E-02 & 7.90E-02 & 0.063 $\pm$ 0.001 & 0.0592 $\pm$ 0.0004\ (1,5) & 9.21E-02 & 9.34E-02 & 0.081 $\pm$ 0.001 & 0.0694 $\pm$ 0.0004\ (1,6) & 8.05E-02 & 8.20E-02 & 0.073 $\pm$ 0.001 & 0.0573 $\pm$ 0.0004\ (1,7) & 5.16E-02 & 5.33E-02 & 0.048 $\pm$ 0.001 & 0.0392 $\pm$ 0.0003\ (1,8) & 2.39E-02 & 2.52E-02 & 0.0202 $\pm$ 0.0008 & 0.0163 $\pm$ 0.0002\ (1,9) & 7.45E-03 & 8.37E-03 & 0.0061 $\pm$ 0.0005 & 0.0055 $\pm$ 0.0001\ (1,10) & 1.24E-03 & 1.78E-03 & 0.0014 $\pm$ 0.0003 & 0.00124 $\pm$ 6E-05\ (1,11) & 1.95E-05 & 1.99E-04 & 0.0006 $\pm$ 0.0002 & 0.00118 $\pm$ 6E-05\ (1,12) & 8.94E-05 & 1.33E-05 & 0.0005 $\pm$ 0.0002 & 0.0037 $\pm$ 0.0001\ (1,13) & 1.56E-04 & 1.69E-05 & 0.0003 $\pm$ 0.0002 & 0.0046 $\pm$ 0.0001\ (1,14) & 1.08E-04 & 2.40E-05 & 0 & 0.0045 $\pm$ 0.0001\ (2,0) & 3.83E-04 & 1.99E-04 & 0.0005 $\pm$ 0.0001 & 0.00161 $\pm$ 7E-05\ (2,1) & 1.23E-03 & 1.20E-03 & 0.0016 $\pm$ 0.0002 & 0.00233 $\pm$ 8E-05\ (2,2) & 1.61E-03 & 3.06E-03 & 0.0037 $\pm$ 0.0004 & 0.0044 $\pm$ 0.0001\ (2,3) & 9.74E-04 & 4.06E-03 & 0.0051 $\pm$ 0.0004 & 0.0052 $\pm$ 0.0001\ (2,4) & 1.41E-04 & 2.92E-03 & 0.0048 $\pm$ 0.0004 & 0.005 $\pm$ 0.0001\ (2,5) & 1.05E-04 & 1.84E-03 & 0.005 $\pm$ 0.0004 & 0.0043 $\pm$ 0.0001\ (2,6) & 7.93E-04 & 3.63E-03 & 0.0068 $\pm$ 0.0005 & 0.006 $\pm$ 0.0001\ (2,7) & 1.49E-03 & 6.90E-03 & 0.0096 $\pm$ 0.0005 & 0.007 $\pm$ 0.0001\ (2,8) & 1.71E-03 & 7.86E-03 & 0.0096 $\pm$ 0.0005 & 0.0078 $\pm$ 0.0002\ (2,9) & 1.50E-03 & 5.59E-03 & 0.0063 $\pm$ 0.0004 & 0.0053 $\pm$ 0.0001\ (2,10) & 1.08E-03 & 2.48E-03 & 0.0028 $\pm$ 0.0003 & 0.00219 $\pm$ 8E-05\ (2,11) & 6.70E-04 & 6.10E-04 & 0.0009 $\pm$ 0.0002 & 0.0012 $\pm$ 6E-05\ (2,12) & 3.63E-04 & 5.24E-05 & 0.0003 $\pm$ 0.0001 & 0.00034 $\pm$ 3E-05\ (2,13) & 1.72E-04 & 1.32E-05 & 0.0001 $\pm$ 8E-05 & 0.00037 $\pm$ 3E-05\ (2,14) & 7.12E-05 & 2.40E-05 & 0.00014 $\pm$ 9E-05 & 0.00074 $\pm$ 5E-05\ (3,0) & 8.70E-05 & 1.50E-04 & 0.00012 $\pm$ 9E-05 & 0.00133 $\pm$ 6E-05\ (3,1) & 1.12E-03 & 9.95E-04 & 0.0003 $\pm$ 0.0001 & 0.00132 $\pm$ 6E-05\ (3,2) & 5.23E-03 & 3.46E-03 & 0.0023 $\pm$ 0.0003 & 0.0036 $\pm$ 0.0001\ (3,3) & 1.33E-02 & 7.92E-03 & 0.0054 $\pm$ 0.0004 & 0.0062 $\pm$ 0.0001\ (3,4) & 2.19E-02 & 1.29E-02 & 0.011 $\pm$ 0.0006 & 0.0117 $\pm$ 0.0002\ (3,5) & 2.53E-02 & 1.53E-02 & 0.014 $\pm$ 0.0007 & 0.0135 $\pm$ 0.0002\ (3,6) & 2.16E-02 & 1.35E-02 & 0.0121 $\pm$ 0.0006 & 0.0115 $\pm$ 0.0002\ (3,7) & 1.39E-02 & 8.77E-03 & 0.0075 $\pm$ 0.0005 & 0.0078 $\pm$ 0.0002\ (3,8) & 6.67E-03 & 4.15E-03 & 0.0039 $\pm$ 0.0004 & 0.00296 $\pm$ 9E-05\ (3,9) & 2.31E-03 & 1.37E-03 & 0.0014 $\pm$ 0.0002 & 0.00103 $\pm$ 6E-05\ (3,10) & 5.13E-04 & 2.86E-04 & 0.0004 $\pm$ 0.0001 & 0.00025 $\pm$ 3E-05\ (3,11) & 4.40E-05 & 3.18E-05 & 0.0002 $\pm$ 0.0001 & 0.00021 $\pm$ 2E-05\ (3,12) & 2.11E-06 & 7.28E-06 & 0.0002 $\pm$ 0.0001 & 0.00071 $\pm$ 5E-05\ (3,13) & 1.79E-05 & 8.77E-06 & 0.00018 $\pm$ 9E-05 & 0.00091 $\pm$ 5E-05\ (3,14) & 1.77E-05 & 7.78E-06 & 0.0001 $\pm$ 8E-05 & 0.00084 $\pm$ 5E-05\ (4,0) & 9.46E-06 & 3.58E-05 & 0 & 0.0004 $\pm$ 3E-05\ (4,1) & 8.74E-06 & 2.20E-04 & 0.0004 $\pm$ 0.0001 & 0.00055 $\pm$ 4E-05\ (4,2) & 2.89E-07 & 5.52E-04 & 0.0006 $\pm$ 0.0002 & 0.00096 $\pm$ 5E-05\ (4,3) & 3.10E-05 & 7.20E-04 & 0.0011 $\pm$ 0.0002 & 0.00115 $\pm$ 6E-05\ (4,4) & 8.57E-05 & 5.26E-04 & 0.0012 $\pm$ 0.0002 & 0.00112 $\pm$ 6E-05\ (4,5) & 9.82E-05 & 3.86E-04 & 0.0014 $\pm$ 0.0002 & 0.00095 $\pm$ 5E-05\ (4,6) & 6.28E-05 & 7.62E-04 & 0.002 $\pm$ 0.0003 & 0.00118 $\pm$ 6E-05\ (4,7) & 2.51E-05 & 1.35E-03 & 0.0023 $\pm$ 0.0003 & 0.0016 $\pm$ 7E-05\ (4,8) & 7.91E-06 & 1.48E-03 & 0.0014 $\pm$ 0.0002 & 0.00149 $\pm$ 7E-05\ (4,9) & 3.98E-06 & 1.03E-03 & 0.0011 $\pm$ 0.0002 & 0.00117 $\pm$ 6E-05\ (4,10) & 5.12E-06 & 4.58E-04 & 0.0004 $\pm$ 0.0001 & 0.00043 $\pm$ 4E-05\ (4,11) & 8.37E-06 & 1.20E-04 & 0.00015 $\pm$ 8E-05 & 0.00027 $\pm$ 3E-05\ (4,12) & 1.07E-05 & 1.67E-05 & 0 & 0.0001 $\pm$ 2E-05\ (4,13) & 1.00E-05 & 5.98E-06 & 0 & 7E-05 $\pm$ 1E-05\ (4,14) & 7.02E-06 & 5.32E-06 & 0.00015 $\pm$ 7E-05 & 0.00013 $\pm$ 2E-05\ ![Photoionization of sulfur dioxide to the cation starting in the vibrationless state $n$ = 0, $m$ = 0.](so2np_spectra_00.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 2.82E-02 & 2.88E-02 & 0.0275 $\pm$ 0.0009 & 0.0497 $\pm$ 0.0003\ (0,1) & 1.14E-01 & 1.15E-01 & 0.112 $\pm$ 0.002 & 0.1167 $\pm$ 0.0005\ (0,2) & 2.13E-01 & 2.15E-01 & 0.213 $\pm$ 0.002 & 0.2017 $\pm$ 0.0006\ (0,3) & 2.47E-01 & 2.47E-01 & 0.245 $\pm$ 0.002 & 0.2136 $\pm$ 0.0006\ (0,4) & 1.97E-01 & 1.95E-01 & 0.202 $\pm$ 0.002 & 0.1652 $\pm$ 0.0006\ (0,5) & 1.14E-01 & 1.11E-01 & 0.113 $\pm$ 0.002 & 0.0947 $\pm$ 0.0005\ (0,6) & 4.85E-02 & 4.72E-02 & 0.051 $\pm$ 0.001 & 0.0452 $\pm$ 0.0003\ (0,7) & 1.54E-02 & 1.48E-02 & 0.0152 $\pm$ 0.0007 & 0.0268 $\pm$ 0.0003\ (0,8) & 3.57E-03 & 3.38E-03 & 0.0037 $\pm$ 0.0004 & 0.00302 $\pm$ 9E-05\ (0,9) & 5.68E-04 & 5.28E-04 & 0.0003 $\pm$ 0.0002 & 0.00091 $\pm$ 5E-05\ (1,0) & 9.40E-06 & 2.89E-05 & 0.0003 $\pm$ 0.0001 & 0.00108 $\pm$ 5E-05\ (1,1) & 4.35E-05 & 1.53E-04 & 0.0007 $\pm$ 0.0002 & 0.00206 $\pm$ 7E-05\ (1,2) & 7.70E-04 & 9.82E-04 & 0.0008 $\pm$ 0.0002 & 0.00318 $\pm$ 9E-05\ (1,3) & 2.67E-03 & 2.84E-03 & 0.0017 $\pm$ 0.0003 & 0.00374 $\pm$ 0.0001\ (1,4) & 4.54E-03 & 4.53E-03 & 0.0023 $\pm$ 0.0003 & 0.004 $\pm$ 0.0001\ (1,5) & 4.78E-03 & 4.60E-03 & 0.0031 $\pm$ 0.0003 & 0.0035 $\pm$ 9E-05\ (1,6) & 3.44E-03 & 3.22E-03 & 0.0022 $\pm$ 0.0003 & 0.00219 $\pm$ 7E-05\ (1,7) & 1.76E-03 & 1.62E-03 & 0.0009 $\pm$ 0.0002 & 0.0014 $\pm$ 6E-05\ (1,8) & 6.53E-04 & 5.89E-04 & 0.0008 $\pm$ 0.0002 & 0.00038 $\pm$ 3E-05\ (1,9) & 1.73E-04 & 1.54E-04 & 0.0003 $\pm$ 9E-05 & 0.00011 $\pm$ 2E-05\ ![Photoionization of sulfur dioxide to the cation starting with zero quanta in the symmetric-stretching mode and one quantum in the bending mode $n$ = 0, $m$ = 1.](so2np_spectra_01.png) [\|c\|c\|c\|c\|c\|]{} &\ $(n',m')$ & ------------- Classically calculated ------------- & ----------------- Master equation simulation ----------------- & ------------ Single-bit extraction ------------ & Sampling\ (0,0) & 8.52E-02 & 8.36E-02 & 0.075 $\pm$ 0.001 & 0.0855 $\pm$ 0.0005\ (0,1) & 1.79E-01 & 1.78E-01 & 0.174 $\pm$ 0.002 & 0.1566 $\pm$ 0.0006\ (0,2) & 1.17E-01 & 1.19E-01 & 0.124 $\pm$ 0.002 & 0.1271 $\pm$ 0.0006\ (0,3) & 8.11E-03 & 1.50E-02 & 0.0306 $\pm$ 0.0009 & 0.0395 $\pm$ 0.0003\ (0,4) & 3.64E-02 & 4.42E-02 & 0.052 $\pm$ 0.001 & 0.0564 $\pm$ 0.0004\ (0,5) & 1.40E-01 & 1.42E-01 & 0.144 $\pm$ 0.002 & 0.1226 $\pm$ 0.0006\ (0,6) & 1.73E-01 & 1.69E-01 & 0.166 $\pm$ 0.002 & 0.1299 $\pm$ 0.0006\ (0,7) & 1.21E-01 & 1.16E-01 & 0.113 $\pm$ 0.002 & 0.0872 $\pm$ 0.0005\ (0,8) & 5.56E-02 & 5.22E-02 & 0.053 $\pm$ 0.001 & 0.0382 $\pm$ 0.0003\ (0,9) & 1.74E-02 & 1.60E-02 & 0.0176 $\pm$ 0.0007 & 0.0133 $\pm$ 0.0002\ (0,10) & 3.62E-03 & 3.24E-03 & 0.0036 $\pm$ 0.0004 & 0.00296 $\pm$ 9E-05\ (0,11) & 4.51E-04 & 3.96E-04 & 0.0016 $\pm$ 0.0003 & 0.00206 $\pm$ 8E-05\ (1,0) & 1.68E-03 & 1.55E-03 & 0.0037 $\pm$ 0.0003 & 0.0047 $\pm$ 0.0001\ (1,1) & 3.27E-03 & 3.16E-03 & 0.0064 $\pm$ 0.0005 & 0.0082 $\pm$ 0.0002\ (1,2) & 3.48E-03 & 3.48E-03 & 0.0058 $\pm$ 0.0004 & 0.0083 $\pm$ 0.0002\ (1,3) & 3.89E-03 & 3.99E-03 & 0.0064 $\pm$ 0.0004 & 0.006 $\pm$ 0.0001\ (1,4) & 5.95E-03 & 6.09E-03 & 0.006 $\pm$ 0.0004 & 0.0062 $\pm$ 0.0001\ (1,5) & 9.44E-03 & 9.38E-03 & 0.007 $\pm$ 0.0005 & 0.0077 $\pm$ 0.0001\ (1,6) & 1.18E-02 & 1.14E-02 & 0.0083 $\pm$ 0.0005 & 0.008 $\pm$ 0.0001\ (1,7) & 1.07E-02 & 9.95E-03 & 0.0072 $\pm$ 0.0005 & 0.0067 $\pm$ 0.0001\ (1,8) & 6.90E-03 & 6.24E-03 & 0.0048 $\pm$ 0.0004 & 0.0041 $\pm$ 0.0001\ (1,9) & 3.18E-03 & 2.79E-03 & 0.0031 $\pm$ 0.0003 & 0.00203 $\pm$ 8E-05\ (1,10) & 1.03E-03 & 8.79E-04 & 0.0005 $\pm$ 0.0001 & 0.00061 $\pm$ 4E-05\ (1,11) & 2.26E-04 & 1.86E-04 & 4E-05 $\pm$ 6E-05 & 0.00035 $\pm$ 3E-05\ [99]{} M. 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--- abstract: 'We use a standard Monte-Carlo algorithm to study the slow dynamics of a binary Lennard-Jones glass-forming mixture at low temperature. We find that Monte-Carlo is by far the most efficient way to simulate a stochastic dynamics since relaxation is about 10 times faster than in Brownian Dynamics and about 30 times faster than in Stochastic Dynamics. Moreover, the average dynamical behaviour of the system is in quantitative agreement with the one obtained using Newtonian dynamics, apart at very short times where thermal vibrations are suppressed. We show, however, that dynamic fluctuations quantified by four-point dynamic susceptibilities do retain a dependence on the microscopic dynamics, as recently predicted theoretically.' address: 'Laboratoire des Collo[ï]{}des, Verres et Nanomat[é]{}riaux, UMR 5587, Universit[é]{} Montpellier II and CNRS, 34095 Montpellier, France' author: - 'L. Berthier and W. Kob' title: 'The Monte-Carlo dynamics of a binary Lennard-Jones glass-forming mixture' --- Introduction ============ Numerical simulations play a major role among studies of the glass transition since, in contrast to experiments, the individual motion of a large number of particles can be followed at all times [@hans]. With present day computers, it is possible to follow the dynamics of a simple glass-forming liquid over more than 8 decades of time, and over a temperature window in which average relaxation timescales increase by more than 5 decades. However, at the lowest temperatures studied, relaxation is still orders of magnitude faster than in experiments performed close to the glass transition temperature. Nevertheless, it is now possible to numerically access temperatures which are low enough that many features associated to the glass transition physics can be observed: Strong decoupling phenomena [@harrowell2; @onuki; @berthier], clear deviations from fits to the mode-coupling theory [@KA] (which are experimentally known to hold only at high temperatures), and crossovers towards activated dynamics [@I; @II]. Computer simulations usually study Newtonian Dynamics (ND) by solving a discretized version of Newton’s equations for a given pair interaction between particles [@at]. Here, we study a glass-forming model in which a binary mixture of small and large particles interact via a Lennard-Jones pair potential, a model introduced by Kob and Andersen (KA) [@KA]. It can be interesting to study also different types of microscopic dynamics for the same pair potential. If dynamics satisfies detailed balance with respect to the Boltzmann distribution, all structural quantities remain unchanged, although the dynamics might be very different. In colloidal glasses, for instance, the particles undergo Brownian motion arising from collisions with the molecules of the solvent, and a stochastic dynamics is more appropriate. Theoretical considerations also suggest the study of different dynamics. Gleim et al. studied a Stochastic Dynamics (SD) to investigate whether the relaxation of the KA binary mixture depend on its microscopic dynamics, their answer being “no” [@gleim]. In SD, a friction term and a random noise are added to Newton’s equations, the amplitude of both terms being related by a fluctuation-dissipation theorem. Szamel and Flenner recently used Brownian Dynamics (BD) to study the same KA mixture [@szamel2]. In this description there are no momenta, and positions evolve with a Langevin dynamics. They again find that relaxation using BD is very similar to the one resulting from ND. They emphasize that even the deviations from mode-coupling fitting are similar in BD and ND and conclude that momenta play no role in avoiding the mode-coupling singularity, contrary to previous claims [@previous], but in agreement with more recent ones [@more]. Recently, it was also discovered that dynamic heterogeneity, that is, spatio-temporal fluctuations around the average dynamical behaviour, sensitively depends upon the microscopic dynamics [@I; @II; @science]. In particular, a major role is played by conservation laws for energy and density. In the case of energy the mechanism can be physically understood as follows. For a rearrangement to take place in the liquid, the system has to locally cross an energy barrier. If the dynamics conserves the energy, particles involved in the rearrangment must borrow energy to the neighboring particles. This ‘cooperativity’ might be unnecessary if energy can be locally supplied to the particles by an external heat bath. Conservation laws, therefore, might introduce dynamic correlations between particles and dynamic fluctuations can be different when changing from Newtonian energy conserving dynamics to a stochastic thermostatted dynamics. This predicted influence of the microscopic dynamics on dynamic fluctuations [@I; @II] was in fact our principal motivation for the present study. In this article, we propose a third type of stochastic dynamics for the KA mixture and study in detail the dynamics of the system subjected to a standard Monte-Carlo (MC) dynamics. We find that MC is particularly efficient at relaxing the system since it is about 10 times faster than BD and 30 times faster than SD, while the average dynamics is still in quantitative agreement with ND. We are therefore in position to study both the very low temperature average dynamics of the model and its dynamic fluctuations in detail, shedding new light on both aspects. The paper is organized as follows. In Section \[mc\] we give details about the simulation technique and compare its efficiency to previously studied dynamics. In Section \[results\] we present our numerical results. Section \[conclusion\] concludes the paper. An efficient simulation technique {#mc} ================================= We study a binary Lennard-Jones mixture made of $N_A=800$ and $N_B=200$ particles of types $A$ and $B$, respectively. Particles interact with the following Lennard-Jones pair potential $$\phi_{\alpha \beta}^{\rm LJ}(r)= 4 \epsilon_{\alpha \beta} \left[ \left(\frac{\sigma_{\alpha \beta}} {r} \right)^{12} - \left(\frac{\sigma_{\alpha \beta}}{r} \right)^{6} \right], \label{pp}$$ where $\alpha, \beta \in [{A}, {B}]$ and $r$ is the distance between the interacting pair of particles. Interaction parameters $\epsilon_{\alpha \beta}$ and $\sigma_{\alpha \beta}$ are chosen to prevent crystallization and can be found in Ref. [@KA]. The length and energy are given in the standard Lennard-Jones units $\sigma_{AA}$ (particle diameter), and $\epsilon_{AA}$ (interaction energy), where the subscript $A$ refers to the majority species. The potential is truncated and shifted at a distance $r = 2.5$. Previous work [@hans; @KA] has shown that the dynamics becomes slow below $T \approx 1.0$, while the fitted mode-coupling temperature for this system is $T_c \approx 0.435$, although deviations from mode-coupling behaviour become noticable already below $T \approx 0.47$. We have implemented a standard Monte-Carlo dynamics [@at] for the pair potential in Eq. (\[pp\]). An elementary move can be described as follows. A particle, $i$, located at the position ${\bf r}_i$ is chosen at random. The energy cost, $\Delta E_i$, to move particle $i$ from position ${\bf r}_i$ to a new position ${\bf r}_i + \delta {\bf r}$ is evaluated, $\delta {\bf r}$ being a random vector comprised in a cube of linear length $\delta_{\rm max}$ centered around the origin. The Metropolis acceptance rate, $p = {\rm min} (1, e^{-\beta \Delta E_i})$, where $\beta =1/T$ is the inverse temperature, is then used to decide whether the move is accepted. In the following, one Monte-Carlo timestep represents $N=N_A + N_B$ attempts to make such an elementary move, and timescales are reported in this unit. The one degree of freedom that remains to be fixed is $\delta_{\rm max}$ which determines the average lengthscale of elementary moves. If chosen too small, energy costs are very small and most of the moves are accepted, but the dynamics is very slow because it takes a long time for particles to explore their cage. On the other hand too large displacements will on average be very costly in energy and acceptance rates can become prohibitively small. We seek a compromise between these two extremes by monitoring the dynamics at a moderately low temperature, $T=0.5$, for several values of $\delta_{\rm max}$. As the most sensitive indicator of the relaxational behaviour we measure the contribution from the majority specie $A$ to the self-intermediate scattering function, $$F_s({\bf k},t) = \left\langle \frac{1}{N_A} \sum_{j=1}^{N_A} e^{i {\bf k} \cdot [{\bf r}_j(t) - {\bf r}_j(0)]} \right\rangle. \label{self}$$ We spherically average over wavectors of comparable magnitude, and present results for $\|{\bf k}\|=7.21$, which corresponds to the first diffraction peak in the static structure factor of the liquid. In Fig. \[delta\] we present our results for $\delta_{\rm max}$ values between 0.05 and 0.4. As expected we find that relaxation is slow both at small and large values of $\delta_{\rm max}$, and most efficient for intermediate values. Interestingly we also note that the overall shape of the self-intermediate scattering function does not sensitively depend on $\delta_{\max}$. We define a typical relaxation time as $F_s(k,\tau_\alpha) = e^{-1}$ and show its $\delta_{\rm max}$ dependence in the inset of Fig. \[delta\]. A clear minimum is observed at the optimal value of $\delta_{\max} \approx 0.15$. In the rest of the paper we only present data obtained for this value. As compared to previously studied dynamics, we find that, when expressed in numbers of integration timesteps, structural relaxation in Monte-Carlo simulations is marginally faster than in Newtonian dynamics, but 30 times faster than in Stochastic Dynamics [@gleim], and 10 times faster than in Brownian Dynamics [@szamel2]. We conclude therefore that MC is by far the most efficient way to perform stochastic molecular simulations of the present glass-forming material. The relative inefficiency of both BD and SD is due to the stochastic nature of their microscopic equations of motion. It is well-known that small integration timesteps are required for accurate integration of stochastic equations of motion, in particular to maintain the delicate balance between friction and noise required for the system to converge towards the correct equilibrium distribution [@at]. No such constraint exists for MC dynamics, where elementary moves can be made arbitrarily large. Equilibrium only requires detailed balance to be fulfilled, and this is always the case with the Metropolis algorithm described above. With larger elementary moves, particles can efficiently explore their cage and relaxation is much faster. This physical interpretation is also supported by the optimal value $\delta_{\max}=0.15$ that we report, which corresponds to a mean-square displacement of 0.225, very close to the plateau observed in the mean-square displacement shown in Fig. \[fs\] (see below), which can be taken as a rough estimate of the cage size. Monte-Carlo simulations can of course be made even more efficient by implementing for instance swaps between particles, or using parallel tempering. The dynamical behaviour, however, is then strongly affected by such non-physical moves and only equilibrium thermodynamics can be studied. Since we want to conserve a physically realistic dynamics, we cannot use such improved schemes. We have performed simulations at temperatures between $T=2.0$ and $T=0.43$, the latter being smaller than the fitted mode-coupling temperature. For each temperature we have simulated 10 independent samples to improve the statistics. Initial configurations were taken as the final configurations obtained from previous work performed with ND [@I; @II], so that production runs could be started immediately. For each sample, production runs lasted at least $15\tau_\alpha$ (at $T=0.43$), much longer for higher temperatures. Results ======= Average dynamics ---------------- The self-intermediate scattering function, Eq. (\[self\]), is shown in Fig. \[fs\] for temperatures decreasing from $T=2.0$ down to $T=0.43$. These curves present well-known features. Dynamics at high temperature is fast and has an exponential nature. When temperature is decreased below $T \approx 1.0$, a two-step decay, the slower being strongly non-exponential, becomes apparent. Upon decreasing the temperature further, the slow process dramatically slows down by about 5 decades, while clearly conserving an almost temperature-independent non-exponential shape, as already reported for ND [@KA]. Finally, as reported for SD [@gleim], we find that also the first process, the decay towards a plateau, slows down considerably when decreasing temperature. This process, called ‘critical decay’ in the language of mode-coupling theory [@mct], is not observed when using ND, because it is obscured by the thermal vibrations occuring at high frequencies. Although the plateau seen in $F_s(k,t)$ is commonly interpreted as ‘vibrations of a particle within a cage’, the data in Fig. \[fs\] discard this view. From direct visualisation of the particles’ individual dynamics it is obvious that vibrations take place in just a few MC timesteps, while the decay towards the plateau can be as long as $10^4$ time units at the lowest temperatures studied here. This decay is therefore necessarily more complex, most probably cooperative in nature. This interpretation is supported by recent theoretical studies where a plateau is observed in two-time correlators of lattice models where local vibrations are indeed completely absent [@bethe]. A detailed atomistic description of this process has not yet been reported, but would indeed be very interesting. Next, we study the mean-squared displacement for the majority specie. It is defined as $$\label{msd} \Delta^2 r(t) = \frac{1}{N_A} \sum_{i=1}^{N_A} \left\langle \|{\bf r}_i(t) - {\bf r}_i(0) \|^2 \right\rangle,$$ and we present its temperature evolution in Fig. \[fs\], which mirrors the evolution of the self-intermediate scattering function in the same figure. Since we are studying a stochastic dynamics, displacements are diffusive at both short and long timescales. The plateau observed in $F_s(k,t)$ now translates into a sub-diffusive regime in the mean-squared displacements separating the two diffusive regimes. At the lowest temperature studied, when $t$ changes by three decades from $2\times10^2$ to $2\times 10^5$, the mean-squared displacement changes by a mere factor 2.2 from 0.02 to 0.044. Particles are therefore nearly arrested for several decades of times, before eventually entering the diffusing regime which allows for the relaxation of the structure of the liquid. Comparison to Newtonian and Stochastic Dynamics ----------------------------------------------- The previous subsection has shown that the Monte-Carlo dynamics of the KA mixture is qualitatively similar to the one reported for ND, apart at relatively short times where the effect of thermal vibrations is efficiently suppressed. We now compare our results more quantitatively with the dynamical behaviour observed using ND. In Fig. \[comp\] we compare the time dependence of the self-intermediate scattering function for three types of dynamics: the present Monte-Carlo data, the Newtonian Dynamics data taken from Ref. [@I], and the Stochastic Dynamics results from Ref. [@gleim], all obtained for the same parameters $k=7.21$ and $T=0.45$. We have rescaled the time to obtain maximum overlap in the long-time relaxation of the three curves. Quite strikingly, SD and MC data perfectly overlap over the complete time-range (8 decades of time) of the simulation. Indeed the SD dotted line is barely visible below the full line of the MC data in Fig. \[comp\]. This confirms our claim that MC defines a physically relevant microscopic dynamics, since it is completely equivalent to SD with the major advantage that it is 30 times faster, at least for the KA mixture. In Fig. \[comp\], we also confirm that the approach to the plateau is different in MC/SD and ND. In the latter, phonon-like vibrations affect the initial decay of $F_s(k,t)$. For instance, a shallow dip, generally attributed to the ‘Boson peak’, is observed at low temperature in ND, see the dashed line in Fig. \[comp\]. The long-time decay of the self-intermediate scattering function, however, is in full quantitative agreement for the three dynamics. This agreement was the main claim of Ref. [@gleim], extended to BD in Ref. [@szamel2] and for MC in the present work. Since all dynamics display similar long-time relaxation, it is sensible to also quantitatively compare the temperature evolution of the relaxation times, $\tau_\alpha(T)$, already defined above. This is done in Fig. \[comp\], where we use a standard representation where an Arrhenius slowing down over a constant energy barrier, $\tau_\alpha \sim \exp( E/T)$, would appear as a straight line. The data clearly show some upwards bending in Fig. \[comp\], which places the KA mixture in the family of fragile (though very weakly) glass-formers. We find that the temperature evolution of the alpha-relaxation time measured in MC simulations is in complete quantitative agreement with the one obtained from ND, over the complete temperature range $T=2.0 \to 0.43$. In particular the quality of a power-law fit of the slowing down, $\tau_\alpha \sim (T-T_c)^{-\gamma}$, as suggested by mode-coupling theory, is similar for both dynamics [@KA; @gleim]. We have shown such a fit through our data, using the value $T_c=0.435$ determined in Ref. [@KA]. The fit describes the data over about 2.5 decades. Deviations from the mode-coupling fit appear below $T \approx 0.47$, and become obvious when $T_c$ is approached further. In Fig. \[comp\] we also show the temperature evolution of the self-diffusion constant, defined from the long-time limit of the mean-square displacement as $$D = \lim_{t \to \infty} \frac{\Delta^2 r(t)}{6 t}.$$ The behaviour of the diffusion constant is qualitatively very close to the one of the alpha-relaxation time, and all the above remarks apply. The well-known difference between the two quantities is a slightly stronger temperature evolution of $\tau_\alpha$, implying a well-studied decoupling between translational diffusion and structural relaxation in this system [@hans; @berthier], which is therefore very similar for different types of dynamics. Theoretically, an identical relaxation within MC/SD/BD/ND is an important prediction of mode-coupling theory [@mct] because the theory uniquely predicts the dynamical behaviour from static density fluctuations. Gleim et al. argue that their finding of a quantitative agreement between SD and ND is a nice confirmation of this non-trivial mode-coupling prediction [@gleim]. Szamel and Flenner [@szamel2] confirmed this claim using BD, and argued further that even deviations from mode-coupling predictions are identical. We confirm the validity of this statement even below $T_c$, showing that the agreement between different dynamics, although indeed predicted by mode-coupling theory, is certainly valid at a much more general level. Similarly to Szamel and Flenner, we note that deviations from a power law divergence cannot be attributed to coupling to currents which are expressed in terms of particle velocities. In our MC simulations we have no velocities, so that avoiding the mode-coupling singularity is not due to the hydrodynamic effects pointed out in Ref. [@previous] (see Ref. [@more] for more recent theoretical viewpoints). Multi-point susceptibility -------------------------- Having established the ability of MC simulations to efficiently reproduce the average slow dynamics obtained from ND simulations we now turn to the study of the dynamic fluctuations around the average dynamical behaviour, i.e. to dynamic heterogeneity. Dynamic fluctuations can be studied through the four-point susceptibility, $\chi_4(t)$, which quantifies the strength of the spontaneous fluctuations around the average dynamics by their variance, $$\chi_4(t) = N_A \left[ \langle f_s^2({\bf k}, t) \rangle - F_s^2({\bf k}, t) \right], \label{chi4lj}$$ where $f_s({\bf k},t) = N_A^{-1} \sum_j \cos ({\bf k} \cdot [{\bf r}_j(t) - {\bf r}_j(0)] )$ represents the real part of the instantaneous value of the self-intermediate scattering function, so that $F_s({\bf k},t) = \langle f_s({\bf k},t) \rangle$. As shown by Eq. (\[chi4lj\]), it is clear that $\chi_4(t)$ will be large if run-to-run fluctuations of the self-intermediate scattering functions are large. This is the case when the local dynamics becomes spatially correlated, as already discussed in several papers [@FP; @silvio2; @glotzer; @lacevic; @toni; @mayer]. We show the time dependence of the dynamic susceptibility $\chi_4(t)$ obtained from our MC simulations for various temperatures in Fig. \[chi4\]. As predicted theoretically in Ref. [@toni] we find that $\chi_4(t)$ presents a complex time evolution, closely related to the time evolution of the self-intermediate scattering function. Overall, $\chi_4(t)$ is small at both small and large times when dynamic fluctuations are small. There is therefore a clear maximum observed for times comparable to $\tau_\alpha$, where fluctuations are most prominent. The position of the maximum then shifts to larger times when temperature is decreased, tracking the alpha-relaxation. The most important physical information revealed by these curves is the fact that the amplitude of the peak grows when the temperature decreases. This is direct evidence that spatial correlations grow when the glass transition is approached. The two-step decay of the self-intermediate scattering function translates into a two-power law regime for $\chi_4(t)$ approaching its maximum. We have fitted these power laws, $\chi_4(t) \sim t^a$, followed by $\chi_4(t) \sim t^b$ with the exponents $a =0.35$ and $b=0.75$ in Fig. \[chi4\]. We have intentionally used the notation $a$ and $b$ for these exponents which are predicted, within mode-coupling theory, to be equal to the standard exponents also describing the time dependence of intermediate scattering functions [@mct]. Our findings are in good agreement with previously reported values for $a$ and $b$. See Refs. [@II; @toni] for a more extensive discussion and comparison to other theoretical predictions. We finally compare the dynamic susceptibility for various dynamics. In Fig. \[chi42\], we present the time evolution of $\chi_4(t)$ for a given temperature, $T=0.45$ and four different dynamics: The present MC data, data from SD obtained in Ref. [@I], data for ND in the microcanonical ($NVE$) ensemble from Ref. [@I], and data for ND in the canonical ($NVT$) ensemble from Ref. [@I]. To perform this comparison, we have again rescaled times to obtain the maximum overlap in the long-time region. In Fig. \[chi42\] it is obvious that three curves are identical: ND-$NVE$, MC and SD data perfectly overlap near the maximum of $\chi_4(t)$ and have similar time dependences, apart at very short-times. On the other hand, ND-$NVT$ data display a different time dependence and reveal considerably larger dynamic fluctuations in the long-time regime. We conclude therefore that, contrary to the average dynamics, the dynamic fluctuations quantified through the four-point susceptibility do retain a dependence upon the microscopic dynamics since canonical estimates of $\chi_4(t)$ are different for ND and for MC/SD/BD. Although perhaps counter-intuitive at first sight we find that dynamics with a stochastic heat-bath display dynamic fluctuations similar to the ones measured using microcanonical ND, while fluctuations are much larger in canonical ND simulations. As mentioned in the introduction, this confirms the idea that the energy conservation (implied by Newton’s equations of motion) might lead to an amplification of dynamic fluctuations. With hindsight, this is not such a surprising result: The specific heat, after all, also behaves differently in different statistical ensembles. The ensemble dependence and dependence upon the microscopic dynamics are the main subjects of two recent papers [@I; @II]. There is an experimentally relevant consequence of these findings. The difference between the microcanonical and canonical values of the dynamic fluctuations in ND can be shown to be equal to [@science] $$\chi_4^{NVT}(t) - \chi_4^{NVE}(t) = \frac{T^2}{c_V} \left( \frac{\partial F_s({\bf k},t)}{\partial T} \right)^2, \label{chiT}$$ where $c_V$ is the constant volume specific heat expressed in $k_B$ units. As shown in Fig. \[chi42\] the temperature derivative in Eq. (\[chiT\]) represents in fact the major contribution to $\chi_4^{NVT}$, meaning that the term $\chi_4^{NVE}$ can be neglected in Eq. (\[chiT\]). Since the right hand side of (\[chiT\]) is more easily accessible in an experiment than $\chi_4$ itself, Eq. (\[chiT\]) opens the possibility of an experimental estimate of the four-point susceptibility. This finding, and its experimental application to supercooled glycerol and hard sphere colloids, constitute the central results of Ref. [@science]. Conclusion ========== We have implemented a standard Monte-Carlo dynamics on the well-known binary Lennard-Jones mixture introduced by KA. We have shown that the resulting average dynamics is in full quantitative agreement with results from Newtonian dynamics, while being considerably faster than previously studied stochastic dynamics, namely Brownian and Stochastic dynamics. We have therefore at our disposal an efficient numerical technique to simulate the stochastic dynamics of the KA mixture at low temperature. This allowed us to show, in particular, that dynamic fluctuations retain a dependence upon the microscopic dynamics since four-point dynamical susceptibilities evaluated in the canonical ensemble for ND and MC quantitatively differ, because the energy conservation of Newton’s equations amplify dynamic fluctuations. We wish to thank J.L. Barrat for useful discussions, and G. Biroli, J.P. Bouchaud, K. Miyazaki, and D. 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--- abstract: 'In this paper we use state-of-the-art N-body hydrodynamic simulations of a cosmological volume of side $100\Mpc$ to produce many galaxy clusters simultaneously in both the standard cold dark matter (SCDM) cosmology and a cosmology with a positive cosmological constant ($\Lambda$CDM). We have performed simulations of the same volume both with and without the effects of radiative cooling, but in all cases neglect the effects of star formation and feedback. With radiative cooling clusters are on average five times [less]{} luminous in X-rays than the same cluster simulated without cooling. The importance of the mass of the central galaxy in determining the X-ray luminosity is stressed.' address: - 'Department of Physics, University of Durham, Durham, DH1 3LE, UK ' - 'Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada, L8S 4M1 ' - 'Astronomy Centre, CPES, University of Sussex, Falmer, Brighton, BN1 9QJ, UK ' author: - 'F. R. Pearce$^{1}$, H. M. P. Couchman$^2$, P. A. Thomas$^3$, A. C. Edge$^{1}$' --- 50 50m1 ¶3M 3M 2[c2]{} 3[c3]{} 4[c4]{} \#1[to 0pt[\#1]{}]{} ß \#1 Introduction ============ Clusters of galaxies are the largest virialised structures in the Universe, evolving rapidly at recent times because in hierarchical cosmologies big objects form last. Even at moderate redshifts the number of large dark matter halos in a cold dark matter Universe with a significant, positive cosmological constant is higher than in a standard cold dark matter Universe and it is precisely because both the number density and size of large dark matter halos evolve at different rates in popular cosmological models that observations of galaxy clusters provide an important discriminator between rival cosmologies. The simulations we have carried out follow 2 million gas and 2 million dark matter particles in a box of side $100\Mpc$. We have performed both a SCDM and a $\Lambda$CDM simulation with the parameters; $\Omega=1.0$, $\Lambda=0.0$, h=0.5, $\sigma_8=0.6$ for the former and $\Omega=0.3$, $\Lambda=0.7$, h=0.7, $\sigma_8=0.9$ for the latter. The baryon fraction was set from Big Bang nucleosynthesis constraints, $\Omega_bh^2=0.015$ and we have assumed an unevolving gas metallicity of 0.3 times the solar value. These parameters produce a gas mass per particle of $2\times10^9\Msun$. These simulations [@P99a] produce a set of galaxies that fit the local K-band number counts [@G97]. The brightest cluster galaxies contained within the largest halos are not excessively luminous for a volume of this size, unlike those found in previous work [@KW93; @L99] and presumably [@SO98] (although they do not state a central galaxy mass or galaxy luminosity). The fraction of the baryonic material that cools into galaxies within the virial radius of the large halos in our simulation is typically around 20 percent, close to the observed baryonic fraction in cold gas and stars. This is much less than the unphysically high value of 40 percent reported by [@KW93]. Results ======= For each of the 20 largest clusters from each simulation we follow [@N95] in using the following estimator for the bolometric X-ray luminosity of a cluster, $$L_X = 4 \times 10^{32} \sum \rho_i T_i^{1 \over 2} \thinspace{\rm erg \thinspace s}^{-1} \label{eq:lx}$$ where the sum is over all the gas particles with temperatures above $12000\K$ within the specified radius. Temperatures are in Kelvin and densities are relative to the mean gas density in the box. We plot these bolometric luminosities as a function of radius for each of our relaxed clusters in figure 1. For the simulation without cooling the clusters are several times more luminous than those from the cooling run. This contradicts previous results [@KW93; @SO98; @L99] who all found the X-ray luminosity increased if cooling was turned on. The cooling clusters are less luminous than their counterparts in the simulation without cooling because they have lower central temperatures and similar central densities. Most of the emission coming from the non-cooling clusters comes from the central regions, with little subsequent rise in the bolometric luminosity beyond 0.3 times the virial radius whereas for the majority of the cooling clusters the bolometric luminosity continues to rise out to the virial radius. There has been much debate in the literature centering on the X-ray cluster $L_X$ versus $T$ correlation. The emission weighted mean temperature in keV is plotted against the bolometric luminosity within the virial radius for all our clusters in figure 2. The filled symbols represent the relaxed clusters and the open symbols denote those clusters that show significant substructure. Clearly the simulation without cooling produces brighter clusters at the same temperature. All 3 sets of objects display an $L_X-T$ relation although there are insufficient numbers to tie the trend down very tightly. Also plotted in figure 2 are the observational data [@D95]. Our clusters are smaller and cooler because they are not very massive (due to our relatively small computational volume) but span a reasonable range of luminosities and temperatures. Discussion ========== Implementing cooling clearly has a dramatic effect on the X-ray properties of galaxy clusters. Without cooling our clusters closely resemble those found by previous authors ([@E98] and references therein). These clusters appear to have remarkably similar radial densities and bolometric X-ray luminosity profiles, especially when those with significant substructure are removed. With cooling implemented the cluster bolometric X-ray luminosity profiles span a broader range. The formation of a central galaxy within each halo acts to steepen the dark matter profile, supporting the conclusion of the lensing studies [@K96] that the underlying potential that forms the lens only has a small core. For the largest cluster, a significant amount of baryonic material has cooled and built up a large central galaxy. This localised mass deepens the potential well and contains hot gas with a steeply rising density ($\rho \propto r^{-2.75}$ in the inner regions). For this cluster around 80 percent of the bolometric X-ray emission comes from the galactic region and this must therefore be viewed as a lower limit as the central emission is unresolved. Such a large central spike to the X-ray emission is already only weakly consistent with the latest observational data [@EF99]. For the remaining 19 clusters the central galaxy is not so dominant and a shallower central potential well is formed. In these cases the slope of the central hot gas is $\rho \propto r^{-0.5}$ and the total X-ray emission is well resolved. In principle, the presence of a large galaxy could resolve the problem of the slope of the X-ray luminosity - temperature relation. In large clusters, large central galaxies are more likely to be present and this galaxy deepens the local potential well, boosting the emission above the theoretically expected $L_X \propto T^2$ regression line. Getting a reasonable amount of material to cool into the central galaxy is seen to be of vital importance. Conclusions =========== We have performed two N-body plus hydrodynamics simulations of structure formation within a volume of side $100\Mpc$, including the effects of radiative cooling but neglecting star formation and feedback. By repeating one of the simulations without radiative cooling of the gas we can both compare to previous work and study the changes caused by the cooling in detail. A summary of our conclusions follows. \(a) The bolometric luminosity for the clusters with radiative cooling is around five times lower than for matching clusters without it. Except for the largest cluster where the massive central galaxy produces a deep potential well the X-ray luminosity profile is less centrally concentrated than in the non-cooling case with a greater contribution coming from larger radii. This effect assists in convergence as we are less dependent upon the very centre of the cluster profile. \(b) The spread of the X-ray luminosity – temperature relation is well reproduced by our clusters. Our non-cooling clusters lie close to the regression line suggested by [@E98] and have a similar slope ($\rho \propto r^{-2}$). We suggest that the increasing dominance of a large central galaxy on the local potential may produce the luminosity excess that drives the observed X-ray luminosity – temperature relation away from the theoretically predicted slope. [99]{}[ David, L. P., Jones, C. & Forman, W. 1995, , 445, 578 Eke, V. R., Navarro, J. F. & Frenk, C. S. 1998, , 503, 569 Ettori, S. & Fabian, A. C. 1999, , 305, 834 Gardner, J. P., Sharples, R.M., Frenk, C.S., Carrasco, E., 1997, , 480, 99 Katz, N. & White, S. D. M. 1993, , 412, 455 Kneib, J. -P., Ellis, R. S., Smail, I., Couch, W. J. & Sharples, R. M. 1996, , 471, 643 Lewis, G. F., Babul, A., Katz, N., Quinn, T., Hernquist, L., Weinberg, D. H., 1999, astroph/9907097 Navarro, J. F., Frenk, C. S. & White, S. D. M. 1995, , 275, 720 Pearce, F. R., Jenkins, A., Frenk, C. S., Colberg, J. M., White, S. D. M., Thomas, P. A., Couchman, H. M. P., Peacock, J. A., Efstathiou, G., 1999, , 521, 99 Pearce, F. R., Couchman, H. M. P., Thomas, P. A., Edge, A. C., 1999, astroph/9908062 Suginohara, T. & Ostriker, J. P. 1998, , 507, 16 ]{}
--- abstract: \| A giant star-forming region in a metal-poor dwarf galaxy has been observed in optical lines with the 10-m Gran Telescopio Canarias and in the emission line of CO(1-0) with the NOEMA mm-wave interferometer. The metallicity was determined to be $12+\log({\rm O/H})=7.83\pm 0.09$, from which we estimate a conversion factor of $\alpha_{\rm CO}\sim100\;M_\odot\;{\rm pc}^{-2}\left( \rm{K \;km\; s}^{-1}\right)^{-1}$ and a molecular cloud mass of $\sim2.9\times10^7\;M_\odot$. This is an enormous concentration of molecular mass at one end of a small galaxy, suggesting a recent accretion. The molecular cloud properties seem normal: the surface density, $120\;M_\odot$ pc$^{-2}$, is comparable to that of a standard giant molecular cloud, the cloud’s virial ratio of $\sim1.8$ is in the star-formation range, and the gas consumption time, $0.5$ Gyr, at the present star formation rate is typical for molecular regions. The low metallicity implies that the cloud has an average visual extinction of only $0.8$ mag, which is close to the threshold for molecule formation. With such an extinction threshold, molecular clouds in metal-poor regions should have high surface densities and high internal pressures. If high pressure is associated with the formation of massive clusters, then metal-poor galaxies such as dwarfs in the early universe could have been the hosts of metal-poor globular clusters. author: - 'Bruce G. Elmegreen, Cinthya Herrera, Monica Rubio, Debra Meloy Elmegreen, Jorge Sánchez Almeida, Casiana Muñoz-Tuñón, Amanda Olmo-García' title: 'NOEMA Observations of a Molecular Cloud in the low-metallicity Galaxy Kiso 5639' --- Introduction {#intro} ============ Kiso 5639 [@kiso-all] is a dwarf galaxy with a kpc-size starburst at one end, giving the system a tadpole or cometary shape [@elm12]. The rotation speed of $\sim 35$ km s$^{-1}$ [@jorge13] combined with a radius of 1.2 kpc in the bright part of the disk implies that the dynamical mass there is $3\times 10^8/\sin^2 i\;M_{\odot}$, which is a factor of $\sim 6$ larger than the stellar mass of $5\times 10^7\;M_{\odot}$ from Sloan Digital Sky Survey photometry [@elm12] and comparable to the total H<span style="font-variant:small-caps;">i</span> mass of $\sim3\times10^8\;M_\odot$ [@salzer]. The galaxy is a member of our spectroscopic survey of 22 low metallicity dwarfs where the metallicity in the starburst “head” appears to be less than in the rest of the galaxy (the “tail”); 16 others in this survey have the same metallicity drop, 3 do not and 2 are ambiguous [@jorge13; @jorge14; @jorge15]. This peculiar pattern of metallicity suggests that the starbursts in these systems were triggered by accreting gas with lower metallicity than in the rest of the galaxy. Other examples of metallicity drops were reported in [@levesque11], [@haurberg13] and [@lagos18]. HST observations of Kiso 5639 [@elmegreen16] in six UV-optical and H$\alpha$ filters were used to resolve the head and derive the star formation properties. The head contains 14 young star clusters more massive than $10^4\;M_\odot$ and an overall clustering fraction for star formation of $25-40$%. The H$\alpha$ luminosity of the core region of the head is $8.8\pm0.16\times10^{39}$ erg s$^{-1}$ inside an area of $3.6\times3.6$ square arcsec. The corresponding star formation rate is $\sim0.04\;M_{\odot}$ yr$^{-1}$. This rate is based on a conversion factor of $SFR=4.7\times10^{-42}L(H\alpha)\;M_\odot$ yr$^{-1}({\rm erg\;s}^{-1})^{-1}$ that is appropriate for low metallicity, which introduces a factor of $0.87$ times the standard value [@hunter10; @kennicutt12]. For a distance of 24.5 Mpc [@elm12], the corresponding area is $0.18$ kpc$^2$ ($430\times430$ square pc), and the star formation rate per unit area is $0.23\;M_\odot$ pc$^{-2}$ Myr $^{-1}$. This is a high rate for a small galaxy and it suggests there is a large reservoir of dense gas in the head. For the conventional molecular gas consumption time of $\sim2$ Gyr [@bigiel08], the molecular mass surface density would be $450\;M_\odot$ pc$^{-2}$, which is $\sim40$ times higher than the average stellar surface density and $\sim7$ times higher than the average dynamical surface density in the bright part of the disk. For the Kennicutt-Schmidt relation in whole galaxies from Figure 11 in [@kennicutt12], the required gas surface density would be $\sim200\;M_\odot$ pc$^{-2}$. These high gas surface densities suggest that Kiso 5639 is lopsided and recently accreted at least $\sim10^8\;M_\odot$ of gas on one side. Simulations of such a process are in [@verbeke14] and [@ceverino16]. Detection of CO emission from the head of Kiso 5639 would help to clarify the situation. The metallicity of this galaxy, determined with the 2.5m Nordic Optical Telescope from the \[NII\]$\lambda6583$ to H$\alpha$ line ratio in [@jorge13], was $12+\log({\rm O/H})\sim7.48\pm0.04$. CO is highly underabundant compared to H$_2$ at low metallicity [@leroy11]. The lowest metallicities in regions detected in CO, i.e., $7.5$ in Sextens A and DDO 70 [@shi15; @shi16] and 7.8 in WLM [@elmegreen13; @rubio15] occur in dwarf Irregular galaxies where the CO clouds are small and in the centers of large HI and H$_2$ clouds. For the large molecular mass expected in Kiso 5639 and in other galaxies of its type, CO emission should be more evident. To search for CO in Kiso 5639, we observed CO(1-0) in the head region with NOEMA in D configuration. The result was a detection corresponding to a gas surface density similar to that estimated above, depending on the assumed $\alpha_{\rm CO}$ conversion factor. Because the metallicity is important for this $\alpha_{\rm CO}$, we re-observed Kiso 5639 in several optical lines including \[OII\]$\lambda$3727 and \[OIII\]$\lambda$4363 using the 10m Gran Telescopio Canarias (GTC) to get the metallicity again. The observations, results, and implications are discussed in the next three sections. Observations ============ Metallicity ----------- The metallicity of Kiso 5639 is central to the discussion on the origin of the CO emission. [@jorge13] found a drop in metallicity coinciding with its largest star-forming region, with the value of $12+\log({\rm O/H})\simeq 7.5$. This estimate was based on the strong-line ratio N2 ($\log([{\rm NII}]\lambda 6583/{\rm H}\alpha$)), which is known to have biases [e.g., @morales14]. On the other hand, the SDSS spectrum of the region gives a metallicity of around 7.8 when using either the direct method (DM) or the HIICM procedure by Perez-Montero (2014) (Ruben García-Benito 2017, private communication). In order to remove uncertainties and secure the metallicity estimate, we obtained long-slit spectra of the galaxy integrating for 2.5 hours with the instrument OSIRIS at the 10-m GTC telescope. The slit was placed along the major axis. The resulting visible spectra cover from 3600 Å to 7200 Å with a spectral resolution around 550, thus containing all the spectral lines listed below that are needed for the metallicity measurement, including the critically important temperature sensitive line \[OIII\]$\lambda$4363. The data were calibrated in flux and wavelength using PyRAF[^1], and Gaussian fits provided the fluxes of the emission lines. We employed the code HIICM to infer O/H, which requires \[OII\]$\lambda$3727, \[OIII\]$\lambda$4363, H$\beta$, \[OIII\]$\lambda$5007, H$\alpha$, \[NII\]$\lambda$6583, and \[SII\]$\lambda\lambda$6717,6737. HIICM was chosen because it is robust and equivalent to the direct method when \[OIII\]$\lambda$4363 is available [@perez14; @sanchez16]. The oxygen abundance thus obtained was $12+\log({\rm O/H}) = 7.83 \pm 0.09$, considering $4^{\prime\prime}$ around the star-forming region. HIICM estimates error bars from the difference between the observed line fluxes and those predicted by the best fitting photoionization model. The inferred metallicity is similar when all the spectra of the galaxy are integrated ($7.78 \pm 0.11$), and even if only spectra from the faint tail are considered ($7.79 \pm 0.18$). HIICM also provides N/O, which turns out to be roughly constant with a value around $\log({\rm N/O})\simeq -1.5\pm 0.1$, typical for the metallicity assigned to Kiso 5639 [e.g., @vincenzo16]. Curiously, the new data still show a slight metallicity drop at the head from the N2 method, down to $12+\log({\rm O/H}) \sim 7.6$, with the same value, $\sim7.8$, as for the HIICM measurements elsewhere. There is also a slight indication of a drop in N/O in the head, by a factor of $\sim3$, which is a $\sim3\sigma$ deviation for the head measurement compared to the average in the tail, but only a $1\sigma$ devitation for the weaker tail measurement in comparison to the head. This apparent drop in head metallicity from the \[NII\] line explains the measurement in [@jorge13], but it is not viewed as relevant for our CO observations, which need the Oxygen abundance. Thus we use the HIICM value of $12+\log({\rm O/H}) = 7.83 \pm 0.09$ to determine $\alpha_{\rm CO}$ in the head where CO is detected. CO Observations --------------- We obtained observations of Kiso 5639 on May, July and August 2017, with the Northern Extended Millimeter Array (NOEMA) at Plateau de Bure, in the CO ($J=1-0$) line, at 114.572 GHz redshifted to the velocity of Kiso 5639. The observations were carried out with either 7 or 8 antennas in Configuration D, with baselines between 15 m and 175 m. We used the Widex correlator, with a total bandwidth of 3.6 GHz and a native spectral resolution of 1.95 MHz (5.1 km s$^{-1}$). Data reduction, calibration and imaging were performed with CLIC and MAPPING softwares of GILDAS[^2], using standard procedures. Images were reconstructed using the natural weighting, resulting in a synthesized beam of 2$\farcs$8$\times$3$\farcs$7 (PA=26$^{\circ}$). The rms noise in the CO cube is 14.3 mK (1.6 mJy beam$^{-1}$) in a 5.1 km s$^{-1}$ channel. The resulting data cube was integrated between LSR velocities of $-62.3$ km s$^{-1}$ and $-26.5$ km s$^{-1}$ where CO emission was observed. The rms was measured in the integrated image and a mask for pixels with emission greater than $ 2\sigma$ was made. The average spectrum of Kiso 5639 was obtained within this mask. A 1D Gaussian fit to this spectrum gives $V_{\rm LSR}= -42.1\pm 2.2$ km s$^{-1}$, FWHM$=35.2\pm5.3$ km s$^{-1}$, peak emission $32.1\pm4.5$ mK, and average integrated line profile, $1203\pm247$ mK km s$^{-1}$, as summarized in Table 1. Figure \[Fig1\_CH\_23apr2018\] shows a map of the main emission and the integrated spectrum. The velocity resolution is 5.11 km s$^{-1}$, the half-power beam size is $2.85^{\prime\prime}\times3.72^{\prime\prime}$, and the rms is $1.6$ mJy/beam. The equivalent radius of the source is $2.35^{\prime\prime}$ (280 pc), which corresponds to the area of the detection contour, $2.45\times10^5$ pc$^{2}$. The emission has two peaks separated by $2^{\prime\prime}$, which is 240 pc. Spectra determined for each peak are shown in Figure \[Fig2\_CH\_23apr2018\]. The blue histogram is from one of the peaks and the green histogram is from the other; the red double peak histogram is the composite. Each histogram has a Gaussian fit, as indicated by the red lines and the black line. The integrated line fits to these spectra and the derived quantities are in Table 1. Several other regions inside the primary beam of the NOEMA telescope were also suspected of containing CO emission, but the brightness temperatures were less than 20% of the peak shown in the figure, and we do not consider them to be definitive detections of molecular gas. These include the contoured regions to the southwest and northwest of the main emission in Figure \[Fig1\_CH\_23apr2018\]. Results ======= Conversion from the observed CO flux to a molecular mass depends on the conversion factor, $\alpha_{\rm CO}$. [@hunt15] suggested an extrapolation of $\alpha_{\rm CO}$ from the solar neighborhood value to low metallicity $Z$ as $\alpha_{\rm CO}=4.3(Z/Z_\odot)^{-2}\;M_\odot\;{\rm pc}^{-2}\left( \rm{K \;km\; s}^{-1}\right)^{-1}$, where $Z_\odot$ is the solar metallicity corresponding to $12+\log({\rm O/H}) = 8.69$ [@asplund09], and $Z$ is the metallicity for Kiso 5639 corresponding to $12+\log({\rm O/H}) = 7.83$. The local conversion factor is $\alpha_{\rm CO,\odot}=4.3 \;M_\odot\;{\rm pc}^{-2}\left( \rm{K \;km\; s}^{-1}\right)^{-1}$ including He and heavy elements [@bolatto13]. Similarly, [@amorin16] suggested an extrapolation as $Z$ to the power $-1.5$. In the first case, the result for Kiso 5639 would be $\alpha_{\rm CO}=225$ in these units, and in the second case it would be $84$. Compared to the WLM galaxy, where $\alpha_{\rm WLM}\sim124\pm60$ [@elmegreen13] and $12+\log({\rm O/H})_{\rm WLM}=7.8$, $\alpha_{\rm CO}=108$ and 112 for the same power law scalings. On the other hand, if we extrapolate between the low-$Z$ $\alpha_{\rm CO}$ values determined by [@shi16], we would get $\alpha_{\rm CO}\sim600$ and 900 at $Z=7.8$, respectively, which results in a much larger molecular mass. Here we take $\alpha_{\rm CO}\sim100$ as a conservative estimate. For the observed CO average integrated line profile, $1.2\pm0.2$ K km s$^{-1}$, estimated $\alpha_{\rm CO}=100\;M_\odot\;{\rm pc}^{-2}\left( \rm{K \;km\; s}^{-1}\right)^{-1}$, and CO source area, $2.45\times10^5$ pc$^2$, the H$_2$ mass is the product of these, $2.9\times10^7\;M_\odot$, with the largest uncertainty in the value of $\alpha_{\rm CO}$, which is probably a factor of 2. The mass divided by the total emitting area is the molecular surface density, $\Sigma_{\rm mol}= 120\;M_\odot$ pc$^{-2}$. In this same region, the star formation rate is $\sim0.04\;M_{\odot}$ yr$^{-1}$ and the star formation surface density is $0.23\;M_\odot$ pc$^{-2}$ Myr $^{-1}$, as given above. These imply that the molecular gas consumption time is $0.5-0.7$ Gyr, which is slightly less than in normal spiral galaxy disks [@bigiel08; @schruba11]. This time is about the same as in the SMC, where the metallicity is also low ($Z\sim0.2Z_\odot$) and $\alpha_{\rm CO}$ is high, $\sim220\;M_\odot\;{\rm pc}^{-2}\left(\rm{K \;km\; s}^{-1}\right)^{-1}$ [@bolatto11]. Following the same method, the masses of the two components of the emission in Figures \[Fig1\_CH\_23apr2018\] and \[Fig2\_CH\_23apr2018\] can be determined (see Table 1). The component in the east has a velocity of $-35$ km s$^{-1}$ and an average integrated line profile of $0.66\pm0.25$ K km s$^{-1}$, and the component in the west has a velocity of $-54$ km s$^{1}$ and an average integrated line profile of $0.47\pm0.22$ K km s$^{-1}$. We assume each has the same total area. Then the masses are these line integrals multiplied by the areas and the $\alpha_{\rm CO}$ factor, which are $1.6\times10^7 \;M_\odot$ in the east and $1.1\times10^7 \;M_\odot$ in the west, with surface densities of $66\;M_\odot$ pc$^{-2}$ and $47\;M_\odot$ pc$^{-2}$, respectively. The velocity difference between the components, $19$ km s$^{-1}$, is nearly half of the rotation velocity of the galaxy, $34.7\pm6.2$ km s$^{-1}$ [@jorge13], suggesting a catastrophic event. The masses, radii, and velocity dispersions can be combined to determine the virial ratio, $5R\sigma^2/(GM)$, where $\sigma=0.42\times{\rm FWHM}$. These ratios require a deconvolved radius, $R$ which is $(2.35^2-2.85\times3.73/4)^{0.5}=1.69^{\prime\prime}$ for the total cloud (using the source and beam sizes from above). The radii of the components are taken to have upper limits equal to the average beam size of $1.63^{\prime\prime}$. The results are in Table 1. The virial ratios are in the range $<1.1$ to 1.8, which suggests that the whole cloud and the two separate parts are gravitationally self-bound. Figure 3 shows the HST image of Kiso 5639 in H$\alpha$, V, and B bands with superimposed CO contours at values of $2\sigma$, $3\sigma$ and $4\sigma$. The main molecular cloud is $2.2^{\prime\prime}=260$ pc to the east of the bright star formation region. This offset means that comparisons to the Kennicutt-Schmidt relation are not exactly appropriate. Still, the molecular gas reservoir for the starburst appears to be of sufficient mass to explain the observed young stars. Figure 4 shows the same CO contours on the HST H$\alpha$ image of the whole galaxy. An expanded view of the head region is on the left, where the black contours are 3%, 10% and 20% of the peak H$\alpha$ emission. Offset positions are as in Fig. \[Fig1\_CH\_23apr2018\]. Implications ============ Subject to uncertainties about the conversion factor between CO emission and molecular gas mass, our observations suggest that a $2.9\times10^7\;M_\odot$ molecular cloud with a surface density of $\sim 120\;M_\odot$ pc$^{-2}$ spanning a region 560 pc in diameter is associated with the starburst head of the tadpole galaxy Kiso 5639, where HST observations previously suggested that the star formation rate is $0.04\;M_\odot$ yr$^{-1}$ and the star formation surface density is $0.23\;M_\odot$ pc$^{-2}$ Myr$^{-1}$. The molecular surface density is comparable to that in giant molecular clouds in local galaxies, which average $100-200\;M_\odot$ pc$^{-2}$ [@heyer15], but the extinction through the molecular cloud in Kiso 5639 should be much less. Considering the local conversion factor between color excess and HI column density [@bohlin78], $A_{\rm V}=N/(1.87\times10^{21}\;{\rm cm}^{-2})$, for a ratio of total to selective extinction $R=3.1$, and using a mean molecular weight of 1.36 times the hydrogen mass, the extinction through a cloud is related to its surface density by $$\Sigma_{\rm gas}=20.2A_{\rm V}\left(Z_\odot/Z\right)\;M_\odot\;{\rm pc}^{-2}.$$ For Kiso 5639, $Z_\odot/Z=7.2$, so the observed $\Sigma_{\rm gas}=120\;M_\odot\;{\rm pc}^{-2}$ corresponds to $A_{\rm V}=0.8$ mag. In the solar neighborhood, this is comparable to the threshold for CO formation [@pineda08; @glover12]. The same extinction threshold was obtained for the WLM galaxy at a metallicity of $12+\log({\rm O/H})=7.8$, where pc-size CO clouds in the core of a giant HI and H$_2$ envelope had a total shielding column density equivalent to $\sim1.5$ mag visual extinction (@rubio15, see also @schruba17). The assumed value of $\alpha_{\rm CO} = 100 \;M_\odot \;{\rm pc}^{-2} \left(\rm{K \;km\; s}^{-1}\right)^{-1}$ implies that the ratio of invisible molecular hydrogen to observed CO is high, $\sim23$ times higher than in the solar neighborhood. A similar situation arises for the low-metallicity galaxy WLM, where the CO clouds resolved by ALMA are pc-scale inside resolved HI and dust clouds that are $\sim200$ pc in size. If the peak CO antenna temperature of $\sim30$ mK in Kiso 5639 corresponds to a beam-diluted thermal temperature of $\sim30$ K, which is not unreasonable for a molecular region of intense star formation [e.g., @glover12], then the beam-dilution factor of the total CO is $\sim10^{-3}$, and their individual radii would be $\sim0.03/\sqrt{N}$ of the overall cloud radius, or $\sim9/\sqrt{N}$ pc for $N$ cores. Such small CO cores would presumably have a collective motion that is observed as the CO emission Gaussian linewidth of $\sigma=15$ km s$^{-1}$. Considering again a typical dense cloud temperature of 30 K, these motions would have a Mach number of $\sim46$ and a compression ratio in the shocked regions of approximately the square of this, $\sim2000$. The average compressed density is then the compression ratio times the average density. The average density is the surface density divided by the cloud thickness. For a self-gravitating slab, the thickness is $2\sigma^2/(\pi G \Sigma_{\rm gas})$, which is 140 pc in our case. Then the average density is $\sim7.6$ cm$^{-3}$. If this average density is compressed by shocks, then the density in the compressed regions, where the CO might actually be located, is $1.6\times10^4$ cm$^{-3}$. For a temperature of $\sim30$ K, the thermal pressure would be $4.8^\times10^5k_{\rm B}$. This derivation of pressure is the same as what we would get from the equation for self-gravitational binding pressure, $(\pi/2)G\Sigma_{\rm gas}^2$, using $\Sigma_{\rm gas}\sim120\;M_\odot$ pc$^{-2}$. Any additional contribution to the surface density from atomic gas in an envelope around the molecular cloud would increase the pressure. The derived pressure inside the Kiso 5639 molecular cloud is comparable to the pressures in local giant molecular clouds, corresponding to similar surface densities. Pressure has been proposed to control the fraction of mass in the form of bound clusters in the star formation process [@kruijssen12]. Kiso 5639 has a relatively high clustering fraction, 30%-45% [@elmegreen16] compared to other dwarf galaxies [@billett02], perhaps because of its relatively high pressure. Its clustering fraction is about the same as in spirals [@adamo15] where the pressure is the same. High pressure might also be necessary to form more massive clusters [@elmegreen01]. The present observation of molecular clouds close to the extinction threshold for molecule formation [and in @rubio15; @schruba17] suggest that the surface densities (and therefore pressures) of star-forming clouds will be higher at lower metallicity. That is, low metallicity and the corresponding low dust opacity per column of gas could diminish the ability of self-gravitating clouds to shield themselves against background starlight. This would delay the formation of molecules and cold thermal temperatures until the cloud surface density is high. Then the pressure, which depends only on the surface density, would become high too. The result would be a greater likelihood of gravitationally bound massive clusters in low metallicity galaxies. Massive concentrations of molecular gas as in Kiso 5639 would be needed too. We thank R. García-Benito for deriving the abundance of Oxygen from the SDSS spectrum. We are grateful to the referee for comments. M.R. wishes to acknowledge support from CONICYT(CHILE) through FONDECYT grant No1140839 and partial support from project BASAL PFB-06. M.R. is a member of UMI-FCA, CNRS/INSU, France (UMI 3386). AOG thanks Fundación La Caixa for financial support in the form of a PhD contract, and SA and CMT acknowledge MINECO for funding through the project AYA2016-79724-C4-2-P. This work is based on observations carried out under project number S17AP with the IRAM NOEMA Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). Based on observations made with GTC, in the Spanish Observatorio del Roque de los Muchachos of the IAC, under DDT. Adamo, A., Kruijssen, J. M. D., Bastian, N., Silva-Villa, E., & Ryon, J. 2015, MNRAS, 452, 246 Amorín, R., Muñoz-Tuñón, C., Aguerri, J. A. L., & Planesas, P. 2016, A&A, 588, 23 Asplund, M., Grevesse, N., Sauval, A.J., & Scott, P. 2009, ARA&A, 47, 481 Bigiel, F., Leroy, A., Walter, F., Brinks, E., de Blok, W. J. 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--- abstract: 'Quantum dots are arguably the best interface between matter spin qubits and flying photonic qubits. Using quantum dot devices to produce joint spin-photonic states requires the electronic spin qubits to be stored for extended times. Therefore, the study of the coherence of spins of various quantum dot confined charge carriers is important both scientifically and technologically. In this study we report on spin relaxation measurements performed on five different forms of electronic spin qubits confined in the very same quantum dot. In particular, we use all optical techniques to measure the spin relaxation of the confined heavy hole and that of the dark exciton – a long lived electron-heavy hole pair with parallel spins. Our measured results for the spin relaxation of the electron, the heavy-hole, the dark exciton, the negative and the positive trions, in the absence of externally applied magnetic field, are in agreement with a central spin theory which attributes the dephasing of the carriers’ spin to their hyperfine interactions with the nuclear spins of the atoms forming the quantum dots. We demonstrate that the heavy hole dephases much slower than the electron. We also show, both experimentally and theoretically, that the dark exciton dephases slower than the heavy hole, due to the electron-hole exchange interaction, which partially protects its spin state from dephasing.' author: - Dan Cogan - Oded Kenneth - 'Netanel H. Lindner' - Giora Peniakov - Caspar Hopfmann - Dan Dalacu - 'Philip J. Poole' - Pawel Hawrylak - David Gershoni bibliography: - 'spin\_relaxation\_dynamics.bib' title: 'Depolarization of Electronic Spin Qubits Confined in Semiconductor Quantum Dots. ' --- \#1[\|\#1]{} \#1[\#1\|]{} Introduction ============ The electronic spin in semiconductor nanostructures can often be described as an isolated physical two level system. As such it has long been considered an excellent qubit with great potential to be used in future quantum information processing based technologies [@Loss1998; @spinsasqubits; @Kimble2008]. Moreover, semiconductor nanostructures, which confine single electrons, are easily integrated into electronic and optical devices and circuits, which dovetail with the contemporary semiconductor based electro-optic technology. Therefore, many efforts have been devoted recently to demonstrate that various forms of the electronic spin in semiconductor nanostructures and in particular in quantum dots (QDs) can be initiated and controlled with relatively high fidelities, using optics and electronics means [@Berezovsky_2008; @Press_2008; @Ramsay_2008; @Michler2017]. An important advantage of semiconductor electronic spin qubits, which are anchored to the device, is their strong interaction with photons, which can be used as flying qubits to communicate quantum information to remote locations [@Imamog_lu_1999; @Gao2012; @Greve2012; @Schaibley2013]. These advantages have been recently used for instance, to demonstrate that a QD confined electronic spin, can be used as an entangler for on demand production of a long string of entangled photons in a cluster state [@Schwartz2016]. The main decoherence mechanism of the confined electronic spin (central spin) in semiconductor QDs is its interaction with the spins of the nuclei in its vicinity [@Gammon_2001; @Efros2002; @Khaetskii2002; @Fischer2008]. Therefore, it is essential, both scientifically and technologically, to study and to characterize these dephasing processes. In this work we comprehensively study, both experimentally and theoretically, the dephasing dynamics of QD confined electronic spins in 5 different forms: a) conduction band electron, b) valence band heavy-hole, c) negative trion, d) positive trion, and e) dark exciton (DE). All in the same single QD. Semiconductor QDs are formed by $\sim10^{5}$ molecules of one semiconductor compound embedded in another semiconductor compound of higher bandgap energy. These formations give rise to nanometer scale three-dimensional (3D) potential traps, which confine single electronic charge carriers (electrons in the conduction bands and holes in the valence bands) and isolate them from their environment. The energy spectrum of these confined carriers is therefore discrete, giving rise to well defined and spectrally sharp optical transitions between these discrete levels [@marzin1994; @Dekel1998] . In Fig. \[fig:Spin-wavefunctions\] we display the electronic spin wavefunctions and Bloch-sphere representations of all the electronic spin qubits used in this work. The confined conduction electron levels have a vanishing atomic orbital momentum and thus their total spin projection on the QD growth direction is $\pm1/2$ . Therefore, they form physical two level systems or qubits [@DiVincenzo_2000]. The spin state of the qubit is represented on the Bloch sphere, where the spin up and the spin down states are located at the north and south poles of the sphere, respectively, and any superposition of these two states is represented by a point on the sphere’s surface. The confined valence-band electron states have total atomic orbital momentum of 1. The spin-orbit interaction, together with the quantum confinement along the growth direction and the biaxial lattice mismatch compressive strain, inherent to our strain induced QDs, results in a large energy splitting between the upper most valence states [@Ivchenko2005]. The highest valence electron states in which the orbital spin and electronic spin are parallel, are few tens meV higher than the states in which the orbital and electronic spins are anti-parallel. At low temperature, the valence band states are fully occupied. Confined positive charge carriers in the QD are therefore formed due to the absence of valence band electrons. Thus, the lowest energy hole states have angular momentum projection of $\pm3/2$ on the growth direction, (heavy-holes). A heavy hole, is yet another form of a QD confined electronic spin qubit [@Brunner2009; @De_Greve_2011] as shown in Fig. \[fig:Spin-wavefunctions\]. Another form of a confined electronic spin qubit is the electron-heavy-hole pair, or the exciton[@Benny2011; @Kodriano2012]. Excitons in which the heavy hole spin and the electron spin are anti-parallel have total spin projection of $\pm1$, they are optically active and therefore called bright excitons (BEs). The qubit that they form [@Benny2011; @Poem2011; @Kodriano2012] recombines within a short radiative lifetime (\~1 ns), which limits their use as a matter spin qubit. In contrast, excitons in which the electron and heavy-hole spins are parallel, are optically inactive since the electromagnetic radiation barely interacts with the electronic spin. These excitons are called dark excitons (DEs). They have total spin projection of $\pm2$ on the QD growth axis and live orders of magnitude longer than the BE [@McFarlane_2009]. Consequently they can be used for implementing sophisticated quantum information protocols [@Poem2010; @Schwartz2015; @Schwartz2016]. In the following we denote these three long lived forms of spin qubits (electron, heavy-hole and DE) - ground level qubits. The ground level qubits are stable, and once generated in the QD they live in it for a very long time. The ground level qubits can be optically excited to their respective excited level qubits by absorbing a single photon, which adds an electron-hole pair to the QD. Moreover, by using a resonantly tuned optical $\pi$-pulse, this excitation can be done deterministicaly. The resonant excitation converts the ground level qubits to their excited level qubits, as schematically described in Fig. \[fig:Spin-wavefunctions\]. In Fig. \[fig:Spin-wavefunctions\], green upward arrows represent the optical laser excitations, which convert the electron spin qubit to the negative trion qubit, the heavy-hole qubit to the positive trion qubit, and the DE to the spin-blockaded biexciton (BiE)-qubit. As can be seen in Fig. \[fig:Spin-wavefunctions\] the negative and positive trion qubits, are formed by three carriers. The negative trion is formed by two ground level conduction band electrons in a singlet state and a single ground level heavy-hole, while the positive trion is formed by two ground level heavy-holes and a single ground level electron. In both cases, the spin state of the trion qubits is determined by the minority carrier, $\pm3/2$ for the negative trion, and $\pm1/2$ for the positive trion. Unlike the trions , which are formed by three carriers, the BiE is formed by four carriers. Two ground level electrons in a singlet spin state, and two heavy holes with parallel spins in the ground and first excited valence band levels. Consequently, the BiE qubit spin states are $\pm3$, and it is determined by the two parallel heavy-holes’ spin directions. Once formed, the excited spin qubits, which are optically active, decay radiatively within the radiative lifetime of a ground level electron-hole pair (\~ 1 ns), by emitting a single photon and the system returns to the ground level qubit. The photon emissions are schematically described by the downward magenta arrows in Fig. \[fig:Spin-wavefunctions\]. If the upper qubit is properly initialized in a coherent superposition of its two spin states, the polarization of the emitted photon (“flying photonic qubit”) is expected to be entangled with the spin state of the ground level spin qubit, which remains in the QD [@Gao2012; @Greve2012; @Schaibley2013; @Schwartz2016]. ![\[fig:Spin-wavefunctions\](a) Spin wavefunctions and Bloch-sphere representations of the six matter spin qubits used in this work. The 6 qubits, represented by their Bloch spheres, are divided into 3 pairs of ground and excited level qubits. The spin wavefunctions of the ground- (excited-) level qubits are schematically described below (above) the respective Bloch spheres, where $\uparrow$ ($\Downarrow$) represents spin up electron (down heavy hole), and the blue (red) color represents a carrier in its ground (excited) energy level. Green upward arrows represent laser pulses which convert the ground level qubit to its respective excited level qubit. Magenta downward arrows represent single photons emitted from the excited qubits thereby returning to the ground level qubit. (b) The optical transitions and polarization selection rules for the electron-trion system, which form a ground level–excited level qubit pair. Note that in this example (as in all other cases) an optical $\Pi$-system is described, but the exciting laser pulse is tuned to an excited trion level, in order to facilitate polarization tomography of the emitted photon (magenta downdward arrows) by spectrally separating the emission from the exciting laser pulse (green upward arrows). The fast (\~70 ps [@Schwartz2015]) phonon-assisted relaxation of the excited trion to the ground trion level is represented by gray curly downward arrows. The right (left) hand circular polarization of the photons which connect the $1/2$ ($-1/2$) spin state of the ground level qubit with the $+3/2$ ($-3/2$) spin state of the excited qubit are marked by $R$ ($L$). ](wavefunctions_4){width="1\columnwidth"} At low temperatures and in the absence of external magnetic field, the main decoherence mechanism of these electronic spin qubits is the hyperfine interaction between the electronic (central) spin and the spin of the nuclei of the $\sim10^{5}$ atoms which form the QDs [@Gammon_2001; @Efros2002; @Khaetskii2002]. The two types of charge carriers in semiconductors, the negative conduction band electrons, and the positive, valence band holes interact differently with the nuclei, since their orbital momentum around the nucleus is different. The conduction electrons have zero atomic orbital momentum, while valence band holes have unit atomic orbital momentum. Consequently, the conduction electron’s wavefunction strongly overlaps with the nucleus and interacts with the nuclear spin via the Fermi contact interaction. In contrast, the valence hole’s wavefunction vanishes at the nucleus site and therefore its spin interacts with the nuclear spin via the weaker dipole-dipole hyperfine interaction [@Fischer2008]. In addition, while the conduction-electron interaction with the nuclei, which we denote by $\gamma_{e}$ is isotropic, the interaction of the valence heavy hole for which the orbital angular momentum and the spin are aligned parallel to the growth direction, is anisotropic. We denote by $\gamma_{h_{z}}$the interaction of the valence heavy-hole spin with the nuclei spin bath along the QD growth axis ($\hat{z}$) and by $\gamma_{h_{p}}$ the interaction with nuclear spins in the plane perpendicular to $\hat{z}$. The dynamics of the electronic central spin can be divided into two different time domains as schematically described in Fig. \[fig:spin-dephasing\] a, b and c for the electron, heavy hole and DE spins respectively [@Efros2002]. ![image](SpinDephasingProcess_3){width="100.00000%"} During the first stage, the central spin precesses around a mean effective magnetic field generated by the frozen fluctuations of the nuclear spins in its vicinity. The electron interacts with the nuclear field via the isotropic Fermi contact hyperfine interaction marked by $\gamma_{e}$, while the heavy-hole interacts via the anisotropic dipole-dipole hyperfine interaction marked by $\gamma_{h_{z}}$ and $\gamma_{h_{p}}$. As the DE is formed by an electron-hole pair with parallel spins, each of these carriers interacts with the nuclear magnetic field, while at the same time they also interact with each other, via the electron-hole exchange interactions. The most important term in this interaction is the isotropic term $\Delta_{0}$ [@Bayer2002; @Ivchenko2005], separating the DE and BE (an antiparallel electron-hole pair) energy levels. Being much stronger than the hyperfine interactions it prevents the separate spin flip of either one of the two individual spins and consequently protects the DE spin from dephasing. It turns out, as we show in Appendix B, below, that the DE nuclear field induced dephasing is caused mainly due to small DE-BE mixing terms (of order $10^{-3}$). During the second stage, at longer times, the fluctuations in the nuclear magnetic field can no longer be considered “frozen” and they slowly evolve in time. This evolution is described as local precession of the effective magnetic field around local directions denoted by $\hat{n}$. A relatively simple model describes this motion as generated by the quadrupole interaction (denoted by $\gamma_{Q}$) of the nuclear spins with the strain induced electric fields gradients in the QD [@Sinitsyn2012; @Bechtold2015]. We adopt this description, since it permits analytic solution to the problem, thereby simplifying the comparison with the measured data, while keeping the generality of our approach. Finally, at yet longer times, which is beyond the scope of this work, the nuclei also interact with each other via the dipole-dipole nuclear interaction [@Erlingsson2004]. During the second stage the central spin continues to interact with the slowly varying effective nuclear magnetic field in the same manner as it does during the first stage. Therefore, the central spin dynamics can be described as a sort of “convolution” between the relatively fast dynamics of the spin around the average nuclear magnetic field, with the dynamics of the slowly varying nuclear field. The details of the model involved in these calculations, which follows references [@Efros2002; @Sinitsyn2012; @Bechtold2015], describing the evolution of the electron, and the generalization of the model to include the heavy-hole evolution, are described in Appendix A. The model which describe the dynamics of the DE is developed in Appendix B. A great deal of effort was devoted to study the coherence properties of the central electronic spin for both, conduction band electrons [@Bluhm2010; @Braun2005], and valence band heavy-holes [@Brunner2009; @Eble2009; @Fras2011; @Li2012; @Gerardot2008], confined in QDs. The temporal evolution of a single electron spin at vanishing external magnetic field was experimentally measured recently by Bechtold and coworkers [@Bechtold2015]. To the best of our knowledge, similar measurements for the heavy-hole as a central spin have not been reported so far. Here, we present comprehensive measurements of the spin depolarization dynamics for both the electron and the heavy hole as well as for their correlated pair – the DE. All these forms of central electronic spin are confined to the same QD. In addition, we show, by measuring the temporal evolution of the positive and negative trions’ spins, that the presence of two additional paired charge carriers, does not affect the central spin depolarization. Our measurements were preformed optically without applying any external magnetic field. In addition, we carried out the experiments in a way which prevented the generation of a steady state nuclear Overhauser field. The experimental methods and measurements are described below and the measured results are compared with the central spin models discussed in the Appendices. The device and experimental methods =================================== The InP nanowire containing a single InAsP quantum dot [@Dalacu2009; @Dalacu2012; @Bulgarini2014] was grown using chemical beam epitaxy with trimethylindium and pre-cracked $\mathrm{PH_{3}}$ and $\mathrm{AsH_{3}}$ sources. The nanowires were grown on a $\mathrm{SiO_{2}}$-patterned (111)B InP substrate consisting of circular holes opened up in the oxide mask using electron-beam lithography and a hydrofluoric acid wet-etch. Gold was deposited in these holes using a self-aligned lift-off process, which allows the nanowires to be positioned at known locations on the substrate. The thickness of the deposited gold is chosen to give 20-nm to 40-nm diameter particles, depending on the size of the hole opening. The nanowires were grown at $420^{\circ}$ C with a trimethylindium flux equivalent to that used for a planar InP growth rate of 0.1 $\text{μm/hr}$ on (001) InP substrates at a temperature of $500^{\circ}$ C. The growth is a two-step process: (i) growth of a nanowire core containing the quantum dot, nominally 200 nm from the nanowire base, and (ii) cladding of the core to realize nanowire diameters (around 200 nm) for efficient light extraction. The quantum dot diameters are determined by the size of the nanowire core. The particular QD reported on here has diameter of $\sim30$ nm. The sample was placed inside a sealed metal tube cooled by a closed-cycle helium refrigerator maintaining a temperature of 4 K. A 60 microscope objective with numerical aperture of 0.85 was placed above the sample and used to focus the laser beams on the sample surface and to collect the emitted PL from it. Pulsed laser excitations were used. The picosecond pulses were generated by two synchronously pumped dye lasers at a repetition rate of 76 MHz. The temporal width of the pulses was 12 ps and their spectral width $\sim100$ μeV. Light from a continuous wave (CW) laser, modulated by an acousto-optic modulator, synchronized with the dye lasers, was used to produce pulses of up to 30 ns duration. These pulses were used to set the average QD charge state [@Benny2012]. A second CW laser, modulated by an electro-optic modulator, was used to produce depletion pulses of 30 ns duration [@Schmidgall2015]. The timing between the two synchronized ps pulses was controlled using 2 cavity dumpers which effectively reduced the repetition rate down to 0.5 MHz. In addition, a computer controlled motorized delay line was used to finely tune the temporal delay between the pulses. The polarizations of the excitation pulses were independently adjusted using polarized beam splitters (PBS) and two pairs of computer-controlled liquid crystal variable retarders (LCVRs) [@Schwartz2016]. The collected PL was equally divided into 2 beams by a non-polarizing beam splitter. Two pairs of LCVRs and a PBS were then used to analyze the polarizations of each beam. This way the emitted PL was divided into four beams, allowing selection of two independent polarization projections and their complementary polarizations. The PL from each beam was spectrally analyzed by either a 1 or 0.5 meter monochromator and detected by a silicon avalanche photodetector coupled to a PicoQuant HydraHarp 400™ time-correlated photon counting and time tagging system, synchronized with the pulsed lasers. This way the arrival times of up to 4 emitted photons have been recorded with respect to the synchronized laser pulses. ![\[fig:Experimental-procedures\]Schematic description of the experiments for measuring the spin dynamics of: a) Positive and negative trions, b) Single electron and single heavy-hole. c) Dark exciton. The optical transitions in each experiment are described by the energy level diagram to the right. The carrier’s spins are marked in the figure using the notations of Fig. \[fig:Spin-wavefunctions\], where blue (red) color represent ground (excited) single carrier states. CW1 and CW2 represent the 20 ns gated CW laser pulses where $P_{1}$ and $P_{2}$ represent the 12 ps pulses produced by the synchronously pumped and cavity dumped dye lasers. $\Delta t$ is the time delay between the two pulses in each repetition period, controlled by two cavity dumpers and a delay line. $D_{1}$ and $D_{2}$ represent the emission and time resolved detection of the two single photons emitted as a result of the $P_{1}$ and $P_{2}$ excitations. ](procedure_3){width="1\columnwidth"} We used the optical transitions between the ground level qubits and the excited level qubits to initialize the spin state of both qubits, and then for probing the spin state of the qubits at a later time. We facilitate the optical transition selection rules of the $\Pi$-systems described in Fig. 1b in order to do that. For initializing the excited qubit, one simply applies an $R$ or $L$ polarized $\pi$-pulse. For probing the excited qubit spin projection, one simply measures the degree of circular polarization of the emitted photons $\hat{S}_{z}=\left(I_{R}-I_{L}\right)/\left(I_{R}+I_{L}\right)$ where $I_{R(L)}$ is the measured emission intensity projected on right (left) hand circular polarization. The initialization of the ground level qubit is provided by detecting $R$ or $L$ polarized single photon, which heralds the spin state of the qubit at the photon emission time. Probing the ground level qubit spin state is done by first converting the state into the state of the excited level qubit, using an horizontally linearly polarized ($H=\left(R+L\right)/\sqrt{2}$) $\pi$-pulse, and then measuring the time resolved degree of circular polarization of the emitted photons. For example, in Fig. \[fig:Spin-wavefunctions\](b) if the electron spin state before the pulse is described by: $\hat{\rho}_{\mathrm{electron}}=p{\left\|\Psi_{\mathrm{electron}}\right\rangle }{\left\langle \Psi_{\mathrm{electron}}\right\|}$+$\left(1-p\right)\frac{1}{2}\mathbb{I}$, where $\mathbb{I}$ is the identity matrix and $p$ is the probability of $\hat{\rho}_{\mathrm{electron}}$ being in a pure state ${\left\|\Psi_{\mathrm{electron}}\right\rangle }=\alpha{\left\|\uparrow\right\rangle }+\beta{\left\|\downarrow\right\rangle }$, then after the pulse the photogenerated trion spin state is given by: $\hat{\rho}_{\mathrm{trion}}=p{\left\|\Psi_{\mathrm{trion}}\right\rangle }{\left\langle \Psi_{\mathrm{trion}}\right\|}$+$\left(1-p\right)\frac{1}{2}\mathbb{I}$, with ${\left\|\Psi_{\mathrm{trion}}\right\rangle }=\alpha{\left\|\uparrow\downarrow\Uparrow\right\rangle }+\beta{\left\|\downarrow\uparrow\Downarrow\right\rangle }$, with the same $\alpha$, $\beta$ and $p$. Here, we assume of course, that the fidelity of the optical excitation by the $H$ polarized $\pi$-pulse is unity and that the experimental deviation from truly $H$ polarization is negligible. The spin projection of excited qubit on the $\hat{z}$-direction is then deduced by measuring the degree of circular polarization of the emitted photons. We conducted 5 different experiments in order to comprehensively study the central spin dynamics for various confined spin qubits in the QD. In the first 2 measurements, schematically described in Fig. \[fig:Experimental-procedures\]a, we measured the depolarization of the negative or positive trions. We first pump the QD to either a negative or a positive charge state by using above bandgap CW1 pulse of about 10ns duration[@Benny2012]. Then, either an excited negative or positive trion was photogenerated by using a short circularly polarized quasi-resonant \~12 ps long laser pulse. The polarization of the excitation pulse determines the spin polarization of the minority carrier in the initialized trion [\[]{}hole (electron) in the negative (positive) trion[\]]{}. After a fast (\~70 ps [@Schwartz2015]) spin preserving phonon assisted relaxation of the excited trion, a ground level trion is formed. When the trion decays radiatively, the polarization of the emitted photon reflects the spin of the minority carrier at the particular time in which the photon is emitted. Thereby, by using time resolved circular polarization sensitive PL measurements we probe the spin relaxation dynamics of the minority carrier in the trion. This technique provides a simple way of measuring the dynamics of the spin of the confined electron (hole) in the presence of a spin singlet pair of two holes (electrons). Unfortunately, this simple method is limited by the relatively short radiative lifetime of the trion. Only the evolution during the first time domain can be measured this way. In order to avoid generating a steady state Overhauser field in the QD due to the repeated circularly polarized quasi-resonant excitation pulse, a second pulse with opposite circular polarization is used to re-excite the trion a few ns after the first pulse, during the same excitation period. The time resolved degree of circular polarization was deduced using the resulted PL from both complementary pulses. The measurement of the spin dynamics of either the single electron or heavy-hole was carried out using the same experimental system but at somewhat different manner, as schematically described in Fig. \[fig:Experimental-procedures\]b. In the inset to this figure we describe the energy levels of the heavy-hole system. Here, after the optical charging, a trion was generated by quasi resonant excitation using a horizontal ($H$) polarized pulse. Either the electron or the hole spin was initialized by detecting the circular polarization of the emitted single photon. In order to probe the temporal dependence of the spin state of the carrier, a second, horizontal polarized delayed 12 ps pulse is used to re-excite the carrier to its respective trion and the resulting circular polarization of the emitted photon is used to measure the spin polarization of the carrier at the re-excitation time. This measurement is not limited by the radiative lifetime of the trion, however, it requires two-photon intensity correlation measurements in a relatively slow repetition rate (\~500 kHz). We achieved this low repetition rate by using the cavity dumpers. The feasible maximal delay time (\~1 μs$)$ between the pulses was defined by the rejection ratio (of about $\sim2\vartimes10^{-3}$ ) of neighboring pulses of the cavity dumpers. Note that in these experiments the generation of an Overhauser field is avoided because the initialization of the central spin is not done deterministically by using circularly polarized excitation, but rather probabilistically by post-selecting the detected circular polarization of the emitted first photon. The spin dynamics of the DE was probed as schematically described in Fig. \[fig:Experimental-procedures\]c. Here, we used above-bandgap optical pumping of about 20 ns to neutralize the QD and then another quasi-resonant pumping of about 20 ns to deplete the QD from the DE [@Schmidgall2015]. After depleting the QD, a quasi-resonant circularly polarized 12 ps pulse initialized the DE in spin up excited state [@Schwartz2015a]. Following this initialization, the DE relaxes to its ground state within \~70 ps by spin-preserving emission of a phonon. In order to probe the DE state, a delayed, linearly polarized resonant 12 ps pulse converted the DE qubit into the BiE qubit. Note that the horizontal polarization of the laser preserves the phase of the qubit. The detection of a circularly polarized photon, which results from the radiative recombination (\~1 ns lifetime) of the BiE is then used to probe the spin state of the DE in the QD, at the converting pulse time. Repetition rates as low as \~500 kHz, allow temporal delays of over 1 μs between initialization and probing of the spin. In this experimental method an Overhauser field is not generated in the sample since the gated CW pulses used to optically pump and deplete the QD are linearly polarized. ![image](Spin_relaxation_10.PNG){width="100.00000%"} Results and discussion ====================== In Fig. \[fig:Results\] we present the measured degree of the average central spin polarization $\left\langle S_{z}(t)\right\rangle $ as a function of time after its initialization, for the 5 spin qubits: the conduction band electron, the valence band heavy-hole, the positive and negative trions, and the dark exciton. The error bars represent one standard deviation of the experimental uncertainty. At time zero the central spin is initialized to spin-up state. Then, the projection of the spin on $\hat{z}$ direction (the QD growth axis) is displayed as a function of time. The conduction band electron spin state (blue rectangles) depolarizes from its initial state within \~2 ns. The spin polarization then revives to about a third of the initial polarization. From this level the polarization continues to decay at a much slower rate, reaching a second minimum at about \~200 ns. Afterwards the spin polarization revives again to about 10% of the initial polarization. This behavior is similar to that reported in Ref. [@Bechtold2015], as predicted by Ref. [@Efros2002]. Roughly speaking, the first fast dephasing step is a measure for the strong Fermi-contact hyperfine interaction of the electron with the nuclear spin bath, while the second step measures the strength of the quadrupole interaction of the nuclear spin bath with the strain induced electric field gradients in the QD. After initialization, the heavy-hole (red triangles) spin depolarizes in about an order of magnitude slower than the electron spin. This is due to the much weaker dipole-dipole hyperfine interaction. The hole spin polarization decreases at about \~20 ns to about one half of its initial polarization. Afterwards it mildly revives followed by a slow decay due to the quadrupole interaction of the nuclear bath. The positive trion spin polarization (blue $\times$ symbols), behaves similarly to that of the electron, while the negatively charged trion spin polarization (red $\times$ symbols) follows that of the heavy-hole. This is not surprising, since the trion polarization reflects the polarization of the unpaired minority carrier, in the presence of the two paired majority carriers. As explained above, the trions spin measurements are limited by their radiative lifetime of about \~1 ns. The dark exciton (black diamonds) decoheres slowly, in a similar rate to the heavy-hole. However, like the electron, after the initial decay, it strongly revives to about two thirds of its initial polarization. This is due to the strong exchange interaction between the electron and hole that protects both carriers from flipping their individual spins. Later, after \~200 ns the dark exciton polarization continues to decay due to the quadrupole interaction. We fit the measured temporal behavior of the electron, heavy hole and dark exciton using one conceptually simple central spin model. For the fitting, only five free parameters are used: 1) The hyperfine Fermi-contact interaction $\gamma_{e}$, 2) the heavy-hole out-of-plain hyperfine dipole-dipole interaction $\gamma_{h_{Z}}$, 3) the heavy-hole in-plain hyperfine dipole-dipole interaction $\gamma_{h_{p}}$, 4) the DE in-plane interaction $\gamma_{\mathrm{DE}_{p}}$ and 5) the quadrupole interaction $\gamma_{Q}$. These parameters are accurately defined in the appendices, where the models are discussed for the electron and the heavy-hole (Appendix A), and for the DE (Appendix B). The best fitted parameters are given in Table I, where they are also compared with the available literature. Our analysis provides an estimation of the number of atoms in the QD: $N_{L}=3\cdot10^{5}.$ With this estimation our fitted hyperfine Fermi contact $\gamma_{e}$ is comparable to that of Ref. [@Gotschy1989]. Characteristic spin depolarization times during the first and second temporal stages can be obtained from our fitting procedure quite straightforwardly. Since the central spins in this work are initialized in z direction, depolarization is caused by the in-plane interaction parameters. Thus, the temporal location of the first minimum is a rough measure of the in-plane interaction parameter: $T_{\min}=\hbar/\gamma_{_{p}}\sim2$, 20 and 14 ns for the electron, heavy-hole and DE, respectively. Thus, $\gamma_{e}$, $\gamma_{h_{p}}$ and $\gamma_{\mathrm{DE}_{p}}$ are given by $0.34,$ $0.031$ and $0.047\,\text{μeV}$, respectively. The central spin interaction with the nuclear field along the $z$-direction, acts as a restraining force, which actually prolongs the spin coherence. Therefore, roughly speaking, the ratio between these interactions ( $R_{\gamma}=\gamma_{_{Z}}/\gamma_{_{P}})$ determines the depth of the first polarization minimum and the maximum value of the polarization after its revival. We thus obtain $R_{\gamma}$=1, 3.5 and 5, for the electron, hole and DE, respectively. Note that for the electron the ratio is by definition 1, and therefore the polarization degree revives to 1/3 of its initial value, while for the hole and DE it revives to higher values. During the second temporal stage, the polarization of all three central spins decays more or less at the same rate, determined by the quadrupole interaction $\gamma_{Q}$. Therefore the temporal location of the second minimum is about the same in all cases given by $T_{\min_{Q}}=\hbar/\gamma_{Q}\sim200\,\mathrm{ns}$ or $\gamma_{Q}\sim0.003\,\text{μeV}$. A common practice for quantifying the depolarization value of a spin qubit is to define the depolarization time as the time it takes for the polarization to reduce to 1/e of its initial state. We adopt this practice, though the measured depolarizations are clearly non-exponential. The measured depolarization times thus obtained are $1.5$, $130$ and $145$ ns for the electron, heavy hole and DE respectively. [llc]{} Interaction & This work (μeV) & Literature (μeV) [\ ]{}$\gamma_{e}\,$ & $0.34\pm0.03$ & $0.33$ [@Bechtold2015][\ ]{} $\gamma_{h_{z}}\,\,$ & $0.11\pm0.03$ & $0.081$ [@Eble2009][\ ]{}$\gamma_{h_{p}}\,\,$ & $0.031\pm0.006$ & $0.047$ [@Eble2009][\ ]{}$\gamma_{\mathrm{DE}_{p}}\,\,\,$ & $0.047\pm0.006$ & —–[\ ]{} $\gamma_{Q}\qquad$ & $0.0031\pm0.001$ & $0.00087$ [@Bechtold2015][\ ]{} Summary ======= We investigated both experimentally and theoretically the depolarization dynamics of five different electronic spin configurations confined in the same semiconductor quantum dot. Our measurements were carried out all optically and in the absence of externally applied magnetic field. We show that the measured temporal spin depolarization is well described by a central spin model which attributes the depolarization to the hyperfine interaction between the electronic spin and the nuclear spin bath of the QD atoms. We divide the depolarization into two temporal stages. During the initial stage the central spin precesses around the effective magnetic fields of the frozen fluctuations of the $10^{5}$ nuclear spins in the QD. During the second stage the central spin precession follows adiabatically the nuclear spin bath dynamics which ceases to be frozen and effectively precesses around strain induced electric fields gradients in the QD. These two processes result in a relatively fast initial depolarization of the central spin reaching a first minimum. The depolarization minimum is then followed by a temporal revival of the polarization degree and finally by a second depolarization reaching a minimum at a much later time which is more or less equal for all the electronic central spin cases. Our model assumes that while the hyperfine interaction between the central spin and the nuclear spins is isotropic for the electron, it is anisotropic for the heavy-hole and therefore also for the DE, which is formed by an electron–heavy-hole pair. The depolarization times that we measured in zero magnetic field show that the electron depolarizes much faster than the heavy-hole This observation is explained by the difference between the strong isotropic electron-nucleous hyperfine contact interaction ($\gamma_{e}$ ) and the anisotropic hole-nucleous dipolar hyperfine interactions ($\gamma_{h_{Z}},\gamma_{h_{p}}$). The heavy hole spin depolarizes faster than the dark exciton spin due to the electron-hole exchange interaction, which protects the dark-exciton spin from depolarizing. The depolarization of the dark-exciton results from residual dark exciton–bright exciton mixing. We believe that this mixing can be significantly reduced by increasing the QD symmetry and by avoiding alloying. In this case the dark-exciton may form an almost non-dephasing electronic spin qubit in a semiconductor environment. Acknowledgement =============== The support of the Israeli Science Foundation (ISF), and that of the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 695188) are gratefully acknowledged. Hyperfine interaction of the electron and the heavy-hole ======================================================== We outline here a model for describing the temporal evolution of the QD confined central spin polarization in the absence of externally applied magnetic field but in the presence of effective magnetic field generated by the nuclear spins, which comprise the QD. As the central spin we consider either the electron or the heavy hole. We then apply the same model also to a central spin formed by the DE – a long lived electron–heavy-hole pair, as will be discussed in Appendix B. As all three cases involve a two level system (a qubit) they may be described using the Pauli matrices $\sigma_{x},\sigma_{y},\sigma_{z}$ and the effective Hamiltonian must take the form $$H=\frac{1}{2}\vec{C}\cdot\vec{\sigma},$$ for some $\vec{C}=(C_{x},C_{y},C_{z})$. The exact expression of $\vec{C}$ will be different, of course, for each type of central spin. The hyperfine Fermi-contact interaction between an electron and all the nuclei in the QD is given by [@Efros2002] : $$H=\frac{\nu_{0}}{2}\sum\|\psi_{\mathrm{env}}(\vec{r}_{i})\|^{2}A_{e}^{i}\vec{I}_{i}\cdot\overrightarrow{\sigma}.$$ Here $\nu_{0}$ is the volume of the unit cell, $\vec{r}_{i}$ and $\vec{I}_{i}$ are the ith nucleus position and its spin operator, $\psi_{\mathrm{env}}(\vec{r})$ describes the electron envelope wavefunction and $A_{e}^{i}$ is an effective hyperfine interaction constant between the electron and the specific nucleus in the $\vec{r}_{i}$ position where the index i runs over all the nuclei in the QD. Since $A_{e}^{i}$ depends on the atomic nuclear spin it is much larger for indium atoms than for all other atoms in the QD. Thus, in principle, one can neglect other nuclei contributions. We proceed by defining an expression for the effective magnetic field, which the nuclei apply on the electron. The field, known also as the Overhauser field, is defined as: $$\vec{B}_{N}=\frac{1}{g_{e}\mu_{B}}\vec{C}_{e}=\frac{\nu_{0}}{g_{e}\mu_{B}}\sum A_{e}^{i}\|\psi_{\mathrm{env}}(\vec{r}_{i})\|^{2}\left\langle \vec{I}_{i}\right\rangle _{N},$$ where $g_{e}$ and $\mu_{B}$ are the electron g-factor and Bohr magneton respectively, and $\left\langle ...\right\rangle _{N}$ denotes a quantum mechanical average over the nuclear spins which interact with the electron. Assuming that different nuclear spins are not correlated allows one to treat $\vec{B}_{N}(t)$ as having isotropic Gaussian random distribution satisfying $$\langle\vec{B}\rangle=0,\quad\langle B_{Nx}^{2}\rangle=\langle B_{Ny}^{2}\rangle=\langle B_{Nz}^{2}\rangle=\sigma^{2},$$ where the width of the distribution $\sigma$ is given by [@Efros2002] $$3\sigma^{2}=\sum\frac{(A_{e}^{i})^{2}}{\mu_{B}^{2}g_{e}^{2}}\nu_{0}^{2}\|\psi_{\mathrm{env}}(\vec{r}_{i})\|^{4}I_{i}(I_{i}+1).$$ It is then convenient to define a modified unitless magnetic field $\vec{\tilde{B}}=\frac{1}{\sigma}\vec{B}_{N}$. In the following we simply mark this modified Overhauser field as $\vec{B}$. The electron spin Hamiltonian can then be expressed by $H=\frac{1}{2}\vec{C}_{e}\cdot\vec{\sigma}$ with $\vec{C}_{e}=\gamma_{e}\vec{B}$ where $\gamma_{e}=g_{e}\mu_{B}\sigma$ is the electron coupling constant in energy units, which we use as a fitting parameter. While for the electron, $s$-wave molecular symmetry results in a scalar effective coupling $A_{e}^{i}$, for the heavy hole it is described by an anisotropic tensor $$\hat{A}_{h}^{i}=\left(\begin{array}{ccc} A_{h,p}^{i}\\ & A_{h,p}^{i}\\ & & A_{h,z}^{i} \end{array}\right).$$ Where the in plane dipole-dipole interaction constant $A_{h,p}^{i}$ does not strictly vanish for the heavy-hole due to mixing with the light-hole [@Eble2009]. Therefore, for the heavy-hole we define $C_{z}=\gamma_{h_{z}}B_{z},C_{x,y}=\gamma_{h_{p}}B_{x,y}$ where $\gamma_{h_{z}}>\gamma_{h_{p}}$, are also fitting parameters. Strictly speaking, the field $\vec{B}$ appearing here is not exactly the same one as in the electron case. This is due to differences in relative weighting of various nuclei between electron and hole wavefunctions. For our purpose, however, it is sufficient that the fields have the same Gaussian statistics. For the moment we allow the functional relation between $\vec{C}$ and $\vec{B}$ to be arbitrary and since our discussion is independent of these relations, it applies to all three cases. At short times $\vec{B}$ and hence also $\vec{C}$ can be treated as time independent and one readily find the solution $$\begin{aligned} \vec{S}(t) & =\frac{\vec{S}_{0}\cdot\vec{C}}{C^{2}}\vec{C}+\left(\vec{S}_{0}-\frac{\vec{S}_{0}\cdot\vec{C}}{C^{2}}\vec{C}\right)\cos\left(\frac{C}{\hbar}t\right)\label{st}\\ & -\frac{\vec{S}_{0}\times\vec{C}}{C}\sin\left(\frac{C}{\hbar}t\right),\nonumber \end{aligned}$$ where $\vec{S}_{0}=\vec{S}(0)$ is the central spin initial value. The first term is time independent and survives for long times. Upon averaging over the random ensemble of possible $\vec{C}$s one typically finds that the oscillating terms turn into exponentially decaying transients, relevant at short times only. In practice the last term usually vanishes by symmetry under $\vec{C}\rightarrow-\vec{C}$. In particular it applies to our experiments, which were carried out in the absence of externally applied magnetic field. Therefore, in the following we disregard this term. At longer times, we use the adiabatic approximation and assume that the central spin follows the direction of $\vec{C}$, while the rapidly rotating components orthogonal to $\vec{C}$ average to zero. We can therefore write $$\vec{S}(t)=\left(\vec{S}_{0}\cdot\hat{C}(0)\right)\hat{C}(t)=\frac{\vec{S}_{0}\cdot\vec{C}(0)}{C(0)C(t)}\vec{C}(t).\label{lt}$$ For small $t$ this clearly coincides with the first term of Eq. (\[st\]). As the other terms of Eq. (\[st\]) vanish at long times one sees that the two relations Eqs. (\[st\],\[lt\]) can be combined into an expression which applies at arbitrary time $t$: $$\begin{aligned} \vec{S}(t) & =\frac{\vec{S}_{0}\cdot\vec{C}(0)}{C(0)C(t)}\vec{C}(t)\\ & +\left(\vec{S}_{0}-\frac{\vec{S}_{0}\cdot\vec{C}(0)}{C(0)^{2}}\vec{C}(0)\right)\cos\left(\frac{C(0)}{\hbar}t\right).\nonumber \end{aligned}$$ The Gaussian probability density corresponding to the dimensionless Overhauser field at a given moment is given by $$dP_{1}=\frac{1}{(2\pi)^{3/2}}\exp\left(-\frac{1}{2}B^{2}\right)d^{3}B.\label{p1}$$ Assuming further that $$\langle B_{i}(t_{1})B_{j}(t_{2})\rangle=\delta_{ij}f(t_{2}-t_{1})$$ (Consistency requires $f(0)=1$) we can write the joint probability density of $\vec{B}_{1}=\vec{B}(0)$ and $\vec{B}_{2}=\vec{B}(t)$ as $$\begin{aligned} dP_{2} & =\frac{d^{3}B_{1}d^{3}B_{2}}{\left(2\pi\sqrt{1-f(t)^{2}}\right)^{3}}\nonumber \\ & \exp\left[-\frac{1}{2}\left(B_{1}^{2}+B_{2}^{2}-2f(t)\vec{B}_{1}\cdot\vec{B}_{2}\right)/(1-f(t)^{2})\right].\label{eq:dp2}\end{aligned}$$ Using the probability distributions Eqs. (\[p1\],\[eq:dp2\]) we can write the average central spin evolution as $$\begin{aligned} \langle\vec{S}(t)\rangle & =\int\frac{\vec{S}_{0}\cdot\vec{C}(0)}{C(0)C(t)}\vec{C}(t)\,dP_{2}\nonumber \\ & +\int\left(\vec{S}_{0}-\frac{\vec{S}_{0}\cdot\vec{C}}{C^{2}}\vec{C}\right)\cos\left(\frac{C}{\hbar}t\right)dP_{1}\label{eq:St}\end{aligned}$$ Actual computation of the integrals requires using the specific functional relation between $\vec{B}$ and $\vec{C}$. For the electron as the central spin, we simply substitute $\vec{C}=\gamma_{e}\vec{B}$ and $\vec{S}_{0}=\hat{z}$ in Eq. (\[eq:St\]) and obtain integrals which can be evaluated analytically [@Efros2002; @Bechtold2015], resulting in $$\begin{aligned} \langle S_{z}\rangle & =\frac{2}{3}\left(1-\left(\frac{\gamma_{e}t}{\hbar}\right)^{2}\right)e^{-\frac{1}{2}\left(\frac{\gamma_{e}t}{\hbar}\right)^{2}}\\ & +\frac{2}{3\pi}\left(\sqrt{\frac{1}{f(t)^{2}}-1}+\left(2-\frac{1}{f(t)^{2}}\right)\arcsin(f(t))\right).\end{aligned}$$ For the heavy-hole as the central spin we have $C_{z}=\alpha B_{z},\;\;C_{x,y}=\beta B_{x,y}$ with $\alpha=\gamma_{h_{z}}$ $\beta=\gamma_{h_{p}}$ In this case $\langle S_{z}\rangle$ is given according to Eq. (\[eq:St\]) by a sum of two rather complicated integrals. The second term of Eq. (\[eq:St\]) can be reduced into a one-dimensional (1D) integral which we than calculate numerically $$\frac{\beta^{2}}{\left(\alpha^{2}-\beta^{2}\right)^{3/2}}\int_{\beta}^{\alpha}d\xi\,\frac{(\alpha^{2}-\xi^{2})(1-\sigma^{2}\xi^{2}t^{2})}{\xi\sqrt{\xi^{2}-\beta^{2}}}e^{-\frac{1}{2}\sigma^{2}\xi^{2}t^{2}}.\label{Hole1}$$ The first term of Eq. (\[eq:St\]) is a more complicated 6D integral. If we use the following shorthands $$a_{0}=\sqrt{\left(\frac{\cos^{2}\theta}{\alpha^{2}}+\frac{\sin^{2}\theta}{\beta^{2}}\right)\left(\frac{\cos^{2}\theta'}{\alpha^{2}}+\frac{\sin^{2}\theta'}{\beta^{2}}\right)},$$ $$a_{1}=f(t)\left(\frac{\cos\theta\cos\theta'}{\alpha^{2}}+\frac{\sin\theta\sin\theta'}{\beta^{2}}\cos\varphi\right),$$ $$A_{0}=\frac{(1-f(t)^{2})^{3/2}}{4\pi^{2}\alpha^{2}\beta^{4}}\sin(2\theta)\sin(2\theta')$$ then the 6D integral can be reduced into a 3D one $$\begin{aligned} \int_{0}^{2\pi}d\varphi\int_{0}^{\pi/2}d\theta\int_{0}^{\pi/2}d\theta'\,A_{0}\biggl[\frac{3a_{1}}{(a_{0}^{2}-a_{1}^{2})^{2}}\nonumber \\ +\frac{a_{0}^{2}+2a_{1}^{2}}{(a_{0}^{2}-a_{1}^{2})^{5/2}}\arcsin(a_{1}/a_{0})\biggr],\label{Hole2}\end{aligned}$$ which we than calculate numerically. The function $f(t)$ is essentially the Overhauser field time correlator. An appropriate model for the evolution of the Overhauser field is required for its evaluation. By using $\vec{B}=\frac{\nu_{0}}{g_{e}\mu_{B}\sigma}\sum A_{i}\|\psi_{env}(\vec{r}_{i})\|^{2}\vec{I}_{i}$ one obtains: $$\begin{aligned} 3\sigma^{2}f(t) & =\langle\vec{B}(0)\cdot\vec{B}(t)\rangle\\ = & \sum\left(\frac{\nu_{0}}{g_{e}\mu_{B}\sigma}A_{i}^{2}\|\psi_{\mathrm{env}}(\vec{r}_{i})\|^{2}\right)^{2}\langle\vec{I}_{i}(0)\cdot\vec{I}_{i}(t)\rangle.\end{aligned}$$ A particularly simple model assumes that the Overhauser field evolution is dominated by the quadrupole interaction of the nuclear spins [@Sinitsyn2012; @Bechtold2015]. Though more complicated models exist as well [@Efros2002; @Bechtold2015; @Al-Hassanieh2006], this model permits analytical solutions. Within this model each nuclear spin $\vec{I}_{k}=\vec{I}$ evolves independently of the others by a Hamiltonian of the form $H_{Q}=V_{ij}I_{i}I_{j}$ with random $V_{ij}=V_{ji}$ which relates to the local electric field gradients [@Sinitsyn2012] (EFG). We take the initial state of the nuclear spin to be random and we average over the corresponding wave function thereby obtaining $$\langle\vec{I}(0)\cdot\vec{I}(t)\rangle\propto{\operatorname{Tr}}\left(\vec{I}\cdot e^{iH_{Q}t}\vec{I}e^{-iH_{Q}t}\right)$$ As different nuclear spins have different EFG we obtain the Overhauser-correlator $f(t)$ by averaging over the$V_{ij}$ terms. We take (as common in random matrix theory) the elements of the symmetric matrix $V_{ij}$ to be independent Gaussian random variables of variance $\gamma_{Q}^{2}$. Up to overall normalization we obtain $$f(t)\propto\int dV\,e^{-TrV^{2}/(2\gamma_{Q}^{2})}{\operatorname{Tr}}\left(\vec{I}\cdot e^{iH_{Q}t}\vec{I}e^{-iH_{Q}t}\right).$$ Noting that $V$ can be taken as a traceless tensor and in addition using its polar decomposition reduce the above expression into a two-dimensional integral which we express as $$\begin{aligned} f(t) & \propto\int dx_{1}dx_{2}dx_{3}\,\delta\left(\sum x_{i}\right)\prod_{i<j}\|x_{i}-x_{j}\|e^{\frac{-(x_{1}^{2}+x_{2}^{2}+X_{3}^{2})}{2\gamma_{Q}^{2}}}\\ & \cdot{\operatorname{Tr}}\left(\vec{I}\cdot e^{i\sum x_{i}I_{i}^{2}t}\vec{I}e^{-i\sum x_{i}I_{i}^{2}t}\right).\end{aligned}$$ For $I=\frac{3}{2}$ we evaluate this expression and obtain: $$f_{\frac{3}{2}}(t)\propto\int_{0}^{\infty}dx\,x^{4}\,e^{-x^{2}/(2\gamma_{Q}^{2})}(3+2\cos(\sqrt{6}xt))$$ $$f_{\frac{3}{2}}(t)=\frac{3}{5}+\frac{2}{5}\left(1-2\left(\frac{\gamma_{Q}t}{\hbar}\right)^{2}+12\left(\frac{\gamma_{Q}t}{\hbar}\right)^{4}\right)e^{-3\left(\frac{\gamma_{Q}t}{\hbar}\right)^{2}}.\label{jygjy}$$ For higher values of the nuclear spin $I$, we calculated $f_{I}(t)$ numerically as a function of the dimensionless product $\gamma_{Q}t/\hbar$. This gave qualitatively similar result to Eq. (\[jygjy\]) with some modifications. Since our QD contains $I=\frac{3}{2}$ , $I=\frac{9}{2}$ and $I=\frac{1}{2}$ we averaged over these values using the relative nuclear abundance multiplied by the squared nuclear moments as weights. In Fig. \[fig:f(t)\] we display the normalized Overhauser-correlator for various types of nuclear spins in the QD. For simplicity we assume the same $\gamma_{Q}$ for all atom types. In practice the Indium contribution dominates the average due to its large magnetic moment. ![\[fig:f(t)\]The calculated normalized Overhauser-correlator $f_{I}(t)$ for the various types of nuclear spins which comprise the QD. $I_{\mathrm{mean}}$ is the mean value of the correlator taking into account the isotopic abundances weighted by their squared magnetic moments.](ft_3){width="1\columnwidth"} ——— Hyperfine interaction of the dark exciton. ========================================== The spin projection ($S_{z}(t)$) of the DE strongly depends on the electron-hole exchange interaction. We describe the DE qubit by its two spin states: $\|\Uparrow\uparrow\rangle$ and $\|\Downarrow\downarrow\rangle$ , with $J_{z}=+2$, and -2 respectively. While the DE interaction with the z-component $B_{z}$ of the Overhauser field is similar to that of a spin $\frac{1}{2}$ central spin (up to multiplicative constant [@Bayer2002]), its interaction with the $B_{x,y}$ components is very different. Strictly speaking, a standard $\vec{B}\cdot\vec{J}$ Hamiltonian would have to act four times in order to flip a $J_{z}=2$ state into $J_{Z}=-2$ state. However, if one fully considers the electron-hole exchange interaction, this is not the case. In the bright and dark excitons basis $\{\|\Uparrow\downarrow\rangle,\|\Downarrow\uparrow\rangle,\|\Uparrow\uparrow\rangle,\|\Downarrow\downarrow\rangle\}$ , the exchange interaction can be expressed as [@Bayer2002; @Don2016] $$\frac{1}{2}\left(\begin{array}{cccc} \Delta_{0} & \Delta_{1}^{} & \Delta_{3} & \Delta_{4}\\ \Delta_{1} & \Delta_{0} & -\Delta_{4}^{} & -\Delta_{3}^{}\\ \Delta_{3}^{} & -\Delta_{4} & -\Delta_{0} & \Delta_{2}^{}\\ \Delta_{4}^{} & -\Delta_{3} & \Delta_{2} & -\Delta_{0} \end{array}\right),$$ where $\Delta_{0}$ is the isotropic exchange interaction. It is a real number, which defines the energy splitting between the DE and BE eigenstates. It was measured to be $\Delta_{0}$=260 eV for the QD under study. The term $$\Delta_{1}=\delta_{1}\exp(i2\theta_{1})$$ is the anisotropic long-range exchange interaction. Here $\delta_{1}$ is a positive number defining the magnitude of the bright exciton (BE) fine structure splitting (FSS) [@Ivchenko2005], and $\theta_{1}$ defines the directions of the two cross linearly polarized components of the BE spectral lines with respect to the crystallographic directions [@Bayer2002]. $$\Delta_{2}=\delta_{2}\exp(i2\theta_{2})$$ describes the FSS of the dark exciton. Here $\delta_{2}$ and $\theta_{2}$ are real numbers mainly given by the short range anisotropic exchange interaction. $$\Delta_{3}=\delta_{3}\exp(i2\theta_{3})$$ and $$\Delta_{4}=\delta_{4}\exp(i2\theta_{4})$$ are also long-range exchange interactions that couple between the DE and BE states. Strictly speaking, for a $C_{3v}$ symmetrical QD, $\delta_{1}$, $\delta_{2}$, $\delta_{3}$ and $\delta_{4}$ are all expected to vanish [@Dupertuis2011]. To within our experimental uncertainty, we found it to be true only for $\delta_{2}(<0.1\,\text{\textmu eV}$), since it results from the short range exchange interaction and therefore affected mainly by the symmetry of the QD’s unit cells [@Bayer2002]. Structural deviations of the QD from symmetry such as composition fluctuations, or faceting, destroy the QD long range symmetry, without affecting its unit cell symmetry. Therefore, they will result in finite $\delta_{1}$ , $\delta_{3}$ and $\delta_{4}$. Indeed, we measured $\delta_{1}=18\,\text{\textmu eV}$ by polarization sensitive spectroscopy, and estimated $\delta_{3}\backsimeq\delta_{4}\backsimeq15\,\text{\textmu eV}$ by measuring the DE radiative lifetime, and verifying the fact that the DE weak absorption line was linearly polarized in-plane [@Schwartz2015] . Since $\|\Delta_{3}\|=\|\Delta_{4}\|$$\neq0$ , these terms induce coupling between the DE and BE states. We define $\left(\Delta_{3}+\Delta_{4}\right)/2\triangleq\Delta_{\mathrm{DB}}$, and since $\|\Delta_{\mathrm{DB}}\|\ll\Delta_{0}$ , the modified DE eigenstates remain almost degenerate such that the symmetric and anti-symmetric spin combinations are expressed as $$\begin{aligned} {\left\|\mathrm{DE}_{AS}\right\rangle } & =N_{AS}\left[\frac{{\left\|\Uparrow\uparrow\right\rangle }-{\left\|\Downarrow\downarrow\right\rangle }}{\sqrt{2}}-\frac{\Delta_{\mathrm{DB}}}{\Delta_{0}}\frac{{\left\|\Uparrow\downarrow\right\rangle }+{\left\|\Downarrow\uparrow\right\rangle }}{\sqrt{2}}\right]\\ {\left\|\mathrm{DE}_{S}\right\rangle } & =\frac{{\left\|\Uparrow\uparrow\right\rangle }+{\left\|\Downarrow\downarrow\right\rangle }}{\sqrt{2}},\end{aligned}$$ where $N_{AS}\sim1$ is a normalization constant. This also agrees with the experimental observation that the DE has only one weak optically active eigenstate, which is linearly polarized like the symmetric BE eigenstate [@zielinsky2014; @Schmidgall_2017; @Don2016]. The mixing term is sufficient to provide a nuclear field dependent flipping of either the heavy hole or the electron in order to change the DE state from the ${\left\|\Uparrow\uparrow\right\rangle }$ to the ${\left\|\Downarrow\downarrow\right\rangle }$ or vice versa. Hence, the interaction is linear in the nuclear magnetic field and the DE Hamiltonian takes the form $H=\frac{1}{2}\vec{C}^{\mathrm{(DE)}}\cdot\vec{\sigma}$ with $$\begin{aligned} C_{x,y}^{\mathrm{(DE)}} & =\frac{2{\operatorname{Im}}[\Delta_{\mathrm{DB}}]}{\Delta_{0}}\left(C_{x,y}^{(e)}+C_{x,y}^{(h)}\right),\\ C_{z}^{\mathrm{(DE)}} & =\left(C_{z}^{(e)}+C_{z}^{(h)}\right).\end{aligned}$$ If we express $\vec{C}_{e},\vec{C}_{h}$ as earlier in terms of the same dimensionless $\vec{B}$, we conclude $$\begin{aligned} \gamma_{\textrm{\textrm{D}E}{}_{p}} & =\frac{2{\operatorname{Im}}[\Delta_{\mathrm{DB}}]}{\Delta_{0}}\left(\gamma_{e}+\gamma_{h_{p}}\right),\label{eq:gamma_DE_p}\\ \gamma_{\textrm{DE}_{z}} & =\gamma_{e}+\gamma_{h_{z}}=\gamma_{e}-\|\gamma_{h_{z}}\|,\label{eq:gamma_DE_z}\end{aligned}$$ where we used the fact that $\gamma_{h_{z}}<0$ [@Witek2011]. ${\operatorname{Im}}[\Delta_{\mathrm{DB}}]\leqq\delta_{3}\approx15\,\text{\textmu eV}$ provides an estimate for $\gamma_{\textrm{\textrm{D}E}{}_{p}}$ (see Table. \[tab:Interaction-energies\]), and we note here that the fields $\vec{C}_{e}$ and $\vec{C}_{h}$ experienced by the electron and by the heavy-hole, respectively, may not be in perfect correlation [@Fischer2008]. This is expected to reduce their interference effects, making $\gamma_{\textrm{DE}_{z}}$ slightly larger and $\gamma_{\textrm{\textrm{D}E}{}_{p}}$ slightly smaller than the above estimations. The DE Hamiltonian as explained above is linear in B and anisotropic, much like the one for the heavy-hole spin. Consequently $\langle S_{z}(t)\rangle$ is derived in a similar way to that of the heavy-hole spin in Eq. (\[Hole1\]) and Eq. (\[Hole2\]) by replacing $\alpha=\gamma_{\textrm{DE}_{z}}$ and $\beta=\gamma_{\textrm{\textrm{D}E}{}_{p}}$.
--- abstract: 'This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin’s finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter.' author: - \| C. Klein,\ Laboratoire de Gravitation et Cosmologie Relativistes,\ Université P. et M. Curie,\ 4, place Jussieu, 75005 Paris, France\ and\ Institut für Theoretische Physik, Universität Tübingen,\ Auf der Morgenstelle 14, 72076 Tübingen,\ Germany title: \| Exact relativistic treatment of stationary counter-rotating dust disks I\ Boundary value problems and solutions --- PACS numbers: O4.20.Jb, 02.10.Rn, 02.30.Jr Introduction {#sec.1} ============ The importance of stationary axisymmetric spacetimes arises from the fact that they can describe stars and galaxies in thermodynamical equilibrium (see e.g. [@hartle; @lindblom]). However the complicated structure of the Einstein equations in the matter region which are apparently not completely integrable has made a general treatment of these equations impossible up to now. Thus only special, possibly unphysical solutions like the one of Wahlquist [@wahl] were found (in [@perjes] it was shown that the Wahlquist solution cannot be the interior solution for a slowly rotating star). Since the vacuum equations in the form of Ernst [@ernst] are known to be completely integrable [@maison; @belzak; @neuglinear], the study of two-dimensional matter models can lead to global solutions of the Einstein equations which hold both in the matter and in the vacuum region: the equations in the matter, which is in general approximated as an ideal fluid, reduce to ordinary non-linear differential equations because one of the spatial dimensions is suppressed. The matter thus leads to boundary values for the vacuum equations. Disks of pressureless matter, so-called dust, are studied in astrophysics as models for certain galaxies and for accretion disks. We will therefore discuss dust disks in more detail, but the used techniques can in principle be extended to more general cases. In the context of galaxy models, relativistic effects only play an important role in the presence of black-holes since the latter are genuinely relativistic objects. A complete understanding of the black-hole disk system even in non-active galaxies is therefore merely possible in a relativistic setting. The precondition to construct exact solutions for stationary black-hole disk systems is the ability to treat relativistic disks explicitly. In this article we will focus on disks of pressureless matter. By constructing explicit solutions, we hope to get a better understanding of the mathematical structure of the field equations and the physics of rapidly rotating relativistic bodies since dust disks can be viewed as a limiting case for extended matter sources. Hence we will discuss relativistic effects for models whose Newtonian limit is of astrophysical importance. We will investigate disks with counter-rotating dust streams which are discussed as models for certain $S0$ and $Sa$ galaxies (see [@galaxies] and references given therein and [@bicak; @ledvinka]). These galaxies show counter-rotating matter components and are believed to be the consequence of the merger of galaxies. Recent investigations have shown that there is a large number of galaxies (see [@galaxies], the first was NGC 4550 in Virgo) which show counter-rotating streams in the disk with up to 50 % counter-rotation. Exact solutions describing relativistic disks are also of interest in the context of numerics. They can be used to test existing codes for stationary axisymmetric stars as in [@eric; @lanza]. Since Newtonian dust disks are known to be unstable against fragmentation and since numerical investigations (see e.g. [@bawa]) indicate that the same holds in the relativistic case, such solutions could be taken as exact initial data for numerical collapse calculations: due to the inevitable numerical error such an unstable object will collapse if used as initial data. In the Newtonian case, dust disks can be treated in full generality (see e.g. [@binney]) since the disks lead to boundary value problems for the Laplace equations which can be solved explicitly. The fact that the complex Ernst equation which takes the role of the Laplace equation in the relativistic case is completely integrable gives rise to the hope that boundary value problems might be solvable here at least in selected cases. The unifying framework for both the Laplace and the Ernst equation is provided by methods from soliton theory, so-called Riemann-Hilbert problems: the scalar problem for the Laplace equation can be always solved with the help of a generalization of the Cauchy integral (see [@jgp2] and references given therein), a procedure which leads to the Poisson integral for distributional densities. Choosing the contour of the Riemann-Hilbert problem appropriately one can construct solutions to the Laplace equation which are everywhere regular except at a disk where the function is not differentiable. Similarly one can treat the relativistic case where the matrix Riemann-Hilbert problem can be related to a linear integral equation. It was shown in [@prd] that the matrix problem for the Ernst equation can be always gauge transformed to a scalar problem on a Riemann surface which can be solved explicitly in terms of Korotkin’s finite gap solutions [@korot1] for rational Riemann-Hilbert data. In this sense these solutions can be viewed as a generalization of the Poisson integral to the relativistic case. Whereas the Poisson integral contains one free function which is sufficient to solve boundary value problems for the scalar gravitational potential, the finite gap solutions contain one free function and a set of complex parameters, the branch points of the Riemann surface. Thus one cannot hope to solve general boundary value problems for the complex Ernst potential within this class because this would imply the choice to specify two free functions in the solution according to the boundary data. This means that one can only solve certain classes of boundary value problems on a given compact Riemann surface. In the first article we investigate the implications of the underlying Riemann surface on the multipole moments and the boundary values taken at a given boundary. The relations will be given for general genus of the surface and will be discussed in detail in the case of genus 1 (elliptic surface) and genus 2, which is the simplest case with generic equatorial symmetry. It is shown that the solution of boundary value problems leads in general to non-linear integral equations. We can identify however classes of boundary data where only one linear integral equation has to be solved. Special attention will be paid to counter-rotating dust disks which will lead us to the construction of the solution for constant angular velocity and constant relative density which was presented in [@prl2]. It contains as limiting cases the static solutions of Morgan and Morgan [@morgan] and the disk with only one matter stream by Neugebauer and Meinel [@neugebauermeinel1]. The potentials of the resulting spacetime at the axis and the disk are presented in the second article, the physical features as the ultrarelativistic limit, the formation of ergospheres, multipole moments and the energy-momentum tensor are discussed in the third article. The present article is organized as follows. In section \[sec.2\] we discuss Newtonian dust disks with Riemann-Hilbert methods and relate the corresponding boundary value problems to an Abelian integral equation. The relativistic field equations and the boundary conditions for counter-rotating dust disks are summarized in section \[sec.3\]. Important facts on hyperelliptic Riemann surfaces which will be used to discuss Korotkin’s class of solutions to the Ernst equation are collected in section \[hyper\]. In section \[sec.4\], we establish relations for the corresponding Ernst potentials on the axis on a given Riemann-surface of arbitrary genus. The found relation limits the possible choice of the multipole moments. We discuss in detail the elliptic and the genus 2 case with equatorial symmetry. This analysis is extended to the whole spacetime in section \[sec.5\] which leads to a set of differential and algebraic equations which is again discussed in detail for genus 1 and 2. The equations for genus 2 are used to study differentially counter-rotating dust disks in section \[sec.6\]: We discuss the Newtonian limit of disks of genus 2. As a first application of this constructive approach we derive the class of counter-rotating dust disks with constant angular velocity and constant relative density of [@prl2]. We prove the regularity of the solution up to the ultrarelativistic limit in the whole spacetime except the disk and conclude in section \[conclusion\]. Newtonian dust disks {#sec.2} ==================== To illustrate the basic concepts used in the following sections, we will briefly recall some facts on Newtonian dust disks. In Newtonian theory, gravitation is described by a scalar potential $U$ which is a solution to the Laplace equation in the vacuum region. We use cylindrical coordinates $\rho$, $\zeta$ and $\phi$ and place the disk made up of a pressureless two-dimensional ideal fluid with radius $\rho_{0}$ in the equatorial plane $\zeta=0$. In Newtonian theory stationary perfect fluid solutions and thus also the here considered disks are known to be equatorially symmetric. Since we concentrate on dust disks, i.e. pressureless matter, the only force to compensate gravitational attraction in the disk is the centrifugal force. This leads in the disk to (here and in the following $f_{x}=\frac{\partial f}{\partial x}$) $$U_{\rho}=\Omega^{2}(\rho)\rho, \label{eq1}$$ where $\Omega(\rho)$ is the angular velocity of the dust at radius $\rho$. Since all terms in (\[eq1\]) are quadratic in $\Omega$ there are no effects due to the sign of the angular velocity. The absence of these so-called gravitomagnetic effects in Newtonian theory implies that disks with counter-rotating components will behave with respect to gravity exactly as disks which are made up of only one component. We will therefore only consider the case of one component in this section. Integrating (\[eq1\]) we get the boundary data $U(\rho,0)$ with an integration constant $U_{0}=U(0,0)$ which is related to the central redshift in the relativistic case. To find the Newtonian solution for a given rotation law $\Omega(\rho)$, we thus have to construct a solution to the Laplace equation which is everywhere regular except at the disk where it has to take on the boundary data (\[eq1\]). At the disk the normal derivatives of the potential will have a jump since the disk is a surface layer. Notice that one only has to solve the vacuum equations since the two-dimensional matter distribution merely leads to boundary conditions for the Laplace equation. In the Newtonian setting one thus has to determine the density for a given rotation law or vice versa, a well known problem (see e.g. [@binney] and references therein) for Newtonian dust disks. The method we outline here has the advantage that it can be generalized to some extent to the relativistic case. We put $\rho_{0}=1$ without loss of generality (we are only considering disks of finite non-zero radius) and obtain $U$ as the solution of a Riemann-Hilbert problem (see e.g. [@jgp2] and references given therein),\ Theorem 2.1:\ Let $\ln G\in C^{1,\alpha}(\Gamma)$ and $\Gamma$ be the covering of the imaginary axis in the upper sheet of $\Sigma_{0}$ between $-\mathrm{ i}$ and $\mathrm{ i}$ where $\Sigma_{0}$ is the Riemann surface of genus 0 given by the algebraic relation $\mu_{0}^{2}(\tau)=(\tau-\zeta)^2+\rho^{2}$. The function $G$ has to be subject to the conditions $G(\bar{\tau})=\bar{G}(\tau)$ and $G(-\tau)=G(\tau)$. Then $$U(\rho,\zeta)=-\frac{1}{4\pi \mathrm{ i}}\int_{\Gamma}^{}\frac{\ln G(\tau) d\tau}{\sqrt{(\tau-\zeta)^2+ \rho^2}} \label{newton2}$$ is a real, equatorially symmetric solution to the Laplace equation which is everywhere regular except at the disk $\zeta=0$, $\rho\leq 1$. The function $\ln G$ is determined by the boundary data $U(\rho,0)$ or the energy density $\sigma$ of the dust ($2\pi \sigma=U_{\zeta}$ in units where the velocity of light and the Newtonian gravitational constant are equal to 1) via $$\ln G(t) = 4\left(U_0+t\int_{0}^{t} \frac{U_{\rho}(\rho)d\rho}{\sqrt{t^2-\rho^2}}\right) \label{eq4}$$ or $$\ln G(t)=4 \int_{t}^{1}\frac{\rho U_{\zeta}}{\sqrt{\rho^{2}-t^{2}}}d\rho \label{eq5}$$ respectively where $t=-\mathrm{i}\tau$.\ The occurrence of the logarithm in (\[newton2\]) is due to the Riemann-Hilbert problem with the help of which the solution to the Laplace equation was constructed. We briefly outline the\ Proof:\ It may be checked by direct calculation that $U$ in (\[newton2\]) is a solution to the Laplace equation except at the disk. The reality condition on $G$ leads to a real potential, whereas the symmetry condition with respect to the involution $\tau\to -\tau$ leads to equatorial symmetry. At the disk the potential takes due to the equatorial symmetry the boundary values $$U(\rho,0)=-\frac{1}{2\pi }\int_{0}^{\rho} \frac{\ln G(t) }{\sqrt{\rho^2-t^{2}}}dt \label{newton3}$$ and $$U_{\zeta}(\rho,0)=-\frac{1}{2\pi }\int_{\rho}^{1}\frac{\partial_{t}(\ln G(t))}{\sqrt{t^{2}-\rho^{2}}}dt \label{eq2}.$$ Both equations constitute integral equations for the ‘jump data’ $\ln G$ of the Riemann-Hilbert problem if the respective left-hand side is known. The equations (\[newton3\]) and (\[eq2\]) are both Abelian integral equations and can be solved in terms of quadratures, i.e. (\[eq4\]) and (\[eq5\]). To show the regularity of the potential $U$ we prove that the integral (\[newton2\]) is identical to the Poisson integral for a distributional density which reads at the disk $$U(\rho)=-2\int_{0}^{1}\sigma(\rho')\rho' d\rho' \int_{0}^{2\pi} \frac{d\phi}{\sqrt{(\rho+\rho')^{2}-4\rho\rho' \cos \phi}} =-4 \int_{0}^{1}\sigma(\rho')\rho' d\rho'\frac{K(k(\rho,\rho'))}{\rho+\rho'}, \label{eq7}$$ where $k(\rho,\rho')=2\sqrt{\rho\rho'}/(\rho+\rho')$ and where $K$ is the complete elliptic integral of the first kind. Eliminating $\ln G$ in (\[newton3\]) via (\[eq5\]) we obtain after interchange of the order of integration $$U=-\frac{2}{\pi}\left(\int_{0}^{\rho}U_{\zeta}\frac{\rho'}{\rho} K\left(\frac{\rho'}{\rho}\right)d\rho'+\int_{\rho}^{1}U_{\zeta} K\left(\frac{\rho}{\rho'}\right)d\rho'\right) \label{eq8}$$ which is identical to (\[eq7\]) since $K(2\sqrt{k}/(1+k))=(1+k)K(k)$. Thus the integral (\[newton2\]) has the properties known from the Poisson integral: it is a solution to the Laplace equation which is everywhere regular except at the disk where the normal derivatives are discontinuous. This completes the proof. Remark: We note that it is possible in the Newtonian case to solve the boundary value problem purely locally at the disk. The regularity properties of the Poisson integral then ensure global regularity of the solution except at the disk. Such a purely local treatment will not be possible in the relativistic case. The above considerations make it clear that one cannot prescribe both $U$ at the disk (and thus the rotation law) and the density independently. This just reflects the fact that the Laplace equation is an elliptic equation for which Cauchy problems are ill-posed. If $\ln G$ is determined by either (\[eq4\]) or (\[eq5\]) for given rotation law or density, expression (\[newton2\]) gives the analytic continuation of the boundary data to the whole spacetime. In case we prescribe the angular velocity, the constant $U_{0}$ is determined by the condition $\ln G(\mathrm{i})=0$ which excludes a ring singularity at the rim of the disk. For rigid rotation ($\Omega=const$), we get e.g.$$\ln G(\tau)=4\Omega^2(\tau^2+1) \label{eq6}$$ which leads with (\[newton2\]) to the well-known Maclaurin disk. Relativistic equations and boundary conditions {#sec.3} ============================================== It is well known (see [@exac]) that the metric of stationary axisymmetric vacuum spacetimes can be written in the Weyl–Lewis–Papapetrou form $$\label{3.1} \mathrm{ d} s^2 =-e^{2U}(\mathrm{ d} t+a\mathrm{ d} \phi)^2+e^{-2U} \left(e^{2k}(\mathrm{ d} \rho^2+\mathrm{ d} \zeta^2)+ \rho^2\mathrm{ d} \phi^2\right) \label{vac1}$$ where $\rho$ and $\zeta$ are Weyl’s canonical coordinates and $\partial_{t}$ and $\partial_{\phi}$ are the two commuting asymptotically timelike respectively spacelike Killing vectors. In this case the vacuum field equations are equivalent to the Ernst equation for the complex potential $f$ where $f=e^{2U}+\mathrm{ i}b$, and where the real function $b$ is related to the metric functions via $$\label{3.2} b_{z}=-\frac{\mathrm{ i}}{\rho}e^{4U}a_{z} \label{vac9}.$$ Here the complex variable $z$ stands for $z=\rho+\mathrm{ i}\zeta$. With these settings, the Ernst equation reads $$\label{3.3} f_{z\bar{z}}+\frac{1}{2(z+\bar{z})}(f_{\bar{z}}+f_z)=\frac{2 }{f+\bar{f}} f_z f_{\bar{z}} \label{vac10}\enspace,$$ where a bar denotes complex conjugation in $\mbox{C}$. With a solution $f$, the metric function $U$ follows directly from the definition of the Ernst potential whereas $a$ can be obtained from (\[vac9\]) via quadratures. The metric function $k$ can be calculated from the relation $$\label{3.4} k_{z} = 2\rho \left(U_{z}\right)^2-\frac{1}{2\rho}e^{4U} \left(a_{z}\right)^2. \label{vac8}$$ The integrability condition of (\[vac9\]) and (\[vac8\]) is the Ernst equation. For real $f$, the Ernst equation reduces to the Laplace equation for the potential $U$. The corresponding solutions are static and belong to the Weyl class. Hence static disks like the counter-rotating disks of Morgan and Morgan [@morgan] can be treated in the same way as the Newtonian disks in the previous section. Since the Ernst equation is an elliptic partial differential equation, one has to pose boundary value problems. The boundary data arise from a solution of the Einstein equations in the matter region. In our case this will be an infinitesimally thin disk made up of two components of pressureless matter which are counter-rotating. These models are simple enough that explicit solutions can be constructed, and they show typical features of general boundary value problems one might consider in the context of the Ernst equation. It is also possible to study explicitly the transition from a stationary to a static spacetime with a matter source of finite extension for these models. Counter-rotating disks of infinite extension but finite mass were treated in [@bicak] and [@pichon], disks producing the Kerr metric and other metrics in [@ledvinka]. To obtain the boundary conditions at a relativistic dust disk, it seems best to use Israel’s invariant junction conditions for matching spacetimes across non-null hypersurfaces [@israel]. Again we place the disk in the equatorial plane and match the regions $V^{\pm}$ ($\pm \zeta>0$) at the equatorial plane. This is possible with the coordinates of (\[vac1\]) since we are only considering dust i.e. vanishing radial stresses in the disk. The jump $\gamma_{\alpha\beta}= K^+_{\alpha\beta}-K^-_{\alpha\beta}$ in the extrinsic curvature $K_{\alpha\beta}$ of the hypersurface $\zeta=0$ with respect to its embeddings into $V^{\pm}=\{\pm\zeta>0\}$ is due to the energy momentum tensor $S_{\alpha\beta}$ of the disk via $$-8\pi S_{\alpha\beta}=\gamma_{\alpha\beta}-h_{\alpha\beta} \gamma_{\epsilon}^{\epsilon} \label{vac16.1}$$ where $h$ is the metric on the hypersurface (greek indices take the values 0, 1, 3 corresponding to the coordinates $t$, $\rho$, $\phi$). As a consequence of the field equations the energy momentum tensor is divergence free, $S^{\alpha\beta}_{;\beta}=0$ where the semicolon denotes the covariant derivative with respect to $h$. The energy-momentum tensor of the disk is written in the form $$S^{\mu\nu}=\sigma_{+} u^{\mu}_+ u^{\nu}_+ +\sigma_{-} u^{\mu}_- u^{\nu}_- \label{vac16.11},$$ where the vectors $u^{\alpha}_{\pm}$ are a linear combination of the Killing vectors, $(u^\alpha_{\pm})=(1, 0, \pm\Omega(\rho))$. This has to be considered as an algebraic definition of the tensor components. Since the vectors $u_{\pm}$ are not normalized, the quantities $\sigma_{\pm}$ have no direct physical significance, they are just used to parametrize $S^{\mu\nu}$. The energy-momentum tensor was chosen in a way to interpolate continuously between the static case and the one-component case with constant angular velocity. An energy-momentum tensor $S^{\mu\nu}$ with three independent components can always be written as $$S^{\mu\nu}=\sigma_{p}^{}v^{\mu}v^{\nu}+p_{p}^{}w^{\mu}w^{\nu} \label{2.31a},$$ where $v$ and $w$ are the unit timelike respectively spacelike vectors $(v^{\mu})=N_{1}(1,0,\omega_{\phi})$ and where $(w^{\mu})=N_{2}(\kappa,0,1)$. This corresponds to the introduction of observers (called $\phi$-isotropic observers (FIOs) in [@ledvinka]) for which the energy-momentum tensor is diagonal. The condition $w_{\mu}v^{\mu}=0$ determines $\kappa$ in terms of $\omega_{\phi}$ and the metric, $$\kappa=-\frac{g_{03}+\omega_{\phi}g_{33}}{g_{00}+\omega_{\phi}g_{03}} \label{2.31a1}.$$ If $p_{p}^{}/\sigma_{p}^{}<1$ the matter in the disk can be interpreted as in [@morgan] either as having a purely azimuthal pressure or as being made up of two counter-rotating streams of pressureless matter with proper surface energy density $\sigma_{p}^{}/2$ which are counter-rotating with the same angular velocity $ \sqrt{p_{p}^{}/\sigma_{p}^{}}$, $$S^{\mu\nu}=\frac{1}{2}\sigma^{}(U_{+}^{\mu}U_{+}^{\nu}+ U_{-}^{\mu}U_{-}^{\nu}) \label{2.31b}$$ where $(U_{\pm}^{\mu})=U^{}(v^{\mu}\pm \sqrt{p^{}_{p}/\sigma^{}_{p}}w^{\mu})$ is a unit timelike vector. We will always adopt the latter interpretation if the condition $p_{p}^{}/\sigma_{p}^{}<1$ is satisfied which is the case in the example we will discuss in more detail in section 7. The energy-momentum tensor (\[2.31b\]) is just the sum of two energy-momentum tensors for dust. Furthermore it can be shown that the vectors $U_{\pm}$ are geodesic vectors with respect to the inner geometry of the disk: this is a consequence of the equation $S^{\mu\nu}_{;\nu}=0$ together with the fact that $U_{\pm}$ is a linear combination of the Killing vectors. In the discussion of the physical properties of the disk we will refer only to the measurable quantities $\omega_{\phi}$, $\sigma_{p}^{}$ and $p_{p}^{}$ which are obtained by the introduction of the FIOs whereas $\sigma_{\pm}$ and $\Omega$ are just used to generate a sufficiently general energy-momentum tensor. To establish the boundary conditions implied by the energy-momentum tensor, we use Israel’s formalism [@israel]. Equation $S^{\alpha\beta}_{;\beta}=0$ leads to the condition $$U_{\rho}\left(1+2\gamma\Omega a+\Omega^{2}a^{2}\right) + \Omega a_{\rho} (\gamma+\Omega a) +\Omega^2 \rho (\rho U_{\rho}-1) e^{-4U}=0, \label{vac20}$$ where $$\gamma(\rho)=\frac{\sigma_{+}(\rho)-\sigma_{-}(\rho)}{\sigma_{+}(\rho) +\sigma_{-}(\rho)} \label{vac20a}.$$ The function $\gamma(\rho)$ is a measure for the relative energy density of the counter-rotating matter streams. For $\gamma\equiv 1$, there is only one component of matter, for $\gamma\equiv 0$, the matter streams have identical density which leads to a static spacetime of the Morgan and Morgan class. As in the Newtonian case, one cannot prescribe both the proper energy densities $\sigma_{\pm}$ and the rotation law $\Omega$ at the disk since the Ernst equation is an elliptic equation. For the matter model (\[vac16.11\]), we get at the disk\ Theorem 3.1:\ Let $\tilde{\sigma}(\rho)=\sigma_{+}(\rho)+\sigma_{-}(\rho)$ and let $R(\rho)$ and $\delta(\rho)$ be given by $$R=\left(a+\frac{\gamma}{\Omega}\right)e^{2U} \label{eq10},$$ and $$\delta(\rho)=\frac{1-\gamma^{2}(\rho)}{\Omega^{2}(\rho)} \label{eq9a}.$$ Then for prescribed $\Omega(\rho)$ and $\delta(\rho)$, the boundary data at the disk take the form $$f_{\zeta}=-\mathrm{i}\frac{R^{2}+\rho^{2}+\delta e^{4U}}{2R\rho}f_{\rho} +\frac{\mathrm{i}}{R}e^{2U} \label{eq9}.$$ Let $\sigma$ be given by $\sigma=\tilde{\sigma } e^{k-U}$. Then for given density $\sigma$ and $\gamma$, the boundary data read, $$(\rho^2+\delta e^{4U}) \left(\left(e^{2U}\right)_{\rho} \left(e^{2U}\right)_{\zeta} +b_{\rho}b_{\zeta}\right)^2 -2\rho e^{2U} \left(e^{2U}\right)_{\zeta}\left(\left(e^{2U}\right)_{\rho} \left(e^{2U}\right)_{\zeta} +b_{\rho}b_{\zeta}\right) +b_{\rho}^2 e^{4U}=0 \label{16.15},$$ and $$\left(b_{\rho}-a\left(\left(e^{2U}\right)_{\rho} \left(e^{2U}\right)_{\zeta} +b_{\rho}b_{\zeta}\right)\right)^2+8\pi \rho\sigma e^{2U}\gamma^{2}\left(\left(e^{2U}\right)_{\rho} \left(e^{2U}\right)_{\zeta} +b_{\rho}b_{\zeta}\right)=0 \label{16.17}.$$* Proof:\ The relations (\[vac16.1\]) lead to $$\begin{aligned} -4\pi e^{(k-U)}S_{00} & = & \left(k_{\zeta}-2U_{\zeta} \right)e^{2U}, \nonumber \\ -4\pi e^{(k-U)}( S_{03}-aS_{00}) & = & -\frac{1}{2}a_{\zeta}e^{2U} \nonumber ,\\ -4\pi e^{(k-U)} (S_{33}-2a S_{03}+a^2 S_{00})& = & -k_{\zeta}\rho^2e^{-2U} \label{16.7},\end{aligned}$$ where $$\begin{aligned} S_{00} & = & \tilde{\sigma} e^{4U}\left(1+\Omega^2 a^2 +2\Omega a \gamma \right), \nonumber \\ S_{03}-a S_{00} & = & -\tilde{\sigma} \rho^2 \Omega \left(\Omega a +\gamma \right), \nonumber \\ S_{33}-2a S_{03}+a^2 S_{00} & = & \tilde{\sigma} \Omega^2 \rho^4 e^{-4U} \label{gegen3}.\end{aligned}$$ One can substitute one of the above equations by (\[vac20\]) in the same way as one replaces one of the field equations by the covariant conservation of the energy momentum tensor in the case of three-dimensional ideal fluids. This makes it possible to eliminate $k_{\zeta}$ from (\[16.7\]) and to treat the boundary value problem purely on the level of the Ernst equation. The function $k$ will then be determined via (\[vac8\]) with the found solution of the Ernst equation. It is straight forward to check the consistency of this approach with the help of (\[vac8\]). If $\Omega$ and $\gamma$ (and thus $\delta$) are given, one has to eliminate $\tilde{\sigma}$ from (\[16.7\]) and (\[gegen3\]). This can be combined with (\[vac20\]) and (\[vac9\]) to (\[eq9\]). If the function $\gamma$ and $\sigma$ are prescribed (this makes it possible to treat the problem completely on the level of the Ernst equation), one has to eliminate $\Omega$ from (\[vac20\]), (\[16.7\]) and (\[gegen3\]) which leads to (\[16.15\]) and (\[16.17\]). This completes the proof. Remark: For given $\Omega(\rho)$ and $\delta(\rho)$, equation (\[vac20\]) is an ordinary non-linear differential equation for $e^{2U}$, $$(R^{2}-\rho^{2})_{\rho}e^{2U}-2R e^{4U} \left(\frac{\gamma}{\Omega}\right)_{\rho} =(R^{2}-\rho^{2}-\delta e^{4U})\left(e^{2U}\right)_{\rho} \label{vac8a}.$$ For constant $\Omega$ and $\gamma$ we get $$R^{2}-\rho^{2}+\delta e^{4U}=\frac{2}{\lambda}e^{2U} \label{eq11},$$ where $\lambda=2\Omega^{2}e^{-2U_{0}}$. For given boundary values as in Theorem 3.1, the task is to to find a solution to the Ernst equation which is regular in the whole spacetime except at the disk where it has to satisfy two real boundary conditions. In the following we will concentrate on the case where the angular velocity $\Omega$ and the relative density $\gamma$ are prescribed. Solutions on hyperelliptic Riemann surfaces {#hyper} =========================================== The remarkable feature of the Ernst equation is that it is completely integrable which means that the Riemann-Hilbert techniques used in the Newtonian case can be applied here, too. This time, however, one has to solve a matrix problem (see e.g. [@prd] and references given therein) which cannot be solved generally in closed form. In [@prd] it was shown that the problem can be gauge transformed to a scalar problem on a four-sheeted Riemann surface. In the case of rational ‘jump data’ of the Riemann-Hilbert problem, this surface is compact and the corresponding solutions to the Ernst equation are Korotkin’s finite gap solutions [@korot1]. In the following we will concentrate on this class of solutions and investigate its properties with respect to the solution of boundary value problems. Theta functions on hyperelliptic Riemann surfaces {#subsec.1} ------------------------------------------------- We will first summarize some basic facts on hyperelliptic Riemann surfaces which we will need in the following. We consider surfaces $\Sigma$ of genus $g$ which are given by the relation $\mu^{2}(K)=(K+\mathrm{ i}z)(K-\mathrm{ i}\bar{z}) \prod_{i=1}^{g}(K-E_{i})(K-\bar{E}_{i})$ where the $E_{i}$ do not depend on the physical coordinates $z$ and $\bar{z}$. We introduce the standard quantities associated with a Riemann surface (see [@farkas]), with respect to the cut system of figure 1 (we order the branch points with $\mbox{Im}E_{i}<0$ in a way that $\mbox{Re}E_{1}<\mbox{Re}E_{2}<\ldots <\mbox{Re}E_{g}$ and assume for simplicity that the real parts of the $E_{i}$ are all different; we write $E_{i}=\alpha_{i}+\beta_{i}$), the $g$ normalized differentials of the first kind $\mathrm{ d}\omega_i$ defined by $\oint_{a_i}\mathrm{ d}\omega_j= 2\pi\mathrm{ i} \delta_{ij}$, and with $P_{0}=-\mathrm{i}z$ the Abel map $\omega_i(P)=\int_{P_0}^{P}\mathrm{ d}\omega_i$ which is defined uniquely up to periods. Furthermore, we define the Riemann matrix $\Pi$ with the elements $\pi_{ij}= \oint_{b_i}\mathrm{ d}\omega_j$, and the theta function $\Theta\left[m\right](z)= \sum_{N\in{Z}^g}^{}\exp\left\{\frac{1}{2}\left\langle\Pi (N+ \frac{m^{1}}{2}),(N+\frac{m^{1}}{2}) \right\rangle+\left\langle (z+\pi \mathrm{ i} m^{2}),(N+\frac{\alpha}{2}) \right\rangle\right\}$ with half integer characteristic $[m]=\left[m^{1} \atop m^{2}\right]$ and $m^{1}_i,m^{2}_i=0,1$ ($\left\langle N,z\right\rangle=\sum_{i=1}^g N_iz_i$). A characteristic is called odd if $\langle m^{1},m^{2}\rangle \neq 0 \mbox{ mod } 2$. The normalized (all $a$–periods zero) differential of the third kind with poles at $P_{1}$ and $P_{2}$ and residue $+1$ and $-1$ respectively will be denoted by $\mathrm{ d} \omega_{P_{1}P_{2}}$. A point $P\in \Sigma$ will be denoted by $P=(K,\pm \mu(K))$ or $K^{\pm}$ (the sheets will be defined in the vicinity of a given point on $\Sigma$, e.g. $\infty$). \ The theta functions are subject to a number of addition theorems. We will need the ternary addition theorem which can be cast in the form\ Theorem 4.1: Ternary addition theorem\ Let $[m_{i}]=[m_{i}^{1},m_{i}^{2}]$ $(i=1,\ldots,4)$ be arbitrary real $2g$-dimensional vectors. Then $$\begin{aligned} &&\Theta[m_1](u+v)\Theta[m_2](u-v)\Theta[m_3](0)\Theta[m_4](0) \label{ternary}\\ &&=\frac{1}{2^g}\sum\limits_{2a \in (Z_2)^{2g}}\exp(-4\pi \mathrm{i} \langle m_1^{1},a^{2}\rangle ) \Theta[n_1+a](u)\Theta[n_2+a](u)\Theta[n_3+a](v)\Theta[n_4+a](v) \nonumber,\end{aligned}$$ where $a=(a^{1},a^{2})$, and $(m_1,\dots,m_4)=(n_1,\dots,n_4) T$ with $$T=\frac{1}{2}\left ( \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{array} \right) \label{T}.$$ Each $1$ in $T$ denotes the $g\times g$ identity matrix. For a proof see e.g. [@algebro]. Let us recall that a divisor $X$ on $\Sigma$ is a formal symbol $X=n_{1}P_{1}+\ldots+ n_{k}P_{k}$ with $P_{i}\in \Sigma$ and $n_{i}\in \mbox{Z}$. The degree of a divisor is $\sum_{i=1}^{k}n_{i}$. The Riemann vector $K_{R}$ is defined by the condition that $\Theta(\omega(W)+K_{R})=0$ if $W$ is a divisor of degree $g-1$ or less. We use here and in the following the notation $\omega(W)=\int_{P_{0}}^{W}\mathrm{d}\omega =\sum_{i=1}^{g-1}\omega(w_{i})$. We note that the Riemann vector can be expressed through half-periods in the case of a hyperelliptic surface. The quotient of two theta functions with the same argument but different characteristic is a so-called root function which means that its square is a function on $\Sigma$. One can prove (see [@algebro] and references therein)\ Theorem 4.2: Root functions\ Let $Q_{i}$, $i=1,\ldots, 2g+2$, be the branch points of a hyperelliptic Riemann surface $\Sigma_{g}$ of genus $g$ and $A_{j}=\omega(Q_{j})$ with $\omega(Q_{1})=0$. Furthermore let $\{i_{1},\ldots, i_{g}\}$ and $\{j_{1},\ldots,j_{g}\}$ be two sets of numbers in $\{1,2,\ldots,2g+2\}$. Then the following equality holds for an arbitrary point $P\in \Sigma_{g}$, $$\frac{\Theta\left[K_{R}+\sum_{k=1}^{g}A_{i_{k}}\right]\left( \omega(P)\right)}{\Theta\left[K_{R}+ \sum_{k=1}^{g}A_{j_{k}}\right]\left( \omega(P)\right)}=c_{1}\sqrt{\frac{(K-E_{i_{1}})\ldots(K-E_{i_{g}})}{ (K-E_{j_{1}})\ldots(K-E_{j_{g}})}} \label{root1},$$ where $c_{1}$ is a constant independent on $K$. Let $X=P_{1}+\ldots+P_{g}$ with $P_{j}=(K_{j},\mu(K_{j})$ be a divisor of degree g on $\Sigma_{g}$ then the following identity exists, $$\frac{\Theta\left[K_{R}+A_{i}\right]\left( \omega(X)\right)}{\Theta\left[K_{R}+ A_{j}\right]\left( \omega(X)\right)}=c_{2}\prod_{k=1}^{g} \sqrt{\frac{(K_{k}-Q_{i})}{(K_{k}-Q_{j}) }} \label{root2},$$ where $c_{2}$ is a constant independent on the $K_{k}$. The notion of divisors makes it possible to state Jacobi’s inversion theorem in a very compact form,\ Theorem 4.3: Jacobi inversion theorem\ Let $A,B \in \Sigma$ be divisors of degree g and $u\in \mathrm{C}^{g}$. Then for given $B$ and $u$, the equation $\omega(A)-\omega(B)=u$ for the divisor $A$ is always solvable. For a proof we refer the reader to the standard literature, e.g. [@farkas]. We remark that the divisor may not be uniquely defined in the general case which means that one or more $P_{i}\in A$ can be freely chosen. We will not consider such special cases in the following and refer the reader for the so-called special divisors to the literature as [@algebro]. For divisors $A-B$ of degree zero, one can formulate Abel’s theorem.\ Theorem 4.4: Abel’s theorem\ Let $A,B\in \Sigma$ be divisors of degree $n$ subject to the relation $\omega(A)-\omega(B)=0$. Then $A$ and $B$ are the set of zeros respectively poles of a meromorphic function $F$. For a proof see [@farkas]. We remark that this function is a rational function on the surface cut along the homology basis. We have the Corollary 4.5:\ Let the condition of Abel’s theorem hold. Then the following identity holds for the integral of the third kind $$\int_{B}^{A}d\omega_{PQ}=\ln \frac{F(P)}{F(Q)} \label{eq20}.$$ Solutions to the Ernst equation {#subsec.4.2} ------------------------------- We are now able to write down a class of solutions to the Ernst equation on the surface $\Sigma$.\ Theorem 4.5:\ Let the Riemann surface $\Sigma$ be given by the relation $\mu^{2}(K)=(K+\mathrm{i}z)(K-\mathrm{i}z)\prod_{i=1}^{g}(K-E_{i}) (K-\bar{E}_{i})$, let $u$ be the vector with the components $u_i=\frac{1}{2\pi \mathrm{ i}} \int_{\Gamma}^{}\ln G d\omega_i$ where $\Gamma$ is as in theorem 2.1, let $G$ be subject to the condition $G(\tau)=\bar{G}(\bar{\tau})$, and let $[m]=[m^{1},m^{2}]$ with $m^{1}_{i}=0$ and $m^{2}_{i}$ arbitrary for $i=1,\ldots,g$ be a theta characteristic. Then the function $f$ given by $$f(\rho,\zeta)=\frac{\Theta[m](\omega(\infty^{+})+u)}{ \Theta[m](\omega(\infty^{-})+u)} \exp\left\{ \frac{1}{2\pi\mathrm{ i}}\int\limits_\Gamma \ln G(\tau)\mathrm{ d}\omega_{\infty^{+}\infty^-}(\tau) \right\} \label{rel1a},$$ is a solution to the Ernst equation.\ This class of solutions was first given by Korotkin [@korot1], the straight forward continuous limit leading to the above form can be found in [@korotmat1; @korotmat2]. For the relation to Riemann-Hilbert problems see [@prd]. In the case genus 0, the Ernst potential is real, and we get a solution of the Weyl class in the form (\[newton2\]). For higher genus, these solutions are in general non-static and thus generalize (\[newton2\]) to the stationary case. In [@prl; @prd2] it was possible to identify a physically interesting subclass. Theorem 4.6:\ Let the conditions of Theorem 4.5 hold, and in addition let $\Sigma$ be a hyperelliptic Riemann surface of even genus $g=2n$ given by $\mu^{2}(K)=(K+\mathrm{i}z)(K-\mathrm{i}z)\prod_{i=1}^{n}(K^{2}-E_{i}^{2}) (K^{2}-\bar{E}_{i}^{2})$, let the function $G$ be subject to the condition $G(-\tau)=G(\tau)$, and let $[n]$ be the characteristic with $n^{1}_{i}=0$ and $n^{2}_{i}=1$. Then the function $f$ given by $$f(\rho,\zeta)=\frac{\Theta[n](\omega(\infty^{+})+u)}{ \Theta[n](\omega(\infty^{-})+u)} \exp\left\{ \frac{1}{2\pi\mathrm{ i}}\int\limits_\Gamma \ln G(\tau)\mathrm{ d}\omega_{\infty^{+}\infty^-}(\tau) \right\} \label{rel1}$$ is an equatorially symmetric solution to the Ernst equation ($f(-\zeta)=\bar{f}(\zeta)$) which is everywhere regular except at the disk if $\Theta(\omega(\infty^{-})+u) \neq 0$. For a proof see [@prl; @prd2] where one can also find how the characteristic can be generalized. In the following we will only use the characteristic of the above theorem. A quantity of special interest is the metric function $a$. In [@korot1] it was shown that one can relate it directly to theta functions without having to perform the integration of (\[vac9\]), $$Z:=(a-a_{0})e^{2U}=D_{\infty^{-}}\ln\frac{\Theta(\omega(\infty^{-})+u)}{ \Theta[n](\omega(\infty^{-})+u)} \label{eq24}$$ where $D_{P}F(\omega(P))$ denotes the coefficient of the linear term in the expansion of the function $F(\omega(P))$ in the local parameter in the neighborhood of $P$, where $\Theta$ is the Riemann theta function with the characteristic $[m]$ and $m^{1}_{i}=m^{2}_{i}=0$, and where the constant $a_{0}$ is defined by the condition that $a$ vanishes on the regular part of the axis. It is possible to give an algebraic representation of the solutions (\[rel1\]) (see [@meinelneugebauer] and [@korotneu]). We define the divisor $X=\sum_{i=1}^{g}K_{i}$ as the solution of the Jacobi inversion problem ($i=1,\ldots,g$) $$\omega_{i}(X)-\omega_{i}(D)=\frac{1}{2\pi \mathrm{i}} \int_{\Gamma}^{}\ln G \frac{\tau^{i-1}d\tau}{\mu(\tau)}=:\tilde{u}_{i} \label{eq18},$$ where the divisor $D=\sum_{i=1}^{g}E_{i}$. With the help of these divisors, we can write (\[rel1\]) in the form $$\ln f=\int_{D}^{X}\frac{\tau^{g}d\tau}{\mu(\tau)}-\frac{1}{2\pi \mathrm{i}} \int_{\Gamma}^{}\ln G \frac{\tau^{g}d\tau}{\mu(\tau)} \label{eq19},$$ Since the $\tilde{u}_{i}$ in (\[eq18\]) are just the periods of the second integral in (\[eq19\]), they are subject to a system of differential equations, the so-called Picard-Fuchs system (see [@prd2] and references given therein). In our case this leads to $$\sum_{n=1}^{g}\frac{(K_n-P_0)K_n^j}{\mu(K_n)}K_{n,z}=0, \quad j=0,..., g-2 \label{ddd17}$$ and $$(\ln f)_{z}=\sum_{n=1}^{g}\frac{(K_n-P_0)K_n^{g-1}}{\mu(K_n)}K_{n,z} \label{ddd18}.$$ Solving for the $K_{n,z}$, $n=1,\ldots,g$, we get $$K_{n,z}=(\ln f)_{z}\frac{\mu(K_{n})}{K_{n}-P_{0}}\frac{1}{\prod_{m=1, m\neq n}^{g}(K_{n}-K_{m})} \label{ddd19}.$$ Additional information follows from the reality of the $\tilde{u}_{i}$ which implies $\omega(X)-\omega(D)=\omega(\bar{X})-\omega(\bar{D})$. Using Abel’s theorem on this condition, we obtain the relation for an arbitrary $K\in \mbox{C}$ $$(1-x^2)\prod_{i=1}^{g}(K-K_i)(K-\bar{K}_i)=\prod_{i=1}^{g}(K-E_i)(K -\bar{E}_i)-(K-P_0)(K-\bar{P}_0)Q_2^2(K) \label{ddd2},$$ where with purely imaginary $x_i$, $x$ $$Q_2(K)=x_0+x_1K+...+x_{g-2}K^{g-2}+xK^{g-1} \label{ddd3}.$$ Since (\[ddd2\]) has to hold for all $K\in\mbox{C}$, it is equivalent to $2g$ real algebraic equations for the $K_{i}$ if the $x_{i}$ are given. Using (\[eq20\]) and (\[eq19\]) we find $$\frac{f}{\bar{f}}=\frac{1+x}{1-x} \label{eq21}$$ which implies $x=\mathrm{i}be^{-2U}$. Remark: To solve boundary value problems with the class of solutions (\[rel1\]), one has two kinds of freedom: the function $G$ as before and the branch points $E_{i}$ of the Riemann surface as a discrete degree of freedom. Since one would need to specify two free functions to solve a general boundary value problem for the Ernst equation, it is obvious that one can only solve a restricted class of problems on a given surface, and that one cannot expect to solve general problems on a surface of finite genus. But once one has constructed a solution which takes the imposed boundary data at the disk, one has to check the condition $\Theta(\omega(\infty^{-})+u) \neq 0$ in the whole spacetime to actually prove that one has found the desired solution: a solution that is everywhere regular except at the disk where it has to take the imposed boundary conditions. There are in principle two ways of generalizing the approach used for the Newtonian case: One can eliminate $\Omega$ from the two real equations (\[eq9\]) and enter the resulting equation with a solution (\[rel1\]) on a chosen Riemann surface. This will lead for given $\gamma$ to a non-linear integral equation for $\ln G$. In general there is little hope to get explicit solutions to this equation (for a numerical treatment of differentially rotating disks along this line in the genus 2 case see [@ansorgmeinel]). Once a function $G$ is found, one can read off the rotation law $\Omega$ on a given Riemann surface from (\[rel1\]). Another approach is to establish the relations between the real and the imaginary part of the Ernst potential which exist on a given Riemann surface for arbitrary $G$. The simplest example for such a relation is provided by the function $w=e^{\mathrm{i}\psi}$ which is a function on a Riemann surface of genus 0, where we have obviously $\|w\|=1$. As we will point out in the following, similar relations also exist for an Ernst potential of the form (\[rel1\]), but they will lead to a system of differential equations. Once one has established these relations for a given Riemann surface, one can determine in principle which boundary value problems can be solved there (in our example which classes of functions $\Omega$, $\gamma$ can occur) by the condition that one of the boundary conditions must be identically satisfied. The second equation will then be used to determine $G$ as the solution of an integral equation which is possibly non-linear. Following the second approach, we want to study the implications of the hyperelliptic Riemann surface for the physical properties of the solutions. Axis Relations {#sec.4} ============== In order to establish relations between the real and the imaginary part of the Ernst potential, we will first consider the axis of symmetry ($\rho=0$) where the situation simplifies decisively. In addition the axis is of interest since the asymptotically defined multipole moments [@geroch; @hansen] can be read off there [@fodor]. On the axis the Ernst potential can be expressed through functions defined on the Riemann surface $\Sigma'$ given by $\mu'{}^{2}=\prod_{i=1}^{g}(K-E_{i})(K-\bar{E}_{i})$, i.e. the Riemann surface obtained from $\Sigma$ by removing the cut $[P_{0},\bar{P}_{0}]$ which just collapses on the axis. We use the notation of the previous section and let a prime denote that the corresponding quantity is defined on the surface $\Sigma'$. The cut system is as in the previous section with $[E_{1}, \bar{E}_{1}]$ taking the role of $[P_{0},\bar{P}_{0}]$ (all $b$-cuts cross $[E_{1},\bar{E}_{1}]$). We choose the Abel map in a way that $\omega'(E_{1})=0$. It was shown in [@prd2] that for genus $g>1$ the Ernst potential takes the form (for $\zeta>0$) $$f(0,\zeta)=\frac{\vartheta \left(\int_{\zeta^+}^{\infty^+}\mathrm{d}\omega'+u'\right) -\exp(-(\omega'_g(\infty^+)+u_g)) \vartheta\left(\int_{\zeta^-}^{\infty^+}\mathrm{d}\omega' +u'\right)}{\vartheta \left(\int_{\zeta^+}^{\infty^+}\mathrm{d}\omega'-u'\right) -\exp(-(\omega'_g(\infty^+)-u_g)) \vartheta\left(\int_{\zeta^-}^{\infty^+}\mathrm{d}\omega' -u'\right)}e^{I+u_{g}} \label{sing7},$$ where $\vartheta$ is the theta function on $\Sigma'$ with the characteristic $\alpha_{i}'=0$, $\beta_{i}'=1$ for $i=1,\ldots, g-1$, where $I=\frac{1}{2\pi \mathrm{i}} \int\limits_\Gamma \ln G(\tau)\mathrm{ d}\omega'_{\infty^{+}\infty^-}(\tau)$, $\mathrm{d}\omega_{g}= \mathrm{d}\omega_{\zeta^{-}\zeta^{+}}$, and where $u_{g}=\frac{1}{2\pi \mathrm{i}} \int\limits_\Gamma \ln G(\tau)\mathrm{ d}\omega_{g}(\tau)$. Notice that the $u_{i}'$ and $I$ are constant with respect to $\zeta$. The only term dependent both on $G$ and on $\zeta$ is $u_{g}$. To establish a relation on the axis between the real and the imaginary part of the Ernst potential independent of $G$, the first step must be thus to eliminate $u_{g}$. We can state the following Theorem 4.1:\ The Ernst potential (\[sing7\]) satisfies for $g> 1$ the relation $$P_{1}(\zeta)f\bar{f}+P_{2}(\zeta)b+P_{3}(\zeta)=0 \label{eq13},$$ where the $P_{i}$ are real polynomials in $\zeta$ with coefficients depending on the branch points $E_{i}$ and the $g$ real constants $\int_{\Gamma}^{}\ln G \tau^{i}d\tau /\mu'(\tau)$ with $i=0,\ldots,g-1$. The degree of the polynomials $P_{1}$ and $P_{3}$ is $2g-3$ or less, the degree of $P_{2}$ is $2g-2$ or less. To prove this theorem we need the fact that one can express integrals of the third kind via theta functions with odd characteristic denoted by $\vartheta_{o}$, $$\exp(-\omega_{g}(\infty^{+}))=-\frac{\vartheta_{o}(\omega'(\infty^{+})- \omega'(\zeta^{+}))}{\vartheta_{o}(\omega'(\infty^{+})- \omega'(\zeta^{-}))}. \label{eq13.3}$$ Proof:\ The first step is to establish the relation $$Af\bar{f}+B\mathrm{i}b+1=0 \label{eq12},$$ where $$Ae^{2I}=-\frac{\vartheta\left(u'+\int_{\zeta^{-}}^{\infty^{-}}\mathrm{d}\omega'\right) \vartheta\left(u'+\int_{\zeta^{+}}^{\infty^{-}}\mathrm{d}\omega'\right)}{ \vartheta\left(u'+\int_{\zeta^{-}}^{\infty^{+}}\mathrm{d}\omega'\right) \vartheta\left(u'+\int_{\zeta^{+}}^{\infty^{+}}\mathrm{d}\omega'\right)} \label{eq14}$$ and $$Be^{I}=\frac{e^{-\omega_{g}(\infty^{+})}\vartheta\left( u'+\int_{\zeta^{-}}^{\infty^{+}}\mathrm{d}\omega'\right) \vartheta\left(u'+\int_{\zeta^{+}}^{\infty^{-}}\mathrm{d}\omega'\right) +e^{\omega_{g}(\infty^{+})}\vartheta\left(u'+ \int_{\zeta^{+}}^{\infty^{+}}\mathrm{d}\omega'\right) \vartheta\left(u'+\int_{\zeta^{-}}^{\infty^{-}}\mathrm{d}\omega'\right)}{ \vartheta\left(u'+\int_{\zeta^{-}}^{\infty^{+}}\mathrm{d}\omega'\right) \vartheta\left(u'+\int_{\zeta^{+}}^{\infty^{+}}\mathrm{d}\omega'\right)} \label{eq15}$$ which may be checked with (\[sing7\]) by direct calculation. The reality properties of the Riemann surface $\Sigma'$ and the function $G$ imply that $A$ is real and that $B$ is purely imaginary. We use the addition theorem (\[ternary\]) with $[m_{1}]=\ldots=[m_{4}]$ equal to the characteristic of $\vartheta$ for (\[eq14\]) to get $$Ae^{2I}=-\frac{\sum\limits_{2a \in (Z_2)^{2g}}\exp(-4\pi \mathrm{i} \langle m_1^{1},a^{2}\rangle )\vartheta^{2}[a](u'+\omega'(\infty^{-})) \vartheta^{2}[a](\omega'(\zeta^{+}))}{\sum\limits_{2a \in (Z_2)^{2g}}\exp(-4\pi \mathrm{i} \langle m_1^{1},a^{2}\rangle ) \vartheta^{2}[a](u'+\omega'(\infty^{+})) \vartheta^{2}[a](\omega'(\zeta^{+}))}. \label{eq16}$$ This term is already in the desired form. Using the relation for root functions (\[root1\]), one can directly see that the right-hand side is a quotient of polynomials of order $g-1$ or lower in $\zeta$. For (\[eq15\]) we use (\[eq13.3\]) with $[\tilde{m}_{1}]=[\tilde{m}_{2}] =[K_{R}]$ as the characteristic of the odd theta function $\vartheta_{o}$ and let $[\tilde{m}_{3}]=[\tilde{m}_{4}]$ be equal to the characteristic of $\vartheta$. The addition theorem (\[ternary\]) then leads to $$\begin{aligned} Be^{I}&=&-\frac{\sum\limits_{2a \in (Z_2)^{2g}}\exp(-4\pi \mathrm{i} \langle \tilde{m}_1^{1},a^{2}\rangle )\Theta'^{2}[n+a](u'+\omega'(\infty^{-})) \vartheta^{2}[a](\omega'(\zeta^{+}))}{\sum\limits_{2a \in (Z_2)^{2g}}\exp(-4\pi \mathrm{i} \langle m_1^{1},a^{2}\rangle ) \vartheta^{2}[a](u'+\omega'(\infty^{+})) \vartheta^{2}[a](\omega'(\zeta^{+}))}\times\nonumber\\ &&\sum\limits_{2a \in (Z_2)^{2g}}\exp(-4\pi \mathrm{i} \langle m_1^{1},a^{2}\rangle )\frac{\vartheta^{2}[a](u')}{\vartheta^{2}(0)} \times\nonumber\\ && \left(\frac{\vartheta^{2}[a](\omega'(\infty^{+})-\omega'(\zeta^{+}))}{ \vartheta_{o}^{2}(\omega'(\infty^{+})-\omega'(\zeta^{+}))} +\frac{\vartheta^{2}[a](\omega'(\infty^{+})-\omega'(\zeta^{-}))}{ \vartheta_{o}^{2}(\omega'(\infty^{+})-\omega'(\zeta^{-}))}\right) \label{eq17},\end{aligned}$$ where $n$ follows from $\tilde{m}$ as in Theorem 4.1. The first fraction in (\[eq17\]) is again the quotient of polynomials of degree $g-1$ in $\zeta$ for the same reasons as above. But since the quotient must vanish for $\zeta\to\infty$, the leading terms in the numerator just cancel. It is thus a quotient of polynomials of degree $g-2$ or less in the numerator and $g-1$ or less in the denominator. To deal with the quotients $ \vartheta^{2}[a](\omega'(\infty^{+})-\omega'(\zeta^{\pm}))/ \vartheta_{o}^{2}(\omega'(\infty^{+})-\omega'(\zeta^{\pm}))$, we define the divisors $T^{\pm}=T_{1}^{\pm}+\ldots+T_{g-1}^{\pm}$ as the solutions of the Jacobi inversion problems $\omega'(T^{\pm})-\omega'(Y)= \omega'(\infty^{+})+\omega'(\zeta^{\pm})$ where $Y$ is the divisor $Y=E_{1}+\ldots+E_{g-1}$. Abel’s theorem then implies for arbitrary $K\in \mbox{C}$ $$\prod_{i=1}^{g-1}(K-T_{i}^{\pm})(K-\zeta)= (K-A^{\pm})^{2}\prod_{i=1}^{g-1}(K-E_{i})-(K-E_{g})\prod_{i=1}^{g}(K-\bar{E}_{i}) \label{eq17a},$$ where $$\zeta-A^{\pm}=\pm \frac{\mu'(\zeta)}{\prod_{i=1}^{g-1}(\zeta-E_{i})} \label{eq17b}.$$ Let $Q_{j}$ be given by the condition $[Q_{j}+K_{R}]=[a]$, i.e. $Q_{j}$ is a branch point of $\Sigma'$. Then we get for the quotient $$\frac{\vartheta^{2}[a](\omega'(\infty^{+})-\omega'(\zeta^{\pm}))}{ \vartheta_{o}^{2}(\omega'(\infty^{+})-\omega'(\zeta^{\pm}))}=const \prod_{i=1}^{g-1}\frac{T^{\pm}_{i}-Q_{j}}{T^{\pm}_{i}-E_{1}} \label{eq17c},$$ where $const$ is a $\zeta$-independent constant. With the help of (\[eq17a\]), it is straight forward to see that for $Q_{j}\in Y$, the theta quotient is just proportional to $(\zeta-E_{1})/(\zeta-Q_{j})$ whereas for $Q_{j}\notin Y$, the term is proportional to $(\zeta-E_{1})(Q_{j}-A^{\pm})^{2}/(\zeta-Q_{j})$. Using (\[eq17b\]) one recognizes that the terms containing roots just cancel in (\[eq17\]). The remaining terms are just quotients of polynomials in $\zeta$ with maximal degree $g$ in the numerator and $g-2$ in denominator. This completes the proof. Remark: The remaining dependence on $G$ through $u'$ and $I$ can only be eliminated by differentiating relation (\[sing7\]) $g$ times. If we prescribe e.g. the function $b$ on a given Riemann surface (this just reflects the fact that the function $G$ can be freely chosen in (\[sing7\])), we can read off $e^{2U}$ from (\[eq13\]). To fix the constants related to $G$ in (\[eq13\]) one needs to know the Ernst potential and $g-1$ derivatives at some point on the axis where the Ernst potential is regular, e.g. at the origin or at infinity, where one has to prescribe the multipole moments. If the Ernst potential were known on some regular part of the axis, one could use (\[eq13\]) to read off the Riemann surface (genus and branch points). Equation (\[sing7\]) is then an integral equation for $G$ for known sources. This just reflects a result of [@hauser] that the Ernst potential for known sources can be constructed via Riemann-Hilbert techniques if it is known on some regular part of the axis. In practice it is difficult to express the coefficients in the polynomials $P_{i}$ via the constants $u_{i}'$ and $I$, and it will be difficult to get explicit expressions. We will therefore concentrate on the general structure of the relation (\[eq13\]), its implications on the multipoles and some instructive examples. Let us first consider the case genus 1 which is not generically equatorially symmetric. In this case the Riemann surface $\Sigma'$ is of genus 0. One can use formula (\[sing7\]) for the axis potential if one replaces the theta functions by 1. We thus end up with $$f\bar{f} -2b \frac{\zeta-\alpha_{1}}{\beta_{1}} e^{I}=e^{2I} \label{bound4}.$$ Here the only remaining $G$-dependence is in $I$. If $f_{0}=f(0,0)$ is given, $e^{I}$ follows from $f_{0}\bar{f}_{0}+2b_{0} \alpha_{1}e^{I}/\beta_{1}=e^{2I}$, if the in general non-real mass $M$ is known, the constant $e^{I}$ follows from $1+4\mbox{Im}M e^{I}/\beta_{1}=e^{2I}$. In the latter case the imaginary part of the Arnowitt-Deser-Misner mass (this corresponds to a NUT-parameter) will be sufficient. Differentiating (\[bound4\]) once will lead to a differential relation between the real and the imaginary part of the Ernst potential which holds for all $G$, which means it reflects only the impact of the underlying Riemann surface on the structure of the solution. Remark: For equatorially symmetric solutions, one has on the positive axis the relation $f(-\zeta)\bar{f}(\zeta)=1$ (see [@panos; @meinel], this is to be understood in the following way: the function $\|\zeta\|$ is even in $\zeta$, but restricted to positive $\zeta$ it seems to be an odd function, and it is exactly this behavior which is addressed by the above formula). This leads to the conditions $$P_{1}(-\zeta)=-P_{3}(\zeta), \quad P_{2}(-\zeta)=P_{2}(\zeta) \label{eq15.1}.$$ The coefficients in the polynomials depend on the $g/2$ integrals $\int_{\Gamma}^{}d\tau\ln G \tau^{2i}/\mu'(\tau)$ ($i=0,\ldots, g/2-1$ and the branch points. The simplest interesting example is genus 2, where we get with $E_{1}^{2}=\alpha+\mathrm{i}\beta$ $$f\bar{f}(\zeta-C_{1})+\frac{\sqrt{2}}{C_{2}}(\zeta^{2}- \alpha-C_{2}^{2})b=\zeta+C_{1} \label{eq16.1},$$ i.e. a relation which contains two real constants $C_{1}$, $C_{2}$ related to $G$. In case that the Ernst potential at the origin is known, one can express these constants via $f_{0}$. A relation of this type, which is as shown typical for the whole class of solutions, was observed in the first paper of [@neugebauermeinel1] for the rigidly rotating dust disk. Differential relations in the whole spacetime {#sec.5} ============================================= The considerations on the axis have shown that it is possible there to obtain relations between the real and the imaginary part of the Ernst potential which are independent of the function $G$ and thus reflect only properties of the underlying Riemann surface. The found algebraic relations contain however $g$ real constants related to the function $G$, which means that one has to differentiate $g$ times to get a differential relation which is completely free of the function $G$. These constants were just the integrals $u'$ and $I$ which are only constant with respect to the physical coordinates on the axis where the Riemann surface $\Sigma$ degenerates. Thus one cannot hope to get an algebraic relation in the whole spacetime as on the axis. Instead one has to deal with integral equations or to look directly for a differential relation. To avoid the differentiation of theta functions with respect to a branch point of the Riemann surface, we use the algebraic formulation of the hyperelliptic solutions (\[eq18\]) and (\[eq19\]). From the latter it can also be seen how one could get a relation independent of $G$ without differentiation: one can consider the equations (\[eq18\]) and (\[eq19\]) as integral equations for $G$. In principle one could try to eliminate $G$ and $X$ from these equations and (\[ddd2\]). We will not investigate this approach but try to establish a differential relation. To this end it proves helpful to define the symmetric (in the $K_{n}$) functions $S_{i}$ via $$\prod_{i=1}^{g}(K-K_i)=:K^g-S_{g-1}K^{g-1}+...+S_0 \label{eq22}$$ i.e. $S_{0}=K_{1}K_{2}\ldots K_{g}$, …, $S_{g-1}=K_{1}+\ldots+K_{g}$. The equations (\[ddd2\]) are bilinear in the real and imaginary parts of the $S_{i}$ which are denoted by $R_{i}$ and $I_{i}$ respectively. With this notation we get Theorem 6.1:\ The $x_{i}$ and the Ernst potential $f$ are subject to the system of differential equations $$\begin{aligned} 0&=&\left(R_{0}-P_{0}R_{1}+\ldots +P_{0}^{g}(-1)^{g}\right)x_{z}- \frac{\mathrm{i}}{2}Q_{2}(P_{0})\nonumber\\ &&-\frac{\mathrm{i}}{2}(1-x^{2})(\ln f\bar{f})_{z}\left(I_{0}-P_{0}I_{1}+\ldots +(-1)^{g-1}I_{g-1}P_{0}^{g-1}\right) \label{eq23}\end{aligned}$$ and for $g>1$ $$\begin{aligned} x_{j,z} & = &x_{z} \left((-1)^{j+1}R_{j+1}+\ldots+P_{0}^{g-j-1}\right) -\mathrm{i}(x_{j+1}+\ldots +xP_{0}^{g-j-2})\nonumber \\ & &-\frac{\mathrm{i}}{2}(1-x^{2})(\ln f\bar{f})_{z}( ((-1)^{j+1}I_{j+1}+\ldots-P_{0}^{g-j-2}I_{g-1}) \label{ddd35}.\end{aligned}$$ Proof:\ Differentiating (\[ddd2\]) with respect to $z$ and eliminating the derivatives of the $K_{i,z}$ via the Picard-Fuchs relations (\[ddd19\]), we end up with a linear system of equations for the derivatives of the $x_{i}$ and $x$ which can be solved in standard manner. The Vandemonde-type determinants can be expressed via the symmetric functions. For $x_{z}$ one gets (\[eq23\]). The equations for the $x_{j,z}$ are bilinear in the symmetric functions. They can be combined with (\[eq23\]) to (\[ddd35\]). Remark: If one can solve (\[ddd2\]) for the $K_{i}$, the equations (\[eq23\]) and (\[ddd35\]) will be a non-linear differential system in $z$ (and $\bar{z}$ which follows from the reality properties) for the $x_{i}$, $x$ and $f$ which only contains the branch points of the Riemann surface as parameters. For the metric function $a$, we get with (\[eq24\]) Theorem 6.2:\ The metric function $a$ is related to the functions $x_{i}$ and $S_{i}$ via $$Z=\frac{\mathrm{i}x_{g-2}}{1-x^{2}}-I_{g-1}-\frac{\mathrm{i}x\zeta}{1-x^{2}} \label{eq26}.$$ for $g>1$ and $$Z=-I_{0}+\frac{\mathrm{i}x(\alpha_{1}-\zeta)}{1-x^{2}} \label{eq26a}$$ for $g=1$. Proof:\ To express the function $Z$ via the divisor $X$, we define the divisor $T=T_{1}+\ldots+T_{g}$ as the solution of the Jacobi inversion problem $\omega(T)=\omega(X)+\omega(P)$ where $P$ is in the vicinity of $\infty^{-}$ (only terms of first order in the local parameter near $\infty^{-}$ are needed). Using the formula for root functions (\[root2\]), we get for the quantity $Z$ in (\[eq24\]) $$Z=\frac{\mathrm{i}}{2}D_{\infty^{-}}\ln\prod_{i=1}^{g}\frac{T_{i}- \bar{P}_{0}}{T_{i}-P_{0}} \label{eq25}.$$ Applying Abel’s theorem to the definition of $T$ and expanding in the local parameter near $\infty^{-}$, we end up with (\[eq26\]) for general $g>1$ and with (\[eq26a\]) for $g=1$. Remark:\ 1. For $g>1$ equation (\[eq26\]) can be used to replace the relation for $x_{g-2,z}$ in (\[ddd35\]) since the latter is identically fulfilled with (\[eq26\]) and (\[vac9\]). 2\. An interesting limiting case is $G\approx 1$ where $f\approx 1$, i.e. the limit where the solution is close to Minkowski spacetime. By the definition (\[eq18\]), the divisor $X$ is in this case approximately equal to $D$. Thus the symmetric functions in (\[ddd35\]) and (\[eq23\]) can be considered as being constant and given by the branch points $E_{i}$. Relation (\[eq26\]) implies that the quantity $Z$ is approximately equal to $I_{g-2}$ in this limit, i.e. it is mainly equal to the constant $a_{0}$ in lowest order. Since the differential system (\[ddd35\]) and (\[eq23\]) is linear in this limit, it is straight forward to establish two real differential equations of order $g$ for the real and the imaginary part of the Ernst potential. In principle this works also in the non-linear case, where sign ambiguities in the solution of (\[eq18\]) can be fixed by the Minkowskian limit. To illustrate the above equations we will first consider the elliptic case. This is the only case where one can establish an algebraic relation between $Z$ and $b$ independent of $G$. Equations (\[ddd2\]) lead to $$\begin{aligned} (1-x^{2})R_{0} & = & \alpha_{1}-\zeta x^{2} \nonumber \\ (1-x^{2})S_{0}\bar{S}_{0} & = & E_{1}\bar{E}_{1}-P_{0}\bar{P}_{0}x^{2} \label{eq27}.\end{aligned}$$ Formula (\[eq26a\]) takes with (\[eq27\]) (the sign of $I_{0}$ is fixed by the condition that $I_{0}=-\beta_{1}$ for $x=0$) the form $$(1-x^{2})Z=\mathrm{ i}x(\alpha_{1}-\zeta)+\sqrt{(1-x^{2})(\beta_{1}^{2}- \rho^{2}x^{2}) -x^{2}(\alpha_{1}-\zeta)^{2}} \label{eq28}.$$ This relation holds in the whole spacetime for all elliptic potentials, i.e. for all possible choices of $G$ in (\[rel1\]). This implies that one can only solve boundary value problems on elliptic surfaces where the boundary data at some given contour $\Gamma_{z}$ satisfy condition (\[eq28\]). In the case genus 2, we get for (\[ddd2\]) $$\begin{aligned} (1-x^2)R_{1} & = & \alpha_1+\alpha_2-\zeta x^2 +xx_0, \nonumber \\ (1-x^2)(R_{1}^{2}+I_{1}^{2}+2R_{0}) & = & (\alpha_1+\alpha_2)^2+2\alpha_1\alpha_2 +\beta_1^2+\beta_2^2- x_0^2-x^2(\rho^2+\zeta^2)+4\zeta xx_0, \nonumber \\ (1-x^2)(R_{1}R_{0}+I_{1}I_{0}) & = & \alpha_1\alpha_2(\alpha_1+\alpha_2)+\alpha_1\beta_2^2 +\alpha_2\beta_1^2-\zeta x_0^2+(\rho^2+\zeta^2)xx_0, \nonumber \\ (1-x^2)(R_{0}^{2}+I_{0}^{2}) & = & (\alpha_1^2+\beta_1^2)(\alpha_2^2+\beta_2^2) -(\rho^2+\zeta^2)x_0^2 \label{bound33}.\end{aligned}$$ The aim is to determine the $S_{i}$ and $x_{0}$ from (\[bound33\]) and $$(1-x^{2})(Z+I_{1})=\mathrm{i}x_{0}-\zeta \mathrm{i}x, \label{bound33.1}$$ and to eliminate these quantities in $$(R_{0}-P_{0}R_{1}+P_{0}^{2})x_{z}=\frac{\mathrm{i}}{2}(x_{0}+P_{0}x) +\frac{\mathrm{i}}{2}(1-x^{2})(\ln f\bar{f})_{z}(I_{0}-P_{0}I_{1}) \label{eq29}$$ which follows from (\[eq23\]). Remark: Boundary value problems\ Since the above relations will hold in the whole spacetime, it is possible to extend them to an arbitrary smooth boundary $\Gamma_{z}$, where the Ernst potential may be singular (a jump discontinuity) and where one wants to prescribe boundary data (combinations of $f$, $f_{z}$). If these data are of sufficient differentiability (at least $C^{g,\alpha}(\Gamma_{z})$), we can check the solvability of the problem on a given surface with the above formulas. The conditions on the differentiability of the boundary data can be relaxed by working directly with the equations (\[eq18\]) and (\[eq19\]) which can be considered as integral equations for $\ln G$. The latter is not very convenient if one wants to construct explicit solutions, but it makes it possible to treat boundary value problems where the boundary data are Hölder continuous. We will only work with the differential relations and consider merely the derivatives tangential to $\Gamma_{z}$ in (\[ddd35\]) to establish the desired differential relations between $a$, $b$ and $U$. One ends up with two differential equations which involve only $U$, $b$ and derivatives. The aim is to construct the spacetime which corresponds to the prescribed boundary data from these relations. To this end one has to integrate the differential relations using the boundary conditions. Integrating one of these equations, one gets $g$ real integration constants which cannot be freely chosen since they arise from applying the tangential derivatives in (\[ddd35\]). Thus they have to be fixed in a way that the integrals on the right-hand side of (\[eq18\]) are in fact the $b$-periods of the second integral on the right-hand side of (\[eq18\]) and that (\[eq19\]) holds. The second differential equation arises from the use of normal derivatives of the Ernst potential in (\[eq23\]). To satisfy the $b$-period condition (\[eq18\]), one has to fix a free function in the integrated form of the corresponding differential equation. Thus one has to complement the two differential equations following from (\[eq23\]) with an integral equation which is obtained by eliminating $G$ from e.g. $\tilde{u}_{1}$ and $\tilde{u}_{2}$ in (\[eq18\]). For given boundary data, the system following from (\[eq18\]) may in principle be integrated to give $e^{2U}$ and $b$ in dependence of the boundary data. Then the in general non-linear integral equation will establish whether the boundary data are compatible with the considered Riemann surface. This is typically a rather tedious procedure. There is however a class of problems where it is unnecessary to use this integral equation. In case that the differential equations hold for an arbitrary function $e^{2U}$, the integral equation will only be used to determine this metric function, but the boundary value problem will be always solvable (locally). This offers a constructive approach to solve boundary value problems without having to consider non-linear integral equations. Counter-rotating disks of genus 2 {#sec.6} ================================= Since it is not very instructive to establish the differential relations for genus 2 in the general case, we will concentrate in this section on the form these equations take in the equatorially symmetric case for counter-rotating dust disks. In this case, the solutions are parametrized by $E_{1}^{2}=\alpha+\mathrm{i}\beta$. We will always assume in the following that the boundary data are at least $C^{2}(\Gamma_{z})$ (the normal derivatives of the metric functions have a jump at the disk, but the tangential derivatives are supposed to exist up to at least second order). Putting $s=be^{-2U}$ and $y=e^{2U}$, we get for (\[eq29\]) for $\zeta=0$, $\rho\leq 1$ $$\begin{aligned} \mathrm{ i}x_{0} & = & \left(R_{0}-\rho^{2}-sI_{0}\right)\frac{b_{\zeta}}{y}- \rho\left(R_{1}-sI_{1}\right)\frac{b_{\rho}}{y}-\left(s(R_{0}-\rho^{2}) +I_{0}\right) \frac{y_{\zeta}}{y}+\rho\left(sR_{1}+I_{1}\right)\frac{y_{\rho}}{y}, \nonumber \\ \rho s & = & \left(R_{0}-\rho^{2}-sI_{0}\right)\frac{b_{\rho}}{y} +\rho\left(R_{1}-sI_{1}\right)\frac{b_{\zeta}}{y}- \left(s(R_{0}-\rho^{2})+I_{0}\right) \frac{y_{\rho}}{y}-\rho\left(sR_{1}+I_{1}\right)\frac{y_{\zeta}}{y} \label{achse66},\quad\quad\end{aligned}$$ where the $S_{i}$ and $\mathrm{i}x_{0}$ are taken from (\[bound33\]) and (\[bound33.1\]). Since counter-rotating dust disks are subject to the boundary conditions (\[eq9\]), we can replace the normal derivatives in (\[achse66\]) via (\[eq9\]) which leads to a differential system where only tangential derivatives at the disk occur. With (\[bound33\]) and (\[bound33.1\]) we get $$\begin{aligned} \mathrm{ i}x_{0}+(Z-\mathrm{i}x_{0})\frac{R_{0}-\rho^{2}}{I_{1}R} & = & \left(\rho-\frac{R^2+\rho^2+\delta y^{2}}{2R\rho} \frac{R_{0}-\rho^{2}}{I_{1}}\right)\left((-Z+ \mathrm{i}x_{0})\frac{y_{\rho}}{y}- sZ\frac{b_{\rho}}{y}\right) \label{achse69}, \\ \rho s\left(1-\frac{Z}{R}\right) & = & \left(\frac{R_{0}-\rho^{2}}{I_{1}}-\frac{R^2+\rho^2+\delta y^{2}}{2R} \right)\left(sZ\frac{y_{\rho}}{y}+(-Z+\mathrm{i}x_{0})\frac{b_{\rho}}{y} \right).\quad\quad \label{achse70}\end{aligned}$$ With $I_{1}=\mathrm{i}x_{0}/(1-x^{2})-Z$ and $$R_{0}=\frac{\mathrm{i}x_{0}Z-\alpha-\frac{\rho^{2}}{2}}{1-x^{2}}- \frac{Z^{2}-\rho^{2}}{2}, \label{eq40}$$ the function $\mathrm{i}x_{0}$ follows from $$R_{0}^{2}+\frac{(R_{0}-\rho^{2})^{2}}{I_{1}^{2}} \frac{x^{2}x_{0}^{2}}{(1-x^{2})^{2}}= \frac{\alpha^{2}+\beta^{2}-\rho^{2}x_{0}^{2}}{1-x^{2}} \label{eq41},$$ i.e. an algebraic equation of fourth order for $\mathrm{i}x_{0}$ which can be uniquely solved by respecting the Minkowskian limit. Thus (\[achse69\]) and (\[achse70\]) are in fact a differential system which determines $b$ and $y$ in dependence of the angular velocity $\Omega$. Newtonian limit {#subsec.6.1} --------------- For illustration we will first study the Newtonian limit of the equations (\[achse70\]) (where counter-rotation does not play a role). This means we are looking for dust disks with an angular velocity of the form $\Omega=\omega q(\rho)$ where $\|q(\rho)\|\leq 1$ for $\rho\leq 1$, and where the dimensionless constant $\omega<<1$. Since we have put the radius $\rho_{0}$ of the disk equal to 1, $\omega=\omega \rho_{0} $ is the upper limit for the velocity in the disk. The condition $\omega<<1$ just means that the maximal velocity in the disk is much smaller than the velocity of light which is equal to 1 in the used units. An expansion in $\omega$ is thus equivalent to a standard post-Newtonian expansion. Of course there may be dust disks of genus 2 which do not have such a limit, but we will study in the following which constraints are imposed by the Riemann surface on the Newtonian limit of the disks where such a limit exists. The invariance of the metric (\[vac1\]) under the transformation $t\to-t$ and $\Omega \to -\Omega$ implies that $U$ is an even function in $\omega$ whereas $b$ has to be odd. Since we have chosen an asymptotically non-rotating frame, we can make the ansatz $y=1+\omega^{2}y_{2}+\ldots$, $b=\omega^{3}b_{3}+\ldots$, and $a=\omega^{3}a_{3}+\ldots$. The boundary conditions (\[eq9\]) imply in lowest order $y_{2,\rho}=2q^{2}\rho$, the well-known Newtonian limit. Since (\[vac10\]) reduces to the Laplace equation for $y_{2}$ in order $\omega^{2}$, we can use the methods of section (\[sec.2\]) to construct the corresponding solution. In order $\omega^{3}$, the boundary conditions (\[eq9\]) lead to $$b_{3,\rho}=2\rho qy_{2,\zeta} \label{diff13a},$$ whereas equation (\[vac10\]) leads to the Laplace equation for $b_{3}$. Again we can use the methods of section \[sec.2\], but this time we have to construct a solution which is odd in $\zeta$ because of the equatorial symmetry. In principle one can extend this perturbative approach to higher order, where the field equations (\[vac10\]) lead to Poisson equations with terms of lower order acting as source terms, and where the boundary conditions can also be obtained iteratively from (\[eq9\]). With this notation we get\ Theorem 7.1:\ Dust disks of genus 2 which have a Newtonian limit, i.e. a limit in which $\Omega=\omega q(\rho)$ where $\|q(\rho)\|\leq 1$ for $\rho\leq 1$, are either rigidly rotating ($q=1$) or $q$ is a solution to the integro-differential equation $$b_{3}=\left((R_{0}^{0}-\rho^{2})2q-\kappa\right)y_{2,\zeta} \label{eq30}$$ where in the first case $I_{1}^{0}/R_{0}^{0}=2\omega$ and in the second $I_{1}=\kappa \omega$ with $R_{0}^{0}$ and $\kappa$ being $\omega$ independent constants. Proof:\ Since the right hand side of (\[eq18\]) vanishes, we have $K_{i}=E_{i}$ for $\omega\to0$, and thus $a_{0}=I_{1}$ up to at least order $\omega^{3}$. Keeping only terms in lowest order and denoting the corresponding terms of the symmetric functions by $S_{i}^{0}$, we obtain for (\[achse70\]) $$\omega^{3}b_{3}=y_{2,\zeta} \left(2q(R^{0}_{0}-\rho^{2})\omega^{3} -\omega^{2}I^{0}_{1} \right) \label{diff17}.$$ The second equation (\[achse70\]) involves $b_{3,\zeta}$ and is thus of higher order. If (\[diff17\]) holds, this equation will be automatically fulfilled. The $\omega$-dependence in (\[diff17\]) implies that $R_{0}^{0}$, $I_{1}^{0}$ and thus the branch points must depend on $\omega$. Since $y_{2,\zeta}$ is proportional to the density in the Newtonian case, it must not vanish identically. The possible cases following from equation (\[diff17\]) are constant $\Omega$ or (\[eq30\]). Using (\[eq2\]) and (\[eq4\]), one can express $U_{\zeta}$ directly via $\Omega$ which leads to $$y_{2,\zeta}=\frac{4}{\pi}\int_{0}^{1}\frac{\mathrm{d}\rho'}{\rho+\rho'} \partial_{\rho'}(q^{2}\rho'{}^{2}) K(k)$$ with $k=2\sqrt{\rho\rho'}/(\rho+\rho')$. Thus (\[eq30\]) is in fact an integro-differential equation for $q$. This completes the proof. Explicit solution for constant angular velocity and constant relative density {#subsec.6.2} ----------------------------------------------------------------------------- The simplifications of the Newtonian equation (\[diff17\]) for constant $\Omega$ give rise to the hope that a generalization of rigid rotation to the relativistic case might be possible which we will check in the following. Constant $\gamma/\Omega$ makes it in fact possible to avoid the solution of a differential equation and leads thus to the simplest example. We restrict ourselves to the case of constant relative density, $\gamma=const$. The structure of equation (\[achse70\]) suggests that it is sensible to choose the constant $a_{0}$ as $a_{0}=-\gamma/\Omega$ since in this case $Z=R$. This is the only freedom in the choice of the parameters $\alpha$ and $\beta$ on the Riemann surface one has for $g=2$ since one of the parameters will be fixed as in the Newtonian case by the condition that the disk has to be regular at its rim. The second parameter will be determined as an integration constant of the Picard-Fuchs system. We get\ Theorem 7.2:\ The boundary conditions (\[eq9\]) and (\[eq11\]) for the counter-rotating dust disk with constant $\Omega$ and constant $\gamma$ are satisfied by an Ernst potential of the form (\[rel1\]) on a hyperelliptic Riemann surface of genus 2 with the branch points specified by $$\alpha=-1+\frac{\delta}{2}, \quad \beta=\sqrt{\frac{1}{\lambda^2}+\delta-\frac{\delta^2}{4}} \label{eq37a}.$$ The parameter $\delta$ varies between $\delta=0$ (only one component) and $\delta=\delta_{s}$, $$\delta_{s}=2\left(1+\sqrt{1+\frac{1}{\lambda^{2}}}\right) \label{eq37b},$$ the static limit. The function $G$ is given by $$G(\tau)=\frac{\sqrt{(\tau^{2}-\alpha)^2+\beta^2}+\tau^{2}+1}{ \sqrt{(\tau^{2}-\alpha)^2+\beta^2}-(\tau^{2}+1)} \label{eq38}.$$ This is the result which was announced in [@prl2]. Proof:\ The proof of the theorem is performed in several steps.\ 1. Since the second factor on the right-hand side in (\[achse70\]) must not vanish in the Newtonian limit, we find that for $Z=R$ $$\frac{R_{0}-\rho^{2}}{I_{1}}=\frac{Z^2+\rho^2+\delta y^{2}}{2Z} \label{eq31}.$$ With this relation it is possible to solve (\[eq40\]) and (\[bound33.1\]), $$\begin{aligned} \mathrm{i}x_{0}&=&\frac{Z(\rho^2+2 \alpha -\delta y^2 (1-x^{2}))}{Z^2-\rho^2-\delta y^2}\label{eq32} \\ \frac{\delta^2 y^2}{2}(1-x^{2}) & = & -\frac{1}{\lambda}\left(\frac{1}{\lambda} -\delta y\right) +\delta\left(\alpha+\frac{\rho^2}{2}\right)+ \frac{\frac{1}{\lambda}-\delta y}{\sqrt{\frac{1}{\lambda^2}+\delta \rho^2}} \sqrt{\left(\frac{1}{\lambda^2} -\alpha\delta\right)^2+\delta^2 \beta^2}. \nonumber\end{aligned}$$ One may easily check that equation (\[achse69\]) is identically fulfilled with these settings. Thus the two differential equations (\[achse69\]) and (\[achse70\]) are satisfied for an unspecified $y$ which implies that the boundary value problem for the rigidly rotating dust disk can be solved on a Riemann surface of genus 2 (the remaining integral equation which we will discuss below determines then $y$). 2\. To establish the integral equations which determine the function $G$ and the metric potential $e^{2U}$, we use equations (\[eq18\]). Since we have expressed above the $K_{i}$ as a function of $e^{2U}$ alone, the left-hand sides of (\[eq18\]) are known in dependence of $e^{2U}$. It proves helpful to make explicit use of the equatorial symmetry at the disk. By construction the Riemann surface $\Sigma$ is for $\zeta=0$ invariant under the involution $K\to -K$. This implies that the theta functions factorize and can be expressed via theta functions on the covered surface $\Sigma_{1}$ given by $\mu_{1}^{2}(\tau)=\tau(\tau+\rho^{2})((\tau-\alpha)^{2}+\beta^{2})$ and the Prym variety $\Sigma_{2}$ (which is here also a Riemann surface) given by $\mu_{2}^{2}(\tau)=(\tau+\rho^{2})((\tau-\alpha)^{2}+\beta^{2})$ (see [@algebro; @prd2] for details). On these surfaces we define the divisors $V$ and $W$ respectively via $$u_v=\frac{1}{\mathrm{i}\pi}\int_{0}^{-\rho^{2}}\frac{\ln G(\sqrt{\tau}) d\tau}{\mu_{1}(\tau)} =:\int_{0}^{V}\frac{d \tau}{\mu_1}, \quad u_w= \frac{1}{\mathrm{i}\pi} \int_{-\rho^{2}}^{-1}\frac{\ln G(\sqrt{\tau}) d\tau}{\mu_{2}(\tau)} =:\int_{\infty}^{W}\frac{d \tau}{\mu_2} \label{achse94}.$$ For the Ernst potential we get $$\ln f\bar{f}=-\ln \left(1-\frac{2\mathrm{i}x_0}{Z(1-x^2)}\right)+ \int_{0}^{V}\frac{\tau d\tau}{\mu_1}-I_v \label{achse99a}.$$ where $I_{v}=\frac{1}{2\pi\mathrm{i}}\int_{0}^{-\rho^{2}}\frac{\ln G(\sqrt{\tau})\tau \mathrm{d}\tau}{\mu_{1}(\tau)}$. 3\. Using Abel’s theorem and (\[eq18\]), we can express $V$ and $W$ by the divisor $X$ which leads to $$V=-\frac{\rho^2 x_0^2}{Z^2(1-x^2)-2Z\mathrm{i}x_0} \label{eq33}$$ and $$W+\rho^2=-\frac{1}{x^2}\left( Z^2(1-x^2)-2Z\mathrm{i}x_0 -x_0^2\right) \label{eq34}.$$ 4\. Since $V$ and $I_{v}$ vanish for $\rho=0$, we can use (\[achse99a\]) for $\rho=0$ to determine the integration constant of the Picard-Fuchs system. We get with (\[eq32\]) $$\beta^2=\frac{1}{\lambda^2}-\delta\alpha+\frac{\delta^2}{4} \label{eq35}.$$ 5\. Since $V$ in (\[eq33\]) is with (\[eq32\]) a rational function of $\rho$, $\alpha$ and $\beta$ and does not depend on the metric function $e^{2U}$, we can use the first equation in (\[achse94\]) to determine $G$ as the solution of an Abelian integral which is obviously linear. With $G$ determined in this way, the second equation in (\[achse94\]) can then be used to calculate $e^{2U}$ at the disk which leads to elliptic theta functions (see also [@prd2]). (In the general case, one would have to eliminate $e^{2U}$ in the relations for $u_{v}$ and $u_{w}$ to end up with a non-linear integral equation for $G$.) The integral equation following from (\[achse94\]), $$\int_{0}^{V}\frac{\mathrm{d}\tau}{\mu_{1}(\tau)}=\frac{1}{\mathrm{i}\pi} \int_{0}^{-\rho^{2}}\frac{\ln G}{\mu_{1}(\tau)}\mathrm{ d}\tau \label{eq36}$$ is an Abelian equation and can be solved in standard manner by integrating both sides of the equation with a factor $1/\sqrt{K-r}$ from $0$ to $r$ where $r=-\rho^{2}$. With (\[eq33\]) we get for what is essentially an integral over a rational function $$G(K)=\frac{\sqrt{(K-\alpha)^2+\beta^2}+K-\alpha+\frac{\delta}{2}}{ \sqrt{(K-\alpha)^2+\beta^2}-(K-\alpha+\frac{\delta}{2})} \label{achse114}.$$ 6\. The condition $G(-1)=1$ excludes ring singularities at the rim of the disk and leads to a continuous potential and density there. It determines the last degree of freedom in (\[achse114\]) to $$\alpha=-1+\frac{\delta}{2} \label{eq37}.$$ 7\. The static limit of the counter-rotating disks is reached for $\beta=0$, i.e. the value $\delta_{s}$. This completes the proof. Remark:\ 1. It is interesting to note that there are algebraic relations between $a$, $b$ and $e^{2U}$ though they are expressed via theta functions, i.e. transcendental functions, also at the disk. 2\. It is an interesting question whether there exist disks with non-constant $\gamma/\Omega$ or $\delta$ for genus 2 in the vicinity of the above class of solutions. Whereas this is rather straight forward for a non-constant $\delta$ if $\gamma/\Omega$ are constant, it is less obvious if the latter does not hold. This means that one looks for given $\delta$ for solutions with $$\frac{\gamma}{\Omega}=C_{0}+\epsilon p(\rho) \label{diff1}$$ where $C_0$ is a constant, $\|p\|\leq 1$ is a function of $\rho$, and where $\epsilon<<1$ is a small dimensionless parameter. We can assume that $p$ is not identically constant since this would only lead to a reparametrisation of the above solution. To check if there are solutions for small enough $\epsilon$, one has to redo the steps in the proof of theorem 7.2 in first order of $\epsilon$ by expanding all quantities in the form $y=\bar{y}+\epsilon \hat{y}+...$. Doing this one recognizes that equation (\[achse69\]) becomes a linear first order differential equation for $p$ of the form $p_{\rho}+F(\rho)p=0$ where $F$ is given by the solution for rigid rotation. For a solution $p$ to this equation, the remaining steps can be performed as above. It seems possible to use the theorem on implicit functions to prove the existence of solutions for genus 2 in the vicinity of constant $\gamma/\Omega$, but this is beyond the scope of this article. Global regularity {#subsec.regularity} ----------------- In Theorem 7.2 it was shown that one can identify an Ernst potential on a genus 2 surface which takes the required boundary data at the disk. One has to notice however that this is only a local statement which does not ensure one has found the desired global solution which has to be regular in the whole spacetime except at the disk. It was shown in [@prl; @prd2] that this is the case if $\Theta(\omega(\infty^{-})+u)\neq 0$. In the Newtonian theory (see section (\[sec.2\])), the boundary value problem could be treated at the disk alone because of the regularity properties of the Poisson integral. Thus one knows that the above condition will hold in the Newtonian limit of the hyperelliptic solutions if the latter exists. For physical reasons it is however clear that this will not be the case for arbitrary values of the physical parameters: if more and more energy is concentrated in a region of spacetime, a black-hole is expected to form (see e.g. the hoop conjecture [@hoop]). The black-hole limit will be a stability limit for the above disk solutions. Thus one expects that additional singularities will occur in the spacetime if one goes beyond the black-hole limit. The task is to find the range of the physical parameters, here $\lambda$ and $\delta$, where the solution is regular except at the disk. We can state the\ Theorem 7.3:\ Let $\Sigma'$ be the Riemann surface given by $\mu'{}^{2}=(K^{2}-E)(K^{2}-\bar{E})$ and let a prime denote that the primed quantity is defined on $\Sigma'$. Let $\lambda_{c}(\delta)$ be the smallest positive value $\lambda$ for which $\Theta'(u')=0$. Then $\Theta(\omega(\infty^{-})+u)\neq 0$ for all $\rho$, $\zeta$ and $0<\lambda <\lambda_{c}(\delta)$ and $0\leq \delta\leq \delta_{s}$.\ This defines the range of the physical parameters where the Ernst potential of Theorem 7.2 is regular in the whole spacetime except at the disk. Since it was shown in [@prl; @prd2] that $\Theta'(u')=0$ defines the limit in which the solution can be interpreted as the extreme Kerr solution, the disk solution is regular up to the black-hole limit if this limit is reached.\ Proof:\ 1. Using the divisor $X$ of (\[eq18\]) and the vanishing condition for the Riemann theta function, we find that $\Theta(\omega(\infty^{-})+u)= 0$ is equivalent to the condition that $\infty^{+}$ is in $X$. The reality of the $\tilde{u}_{i}$ implies that $X=\infty^{+}+(-\mathrm{i}z)$. Equation (\[eq18\]) thus leads to $$\begin{aligned} \int_{E_1}^{\infty^+}\frac{d\tau}{\mu}+\int_{E_2}^{-\mathrm{i}z}\frac{d\tau}{\mu} -\frac{1}{2\pi \mathrm{i}}\int_{\Gamma}^{}\frac{\ln G d\tau}{\mu}& \equiv & 0 , \nonumber \\ \int_{E_1}^{\infty^+}\frac{\tau d\tau}{\mu}+\int_{E_2}^{-\mathrm{i}z}\frac{\tau d\tau}{\mu} -\frac{1}{2\pi \mathrm{i}}\int_{\Gamma}^{} \frac{\ln G \tau d\tau}{\mu} & \equiv &0 \label{count55} ,\end{aligned}$$ where $\equiv$ denotes equality up to periods. The reality and the symmetry with respect to $\zeta$ of the above expressions limits the possible choices of the periods. It is straight forward to show that $\Theta(\omega(\infty^{-})+u)= 0$ if and only if the functions $F_{i}$ defined by $$\begin{aligned} F_{1}&:=&\int_{E_1}^{\infty^+}\frac{d\tau}{\mu} +\int_{E_2}^{-\mathrm{i}z}\frac{d\tau}{\mu} -n_{1}\left(2\oint_{b_{1}}\frac{d\tau}{\mu}+2\oint_{b_{2}}\frac{d\tau}{\mu} +\oint_{a_{1}}\frac{d\tau}{\mu} +\oint_{a_{2}}\frac{d\tau}{\mu}\right) -\frac{1}{2\pi \mathrm{i}}\int_{\Gamma}^{} \frac{\ln G d\tau}{\mu} , \nonumber\\ F_{2}&:=&\int_{E_1}^{\infty^+}\frac{\tau d\tau}{\mu} +\int_{E_2}^{-\mathrm{i}z}\frac{\tau d\tau}{\mu} -n_{2}\left(2\oint_{b_{1}}\frac{\tau d\tau}{\mu} +2\oint_{b_{2}}\frac{\tau d\tau}{\mu} +\oint_{a_{1}}\frac{\tau d\tau}{\mu} +\oint_{a_{2}}\frac{\tau d\tau}{\mu}\right) \nonumber\\ &&-\frac{1}{2\pi \mathrm{i}}\int_{\Gamma}^{} \frac{\ln G \tau d\tau}{\mu} \label{count55a} \end{aligned}$$ with the cut system of Fig. 1 and with $n_{1,2}\in \mathrm{Z}$ vanish for the same values of $\rho$, $\zeta$, $\lambda$, $\delta$. The functions $F_{i}$ are both real, $F_{1}$ is even in $\zeta$ whereas $F_{2}$ is odd. Thus $F_{2}$ is identically zero in the equatorial plane outside the disk.\ 2. In the Newtonian limit $\lambda \approx 0$, the above expressions take in leading order of $\lambda$ the form $$F_{1}=\lambda\left((-8n_{1}+1)c_{1}(\rho,\zeta) \ln \lambda-d_{1}(\rho,\zeta) \lambda\right) \label{count55b},$$ and $$F_{2}=\sqrt{\lambda}\left((-8n_{2}+1)c_{2}(\rho,\zeta) \ln \lambda- d_{2}(\rho,\zeta) \lambda^{\frac{3}{2}}\right) \label{count55b1},$$ where we have used the same approach as in the calculation of the axis potential in (\[sing7\]) (see [@prd2] and references given therein); the functions $c_{1}$, $d_{1}$ are non-negative whereas $c_{2}/d_{2}$ is positive in $\mathrm{C}/\{\zeta=0\}$. Thus the $F_{i}$ are zero for $\lambda=0$ which is Minkowski spacetime $f=1$, but they are not simultaneously zero for small enough $\lambda$, i.e. $f$ is regular in the Newtonian regime in accordance with the regularity properties of the Poisson integral. The $F_{i}$ may vanish however at some value $\lambda_{s}$ for given $\rho$, $\zeta$ and $\delta$. Since we are looking for zeros of the $F_{i}$ in the vicinity of the Newtonian regime, we may put $n_{1,2}=1$ here.\ 3. Let $\mathcal{G}$ be the open domain $C/\{\zeta=0,\rho\leq1 \vee \rho=0\}$. It is straight forward to check that the $F_{i}$ are a solution to the Laplace equation $\Delta F_{i}=0$ with $\Delta = 4\left( \partial_{z\bar{z}}+\frac{1}{2(z+\bar{z})} (\partial_{z}+\partial_{\bar{z}})\right)$ for $z,\bar{z}\in\mathcal{G}$. Thus by the maximum principle the $F_{i}$ do not have an extremum in $\mathcal{G}$.\ 4. At the axis for $\zeta>0$, the $\tilde{u}_{i}$ are finite whereas the $F_{i}$ diverge proportional to $-\ln \rho$ for all $\lambda$, $\delta$. Thus $f$ is always regular at the axis.\ 5. Relation (\[eq23\]) at the disk can be written in the form $(y+A)^2+b^2=B^2$ where $A$ and $B$ are finite real quantities. Thus the Ernst potential is always regular at the disk. Due to symmetry reasons $F_{2}\equiv \tilde{u}_{2}$ which is non-zero except at the rim of the disk. For $F_{1}$ one gets at the disk $$F_{1}=\int_{-\rho^{2}}^{\infty^{+}}\frac{d\tau}{\mu_{1}(\tau)}+\int_{0}^{E} \frac{d\tau}{\mu_{1}(\tau)}+\int_{0}^{\bar{E}}\frac{d\tau}{\mu_{1}(\tau)} -u_{v} \label{v1}.$$ With (\[eq36\]) one can see that $F_{1}$ is always positive at the disk.\ 6. Since $F_{1}$ is strictly positive on the axis and the disk and a solution to the Laplace equation in $\mathcal{G}$, it is positive in $\bar{\mathrm{C}}$ if it is positive at infinity. $F_{1}$ is regular for $\|z\|\to\infty$ and can be expanded as $F_{1}=F_{11}/\|z\| +o(1/\|z\|)$ where $F_{11}$ can be expressed via quantities on $\Sigma'$. We get $$F_{11}=\frac{1}{2} \oint_{b_{1}'}^{}\frac{d\tau}{\mu'}-\frac{1}{2\pi \mathrm{i}} \int_{-\mathrm{i}}^{\mathrm{i}}\frac{\ln Gd\tau}{\mu'} \label{count55c}.$$ The quantity $F_{11}\equiv 0$ iff $\Theta'(u')=0$. The condition $F_{11}>0$ is thus equivalent to the condition that $\lambda<\lambda_{c}(\delta)$ where $\lambda_{c}(\delta)$ is the first positive zero of $\Theta'(u')$. This completes the proof. Remark:\ 1. In the second part of the paper we will show that the ultrarelativistic limit (vanishing central redshift) in the case of a disk with one component is given by $\Theta'(u')=0$ for a finite value of $\lambda$. In the presence of counter-rotating matter, this limit is however not reached, the central redshift diverges for $\lambda=\infty$ and $\Theta'(u')\neq0$. This supports the intuitive reasoning that counter-rotation makes the solution more static, i.e. it behaves more like a solution of the Laplace equation with the regularity properties of the Poisson integral.\ 2. Since $F_{2}(\rho,0)=0$ for $\rho\geq 1$, the reasoning in 6. of the above proof shows that there will be a zero of $\Theta(\omega(\infty^{-})+u)$ and thus a pole of the Ernst potential in the equatorial plane for $\lambda>\lambda_{c}(\delta)$ if the theta function in the numerator does not vanish at the same point. In the equatorial plane the Ernst potential can be expressed via elliptic theta functions (see [@prd2]) which have first order zeros. Thus $F_{11}$ will be negative for $\lambda>\lambda_{c}$ in the vicinity of $\lambda_{c}$, and consequently the same holds for $F_{1}$ in the equatorial plane at some value $\rho>1$. It will be shown in the third article that the spacetime has a singular ring in the equatorial plane in this case. The disk is however still regular and the imposed boundary conditions are still satisfied. This provides a striking example that one cannot treat boundary value problems locally at the disk alone in the relativistic case. Instead one has to identify the range of the physical parameters where the solution is regular except at the disk. Conclusion ========== We have shown in this paper how methods from algebraic geometry can be successfully applied to construct explicit solutions for boundary value problems to the Ernst equation. We have argued that there will be differentially rotating dust disks for genus 2 of the underlying Riemann surface in addition to the one we could identify explicitly. To prove existence theorems for solutions to boundary value problems, the methods of [@up; @urs] seem to be better suited since the hyperelliptic techniques are limited to finite genus of the Riemann surface. Moreover the used techniques at the boundary have to be complemented by a proof of global regularity. The finite genus of the Riemann surface also restricts the usefulness in the numerical treatment of boundary value problems. The methods of [@eric] and [@lanza] are not limited in a similar way and have proven to be highly efficient. Thus the real strength of the approach we have presented here is the possibility to construct explicit solutions whose physical features can then be discussed in analytic dependence on the physical parameters up to the ultrarelativistic limit. Whether this approach can be generalized to more sophisticated matter models or whether the equations can still be handled for higher genus will be the subject of further research. Acknowledgement\ I thank J. 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--- abstract: \| We formulate a kinetic model of DNA replication that quantitatively describes recent results on DNA replication in the [in vitro]{} system of [Xenopus laevis]{} prior to the mid-blastula transition. The model describes well a large amount of different data within a simple theoretical framework. This allows one, for the first time, to determine the parameters governing the DNA replication program in a eukaryote on a genome-wide basis. In particular, we have determined the frequency of origin activation in time and space during the cell cycle. Although we focus on a specific stage of development, this model can easily be adapted to describe replication in many other organisms, including budding yeast.\ Journal Ref: [J.Mol.Biol.]{}, [320]{}, 741-750 (2002) author: - 'John Herrick${}^1$, Suckjoon Jun${}^2$, John Bechhoefer${}^{2}$, Aaron Bensimon${}^{1}$' title: Kinetic model of DNA replication in eukaryotic organisms --- Introduction {#introduction .unnumbered} ============ Although the organization of the genome for DNA replication varies considerably from species to species, the duplication of most eukaryotic genomes shares a number of common features: 1\) DNA is organized into a sequential series of replication units, or replicons, each of which contains a single origin of replication.[@Hand; @Friedman] 2\) Each origin is activated not more than once during the cell-division cycle. 3\) DNA synthesis propagates at replication forks bidirectionally from each origin.[@Cairns] 4\) DNA synthesis stops when two newly replicated regions of DNA meet. Understanding how these parameters are coordinated during the replication of the genome is essential for elucidating the mechanism by which $S$-phase is regulated in eukaryotic cells. In this article, we formulate a stochastic model based on these observations that yields a mathematical description of the process of DNA replication and provides a convenient way to use the full statistics gathered in any particular replication experiment. It allows one to deduce accurate values for the parameters that regulate DNA replication in the [Xenopus laevis]{} replication system, and it can be generalized to describe replication in any other eukaryotic system. This type of model has also been shown to apply for the case of RecA polymerizing on a single molecule of DNA.[@Shivashankar] The model, as described in the methods section below, turns out to be formally equivalent to a well-known stochastic description of the kinetics of crystal growth, which allows us to draw on a number of previously derived results and, perhaps equally important, suggests a vocabulary that we find useful and intuitive for understanding the process of replication. Since the kinetics of DNA replication in any cell system depends on two fundamental quantities, replication fork velocity and initiation frequency, one of the principal goals of this kind of analysis is to derive accurate values for these quantities, including any variation, during the course of $S$-phase. As replicon size and the duration of $S$-phase depend on the values of these parameters, this information is indispensable for understanding the mechanisms regulating $S$-phase in any given cell system.[@Pierron; @Walter_Newport; @Hyrien_Mechali; @Coverly_Laskey; @Blow_Chong; @Shinomiya; @Brewer_Fangman; @Gomez] Results {#results .unnumbered} ======= Summary of the [X. Laevis]{} replication experiment {#summary-of-the-x.-laevis-replication-experiment .unnumbered} ----------------------------------------------------- Here, we describe recent experimental results obtained on the kinetics of DNA replication in the well-characterized [Xenopus laevis]{} cell-free system.[@Herrick_JMB; @Lucas] One of the main goals of this paper will be to show that using the theoretical approach described below, one can extract more information – and more reliably – than before from such experiments. In the [Xenopus]{} replication experiments, fragments of DNA that have completed one cycle of replication are stretched out on a glass surface using molecular combing.[@Bensimon; @Michalet; @Herrick_PNAS] Typical two-color epifluorescence images of the combed DNA are shown in Fig. \[fig:fluorescence\]. The DNA that has replicated prior to some chosen time $t$ is labeled with a single fluorescent dye, while DNA that replicated after that time is labeled with two dyes. The result is a series of samples, each of which corresponds to a different time $t$ during $S$-phase. Using an optical microscope, one can directly measure eye, hole, and eye-to-eye lengths at that time. We can thus monitor the evolution of genome duplication from time point to time point, as DNA synthesis advances. (See Fig. \[fig:model\].) Cell-free extracts of eggs from [Xenopus laevis]{} support the major transitions of the eukaryotic cell cycle, including complete chromosome replication under normal cell-cycle control and offers the opportunity to study the way that DNA replication is coordinated within the cell cycle. In the experiment, cell extract was added at $t =$ 2’, and $S$-phase began 15 to 20’ later. DNA replication was monitored by incorporating two different fluorescent dyes into the newly synthesized DNA. The first dye was added before the cell enters $S$-phase in order to label the entire genome. The second dye was added at successive time points $t =$ 25, 29, 32, 35, 39, and 45’, in order to label the later replicating DNA. DNA taken from each time point was combed, and measurements were made on replicated and unreplicated regions. The experimental details are described elsewhere[@Herrick_JMB], but the approach is similar to DNA fiber autoradiography, a method that has been in use for the last 30 years.[@Huberman; @Jasny] Indeed, the same approach has recently been adapted to study the regulatory parameters of DNA replication in HeLa cells.[@Jackson_Pombo] Molecular combing, however, has the advantage that a large amount of DNA may be extended and aligned on a glass slide which ensures significantly better statistics (over several thousand measurements corresponding to several hundred genomes per coverslip). Indeed, the molecular combing experiments provide, for the first time, easy access to the quantities of data necessary for testing models such as the one advanced in this paper. Generalization of the model to account for specific features of the [X. laevis]{} experiment {#generalization-of-the-model-to-account-for-specific-features-of-the-x.-laevis-experiment .unnumbered} ---------------------------------------------------------------------------------------------- The experimental results obtained on the kinetics of DNA replication in the [in vitro]{} cell-free system of [Xenopus laevis]{} [@Herrick_JMB; @Lucas] were analyzed using the kinetic model developed below. In formulating that model, we found that we had to take into account explicitly a number of observations that are peculiar to the particular experiment analyzed: 1\) One goal of the experiment is to measure the initiation function $I(\tau)$, which is the probability of initiating an origin at time $\tau$, per unit length of unreplicated DNA. The simplest assumptions, in terms of our model, would be that either $I$ is peaked at or near $\tau=0$ (all origins initiated at the beginning of $S$-phase) or $I(\tau) = $ constant, (origins initiated at constant rate throughout $S$-phase). However, neither assumption turns out to be consistent with the data analyzed here; thus, we formulated our model to allow for arbitrary initiation patterns and deduced an estimate for $I(\tau)$ directly from the data. We note that initiation is believed to occur synchronously during the first half of $S$-phase in [Drosophila melanogaster]{} early embryos.[@Shinomiya; @Blumenthal] Initiation in the myxomycete [Physarum polycephalum,]{} on the other hand, occurs in a very broad temporal window, suggesting that initiation occurs continuously throughout $S$-phase.[@Pierron] Finally, recent observations suggest that in [Xenopus laevis]{}, early embryos nucleation may occur with increasing frequency as DNA synthesis advances.[@Herrick_JMB; @Lucas] By choosing an appropriate form for $I(\tau)$, one can account for any of these scenarios. Below, we show how measured quantities may, using the model, be inverted to provide an estimate for $I(\tau)$. 2\) The basic form of the model assumes implicitly that the DNA analyzed began replication at $\tau = 0$, but this may not be so, for two reasons: i\) In the experimental protocols, the DNA analyzed comes from approximately 20,000 independently replicating nuclei. Before each genome can replicate, its nuclear membrane must form, along with, presumably, the replication factories. This process takes 15-20 minutes.[@Blow_Laskey; @Blow_Watson; @Wu] Because the exact amount of time can vary from cell to cell, the DNA analyzed at time $t$ in the laboratory may have started replicating over a relatively wide range of times. ii\) In eukaryotic organisms, origin activation may be distributed in a programmed manner throughout the length of $S$-phase, and, as a consequence, each origin is turned on at a specific time (early and late).[@Simon] In the current experiment, the lack of information about the locations of the measured DNA segments along the genome means that we cannot distinguish between asynchrony due to reasons (i) or (ii). We can however account for their combined effects by introducing a starting-time distribution $\phi(t')$, which is the probability—for whatever reason—that a given piece of analyzed DNA began replicating at time $t'$ in the lab. Using our model, we can directly extract the starting time distribution from the data. 3\) The models described above assumed that statistics could be calculated on infinitely long segments of DNA. In the experimental approach, the combed DNA is broken down into relatively short segments (100 kb, typically). Although it is difficult to account for this effect analytically, we wrote a Monte-Carlo simulation that can mimic such “finite-size” effects. 4\) The experiments are all analyzed using an epifluorescence microscope to visualize the fluorescent tracks of combed DNA on glass slides. The spatial resolution ($\approx$ 0.3 $\mu$m) means that smaller signals will not be detectable. Thus, two replicated segments separated by an unreplicated region of size $<$ 0.3 $\mu$m will be falsely assumed to be one longer replicated segment. We accounted for this in the Monte-Carlo simulations by calculating statistics on a coarse lattice whose size equalled the optical resolution, while the simulation itself takes place on a finer lattice. Application of the kinetic model to the analysis of DNA replication in [X. Laevis]{} {#application-of-the-kinetic-model-to-the-analysis-of-dna-replication-in-x.-laevis .unnumbered} -------------------------------------------------------------------------------------- Using the generalizations discussed above, we analyzed recent results obtained on DNA replication in the [Xenopus laevis]{} cell-free system. DNA taken from each time point was combed, and measurements were made on replicated and unreplicated regions. Statistics from each time point were then compiled into six histograms (one for each time point) of the distribution $\rho(f,t)$ of replicated fractions $f$ at time $t$ (Fig. \[fig:rho\]). One can immediately see from Fig. \[fig:rho\] the need to account for the spread in starting times. If all the segments of DNA that were analyzed had started replicating at the same time, then the distributions would have been concentrated over a very small range of $f$. But, as one can see in Fig. \[fig:rho\]C, some segments of DNA (within the same time point) have already finished replicating ($f = 1$) before others have even started ($f = 0$). This spread is far larger than would be expected on account of the finite length of the segments analyzed. Because of the need to account for the spread in starting times, it is simpler to begin by sorting data by the replicated fraction $f$ of the measured segment. We thus assume that all segments with a similar fraction $f$ are at roughly the same point in $S$-phase, an assumption that we can check by partitioning the data into subsets and redoing our measurements on the subsets. In Fig. \[fig:mean-curves\]A-C, we plot the mean values $\ell_h$, $\ell_i$, and $\ell_{i2i}$ against $f$. We then find $f(\tau)$, $I(\tau)$, and the cumulative distribution of lengths between activated origins of replication, $I_{tot}(\tau)$. (See Fig. \[fig:misc\].) The direct inversion for $I(\tau)$ (Fig. \[fig:misc\]B) shows several surprising features: First, origin activation takes place throughout $S$-phase and with increasing probability (measured relative to the amount of unreplicated DNA), as recently inferred from a cruder analysis of data from the same system using plasmid DNA.[@Lucas] Second, about halfway through $S$-phase, there is a marked increase in initiation rate, an observation that, if confirmed, would have biological significance. It is not known what might cause a sudden increase (break point) in initiation frequency halfway through $S$-phase. The increase could reflect a change in chromatin structure that may occur after a given fraction of the genome has undergone replication. This in turn may increase the number of potential origins as DNA synthesis advances.[@Pasero] The smooth curves in Fig \[fig:mean-curves\]A-C are fits based on the model, using an $I(\tau)$ that has two linearly increasing regions, with arbitrary slopes and “break point” (three free parameters). The fits are quite good, except where the finite size of the combed DNA fragments becomes relevant. For example, when mean hole, eye, and eye-to-eye lengths exceed about 10% of the mean fragment size, larger segments in the distribution for $\ell_h(f)$, etc., are excluded and the averages are biased down. We confirmed this with the Monte-Carlo simulations, the results of which are overlaid on the experimental data. The finite fragment size in the simulation matches that of the experiment, leading to the same downward bias. In Fig. \[fig:misc\], we overlay the fits on the experimental data. We emphasize that we obtain $I(\tau)$ directly from the data, with no fit parameters, apart from an overall scaling of the time axis. The analytical form is just a model that summarizes the main features of the origin-initiation rate we determine via our model, from the experimental data. The important result is $I(\tau)$. From the maximum of $I_{tot}(\tau)$, we find a mean spacing between activated origins of 6.3 $\pm$ 0.3 kb, which is much smaller than the minimum mean eye-to-eye separation 14.4 $\pm$ 1.5 kb. In our model, the two quantities differ if initiation takes place throughout S-phase, as coalescence of replicated regions leads to fewer domains, and hence fewer inferred origins (see the note below Eq. \[eq:ell-i2i-tau\] on p. 16). The mean eye-to-eye separation is of particular interest because its inverse is just the domain density (number of active domains per length), which can be used to estimate the number of active replication forks at each moment during $S$-phase. For example, the saturation value of $I_{tot}$ corresponds to the maximum number (about 480,000/genome) of active origins of replication. Since there are about 400 replication foci/cell nucleus, this would indicate a partitioning of approximately 1,200 origins (or, equivalently, about 7.5 Mb) per replication focus.[@Blow_Laskey; @Mills] The distribution of $f$ values in the $\rho(f,t)$ plots can be used to deduce the starting-time distribution ($\phi(t')$), along with the fork velocity $v$. (Fig. \[fig:starting-time\]). The spread in starting times $\phi$ is consistent with a Gaussian distribution, with a mean of $15.9 \pm 0.6$ min. and a standard deviation of $6.1 \pm 0.6$ min. For the fork velocity, we find $v = $ 615 $\pm$ 35 bases/min., in excellent agreement with previous estimates.[@Mahbubani; @Lu] As with the $f$ data, we extracted $\phi(t)$ and $v$ from a global fit to data from all six time points. Discussion {#discussion .unnumbered} ========== Initiation throughout $S$-phase {#initiation-throughout-s-phase .unnumbered} ------------------------------- The view that we are led to here, of random initiation events occurring continuously during the replication of [Xenopus]{} sperm chromatin in egg extracts, is in striking contrast to what has until recently been the accepted view of a regular periodic organization of replication origins throughout the genome.[@Buongiorno-Nardelli; @Laskey; @Coverly_Laskey; @Blow_Chong] For a discussion of experiments that raise doubts on such a view, see Berezney.[@Berezney] The application of our model to the results of Herrick [et al.]{} indicates that the kinetics of DNA replication in the [X. laevis]{} [in vitro]{} system closely resembles that of genome duplication in early embryos. Specifically, we find that the time required to duplicate the genome [in vitro]{} agrees well with what is observed [in vivo]{}. In addition, the model yields accurate values for replicon sizes and replication fork velocities that confirm previous observations.[@Mahbubani; @Hyrien_Mechali] Though replication [in vitro]{} may differ biologically from what occurs [in vivo]{}, the results nevertheless demonstrate that the kinetics remains essentially the same. Of course, the specific finding of an increasing rate of initiation invites a biological interpretation involving a kind of autocatalysis, whereby the replication process itself leads to the release of a factor whose concentration determines the rate of initiation. This will be explored in future work. Directions for future experiments in [X. laevis]{} {#directions-for-future-experiments-in-x.-laevis .unnumbered} ---------------------------------------------------- One effect that we did not include in our analysis is a variable fork velocity. For example, $v$ might decrease as forks coalesce or as replication factor becomes limiting toward the end of $S$ phase.[@Blow_Laskey; @Blow_Watson; @Wu; @Pierron] Such effects, if present, are too small to see in the data analyzed here. Another important question is to separate the effects of any intrinsic distribution due to early and late-replicating regions of the genome of a single cell from the extrinsic distribution caused by having many cells in the experiment. One approach would be to isolate and comb the DNA from a [single]{} cell. Although difficult, such an experiment is technically feasible. The latter problem could be resolved by [in situ]{} fluorescence observations of the chosen cell. Applications to other systems {#applications-to-other-systems .unnumbered} ----------------------------- One can entertain many further applications of the basic model discussed above, which can be generalized, if need be. For example, Blumenthal [et al.]{} interpreted their results on replication in [Drosophila melanogaster]{} for $\rho_{i2i}(\ell,f)$ to imply periodically spaced origins in the genome.[@Blumenthal] (See their Fig. 7.) It is difficult to judge whether their peaks are real or statistical happenstance, but if the conclusion is indeed that the origins in that system are arranged periodically, the kinetics model could be generalized in a straightforward way (introducing an $I(x,\tau)$ that was periodic in $x$). Very recently, detailed data on the replication of budding yeast ([Saccharomyces cerevisiae]{}) have become available.[@Raghuraman] The data provide information on the locations of origins and the timings of their initiation during $S$-phase. These data support the view of origin initiation throughout $S$-phase. Unlike replication in [Xenopus]{} prior to the mid-blastula transition, origins in budding yeast are associated with highly conserved sequence elements (autonomous replication sequence elements, or ARSs). Raghuraman [et al.]{}[@Raghuraman] also give the first estimates of the [distribution]{} of fork velocities during replication. Although broad, the distribution is apparently stationary, and there is no correlation between velocities and the time in $S$-phase when the forks are initiated. The model developed here could be generalized in a straightforward way to the case of budding yeast. Knowing the sequence of the genome and hence the location of potential origins means that the initiation function would be an explicit function of position $x$ along the genome, with peaks of varying heights at each potential origin. The advantage of the kind of modeling advanced here would be the opportunity to derive quantities such as the replication fraction as a function of time in $S$-phase. Raghuraman [et al.]{} fit their data for this “timing curve” to an arbitrarily chosen sigmoidal function. (See their supplementary data, Section II-5.) Such modeling will make it easier to find meaningful biological explanations of the programming of $S$-phase evolution. The origin-spacing problem {#the-origin-spacing-problem .unnumbered} -------------------------- One outstanding issue in DNA replication in eukaryotes is the observation that the replication origins cannot be too far apart, as this would prevent the genome from being replicated completely within the length of a single $S$-phase.[@Gilbert] One solution that has been proposed is that there is an excess of pre-replication complexes (pre-RCs) of highly conserved proteins, which assemble at ORC-bound DNA sites before the cell enters $S$-phase (e.g., Lucas [et al.]{}[@Lucas], and references therein). In this case, the position of each potential origin of replication (POR) can be distributed randomly, with a statistically insignificant probability of having large gaps between PORs. The problem with this solution is that the average POR spacing must be much smaller (less than 1-2 kb) than the reported values of XORC spacing of 7-16 kb.[@Walter_Newport; @Rowles] A second proposed solution to the origin-spacing problem is to invoke correlations in POR spacings. In other words, instead of assuming a purely random pre-RC distribution, one imposes constraints that force a partial periodicity on the POR spacing, so that most of the origins are spaced 5-15 kb apart (Blow [et al.]{},[@Blow_etal] and references therein). This suppresses the formation of large gaps but raises other issues. First, it requires an unknown mechanism to achieve this periodicity of POR spacing. Second, it assumes implicitly that most of the PORs fire during $S$-phase, to prevent the 30 kb gap that could arise from a originÕs failure to initiate, which is not obvious at all. Third, if origins initiate throughout $S$-phase, then there needs to be some kind of correlation that forces the more widely spaced origin groups to initiate early enough in $S$-phase to complete replication in the required time. Implicitly, our model adopts language consistent with the first solution, but it is straightforward to consider the correlations assumed in the second solution. The presence of significant correlations in PORs would not invalidate the results presented here, which pertain to mean quantities (e.g., Fig. \[fig:mean-curves\]); however, it would change their interpretation and could change biological models that one might try to make to explain the observed kinetic parameters we extract using the KJMA model. We plan to investigate these questions, along with the effect of origin efficiency on DNA replication kinetics, in future work. Conclusion {#conclusion .unnumbered} ========== In this article, we have introduced a class of theoretical models for describing replication kinetics that is inspired by well-known models of crystal-growth kinetics. The model allows us to extract the rate of initiation of new origins, a quantity whose time dependence has not previously been measured. With remarkably few parameters, the model fits quantitatively the most detailed existing experiment on replication in [Xenopus]{}. It reproduces known results (for example, the fork velocity) and provides the first reliable description of the temporal organization of replication initiation in a higher eukaryote. Perhaps most important, the model can be generalized in a straightforward way to describe replication and extract relevant parameters in essentially any organism. Methods {#methods .unnumbered} ======= Mathematical analogy between crystal growth and the kinetics of DNA replication {#mathematical-analogy-between-crystal-growth-and-the-kinetics-of-dna-replication .unnumbered} ------------------------------------------------------------------------------- In this section, we describe how certain features of the mathematics describing crystal growth may be mapped onto a model describing the kinetics of DNA replication. We emphasize that the analogy is a formal one – the underlying processes are completely different. However, by mapping our problem onto one that has been long studied in a different context, we can take over a number of results that have already been derived, and we can develop useful intuitions about how to look at experimental results about DNA replication. In the 1930s, several scientists independently derived a stochastic model that described the kinetics of crystal growth.[@Kolmogorov; @Johnson_Mehl; @Avrami] The “Kolmogorov-Johnson-Mehl-Avrami” (KJMA) model has since been widely used by metallurgists and other scientists to analyze thermodynamic phase transformations.[@Christian] In the KJMA model, freezing kinetics result from three simultaneous processes: 1\) nucleation, which leads to discrete solid domains. 2\) growth of the domain. 3\) coalescence, which occurs when two expanding domains merge. Each of these processes has an analog in DNA replication in higher eukaryotes, and more specifically embryos: 1) The activation of an origin of replication is analogous to the nucleation of the solid domains during crystal growth. 2) Symmetric bidirectional DNA synthesis initiated (nucleated) at the origin corresponds to solid-domain growth. 3) Coalescence in crystal growth is analogous to multiple dispersed sites of replicating DNA (replication fork) that advance from opposite directions until they merge. Simple version of the KJMA model for DNA replication {#simple-version-of-the-kjma-model-for-dna-replication .unnumbered} ---------------------------------------------------- In the simplest form of the KJMA model, solids nucleate anywhere in the liquid, with equal probability for all spatial locations (“homogeneous nucleation”), although it is straightforward to describe nucleation at pre-specified sites (“heterogeneous nucleation”), which would correspond to a case where replication origins are specified by fixed genetic sites along the genome. Once a solid domain has been nucleated, it grows out as a sphere at constant velocity $v$. When two solid domains impinge, growth ceases at the point of contact, while continuing elsewhere. KJMA used elementary methods to calculate quantities such as $f(\tau)$, the fraction of the volume that has crystallized by time $(\tau)$. Much later, more sophisticated methods were developed to describe the detailed statistics of domain sizes and spacings.[@Sekimoto; @Ben-Naim] DNA replication, of course, corresponds to one-dimensional crystal growth; the shape in three dimensions of the one-dimensional DNA strand does not directly affect the kinetics modeling. (In the model, replication is one dimensional along the DNA. The configuration of DNA in three dimensions is not directly relevant to the model but can enter indirectly via the nucleation function $I(x, \tau)$. For example, if, for steric reasons, certain regions of the DNA are inaccessible to replication factories, those regions would have a lower (or even zero) value of $I$.) The one-dimensional version of the KJMA model assumes that domains grow out at velocity $v$, assumed to remain constant. The nucleation rate $I(x,\tau) = I_0$ is defined to be the probability of domain formation per unit length of unreplicated DNA per unit time, at the position $x$ and time $\tau$. Following the analogy to the one-dimensional KJMA model, we can calculate the kinetics of DNA replication during $S$-phase. This requires determining the fraction of the genome $f(\tau)$ that has already been replicated at any given moment during $S$-phase. One finds $$f(\tau) = 1 - e^{-I_{0}v\tau^{2}} , \label{eq:ft}$$ which defines a sigmoidal curve. (Eq. \[eq:ft\] assumes an infinite genome length. The relative importance of the finite size of chromosomes is set by the ratio (fork velocity \* duration of $S$-phase) / chromosome length (Cahn, 1996). In the case of the experiment analyzed in this paper, this ratio is $\approx$ 10 bases/sec \* 1000 sec / $10^7$ bases/chromosome $\approx 10^{-3}$, which we neglect.) A more complete description of replication kinetics requires detailed analysis of different statistical quantities, including measurements made on replicated regions (eyes), unreplicated regions (holes), and eye-to-eye sizes (the eye-to-eye size is defined as the length between the center of one eye and the center of a neighboring eye.) The probability distributions may be expressed as functions either of time $\tau$ or replicated fraction $f$. For example, the distribution of holes of size $\ell$ at time $\tau$, $\rho_h (\ell,\tau)$ can be derived by a simple extension of the argument leading to Eq. \[eq:ft\]: $$\rho_h (\ell, \tau) = I_0\tau \cdot e^{-I_0\tau\ell} . \label{meanhole}$$ From Eq. \[meanhole\], the mean size of holes at time $\tau$ is $$\ell_h (\tau) = {1 \over I_0\tau} . \label{eq:i-tau}$$ Determining the probability distributions of replicated lengths (eye sizes) is complicated because a given replicated length may come from a single origin or it may result from the merger of two or more replicated regions. Thus, one must calculate in effect an infinite number of probabilities; by contrast, holes of a given length arise in only one way.[@Ben-Naim] One can nonetheless derive a simple expression for $\ell_i (\tau)$, the mean replicated length at time $\tau$, from a [“mean-field”]{} hypothesis[@Plischke_Bergersen]: the probability distribution of a given replicated length is assumed to be independent of the actual size of its neighbor. One can show that this mean-field hypothesis must always be true in one-dimensional growth problems, but not necessarily in the ordinary three-dimensional setting of crystal growth. In particular, if $I(\tau)$ depends on space, one expects correlations to be important. Using the mean-field hypothesis, we find $$\ell_i(\tau) = \ell_h(\tau) {f \over 1 - f} = {e^{I_0 v\tau^2} - 1 \over I_0 \tau} \label{eq:ell-i-tau}$$ and $$\ell_{i2i}(\tau) = \ell_i(\tau) + \ell_h(\tau) = {\ell_h(\tau) \over 1-f} = {e^{I_0 v\tau^2} \over I_0 \tau} . \label{eq:ell-i2i-tau}$$ The minimum average eye-to-eye size, obtained by differentiating Eq. \[eq:ell-i2i-tau\], is $\ell_{i2i}^{} = \sqrt{2e} \cdot \sqrt{v/I_0}$. These expressions for $\ell_i(\tau)$ and $\ell_{i2i}(\tau)$ allow one to collapse the experimental observations of $\ell_h$, $\ell_i$, and $\ell_{i2i}$ (the mean eye-to-eye separation) onto a single curve. (See Fig. \[fig:mean-curves\]D, below.) Finally, we can calculate the average distance between origins of replication that were activated at different times during the replication process, which is just the inverse of $I_{tot}$, the time-integrated nucleation probability per unit length: $$\ell_0 \equiv I_{tot}^{-1} = {2\over\sqrt{\pi}} \cdot \sqrt{v\over I_0}$$ The last expression shows that, as might have been guessed by dimensional analysis of the model parameters ($I_0$ and $v$), the basic length scale in the model is set by $\ell^ \equiv \sqrt{v/I_0}$. Note that because initiation in the model is occurring throughout $S$-phase, the minimum eye-to-eye distance $\ell_{i2i\_min}$ is not the same as the average separation between origins, $\ell_{0}$. For this simple case, $\ell_{i2i\_min}/ \ell_{0} = \sqrt{e \pi / 2} \approx 2.1$. Generalizations of the KJMA model {#generalizations-of-the-kjma-model .unnumbered} --------------------------------- Based on the specific results of the [Xenopus]{} experiments discussed above, we generalized the simple version of the KJMA model in several ways: The first generalization is to allow for arbitrary $I(\tau)$. Eq. \[eq:ft\] then becomes $$f(\tau) = 1 - e^{-g(\tau)}~~{\rm with}~~g(\tau) = {2v \int_0^\tau I(\tau')(\tau-\tau') \, d\tau'} , \label{eq:ft-general}$$ and, similarly, Eq. \[eq:i-tau\] becomes $$\ell_h(\tau) = \left[{\int_0^\tau I(\tau') \, d\tau'} \right]^{-1} . \label{eq:ell-h-tau-general}$$ The other mean lengths, $\ell_i(\tau)$ and $\ell_{i2i}(\tau)$, continue to be related to $\ell_h(\tau)$ by the general expressions given in Eqs. \[eq:ell-i-tau\] and \[eq:ell-i2i-tau\]. In the experiment, one measures $\ell_h$, $\ell_i$, and $\ell_{i2i}$ as functions of both $\tau$ and $f$. (Because of the start-time ambiguity, the $f$ data are easier to interpret.) The goal is to invert this data to find $I(\tau)$. Using Eqs. \[eq:ft-general\] and \[eq:ell-h-tau-general\], we find $$\tau(f) = {1\over 2v} \int_0^f \ell_{i2i}(f') \, df' = {1\over 2v} \int_0^f {{\ell_h(f')} \over {1-f'}} \, df' .$$ Because $\tau(f)$ increases monotonically, one can numerically invert it to find $f(\tau)$. From $f(\tau)$, one can derive all quantities of interest, including $I(\tau)$. The starting time distribution $\phi(t)$ can be deduced looking at each molecular fragment, measuring its replication fraction $f$, and extrapolating back to a starting time using the experimentally determined $f(\tau)$ curve. (Fragments that are fully replicated ($f = 1$ are excluded.) The starting times are then binned to give $\phi(t)$ directly. Monte-Carlo simulations {#monte-carlo-simulations .unnumbered} ----------------------- We wrote a Monte-Carlo simulation using the programming language of Igor Pro (Wavemetrics) to test various experimental effects that were difficult to model analytically. These included the effects of finite sampling of DNA fragments (on average, 190 molecules per time point), the finite optical resolution of the scanned images, and – most important – the effect of the finite size of the combed DNA fragments. The size of each molecular fragment in the simulation was drawn randomly from an estimate of the actual size distribution of the experimental data. This distribution was approximately log-normal, with an average length of 102 kb. and a standard deviation of 75 kb. In the simulations, we consider each DNA molecular fragment as a one-dimensional lattice, and each lattice site is updated with a timestep $\Delta t$ = 0.2 min. An origin is initiated (lattice site changed from 0 to 1) with a probability determined by the initiation rate $I(\tau)$. Once an origin has been initiated, replication forks grow bidirectionally at a constant rate $v$. The natural size of lattice then would be $v \Delta t$, which is 123 bp for the measured fork velocity $v = 615$ bp/min and chosen time step $\Delta t$. The lattice scale is then roughly the size of origin recognition complex proteins. We sample the simulation results at the same time points as the actual experiments ($t$ = 25, 29, 32, 35, 39, 45 min.) Each sampled molecule is cut at random site to simulate the combing process. The lattice is then “coarse grained” by averaging over approximately four pixels. The coarse lattice length scale is then 0.24 $\mu$m, which roughly corresponds to that of the scanned optical images. Finally, the coarse-grained fragments were analyzed to compile statistics concerning replicon sizes, eye-to-eye sizes, etc. that were directly compared to experimental data. In a first version of the simulation, the lattice was directly simulated using a vector with one element for each lattice site. In a more refined version of the simulation, we noted only the position of the replication forks, which greatly increased the speed of the simulations. We also used the simulation to test a previous algorithm for extracting $I(f)$, the initiation rate as a function of overall replication fraction. The previous algorithm[@Herrick_JMB; @Marheineke] looked for small replicated regions and extrapolated back to an assumed initiation time. We tested this algorithm using our Monte-Carlo analysis and found significant bias in the inferred $I(f)$, while the algorithms we introduce here showed no such bias. Parameter extraction from data {#parameter-extraction-from-data .unnumbered} ------------------------------ We extracted data from both the real experiments and the Monte-Carlo simulations by a global least-squares fit that took into account simultaneously the different data collected (i.e., the different curves in Figs. \[fig:rho\] and \[fig:mean-curves\]). As discussed above, we fit a two-segment straight line to the $I(\tau)$ curve extracted directly from the data for analytic simplicity. Assuming this form for $I(\tau)$, we derive explicit formulae for the curves in Figs. \[fig:rho\] and \[fig:mean-curves\]. The finite size of the molecular fragments studied ($102 \pm 75$ kb) causes systematic deviation from the “infinite-length” formulae we can calculate. Such deviations could be detected using the Monte-Carlo simulations by comparing the extracted values of parameters with those input. The deviations show themselves mainly in two settings: First, whenever the mean length of holes, eyes, or eye-to-eye distances approaches the mean segment length, the observed mean lengths will be systematically too small because the larger end of the experimental distributions is cut off by the finite fragment length. We dealt with this complication by restricting our fit to areas where the mean length being measured is less than 10% of the mean fragment size. The second complication is that the inferred fork velocity is systematically reduced (by about 5% for the fragment size in the experiments analyzed here). We measured this bias using the Monte-Carlo simulations and then corrected the “raw” fork velocity that is given by our least-squares fits. One further subtle point in a global fit is the relative weighting to be given to the data in the $\rho(f)$ curves (Fig. \[fig:rho\]) relative to the data in the mean-value curves (Fig. \[fig:mean-curves\]). 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--- abstract: \| Let $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ be the oscillatory integral operators defined by $\;\displaystyle T^{j,k}_{N}f(s):=\int_{B} \,f(x)\,e^{\imath N{\|x\|}^{j}s^{k}}\,dx, \quad (j,k)\in\{1,2\}^{2},\,$ where $\,B\,$ is the unit ball in ${\mathbb{R}}^{n}\,$ and $\,N\,>>1.$ We compare the asymptotic behaviour as $\,N\rightarrow +\infty\,$ of the operator norms $\,\parallel T^{j,k}_{N} \parallel_ {L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,\,q\in [1,+\infty].\,$ We prove that, except for the dimension $n=1,\,$ this asymptotic behaviour depends on the linearity or quadraticity of the phase in $s$ only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schrödinger equation. author: - 'Ahmed A. Abdelhakim' bibliography: - 'mybibfile.bib' title: '$L^p$-$L^q$ boundedness of integral operators with oscillatory kernels: Linear versus quadratic phases' --- Strichartz estimates for the Schrödinger equation ,Oscillatory integrals,$L^{p}-L^{q}$ boundedness 35B45, 35Q55, 42B20 ### 1. A remark on a counterexample to inhomogeneous Strichartz estimates for the Schrödinger equation and motivation {#a-remark-on-a-counterexample-to-inhomogeneous-strichartz-estimates-for-the-schrödinger-equation-and-motivation .unnumbered} Consider the Cauchy problem for the inhomogeneous free Schrödinger equation with zero initial data $$\begin{aligned} \label{shreq} \imath \partial_{t}u+\Delta u\,=\, F(t,x),\qquad (t,x)\in (0,\infty)\times{\mathbb{R}}^{n},\qquad u(0,x)\,=\,0.\end{aligned}$$ Space time estimates of the form $$\begin{aligned} \label{est1} \|\|u\|\|_{L^{q}_{t}\left( \mathbb{R};L^{r}_{x}({\mathbb{R}}^{n})\right)}\;\lesssim\; \|\|F\|\|_{L^{{\widetilde{q}}^{\prime}}_{t} \left(\mathbb{R};L^{{\widetilde{r}} ^{\prime}}_{x}({\mathbb{R}}^{n})\right)},\end{aligned}$$ have been known as inhomogeneous Strichartz estimates. The results obtained so far (see [@damianoinhom; @Kato; @keeltao; @vilela; @YoungwooKoh]) are not conclusive when it comes to determining the optimal values of the Lebesue exponents $\,q$, $r$, $\tilde{q}\,$ and $\,\tilde{r}\,$ for which the estimate (\[est1\]) holds. Trying to further understand this problem, we [@ahmed1] found new necessary conditions on these exponents values. The counterexample in [@ahmed1], like Example 6.10 in [@damianoinhom], contains an oscillatory factor with high frequency. More precisely, we used a forcing term given by $$\begin{aligned} \label{myphase} F(t,x)= e^{-\imath\, N^2\,t } \,\chi_{[0,\frac{\eta}{N}]}(t)\, \chi_{B\left(\frac{\eta}{N}\right)}{(x)}\end{aligned}$$ where $\,\eta>0\,$ is a fixed small number, $\,N>>1\,$ and $ B\left(\frac{\eta}{N}\right)$ is the ball with radius $\,\eta/N\,$ about the origin. While in [@damianoinhom] the stationary phase method is applied to the inhomogeneity $$\begin{aligned} \label{fphase} F (t,x)= e^{-2\imath\, N^2\,t^{2} }\,\chi_{[0,1]}(t)\,\chi_{B\left(\frac{\eta}{N}\right)}{(x)}.\end{aligned}$$ When $\,t\in [2,3],\,$ both data in (\[myphase\]) and (\[fphase\]) force the corresponding solution $u(t,x)$ to concentrate in a spherical shell centered at the origin with radius about $N.$ This agrees with the dispersive nature of the Schrödinger operator. The shell thickness is different in both cases though. It is about $1$ in the case of the data (\[myphase\]) but about $N$ in the case of (\[fphase\]). The necessary conditions obtained are respectively $$\begin{aligned} \frac{1}{q}\geq\frac{n-1}{\widetilde{r}}-\frac{n}{r}, \qquad \quad\frac{1}{\widetilde{q}}\geq\frac{n-1}{r}- \frac{n}{\widetilde{r}}\end{aligned}$$ and $$\begin{aligned} \label{necess1} \|\frac{1}{r}-\frac{1}{\widetilde{r}}\|\leq \frac{1}{n}.\end{aligned}$$ Observe that the oscillatory function in (\[myphase\]) has a linear phase and is applied for the short time period of length $\:1/\sqrt{\text{frequency}}.\,$ The oscillatory function in (\[fphase\]) on the other hand has a quadratic phase and the oscillation is put to work for a whole time unit. We noticed that the phase in [@damianoinhom] need not be quadratic and we can get the necessary condition (\[necess1\]) using the data $$\begin{aligned} \label{fphase1} F_{l} (t,x)= e^{-\imath\, N^2\,t}\, \chi_{[0,1]}(t)\,\chi_{B\left(\frac{\eta}{N}\right)}{(x)}\end{aligned}$$ where the phase in the oscillatory function is linear. Before we show this, we recall the following approximation of oscillatory integrals according to the principle of stationary phase. \[stationary\] (see [@stein], Proposition 2 Chapter VIII and Lemma 5.6 in [@damianorem]) Consider the oscillatory integral $\;I(\lambda)=\displaystyle \int_{a}^{b}e^{\imath \lambda \phi(s)}\psi(s)d s.\;$ Let the phase $\,\phi \in C^5([a,b])\,$ and the amplitude $\psi\in C^3([a,b])$ such that (i) : $\;\phi^{\prime} (z)=0\,$ for a point $\;z\,\in\, ]a+c, b-c[\;$ with $\,c\,$ a positive constant, (ii) : $\;\|\phi^{\prime} (s)\|\,\gtrsim\, 1,\;$ for all $\;s\,\in\, [a, a+c]\,\cup\, [b-c, b],$ (iii) : $\;\|\phi^{\prime\prime} (s)\|\,\gtrsim\, 1,$ (iv) : $\;\psi^{(j)}\,$ and $\,\phi^{(j+3)}\;$ are uniformly bounded on $[a,b]$ for all $j=0,1,2$. $$\begin{aligned} \hspace{-1 cm}\mbox{Then}\qquad \qquad I(\lambda)\,=\, {\,\sqrt{\frac{2 \pi}{\lambda\|\phi^{\prime\prime} (z)\|}} \,\psi(z)\,e^{\imath \,\lambda\, \phi(z)+\imath\,\mbox{\small sgn}\left( \phi^{\prime\prime} (z)\right)\, \frac{\pi}{4}}}+\mathcal{O}\left(\frac{1}{\lambda}\right), \vspace{-0.22 cm}\end{aligned}$$ where the implicit constant in the $\mathcal{O}-$symbol is absolute. The norm of the inhomogeneous term $F_{l} $ in (\[fphase1\]) has the estimate $$\begin{aligned} \label{normf} \parallel F_{l} \parallel_{L^{\tilde{q}^{\prime}} ([0,1];L^{\tilde{r}^{\prime}}(\mathbb{R}^{n}))}\,\approx\, {\eta}^{n-\frac{n}{\tilde{r}}}\, {N}^{-n}\,{N}^{\frac{n}{\tilde{r}}}.\end{aligned}$$ For the solution of (\[shreq\]), we have the explicit formula $$\begin{aligned} \label{solshro} u(t,x) \,=\, (4 \pi)^{-\frac{n}{2}}\int_{0}^{t} (t-s)^{-\frac{n}{2}}\int_{{\mathbb{R}}^{n}} e^{\imath\frac{\|x-y\|^2}{4(t-s)}}\,F(s,y)\,dy\, ds.\end{aligned}$$ Let us estimate the solution $u_{l}(t,x)$ that corresponds to $F_{l}.$ We shall restrict our attention to the region $$\begin{aligned} \Omega_{\eta,N}=\left\{ (t,x)\in [2,3] \times \mathbb{R}^{n}\!\!:\, 2(t-{3}/{4})N+\eta N^{-1}\,<\|x\|<\,2(t-{1}/{4})N-\eta N^{-1}\right\}.\end{aligned}$$ It will be momentarily seen that this is the region where we can exploit Lemma \[stationary\] to approximate $\, u_{l}(t,x).$ Substituting from (\[fphase1\]) into (\[solshro\]) then applying Fubini’s theorem we get $$\begin{aligned} \label{corsol} u_{l}(t,x)\:=\: (4\pi)^{-\frac{n}{2}} \,\int_{B({\eta}/{N})} \,I_{N}(t,x,y)\, d y\end{aligned}$$ where $\,I_{N}(t,x,y)\,$ is the oscillatory integral $$\begin{aligned} \label{inoscints} I_{N}(t,x,y)\;=\;\int_{0}^{1}\, e^{\imath N^2\, \phi_{N}{(s,t,x,y)}}\,\psi{(s,t)}\, d s,\end{aligned}$$ with the phase $\; \displaystyle \phi_{N}{(s,t,x,y)} =\frac{ \|x-y\|^2}{4\,N^2}\frac{1}{t-s}-s\,$ and amplitude $\, \psi{(s,t)}=(t-s)^{-\frac{n}{2}}.$ For simplicity, we write $\phi(.)$ and $\psi(.)$ in place of $\,\phi_{N}{(.,t,x,y)}\,$ and $\,\psi{(.,t)}\,$ respectively. Next, we verify the conditions (i) - (iv) for $\,\phi\,$ and $\,\psi.$ Let $\,(t,x)\in \Omega_{\eta,N}\,$ and $\,y\in B(\eta/N).$ Observe then that $\;\displaystyle t-{3}/{4}<{\|x-y\|}/{2N}<t-{1}/{4}\;$ and $\; t-s \in [1,3]. $ Therefore (i) : If $\,z\,$ is such that $\,\phi^{\prime}(z)=0\,$ then $\,\displaystyle z =t-{\|x-y\|}/{2N}.\,$ Moreover, $\displaystyle \,z\in\:]{1}/{4},{3}/{4} [.$ (ii) : $\phi^{\prime}$ is monotonically increaing so $\;\displaystyle \min_{s\in[0,1]}{\phi^{\prime}(s)} = \phi^{\prime}(0)=\frac{\|x-y\|^2}{4 N^2\, t^2} > \left(1-\frac{3}{4t}\right)^{2} \gtrsim 1.$ (iii) : $\,\displaystyle \phi^{\prime\prime}(s)\,=\, \frac{\|x-y\|^2}{2 N^2}\frac{1 }{(t- s)^3}\,\approx\,1.$ (iv) : $\;\displaystyle \phi^{(j)}(s)\,=\, \frac{\|x-y\|^2}{4N^2}\frac{ j!}{(t- s)^{(j+1)}} \,\approx\,1,\; j=3,4,5$, $\;\;\;\psi(s)\,=\, (t- s)^{-\frac{n}{2}} \,\approx\,1$,\ $\psi^{\prime}(s)\,=\, \frac{n}{2}(t- s)^{-\frac{n}{2}-1} \,\approx\,1$, $\qquad \psi^{\prime\prime}(s)\,=\, \frac{n}{2}(\frac{n}{2}+1)(t- s)^{-\frac{n}{2}-2}\,\approx\,1.$ Now, applying Lemma \[stationary\] to the oscillatory integral $\,I_{N}(t,x,y)\,$ in (\[inoscints\]) yields $$\begin{aligned} \label{noosciny0} I_{N}(t,x,y)\,=\, {\,\sqrt{\frac{2 \pi}{\phi_N^{\prime\prime}(z,t,x,y)}} \,\psi(z,t)\, \frac{e^{ \frac{\pi}{4} \imath}}{N} \,e^{\imath N^2 \phi_N(z,t,x,y)}}+ \mathcal{O}\left(\frac{1}{N^2}\right).\end{aligned}$$ Since $\,\phi_N(z,t,x,y)+t=\|x-y\|/N\, $ and since $\, N\left(\|x-y\|-\|x\|\right)= \mathcal{O}\left(\eta\right)\,$ whenever\ $\,(t,x)\in \Omega_{\eta,N},\;y\in B(\eta/N).$ Then $\,\,N^2\,\phi_N(z,t,x,y)+N^2\,t =N\,\|x\|+\mathcal{O}\left(\eta\right).$ Hence $$\begin{aligned} \label{noosciny} e^{\imath N^2\,\phi_N(z,t,x,y)}= e^{\imath\left(N\,\|x\|-N^2\,t\right)} \,e^{\mathcal{O}\left(\eta\right)} =e^{\imath\left(N\,\|x\|-N^2\,t\right)}\, \left(1+\mathcal{O}\left(\eta\right)\right).\end{aligned}$$ Inserting (\[noosciny\]) into (\[noosciny0\]) then returning to (\[corsol\]), we discover $$\begin{aligned} u_{l}(t,x)\:=\:& \frac{(4\pi)^{\frac{1-n}{2}} }{\sqrt{2}}\frac{e^{ \frac{\pi}{4} \imath}}{N} \,e^{\imath\left(N\,\|x\|-N^2\,t\right)}\, \int_{B({\eta}/{N})}\,{ \,\frac{\psi(z,t)}{\sqrt{\phi_N^{\prime\prime}(z, t,x,y)}} \, \,\left(1+\mathcal{O}\left(\eta\right)\right)} \,d y\\&\;+ \mathcal{O}\left(\frac{1}{N^2}\right)\, \int_{B({\eta}/{N})}\,\, d y.\end{aligned}$$ Recalling that $\,\psi,\: \phi^{\prime\prime}\approx 1,\,$ we immediately deduce the estimate $$\begin{aligned} &\| u_{l}(t,x)\|\,\gtrsim\,\frac{\|B(\eta/N)\|}{N} \,\approx\,\eta^{n}\,N^{-(1+n)},\quad (t,x)\in \Omega_{\eta,N}.\quad \text{Thus, for all}\;\; t\in [2,3],\\ &\hspace{-1 cm} \|\|u_{l}(t,x)\|\|_{L^{r}_{x}\left({\mathbb{R}}^{n}\right)} \,\geq\, \left( \int_{2(t-{3}/{4})N+\eta N^{-1}\,<\,\|x\|\,<\,2(t-{1}/{4})N-\eta N^{-1}}\, \| u_{l}(t,x)\|^{r}\,dx\,\right)^{\frac{1}{r}} \,\gtrsim\,\eta^{n}\,N^{-(1+n)+\frac{n}{r}}.\end{aligned}$$ Consequently $$\begin{aligned} \label{normul} \|\|u_{l}\|\|_{L^{q}_{t}\left( \mathbb{R};L^{r}_{x}({\mathbb{R}}^{n})\right)} \,\geq\,\|\|u_{l}\|\|_{L^{q}_{t}\left( [2,3];L^{r}_{x}({\mathbb{R}}^{n})\right)} \,\gtrsim\,\eta^{n}\,N^{-(1+n)+\frac{n}{r}}.\end{aligned}$$ Lastly, it follows from (\[normf\]) and (\[normul\]) that $$\begin{aligned} \|\|u_{l}\|\|_{L^{q}_{t}\left( \mathbb{R};L^{r}_{x}({\mathbb{R}}^{n})\right)}/ \parallel F_{l} \parallel_{L^{\tilde{q}^{\prime}} ([0,1];L^{\tilde{r}^{\prime}}(\mathbb{R}^{n}))}\; \gtrsim\; \eta^{\frac{n}{\tilde{r}}}\, N^{\frac{n}{r}-\frac{n}{\tilde{r}}-1}\end{aligned}$$ which, for a fixed $\,\eta,\,$ blows up as $\,N\rightarrow +\infty\,$ if $\,\displaystyle \frac{n}{r}-\frac{n}{\tilde{r}}>1.\,$ In the light of duality this implies the necessary condition (\[necess1\]). These examples made us wonder how exactly different are linear oscillations from quadratic ones if we capture the cancellations in Lebesgue spaces. One way to see this is to consider the operators $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ defined by $$\begin{aligned} \label{intop} T^{j,k}_{N}f(s):=\int_{B} \,f(x)\,e^{\imath N{\|x\|}^{j}s^{k}}\,dx, \qquad (j,k)\in\{1,2\}^{2},\end{aligned}$$ where $\,B\,$ is the unit ball in ${\mathbb{R}}^{n},\,$ and compare the asymptotic behaviour as $\,N\rightarrow +\infty\,$ of their operator norms for all $\,p,\,q\in [1,+\infty].\,$ Let $\,C_{j,k,n}:[0,1]^{2}\rightarrow \mathbb{R}\,$ be the functions defined by $$\begin{aligned} C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)\,:=\, \alpha \quad \text{if}\qquad \parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])} \;\approx\; N^{ - \alpha}.\end{aligned}$$ We discover that $\,C_{j,k,n}\,$ is a continuous function with range $\,[0,{1}/{4}]\,$ when $n=1,$ $j=2$ and $\,[0,{1}/{2}{k}]\,$ otherwise (see the figure below). We actually prove that \[mainthm\] $$C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)\;=\; \left\{ \begin{array}{ll} \frac{1}{4}\, \sigma\left(\frac{1}{p},\frac{1}{q}\right), & \hbox{$n=1,\;$ $j=2$;} \\\\ \frac{1}{2\,k}\, \sigma\left(\frac{1}{p},\frac{1}{q}\right), & \hbox{ $n\geq j$.} \end{array} \right.$$ where $$\label{sgmab} \sigma(a,b):=\left\{ \begin{array}{ll} 2b , & \hbox{$\; 0\leq a \leq 1-b,\;\; 0\leq b \leq \frac{1}{2}$;} \\ 2(1-a) , & \hbox{$\; \frac{1}{2}\leq a \leq 1,\;\;a+ b \geq 1$;} \\ 1 , & \hbox{$\;0\leq a \leq \frac{1}{2},\;\;\frac{1}{2}\leq b \leq 1$.} \end{array} \right.$$ $$\begin{aligned} \begin{tikzpicture} [scale=9] \draw[->] (0.0, 0) -- (0.6, 0) node[below] {$\frac{1}{p}$}; \draw[->] (0,0.0) -- (0, 0.6) node[left] {$\frac{1}{q}$}; \draw (0.5, 0) node[below] {${1}$}; \draw (0, 0.5) node[left] {${1}$}; \draw (0.25, 0) node[below] {$\frac{1}{2}$}; \draw (0, 0.25) node[left] {$\frac{1}{2}$}; \draw (0, 0) -- (0.5, 0.0) -- (0.5, 0.5) -- (0.0, 0.5) -- cycle; \draw (0.5, 0.0) -- (0.25, 0.25)-- (0.25, 0.5); \draw (0.25, 0.25) -- (0.0, 0.25); \draw [loosely dotted] (0.25, 0.0) -- (0.25, 0.25); \draw (0.165,0.125) node{$\;\frac{1}{2}\frac{1}{q}$}; \draw (0.125,0.375) node{$\;\frac{1}{4}$}; \draw (0.375,0.3) node{$\; \frac{1}{2}(1-\frac{1}{p})$}; \draw (0.3, -0.1) node[below] {$C_{2,k,1}$}; \end{tikzpicture}\qquad\qquad\quad \begin{tikzpicture} [scale=9] \draw[->] (0.0, 0) -- (0.6, 0) node[below] {$\frac{1}{p}$}; \draw[->] (0,0.0) -- (0, 0.6) node[left] {$\frac{1}{q}$}; \draw (0.5, 0) node[below] {${1}$}; \draw (0, 0.5) node[left] {${1}$}; \draw (0.25, 0) node[below] {$\frac{1}{2}$}; \draw (0, 0.25) node[left] {$\frac{1}{2}$}; \draw (0, 0) -- (0.5, 0.0) -- (0.5, 0.5) -- (0.0, 0.5) -- cycle; \draw (0.5, 0.0) -- (0.25, 0.25)-- (0.25, 0.5); \draw (0.25, 0.25) -- (0.0, 0.25); \draw [loosely dotted] (0.25, 0.0) -- (0.25, 0.25); \draw (0.165,0.125) node{$\;\frac{1}{k}\frac{1}{q}$}; \draw (0.125,0.375) node{$\;\frac{1}{2}\frac{1}{k}$}; \draw (0.375,0.3) node{$\; \frac{1}{k}(1-\frac{1}{p})$}; \draw (0.3, -0.1) node[below] {$C_{j,k,n}$}; \end{tikzpicture}\end{aligned}$$ For each $\,p,q\in [1,\infty],$ and all dimension $\,n>1,\,$ the asymptotic behaviour of $\,\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])}\,$ as $\,n\rightarrow +\infty\,$ is determined only by the linearity or quadraticity of the phase in $s$. The role of the power $j$ of $x$ appears exclusively in the dimension $n=1.$ There is nothing special about neither the unit interval nor the unit ball in defining the operators $T^{j,k}_{N}$. Actually we shall make use of Hölder inclusions of $L^{p}$ spaces on measurable sets of finite measure (see Lemma \[holder\] below). So we may take any suitable two such sets provided their finite measures are asymptotically equivalent to a constant independent of $ N $ as $ N\rightarrow +\infty.$ Foschi [@damianorem] studied a discrete version of an operator a little simpler than the integral operator $\,T^{1,1}_{N}.\,$ He considered the operator $\,D_{N}:\ell^{p}(\mathbb{C}^{N}) \rightarrow L^{q}(-\pi,\pi)\,$ that assigns to each vector $\,a=(a_{0},a_{1},...a_{N-1})\in {\mathbb{C}}^{N}\,$ the $\,2\pi$-periodic trigonometric polynomial $\,D_{N}a(t)=\sum_{m=0}^{N-1}a_{m}\,e^{\imath\,m\,t}\,$ and described the asymptotic behaviour of $\, \displaystyle\sup_{a\in \mathbb{C}^{N}-\{0\}} {{\parallel D_{N}a\parallel_{L^{q}([-\pi,\pi])} }/ {\parallel a\parallel_{\ell^{p}\left(\mathbb{C}^{N} \right)}}}$ as $N\rightarrow+\infty,$ for all $ 1\leq p,\,q\leq+\infty.$ The norms there are defined by $$\begin{aligned} &\parallel a \parallel_{\ell^{p}}= \left( \sum_{m=0}^{N-1}\|a_{m}\|^{p}\right)^{\frac{1}{p}}, \quad 1\leq p <\infty, \qquad \parallel a \parallel_{\ell^{\infty}}= \max_{0\leq m\leq N-1}\|a_{m}\|,\\ & \parallel f \parallel_{L^{q}}= \left(\frac{1}{2\pi} \int_{-\pi}^{\pi}\|f(t)\|^{q}dt\right)^{\frac{1}{q}}, \quad 1\leq q <\infty, \qquad \parallel f \parallel_{L^{\infty}}= \max_{\|t\|\leq \pi}\|f(t)\|.\end{aligned}$$ This was followed by a similar investigation (see Section 5 in [@damianorem]) of a linear integral operator with an oscillatory kernel $\, L_{N}: L^{p}([0,1])\rightarrow L^{q}([0,1])\,$ defined by $$\begin{aligned} L_{N}f(t)\,:=\,\int_{0}^{1}\, e^{\imath N/(1+t+s)}\,\frac{f(s)}{(1+t+s)^{\gamma}}\,ds, \quad\text{\small for some fixed}\;\; \gamma \geq 0.\end{aligned}$$ ### 2. Proof of Theorem \[mainthm\] {#proof-of-theorem-mainthm .unnumbered} In order to show Theorem \[mainthm\], we shall go through the following steps.\ ***. Find lower bounds for $\,\parallel T^{j,k}_{N} \parallel_ {L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,q\in [1,+\infty]\,$:\ Test the ratio $ \parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}/ \parallel f \parallel_{L^{p}(B)} $ for functions $f \in {L^{p}(B)}$ that kill or at least slow down the oscillations in the integrals $T^{j,k}_{N}f.\,$ Of course this ratio is majorized by $\displaystyle \parallel T^{j,k}_{N} \parallel_ {L^{p}(B)\rightarrow L^{q}([0,1])}=\sup_{f\in L^{p}(B)-\{0\}} {{\parallel T^{j,k}_{N}f\parallel_{L^{q}([0,1])} }/{\parallel f\parallel_{L^{p}\left(B\right)}}}. $ But what is really interesting is the fact that such functions likely maximize the ratio as well.\ ***. We find upper bounds for $\,\parallel T^{j,k}_{N} \parallel_ {L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,q\in [1,+\infty].$ Thanks to interpolation and Hölder’s inequality, we merely need an upper bound for $\parallel T^{j,k}_{N} \parallel_ {L^{2}(B)\rightarrow L^{2}([0,1]).}$ \[holder\] Let $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ be as in (\[intop\]). Assume that $$\begin{aligned} \label{en11} \parallel T^{j,k}_{N}f \parallel_{L^{2}([0,1])} \,\leq\, c_{j,k,N} \parallel f \parallel_{L^{2}(B)}.\end{aligned}$$ Then $$\begin{aligned} \label{consigma} \parallel T^{j,k}_{N} \parallel_{L^{p}(B) \rightarrow L^{q}([0,1])} \;\lesssim_{p,q,n}\; c^{\sigma\left(\frac{1}{p},\frac{1}{q} \right)}_{j,k,N}\end{aligned}$$ where $\,\sigma:[0,1]^{2}\rightarrow [0,1]\,$ is the continuous function in (\[sgmab\]). If we take absolute values of both sides of (\[intop\]) we get the trivial estimate\ $\;\parallel T^{j,k}_{N}f\parallel_{L^{\infty}([0,1])} \,\leq\,\parallel f\parallel_{L^{1}\left(B\right)}.$ Interpolating this with (\[en11\]) using Riesz-Thorin theorem ([@loukas]) implies $$\begin{aligned} \label{int1} \parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])} \,\leq\, c^{2\left(1-\frac{1}{p}\right)}_{j,k,N} \parallel f \parallel_{L^{p}(B)}, \qquad \frac{1}{2}\leq\frac{1}{p}\leq 1,\;\; \frac{1}{q}=1-\frac{1}{p}.\end{aligned}$$ Since, by Hölder’s inequality, $\; \parallel T^{j,k}_{N}f \parallel_{L^{\bar{q}}([0,1])} \,\leq\,\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])}\;$ whenever\ $\,1\leq \bar{q}\leq q\leq \infty,$ then $$\begin{aligned} \label{int2} \parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])} \,\leq\, c^{2\left(1-\frac{1}{p}\right)}_{j,k,N} \parallel f \parallel_{L^{p}(B)}, \qquad \frac{1}{2}\leq\frac{1}{p}\leq 1,\;\; 1-\frac{1}{p}\leq\frac{1}{q}\leq 1.\end{aligned}$$ Applying Hölder’s inequality once more we find that if $\;1\leq p\leq\bar{p}\leq \infty,\,$ then $$\begin{aligned} \nonumber&\hspace{-1 cm}\parallel f \parallel_{L^{p}(B)} \,\leq\,\|B\|^{\frac{1}{p}-\frac{1}{\bar{p}}}\, \parallel f \parallel_{L^{\bar{p}}(B)}. \;\; \text{Therefore by}\;(\ref{int1})\; \text{we have}\\ \label{int3} &\hspace{-0.6 cm}\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])} \,\leq\, \|B\|^{1-\frac{1}{p}-\frac{1}{q}}\, c^{2/q}_{j,k,N} \parallel f \parallel_{L^{p}(B)}, \quad 0\leq\frac{1}{q}\leq \frac{1}{2},\;\; 0\leq\frac{1}{p}\leq 1-\frac{1}{q}.\end{aligned}$$ Moreover, since we know from (\[int2\]) that $$\begin{aligned} \nonumber &\hspace{-1 cm}\parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])} \,\leq\, c_{j,k,N} \parallel f \parallel_{L^{2}(B)}, \quad \frac{1}{2}\leq\frac{1}{q}\leq 1,\quad \text{then}\\ &\label{int4} \parallel T^{j,k}_{N}f \parallel_{L^{q}([0,1])} \,\leq\, \|B\|^{\frac{1}{2}-\frac{1}{p}}\, c_{j,k,N} \parallel f \parallel_{L^{p}(B)}, \quad 0\leq\frac{1}{p}\leq \frac{1}{2},\;\; \frac{1}{2}\leq\frac{1}{q}\leq 1.\end{aligned}$$ If the constants in inequalities (\[int1\]) - (\[int4\]) were sharp, they would be precisely the values of the corresponding norms $\,\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])}.$ Unfortunately, we are not able to compute the optimal constant $\,c_{j,k,N}\,$ in the energy estimate (\[en11\]). Nevertheless, the constants $\,c^{\sigma\left(\frac{1}{p},\frac{1}{q} \right)}_{j,k,N}\,$ in (\[consigma\]) would be good enough for our purpose if, for each $p,q\in [1,+\infty],$ they were asymptotically equivalent, as $ N\rightarrow +\infty$, to the corresponding lower bounds of $\,\parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])}\,$ that we compute in Step 1.\ **.\ (i) Focusing data\ When $\,x\in B(\eta /N^{\frac{1}{j}})\,$ we have $\;\displaystyle e^{\imath N{\|x\|}^{j}s^{k}}= e^{\mathcal{O}\left(\eta\right)}= 1+\mathcal{O}\left(\eta\right), \;\; \text{\small for all}\;s\in [0,1].$ Thus, if we take $f_{j}$ to be the focusing functions $\,f_{j}=\displaystyle \chi_{B(\eta /N^{\frac{1}{j}})}\,$ then $\; \displaystyle \parallel f \parallel_{L^{p}(B)}\,=\, \|B(\eta /N^{\frac{1}{j}})\|^{\frac{1}{p}}\;$ and $$\begin{aligned} T^{j,k}_{N}f_{j}(s)\,=\,\int_{B(\eta /N^{\frac{1}{j}})} \,e^{\imath N{\|x\|}^{j}s^{k}}\,dx= \int_{B(\eta /N^{\frac{1}{j}})} \,\left(1+\mathcal{O}\left(\eta\right)\right)\,dx \,\gtrsim \,\|B(\eta /N^{\frac{1}{j}})\|\end{aligned}$$ for all $ \;0\leq s\leq 1.\;$ Consequently, since $\eta$ is fixed, $$\begin{aligned} \label{lb1} \parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])} \,\geq\,\frac{\parallel T^{j,k}_{N}f_{j} \parallel_{L^{q}([0,1])}}{ \parallel f_{j} \parallel_{L^{p}(B)}} \;\gtrsim\; N^{-\frac{n}{j}\left(1-\frac{1}{p}\right)}.\end{aligned}$$ The figure below illustrates the one dimensional case. $$\begin{aligned} &\hspace{-1 cm} \begin{tikzpicture} [scale=10] \fill[fill = black!10] (0,0.35)--(0.048,0.35)--(0.048,0.0)-- (0,0)--cycle; \draw[->] (0, 0) -- (0.38, 0) node[below] {\small$x$}; \draw[->] (0,0.0) -- (0, 0.4) node[left] {\small$ f_{j}(x)$}; \draw (0.35, 0) node[below] {\small ${1}$}; \draw (0,0.35) node[left] {\small ${1}$}; \draw[thick] (0,0.35)--(0.05,0.35); \draw[thick] (0.05,0.0001)--(0.35,0.0001); \draw (0.07,0.0)node[below] { \footnotesize {$N^{-\frac{1}{j}}$}}; \draw[dotted] (0.35, 0) -- (0.35,0.35)--(0,0.35); \end{tikzpicture}\quad \begin{tikzpicture} [scale=3.5] \draw[->] (0.0, 0) -- (1.1, 0) node[below] {\small $s$}; \draw[->] (0.0,0.0) -- (0.0,1.1); \draw (0.9,1.2) node[left] { \footnotesize {$N^{1/j}$\text{Re}$\left\{T^{j,k}_{N}f_{j}(s)\right\}$}}; \draw (1,0) node[below] {\small${1}$}; \draw (1,-0.001) --(1,0.003); \draw (0.0,0.95) node[left] {\small${1}$}; \draw [blue,samples=500,domain=0.0:1] plot (\x, {cos(0.65\x r)}); \draw [red,samples=500,domain=0.0:1] plot (\x, {cos(0.65\x\x r)}); \draw(0.7,1) node[right,thick] {\tiny {$k=2$}}; \draw(0.6,0.8) node[right,thick] {\tiny {$k=1$}}; \draw(0.0,-0.15) node {}; \end{tikzpicture}\qquad \begin{tikzpicture} [scale=3.5] \draw[->] (0.0, 0) -- (1.1, 0) node[below] {\small $s$}; \draw[->] (0.0,0.0) -- (0.0,1.1); \draw (0.9,1.2) node[left] { \footnotesize {$N^{1/j}$\text{Im}$\left\{T^{j,k}_{N}f_{j}(s)\right\}$}}; \draw (1,0) node[below] {\small${1}$}; \draw (1,-0.001) --(1,0.003); \draw (-0.001,1) --(0.003,1); \draw (0.0,0.95) node[left] {\small${1}$}; \draw [blue,samples=500,domain=0.0:1] plot (\x, {sin(0.65\x r)}); \draw [red,samples=500,domain=0.0:1] plot (\x, {sin(0.65\x\x r)}); \draw(0.7,0.3) node[right,thick] {\tiny {$k=2$}}; \draw(0.6,0.6) node[right,thick] {\tiny {$k=1$}}; \draw(0.0,-0.15) node {}; \end{tikzpicture}\\ &\text{\small \emph{Both real and imaginary parts of the functions}} \;T^{1,k}_{N}f_{1}\; \text{\small\emph{and}}\; T^{2,k}_{N}f_{2}\; \text{\small \emph{have the same profile}}.\end{aligned}$$ (ii) Constant data\ Let $\,g(x)=1.\,$ Whenever $\,\displaystyle s \in [0,\eta/N^{\frac{1}{k}}]\,$ we have $\;\imath N{\|x\|}^{j}s^{k}\,=\,\mathcal{O}\left(\eta\right)\;$ for all $\;x\in B\;$ and it follows that $\;\displaystyle e^{\imath N{\|x\|}^{j}s^{k}}= 1+\mathcal{O}\left(\eta\right).\;$ Hence, when $\,\displaystyle s \in [0,\eta/N^{\frac{1}{k}}],\,$ $$\begin{aligned} T^{j,k}_{N}g(s)\,=\, \int_{B}\,e^{\imath N{\|x\|}^{j}s^{k}}\,dx\,=\, \int_{B}\,\left(1+\mathcal{O}\left(\eta\right)\right)\,dx \,\gtrsim 1.\end{aligned}$$ Therefore, recalling that $\eta$ is fixed, $$\label{lb20} \int_{0}^{1}\|T^{j,k}_{N}g(s)\|^{q}\,ds \,\geq\, \int_{0}^{\eta/N^{\frac{1}{k}}}\|T^{j,k}_{N}g(s)\|^{q}\,ds \,\gtrsim\, \int_{0}^{\eta/N^{\frac{1}{k}}}\,ds \,\approx\,N^{-\frac{1}{k}}.$$ In view of (\[lb20\]), we deduce that $$\begin{aligned} \label{lb2} \parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])} \,\geq\,\frac{\parallel T^{j,k}_{N} g \parallel_{L^{q}([0,1])}}{ \parallel g \parallel_{L^{p}(B)}} \;\gtrsim\; N^{-\frac{1}{k}\frac{1}{q}}.\end{aligned}$$ By rescaling, it is easy to verify that the estimate (\[lb2\]) follows for any complex-valued constant function $g$. The figure below shows the behaviour of $ T_{N}^{j,k}g $ on $[0,1]$ in the dimension $n=1.$ $$\begin{aligned} \hspace{-0.5 cm} \begin{tikzpicture} [scale=10] \draw[->] (0.0, 0.0) -- (0.48,0.0) node[below right] {$s$}; \draw[->] (0.0,0.0) -- (0.0,0.52); \draw(0.25,0.44) node[above] {\scriptsize $ \text{Re}\left\{T^{1,k}_{N}g(s)\right\}=2$}; \draw(0.45,0.44) node[above] {\large $ \frac{\sin{(Ns^{k})}}{N\,s^{k}}$}; \draw (0.0,0.49) node[left] {\scriptsize ${2}$}; \draw (0.1,0.2) node[left] {\scriptsize $k=1$}; \draw (0.175,0.3) node[left] {\scriptsize $k=2$}; \draw [very thin,blue,samples=500,domain=0.001:0.45] plot (\x, {2sin((250\x) r)/((1000\x) )}); \draw [very thin,red,samples=500,domain=0.0012:0.45] plot (\x, {2sin((250\x\x) r)/((1000\x\x) )}); \end{tikzpicture}\qquad\qquad\qquad \begin{tikzpicture} [scale=10] \draw(0.23,0.44) node[above] {\scriptsize $ \text{Im}\left\{T^{1,k}_{N}g(s)\right\}=4$}; \draw(0.46,0.44) node[above] {\large $ \frac{\sin^{2}{(Ns^{k}/2)}}{N\,s^{k}}$}; \draw (0.0,0.49) node[left] {\scriptsize ${2}$}; \draw (0.0,0.49) --(0.001,0.49); \draw (0.09,0.38) node[left] {\scriptsize $k=1$}; \draw (0.2,0.38) node[left] {\scriptsize $k=2$}; \draw [very thin,red,samples=500,domain=0.001:0.45] plot (\x, {4sin((100\x\x) r)sin((100\x\x) r)/((800\x\x) )}); \draw [very thin,blue,samples=500,domain=0.0001:0.45] plot (\x, {4sin((50\x) r)sin((50\x) r)/((400\x) )}); \draw (-0.084, -0.084) node[below] {}; \draw[->] (0.0, 0.0) -- (0.48,0.0) node[below right] {$s$}; \draw[->] (0.0,0.0) -- (0.0,0.52); \end{tikzpicture}\end{aligned}$$ $$\begin{aligned} &\hspace{-1.58 cm}\begin{tabular}{c c} \hspace{-4.75 cm}\vspace{0.25 cm} \scriptsize $2$& \\ \hspace{-1.7 cm} \scriptsize $ k=2$&\\ \hspace{-3.3 cm} \scriptsize $ k=1$&\\ &\hspace{-1 cm} \scriptsize $ k=1\quad k=2$ \vspace{-2.5 cm}\\ \hspace{0.5 cm}\scriptsize{ \text{Re}$\left\{T^{2,k}_{N}g(s)\right\}$} &\hspace{2 cm}\scriptsize {\text{Im}$\left\{T^{2,k}_{N}g(s) \right\}$} \vspace{-1.5 cm}\\ \includegraphics[scale=0.32]{t1001.pdf} &\qquad\quad\; \includegraphics[scale=0.32]{t2001.pdf}\vspace{-1 cm}\\ \hspace{5.5 cm} \small$ s$& \hspace{7 cm} \small $ s$ \end{tabular}\\ &\hspace{-2.3 cm} \text{\small\emph{Functions}}\; \text{\small \emph{Re}}\small \{ T^{1,k}_{N}g(s)\}\; \text{\small\emph{vanish and}}\;\text{\small \emph{Re}}\small \{T^{2,k}_{N}g(s)\}\; \text{\small \emph{change monotonicity, for the first time, when }}\;s=\sqrt[k]{\pi/N}\end{aligned}$$ (iii) Oscillatory data\ Consider the oscillatory function $\,h(x)=e^{2\imath N \left(\|x\|^2-\|x\|\right)}.\,$ Using polar coordinates we can write $$\begin{aligned} T^{j,k}_{N}h(s)\,=\, \int_{S^{n-1}} \int_{0}^{1}\,e^{\imath N \,\left({\rho}^{j}s^{k}+2\rho^2-2\rho\right)}\, \rho^{n-1}\,d\rho \,d\omega\,=\, \omega_{n-1} \:I^{j,k}_{N}(s)\end{aligned}$$ where $I^{j,k}_{N}(s) $ is the oscillatory integral given by $$\begin{aligned} \label{inoscint} I^{j,k}_{N}(s) = \int_{0}^{1}\,e^{\imath N \,\phi_{j,k}(\rho;s)}\, \rho^{n-1}\,d\rho\end{aligned}$$ with the phase $\displaystyle \phi_{j,k}(\rho;s) ={\rho}^{j}s^{k}+2\rho^2-2\rho. $\ The quadratic function $\displaystyle \rho\rightarrow\phi_{j,k}(\rho;s)$, after a suitable translation along the vertical axis, has a single nondegenerate stationary point that happens to lie well inside $]\frac{1}{5},\frac{4}{5}[.$ Indeed, one can simply write $$\begin{aligned} \phi_{j,k}(\rho;s)=\left\{ \begin{array}{ll} 2\left(\rho-\frac{2-s^{k}}{4}\right)^{2}- \frac{\left(2-s^{k}\right)^{2}}{8}, & \hbox{$j=1$;} \\ \left(2+s^{k}\right)\left(\rho-\frac{1}{2+s^{k}}\right)^{2}- \frac{1}{\left(2+s^{k}\right)^{2}}, & \hbox{$j=2$.} \end{array} \right.\end{aligned}$$ Notice also that $\, \left(2-s^{k}\right)/4\in[\frac{1}{4},\frac{1}{2}]\,$ and $\,\left(2+s^{k}\right)^{-1}\in[\frac{1}{3}, \frac{1}{2}]\,$ when $\,s\in [0,1].$ In fact, this is what we were after when we used the oscillatory function $h$ with its particular quadratic phase. Let us see how we benefit from this. We shall work on the integral $\,I^{1,k}_{N}(s)\,$ and the applicability of the same procedure to the integral $\,I^{2,k}_{N}(s)\,$ will be obvious. For simplicity, let $z$ denote $\,\left(2-s^{k}\right)/4.\,$ Then $$\begin{aligned} \nonumber e^{2\imath N\,z^2}\,I^{1,k}_{N}(s) =& \,\int_{0}^{1}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\, \rho^{n-1}\,d\rho\\ \label{feq}=&\,z^{n-1}\,\int_{0}^{1}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho+ \int_{0}^{1}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\, \left(\rho^{n-1}-z^{n-1}\right)\,d\rho.\end{aligned}$$ We compute $$\begin{aligned} \label{t1} \hspace{-1 cm} \int_{0}^{1}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho= \int_{-\infty}^{+\infty}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho- \int_{-\infty}^{0}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho- \int_{1}^{+\infty}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho.\end{aligned}$$ Using the identity (See Exercise 2.26 in [@taobook]) $$\begin{aligned} \int_{-\infty}^{+\infty} \,e^{-ax^2}\,e^{bx}\,dx=\sqrt{\frac{\pi}{a}} \,e^{b^2/4a},\quad a,b \in \mathbb{C},\; \textrm{Re}(a) >0 \qquad \text{we get}\end{aligned}$$ $$\begin{aligned} \label{11} \hspace{-0.5 cm}\int_{-\infty}^{+\infty}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho =\sqrt{\frac{\pi}{2N}}\,e^{\frac{\pi}{4}\imath}.\end{aligned}$$ And since $$\begin{aligned} \hspace{-1 cm} \left\|\;\int_{-\infty}^{0}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,\partial_{\rho} \left(\rho-z\right)^{-1}\,d\rho \right\|\,\leq\, \frac{1}{z},\quad \left\|\; \int_{1}^{+\infty}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,\partial_{\rho} \left(\rho-z\right)^{-1}\,d\rho \right\|\,\leq\, \frac{1}{1-z},\end{aligned}$$ then integration by parts implies $$\begin{aligned} \label{22}&\int_{-\infty}^{0}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho= \frac{\imath\, e^{2\imath N z^{2}}}{4 N z} +\mathcal{O}\left(\frac{1}{Nz}\right),\\ \label{33}&\int_{1}^{+\infty}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho= \frac{\imath\, e^{2\imath N\left(1-z\right)^{2}}}{4 N\left(1-z\right)} +\mathcal{O}\left(\frac{1}{N\left(1-z\right)}\right).\end{aligned}$$ Recalling that $\,\frac{1}{4}\leq z\leq \frac{1}{2}\;$ and using (\[11\]), (\[22\]), (\[33\]) in (\[t1\]) we obtain $$\begin{aligned} \label{44} \int_{0}^{1}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho\,=\, \sqrt{\frac{\pi}{2N}}\,e^{\frac{\pi}{4}\imath} +\mathcal{O}\left(\frac{1}{N}\right).\end{aligned}$$ This gives us an estimate for the first integral on the right hand side of (\[feq\]). The second integral is $\;\mathcal{O}\left({1}/{N}\right).\;$ This follows from integration by parts and the smoothness of the polynomial $\;P(\rho;z):={\left(\rho^{n-1}-z^{n-1}\right)}/{\left(\rho-z\right)}= \sum_{\ell=0}^{n-2}\,\rho^{n-2-\ell}\,z^{\ell}\;$ as we can write $$\begin{aligned} \int_{0}^{1}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\, \left(\rho^{n-1}-z^{n-1}\right)\,d\rho\,=\, \frac{1}{4\imath N} \int_{0}^{1}\, P(\rho;z)\, \partial_{\rho}\,e^{2\imath N \,\left(\rho-z\right)^{2}}\,d\rho.\end{aligned}$$ Plugging (\[44\]) together with the latter estimate into (\[feq\]) we get that $$\begin{aligned} \label{sm} e^{2\imath N\,z^2}\,I^{1,k}_{N}(s) \,=\,z^{n-1}\, \sqrt{\frac{\pi}{2N}}\,e^{\frac{\pi}{4}\imath} +\mathcal{O}\left(\frac{1}{N}\right).\end{aligned}$$ From (\[sm\]) follows the estimate $$\begin{aligned} \left\|I^{1,k}_{N}(s)\right\| \,\gtrsim\,N^{-1/2}.\end{aligned}$$ An explanation for the estimate above comes from the fact that the function $\,\lambda_{N}(\rho;z)= \cos{\left(2N\,\left(\rho-z\right)^{2}\right)}\,$ remains positive for $\;\|\rho-z\|<\sqrt{\left(\pi/4N\right)}\;$ and the further we move from the stationary point $\rho=z$ it, unlike the slowly varying factor $\rho^{n-1}$, oscillates rapidly for large $N$ so that, when summing over $\rho$, integrals over neighbouring halfwaves where $\lambda_{N}$ changes sign almost cancel. See the figure below. An identical estimate for $\,I^{2,k}_{N}(s)\,$ follows applying the same argument above. The approach adopted here is standard. It represents the key idea of the proof of the stationary phase method illustrated by Lemma \[stationary\]. $$\begin{aligned} \begin{tikzpicture}[yscale=1.5] \fill[fill = black!50] (3pi/8,0) -- plot [domain=3pi/8:11pi/13] (\x,{cos(64\x\x )}) -- (11pi/13,0) -- cycle; \fill[fill = black!50] (-3pi/8,0) -- plot [domain=-3pi/8:-11pi/13] (\x,{cos(64\x\x )}) -- (-11pi/13,0) -- cycle; \fill[fill = black!25] (11pi/13,0) -- plot [domain=11pi/13:235pi/208] (\x,{cos(64\x\x )}) -- (235pi/208,0) -- cycle; \fill[fill = black!25] (-11pi/13,0) -- plot [domain=-11pi/13:-235pi/208] (\x,{cos(64\x\x )}) -- (-235pi/208,0) -- cycle; \fill[fill = black!5] (235pi/208,0) -- plot [domain=235pi/208:141pi/104] (\x,{cos(64\x\x )}) -- (141pi/104,0) -- cycle; \fill[fill = black!5] (-235pi/208,0) -- plot [domain=-235pi/208:-141pi/104] (\x,{cos(64\x\x )}) -- (-141pi/104,0) -- cycle; \draw [ <->] (-6.5,0) -- (6.5,0); \draw [help lines,dashed,<-] (0,1.3) -- (0,0); \draw (0,0) node[below] {$\rho=z$}; \draw (-5,1.3) node[above]{$ \cos{\left(N\,\left(\rho-z\right)^{2}\right)}$}; ; \draw [thick,samples=500,domain=-2pi:2pi] plot (\x, {cos(64\x\x )}); \draw [ <-](-3pi/8+0.01,-0.7) --(-2pi/8+0.18,-0.7); \draw [ ->](2pi/8-0.1,-0.7) --(3pi/8-0.01,-0.7); \draw (6.5,0) node[right] {$\rho$}; \draw (0,-0.7) node {$\sqrt{{\pi}/{2N}}$}; \draw [help lines,dashed] (-3pi/8,1) -- (-3pi/8,-1); \draw [help lines,dashed] (3pi/8,1) -- (3pi/8,0-1); \end{tikzpicture}\end{aligned}$$ Finally, since $\;\displaystyle \parallel h \parallel_{L^{p}(B)}\,=\, \|B\|^{{1}/{p}}\,\approx\,1,\;$ then $$\begin{aligned} \label{lb3} \parallel T^{j,k}_{N}\parallel_{L^{p}\left(B\right) \rightarrow L^{q}([0,1])} \,\geq\,\frac{\parallel T^{j,k}_{N} h \parallel_{L^{q}([0,1])}}{ \parallel h \parallel_{L^{p}(B)}} \;\gtrsim\; N^{-\frac{1}{2}}.\end{aligned}$$ Putting (\[lb1\]), (\[lb2\]) and (\[lb3\]) together we deduce $$\begin{aligned} \parallel T^{j,k}_{N} \parallel_{L^{p}(B) \rightarrow L^{q}([0,1])} \;\;\gtrsim\; N^{-\min\left\{\frac{n}{j}\left(1-\frac{1}{p}\right), \,\frac{1}{k}\frac{1}{q},\,\frac{1}{2}\right\}} \,=\,N^{- C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)}.\end{aligned}$$ **. The $\,L^{2} - L^{2}\,$ estimate takes the form: $$\begin{aligned} \label{energy} \left. \begin{array}{ll} \vspace{0.3 cm} \parallel T^{j,k}_{N}f\parallel_{L^{2}([0,1])} \;\lesssim \; N^{-1/2k}\,\parallel f \parallel_{L^{2}\left(B\right)}, & \hbox{$n\geq j$,} \\ \parallel T^{2,k}_{N}f\parallel_{L^{2}([0,1])} \;\lesssim \; N^{-n/2j}\,\parallel f \parallel_{L^{2}\left(B\right)}, & \hbox{$n=1$.} \end{array} \right \}\end{aligned}$$ Besides (\[lb2\]), the estimate (\[energy\]) demonstrates the difference between linear ($k=1$) and quadratic ($k=2$) oscillations. Let $\,x\in {\mathbb{R}}^{n}-\{0\}.\,$ The phase $\;s \longrightarrow {\|x\|}^{j}\,s^{k}\;$ of the oscillatory factor in (\[intop\]) is non-stationary when $\,k=1.\,$ While in the case $\,k=2,\,$ it is stationary with the nondegenerate critical point $s=0.\,$ This is where non-stationary and stationary phase methods (see lemmas \[nonstationary\] and \[stationary0\] below) for estimating oscillatory integrals come into play. As expected from (\[lb1\]), the role of $j$ appears only in the dimension $n=1.$ Using the estimate (\[energy\]) in Lemma \[holder\] we infer $$\begin{aligned} \parallel T^{j,k}_{N} \parallel_{L^{p}(B) \rightarrow L^{q}([0,1])} \;\;\lesssim\; N^{- C_{j,k,n}\left(\frac{1}{p},\frac{1}{q}\right)}.\end{aligned}$$ ### 3. Proof of the energy estimate (\[energy\]) {#proof-of-the-energy-estimate-energy .unnumbered} To prove the estimate (\[energy\]) we need lemmas \[kernelsk\], \[kernelsq\] and \[even\] that we give below. Lemma \[kernelsk\] is based on the assertions of lemmas \[nonstationary\] and \[stationary0\]. \[nonstationary\] ([@stein], Proposition 1 Chapter VIII) Let $\,\psi \in C^{\infty}_{c}\left(\mathbb{R} \right)\,$ and let $\displaystyle\; I(\lambda)= \int_{\mathbb{R}}\,\psi(s)\,e^{\imath \,\lambda\,s}\,ds. \,$ Then $\;\displaystyle \|I(\lambda)\|\;\lesssim\; \min{\left\{ \frac{1}{1+\|\lambda\|}, \frac{1}{1+\lambda^{2}}\right\}}.$ Observing that $\;\displaystyle \int_{0}^{1}\,e^{\imath \,\lambda\,s^{2}}\,ds\,=\, \frac{1}{2}\int_{-1}^{1}\,e^{\imath \,\lambda\,s^{2}}\,ds\;$ and arguing as in (\[t1\])-(\[44\]) implies the estimate in Lemma \[stationary0\]. \[stationary0\] $$\begin{aligned} \left\|\int_{0}^{1}\,e^{\imath \,\lambda\,s^{2}}\,ds\right\|\;\lesssim \; \max{\left\{\frac{1}{1+\sqrt{\|\lambda\|}}, \frac{1}{1+\|\lambda\|}\right\}}.\end{aligned}$$ \[kernelsk\] Let $\,\psi \in C^{\infty}_{c}\left(\mathbb{R} \right)\,$ and let $\;K_{N}^{j,k}:{\mathbb{R}}^{n}\times{\mathbb{R}}^{n} \longrightarrow {\mathbb{C}}\;$ be defined by $$\begin{aligned} K_{N}^{j,k}(x,y):= \left\{ \begin{array}{ll} \displaystyle \int_{\mathbb{R}}\,\psi(s)\,e^{\imath N \left({\|x\|}^{j}-{\|y\|}^{j}\right)s}\,ds, & \hbox{$k=1$;} \\\\ \displaystyle \int_{0}^{1}\,\,e^{\imath N \left({\|x\|}^{j}-{\|y\|}^{j}\right)s^{2}}\,ds , & \hbox{$k=2$.} \end{array} \right.\end{aligned}$$ Then $$\begin{aligned} \label{kernelsk1} &\hspace{-1 cm} \|K_{N}^{j,1}(x,y)\|\;\lesssim\; \min{\left\{ \left(1+N\,\left\|{\|x\|}^{j}-{\|y\|}^{j}\right\|\right)^{-1}, \left(1+N^{2} \, \left({\|x\|}^{j}-{\|y\|}^{j}\right)^{2}\right)^{-1} \right\}}, \\ \label{kernelsk2} &\hspace{-1 cm} \|K_{N}^{j,2}(x,y)\|\;\lesssim\; \max{\left\{\left( 1+\sqrt{N}\, \sqrt{\left\|{\|x\|}^{j}-{\|y\|}^{j}\right\|}\right)^{-1}, \left(1+N\, \left\|{\|x\|}^{j}-{\|y\|}^{j}\right\|\right)^{-1}\right\}}.\end{aligned}$$ The next lemma is mainly a consequence of Young’s inequality. \[kernelsq\] Let $\,p,q,r\geq 1\,$ and $\,1/p+1/q+1/r =2.\,$ Let $\,f\in L^{p}(B),\,$ $\,g\in L^{q}(B)\,$ and $\,h\in L^{r}([0,1]).\,$ Then $$\begin{aligned} \left\|\,\int_{{B}}\,\int_{{B}}\, f(x)\,f(y)\,h(\|x\|^{m}-\|y\|^{m})\,dx\,dy\,\right\|\;\lesssim \;\parallel f \parallel_{L^{p}(B)}\, \parallel g \parallel_{L^{q}(B)}\, \parallel h \parallel_{L^{r}([0,1])}\end{aligned}$$ provided $\,m\leq n$. Switching to polar coordinates by setting $\,x=r_{1}\theta_{1}\,$ and $\,y=r_{2}\theta_{2}\,$ then applying Fubini’s theorem gives $$\begin{aligned} \label{newlemma1} \left\|\,\int_{{B}}\,\int_{{B}}\, f(x)\,f(y)\,h(\|x\|^{m}-\|y\|^{m})\,dx\,dy\,\right\| \,\leq\,\int_{S^{n-1}}\,\int_{S^{n-1}}\, \|Q(\theta_{1},\theta_{2})\| \,d\theta_{1}\,d\theta_{2}\end{aligned}$$ where $$\begin{aligned} Q(\theta_{1},\theta_{2})\,=\, \int_{0}^{1}\,\int_{0}^{1}\, f(r_{1}\theta_{1})\,g(r_{2}\theta_{2}) \,h\left({r_{1}}^{m}-{r_{2}}^{m}\right) \,r_{1}^{n-1}\,r_{2}^{n-1}\,dr_{1}\,dr_{2}.\end{aligned}$$ Changing variables $\:r_{i}^{m}\,\longrightarrow\, \rho_{i}\:$ then using Young’s inequality we get $$\begin{aligned} \hspace{-1 cm} \|Q(\theta_{1},\theta_{2})\|\,\lesssim\, \left(\int_{0}^{1}\left\|f(\sqrt[m]{\rho_{1}}\,\theta_{1}) \right\|^{p}\,\rho_{1}^{p\frac{n-m}{m}}\,d\rho_{1}\right) ^{\frac{1}{p}} \left(\int_{0}^{1}\left\|g(\sqrt[m]{\rho_{2}}\,\theta_{2}) \right\|^{q}\,\rho_{2}^{q\frac{n-m}{m}}\,d\rho_{2}\right) ^{\frac{1}{q}} \parallel h \parallel_{L^{r}([0,1])}.\end{aligned}$$ Reversing the variables change in the first two integrals on the right-hand side of the latter estimate we obtain $$\begin{aligned} \label{newlemma2} \hspace{-1 cm} \nonumber \|Q(\theta_{1},\theta_{2})\|\,\lesssim&\, \left(\int_{0}^{1}\left\|f({r_{1}}\,\theta_{1}) \right\|^{p}\,r_{1}^{(p-1)(n-m)}\, r_{1}^{n-1}\,dr_{1}\right) ^{\frac{1}{p}}\\&\;\nonumber \left(\int_{0}^{1}\left\|g({r_{2}}\,\theta_{2}) \right\|^{q}\,r_{2}^{(p-1)(n-m)}\, r_{2}^{n-1}\,dr_{2}\right) ^{\frac{1}{q}}\, \parallel h \parallel_{L^{r}([0,1])}\\ \leq& \left(\int_{0}^{1}\left\|f({r_{1}}\,\theta_{1}) \right\|^{p}\,r_{1}^{n-1}\,dr_{1}\right) ^{\frac{1}{p}} \left(\int_{0}^{1}\left\|g({r_{2}}\,\theta_{2}) \right\|^{q}\,r_{2}^{n-1}\,dr_{2}\right) ^{\frac{1}{q}} \,\parallel h \parallel_{L^{r}([0,1])}\end{aligned}$$ as long as $\,m\leq n.$ Invoking Hölder’s inequality it follows that $$\begin{aligned} \nonumber &\int_{S^{n-1}}\, \left(\int_{0}^{1}\left\|f({r_{1}}\,\theta_{1}) \right\|^{p}\, r_{1}^{n-1}\,dr_{1}\right)^{\frac{1}{p}}\,d\theta_{1}\\ \label{newlemma3} &\hspace{0.8 cm}\leq\;\omega_{n-1}^{1-\frac{1}{p}}\; \left( \int_{S^{n-1}}\,\int_{0}^{1} \left\|f({r_{1}}\,\theta_{1}) \right\|^{p}\,r_{1}^{n-1}\,dr_{1}\,d \theta_{1}\right)^{\frac{1}{p}}\;= \;\omega_{n-1}^{1-\frac{1}{p}}\; \parallel f \parallel_{L^{p}(B)},\\ \nonumber & \int_{S^{n-1}}\, \left(\int_{0}^{1}\left\|g({r_{2}}\,\theta_{2}) \right\|^{q}\, r_{2}^{n-1}\,dr_{2}\right)^{\frac{1}{q}}\,d\theta_{2}\\ \label{newlemma4} & \hspace{0.8 cm}\leq\;\omega_{n-1}^{1-\frac{1}{q}}\; \left( \int_{S^{n-1}}\,\int_{0}^{1} \left\|g({r_{2}}\,\theta_{2}) \right\|^{q}\,r_{2}^{n-1}\,dr_{2}\,d \theta_{2}\right)^{\frac{1}{q}}\;= \;\omega_{n-1}^{1-\frac{1}{q}}\; \parallel g \parallel_{L^{q}(B)}.\end{aligned}$$ Returning to (\[newlemma1\]) with the estimates (\[newlemma2\]), (\[newlemma3\]) and (\[newlemma4\]) concludes the proof. Remark \[even0\] together with Lemma \[homogeneous\] are needed to show Lemma \[even\]. \[even0\] Suppose that the integral $$\begin{aligned} J\,=\, \int_{-b_{1}}^{b_{1}}...\int_{-b_{m}}^{b_{m}} \,K(t_{1},...,t_{m})\, f_{1}(t_{1})...f_{m}(t_{m})\,dt_{1}...dt_{m}\end{aligned}$$ exists. If $\,K\,$ is even in all its variables then $$\begin{aligned} J\,=\, \int_{0}^{b_{1}}... \int_{0}^{b_{m}} \,K(t_{1},...,t_{m})\, \prod_{i=1}^{m}\left(f_{i}(t_{i})+f_{i}(-t_{i})\right) \,dt_{1}...dt_{m}.\end{aligned}$$ This follows easily from the fact that the integrand in the second expression for $\,J\,$ is even in all variables. Lemma \[homogeneous\] discusses the boundedness of a bilinear form with a homogeneous kernel. \[homogeneous\] Let $\,f\in L^{p}([0,1])\,$ and $\,g\in L^{q}([0,1])\,$ with $\,1\leq p \leq +\infty\,$ and $\,1/p\, +\, 1/q=1.\,$ Assume that $\,K:{[0,1]}\times {[0,1]}\longrightarrow {\mathbb{R}}\,$ is homogeneous of degree $-1,\,$ that is, $\,K(\lambda x, \lambda y)= \lambda^{-1} K(x,y),\,$ for $\,\lambda>0.\,$ Assume also that $$\begin{aligned} \int_{0}^{+\infty} \,\left\|K(x,1)\right\|\,{x}^{-\frac{1}{p}}\,dx \,\lesssim\,1 \qquad \text{or } \qquad \int_{0}^{+\infty} \,\left\|K(1,y)\right\|\,{y}^{-\frac{1}{q}}\,dy \,\lesssim\,1.\end{aligned}$$ Then $$\begin{aligned} \left\|\int_{0}^{1}\int_{0}^{1}\, K(x,y)\,f(x)\,g(y)\,dx\,dy\right\|\;\lesssim\; \parallel f \parallel_{L^{p}([0,1])}\, \parallel g \parallel_{L^{q}([0,1])}.\end{aligned}$$ In [@hardy], one can find a proof for the case when the integrals that define the bilinear form are taken over $\,[0,+\infty[.\,$ We treat this slightly trickier case of finite range without using the result in [@hardy]. Let $\,\displaystyle Q(f,g)\,=\, \int_{0}^{1}\int_{0}^{1}\, K(x,y)\,f(x)\,g(y)\,dx\,dy.\,$ Using a change of variables, $\,x\rightarrow y.u,\,$ and exploiting the homogeneity of the kernel we have $$\begin{aligned} \hspace{-0.8 cm} Q(f,g)\,=\, \int_{0}^{1}y\,g(y)\int_{0}^{\frac{1}{y}} K(y.u,y)\,f(y.u)\,du\,dy\,=\, \int_{0}^{1}g(y)\int_{0}^{\frac{1}{y}} K(u,1)\,f(y.u)\,du\,dy.\end{aligned}$$ By Fubini’s theorem we may write $$\begin{aligned} \label{qfg} \hspace{-1 cm} Q(f,g)=\int_{0}^{1} K(u,1) \int_{0}^{1} f(y.u)\,g(y)\,dy\,du+ \int_{1}^{+\infty} K(u,1) \int_{0}^{\frac{1}{u}} f(y.u)\,g(y)\,dy\,du.\end{aligned}$$ But by Hölder’s inequality we have $$\begin{aligned} \hspace{-0.8 cm} \left\|\int_{0}^{1}\,f(y.u)\,g(y)\,dy\right\| \;\leq&\;\left(\int_{0}^{1}\,\|f(y.u)\|^{p}\,dy\right) ^{\frac{1}{p}} \left(\int_{0}^{1}\,\|g(y)\|^{q}\,dy\right)^{\frac{1}{q}}\\ \,&\hspace{-2 cm}=u^{-\frac{1}{p}}\, \left(\int_{0}^{u}\,\|f(x)\|^{p}\,dx\right) ^{\frac{1}{p}}\, \parallel g \parallel_{L^{q}([0,1])} \;\leq\;u^{-\frac{1}{p}}\,\parallel f \parallel_{L^{q}([0,1])}\, \parallel g \parallel_{L^{q}([0,1])}\end{aligned}$$ for all $\,0 < u < 1.\,$ Similarly $$\begin{aligned} \hspace{-0.8 cm} \left\|\int_{0}^{\frac{1}{u}}\,f(y.u)\,g(y)\,dy\right\| \;\leq&\;\left(\int_{0}^{\frac{1}{u}} \,\|f(y.u)\|^{p}\,dy\right) ^{\frac{1}{p}}\, \left(\int_{0}^{\frac{1}{u}}\,\|g(y)\|^{q} \,dy\right)^{\frac{1}{q}}\\ \,&\hspace{-3.4 cm}=u^{-\frac{1}{p}}\, \left(\int_{0}^{1}\,\|f(x)\|^{p}\,dx\right) ^{\frac{1}{p}}\, \left(\int_{0}^{\frac{1}{u}}\,\|g(y)\|^{q} \,dy\right)^{\frac{1}{q}} \;\leq\;u^{-\frac{1}{p}}\,\parallel f \parallel_{L^{q}([0,1])}\, \parallel g \parallel_{L^{q}([0,1])}\end{aligned}$$ for all $\,1< u < +\infty.\,$ Using the last two inequalities together with the triangle inequality in (\[qfg\]) we get $$\begin{aligned} \hspace{-1 cm} \|Q(f,g)\|\leq& \, \,\parallel f \parallel_{L^{q}([0,1])}\, \parallel g \parallel_{L^{q}([0,1])}\, \left(\int_{0}^{1} \|K(u,1)\| \,u^{-\frac{1}{p}}\,du+ \int_{1}^{+\infty} \|K(u,1)\|\,u^{-\frac{1}{p}}\,du \right)\\ \lesssim&\; \parallel f \parallel_{L^{q}([0,1])}\, \parallel g \parallel_{L^{q}([0,1])}, \qquad \text{when} \quad \int_{0}^{+\infty} \|K(x,1)\|\,x^{-\frac{1}{p}}\,dx \,\lesssim\,1.\end{aligned}$$ When $\;\displaystyle \int_{0}^{+\infty}\left\|K(1,y)\right\| {y}^{-\frac{1}{q}}dy\lesssim 1\;$ the assertion follows analogously. If $\,K(x,y)=\left( x + y \right)^{-1}\,$ in Lemma \[homogeneous\] we get Hilbert’s inequality. \[even\] Let $\,f,g \in L^2([-1,1]).$ Then $$\begin{aligned} \label{even1}&\int_{-1}^{1}\,\int_{-1}^{1}\, \frac{\|f(x)\|\|g(y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy \;\lesssim\; \frac{1}{\sqrt{N}}\, \parallel f \parallel_{L^{2}([-1,1])}\,\parallel g \parallel_{L^{2}([-1,1])}, \\ \label{even2} &\int_{-1}^{1}\,\int_{-1}^{1}\, \frac{\|f(x)\|\,\|g(y)\|}{ \sqrt{\left\|{x}^{2}-{y}^{2}\right\|}} \,dx\,dy \;\lesssim\; \parallel f \parallel_{L^{2}([-1,1])}\,\parallel g \parallel_{L^{2}([-1,1])}.\end{aligned}$$ Beginning with the estimate (\[even1\]), Remark \[even0\] suggests estimating\ \ $\; \displaystyle \int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(\pm x)\|\|g(\pm y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy.\;$ Let $\;\displaystyle W_{N}(f,g):= \int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(x)\|\|g(y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy.$\ \ If $\,x,y \geq 0\,$ and $\,\|x-y\|>>1/\sqrt{N}\,$ then we also have $\,x+y>>1/\sqrt{N}\,$ and consequently $\,N\left\|x^2-y^2\right\|>>1.\,$ Therefore $$\begin{aligned} \hspace{-0.8 cm} \nonumber W_{N}(f,g)&\approx \int\,\int_{ \substack{0\leq x,y\leq1,\\ \|x-y\|\lesssim\; 1/\sqrt{N}}} \frac{\|f(x)\|\|g(y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy+ \int\,\int_{ \substack{0\leq x,y\leq1,\\ \|x-y\|>> 1/\sqrt{N}}} \frac{\|f(x)\|\|g(y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy\\ \nonumber &\lesssim \int\,\int_{ \substack{0\leq x,y\leq1,\\ \|x-y\|\lesssim\; 1/\sqrt{N}}} {\|f(x)\|\|g(y)\|}\,dx\,dy+ \frac{1}{N}\,\int\,\int_{ \substack{0\leq x,y\leq1,\\ \|x-y\|>> 1/\sqrt{N}}} \frac{\|f(x)\|\|g(y)\|}{\left\|x^2-y^2\right\|} \,dx\,dy\\ \label{h1} &\lesssim \int_{0}^{1}\,\int_{0}^{1}\, \chi_{N}{\left(\|x-y\|\right)}{\|f(x)\|\|g(y)\|}\,dx\,dy+ \frac{1}{\sqrt{N}}\,\int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(x)\|\|g(y)\|}{x+y} \,dx\,dy\end{aligned}$$ where $\,\chi_{N}\,$ is the characteristic function of the interval $\,[0,1/\sqrt{N}\,].$ By Young’s inequality we have $$\begin{aligned} \label{h2} \int_{0}^{1}\,\int_{0}^{1}\, \chi_{N}{\left(\|x-y\|\right)}{\|f(x)\|\|g(y)\|} \,dx\,dy\,\leq\, \frac{1}{\sqrt{N}}\, \parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.\end{aligned}$$ And by Hilbert’s inequality $$\begin{aligned} \label{h3} \int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(x)\|\|g(y)\|}{x+y} \,dx\,dy\,\lesssim \;\parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.\end{aligned}$$ Using (\[h2\]) together with (\[h3\]) in (\[h1\]) we obtain $$\begin{aligned} \int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(x)\|\|g(y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy\,\lesssim\,\frac{1}{\sqrt{N}}\, \parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.\end{aligned}$$ In obtaining (\[h1\]), we worked only on the kernel of $W_{N}.$ It is therefore easy to see that replacing the function $\,x\rightarrow f(x)\,$ by the function $\,x\rightarrow f(-x)\,$ or $\,y\rightarrow g(y)\,$ by $\,y\rightarrow g(-y)\,$ then repeating the routine above eventually leads to the estimate $$\begin{aligned} \int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(\pm x)\|\|g(\pm y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy\,\lesssim\,\frac{1}{\sqrt{N}}\, \parallel f \parallel_{L^{2}([-1,1])}\,\parallel g \parallel_{L^{2}([-1,1])}.\end{aligned}$$ This proves (\[even1\]). Taking advantage of Remark \[even0\] again and arguing like before, it suffices to\ \ estimate $\displaystyle V(f,g)=\int_{0}^{1}\,\int_{0}^{1}\, \frac{\|f(x)\|\,\|g(y)\|}{\sqrt{\left\|{x}^{2}-{y}^{2}\right\|}} \,dx\,dy.\;$ Since $\; \displaystyle \int_{0}^{+\infty}\frac{dz}{\sqrt{z}\,\sqrt{\|1-z^2\|}} \,\approx\,1,$\ \ a direct application of Lemma \[homogeneous\] then gives $\,V(f,g)\,\lesssim\,\parallel f \parallel_{L^{2}([0,1])}\,\parallel g \parallel_{L^{2}([0,1])}.$ We are now ready to prove (\[energy\]). We do this for each of the cases $k=1$ and $k=2$ separately.\ The phase is linear in $\textbf{s}\,$ $\,(k=1)$:\ Let $\psi$ be a nonnegative smooth cutoff function such that $\,{supp}\:\psi \subset\;]-1,2[\,$ and $\,\psi(s)=1\,$ on $\,[0,1]$. Since $\,\|T^{j,1}_{N} f \|^2 \,=\, T^{j,1}_{N} f\;\;\overline{T^{j,1}_{N} f}.\,$ Then $$\begin{aligned} &\hspace{-1 cm}\parallel T^{j,1}_{N} f \parallel^{2}_{L^{2}([0,1])} \,= \int_{0}^{1}\,\|T^{j,1}_{N} f(s)\|^{2}\,ds \,\leq\,\int_{\mathbb{R}}\psi(s)\,\|T^{j,1}_{N} f(s)\|^{2}\,ds\\ &\hspace{-1 cm}=\;\int_{\mathbb{R}}\psi(s)\, T^{j,1}_{N} f(s)\;\overline{T^{j,1}_{N} f(s)}\,ds\, =\; \int_{\mathbb{R}}\psi(s)\, \int_{{B}}\,\int_{{B}}\,e^{\imath N \left({\|x\|}^{j}-{\|y\|}^{j}\right)s}\, f(x)\,\overline{f(y)}\,dx\,dy\,ds.\end{aligned}$$ Let $f\in L^{2}(B)$. Applying Fubini’s theorem we get $$\begin{aligned} \label{energy01} \parallel T^{j,1}_{N} f \parallel^{2}_{L^{2}([0,1])}\;\leq\; \int_{{B}}\,\int_{{B}}\,K_{N}^{j,1}(x,y)\, f(x)\,\overline{f(y)}\,dx\,dy.\end{aligned}$$ In the light of the estimate (\[kernelsk1\]) of Lemma \[kernelsk\], it follows that $$\begin{aligned} \label{energy11} \parallel T^{j,1}_{N} f \parallel^{2}_{L^{2}([0,1])}\;\lesssim\; \int_{{B}}\,\int_{{B}}\, \frac{\|f(x)\|\,\|f(y)\|}{1+N^{2} \, \left({\|x\|}^{j}-{\|y\|}^{j}\right)^{2}} \,dx\,dy.\end{aligned}$$ Since $\displaystyle \int_{0}^{1}\frac{dz}{1+N^2 z^2}\approx \frac{1}{N},\,$ then, applying Lemma \[kernelsq\] with $\,h(z)=\left(1+N^2 z^2\right)^{-1}\,$ to the\ \ estimate (\[energy11\]), we obtain $$\begin{aligned} \label{e1} \parallel T^{j,1}_{N} f \parallel_{L^{2}([0,1])}\;\lesssim\; \frac{1}{\sqrt{N}}\,\parallel f \parallel_{L^{2}(B)}, \qquad\text{for all dimensions}\;\;n\geq j.\end{aligned}$$ To finish this case, it remains to estimate $\,T^{2,1}f\,$ in the dimension $\,n=1.$ In view of (\[kernelsk1\]) and (\[energy01\]), we have $$\begin{aligned} \hspace{-1 cm} \parallel T^{2,1}_{N} f \parallel^{2}_{L^{2}([0,1])}\:\lesssim\, \int_{-1}^{1}\,\int_{-1}^{1}\, \frac{\|f(x)\|\,\|f(y)\|}{1+N\, \left\|x^2-y^2\right\|} \,dx\,dy.\end{aligned}$$ Hence, by (\[even1\]) of Lemma \[even\], $$\begin{aligned} \label{e2} \parallel T^{2,1}_{N} f \parallel_{L^{2}([0,1])}\: \lesssim\,\frac{1}{N^{1/4}}\, \parallel f \parallel_{L^{2}([-1,1])}.\end{aligned}$$ The phase is quadratic in $\textbf{s}\,$ $\,(k=2)$:\ For $f\in L^{2}(B)$, using Fubini’s theorem then employing the estimate (\[kernelsk2\]) implies $$\begin{aligned} \label{energy21} \hspace{-0.6 cm} \parallel T^{j,2}_{N} f \parallel^{2}_{L^{2}([0,1])}\:=\, \int_{{B}}\,\int_{{B}}\,K_{N}^{j,2}(x,y)\, f(x)\,\overline{f(y)}\,dx\,dy \,\lesssim\, G^{j}_{N}(f)+H^{j}_{N}(f)\end{aligned}$$ where $$\begin{aligned} G^{j}_{N}(f)\,=&\, \int_{{B}}\,\int_{{B}}\, \frac{\|f(x)\|\,\|f(y)\|}{1+\sqrt{N}\, \sqrt{\left\|{\|x\|}^{j}-{\|y\|}^{j}\right\|}} \,dx\,dy,\\ H^{j}_{N}(f)\,=&\, \int_{{B}}\,\int_{{B}}\, \frac{\|f(x)\|\,\|f(y)\|}{1+N\, \left\|{\|x\|}^{j}-{\|y\|}^{j}\right\|} \,dx\,dy.\end{aligned}$$ Since $\;\displaystyle \int_{0}^{1}\,\frac{dz}{1+\sqrt{N}\,\sqrt{z}} \,\approx\, \frac{1}{\sqrt{N}},\quad \int_{0}^{1}\,\frac{dz}{1+N\,z} \,=\, \text{\large o}\left(\frac{1}{\sqrt{N}}\right), \quad \text{as}\;\;\; N\longrightarrow+\infty, $\ \ then applying Lemma \[kernelsq\] to both $\,G^{j}_{N}(f)\,$ and $\,H^{j}_{N}(f)\,$ gives the estimate $$\begin{aligned} \label{energy22} G^{j}_{N}(f)+ H^{j}_{N}(f)\;\lesssim\; \frac{1}{\sqrt{N}} \parallel f \parallel^{2}_{L^{2}(B)}, \qquad n\geq j.\end{aligned}$$ It remains to control $\:G^{2}_{N}(f)\,$ and $\, H^{2}_{N}(f)\:$ in the dimension $\,n=1.\,$ But when $\,n=1,$ $$\begin{aligned} \hspace{-0.4 cm} G^{2}_{N}(f)\,=&\, \int_{-1}^{1}\,\int_{-1}^{1}\, \frac{\|f(x)\|\,\|f(y)\|}{1+\sqrt{N}\, \sqrt{\left\|{x}^{2}-{y}^{2}\right\|}} \,dx\,dy\\ \leq&\,\frac{1}{\sqrt{N}}\, \int_{-1}^{1}\,\int_{-1}^{1}\, \frac{\|f(x)\|\,\|f(y)\|}{ \sqrt{\left\|{x}^{2}-{y}^{2}\right\|}} \,dx\,dy \,\lesssim\, \frac{1}{\sqrt{N}}\,\parallel f \parallel^{2}_{L^{2}([-1,1])}\quad \text{by}\;\; (\ref{even2})\; \text{of}\; \text{Lemma} \;\ref{even}.\end{aligned}$$ An identical estimate holds for $H^{2}_{N}(f)$ in the dimension $n=1$ because of (\[even1\]). Combining this with (\[energy22\]) and using them in (\[energy21\]) yields $$\begin{aligned} \label{e3} \parallel T^{j,2}_{N} f \parallel_{L^{2}([0,1])} \;\lesssim\;\frac{1}{{N}^{1/4}} \parallel f \parallel_{L^{2}(B)}.\end{aligned}$$ Finally, bringing the estimates (\[e1\]), (\[e2\]) and (\[e3\]) together results in (\[energy\]). References {#references .unnumbered} ========== [10]{} Ahmed A. Abdelhakim, A counter example to Strichartz estimates for the inhomogeneous Schrödinger equation, Journal of Mathematical Analysis and Applications, 414 (2014), 767-772. Damiano Foschi, Some remarks on the $L^{p}-L^{q}$ boundedness of trigonometric sums and oscillatory integrals, Communications on pure and applied analysis, 4 (2005), 569-588. Damiano Foschi, Inhomogeneous Strichartz estimates, Journal of Hyperbolic Differential Equations, 2 (2005), 1–24. Loukas Grafakos, Classical Fourier Analysis, 2nd ed., Springer, 2008. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, UK, 1952. T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 223–238. M. Keel and T. Tao, Endpoint Strichartz estimates, American Journal of Mathematics, 120 (1998), 955–980. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993. T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 2006. M. C. Vilela, Strichartz estimates for the nonhomogeneous Schrödinger equation, Transactions of the American Mathematical Society, 359 (2007), 2123–2136. Youngwoo Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, Journal of Mathematical Analysis and Applications, 373 (2011), 147–160. Mathematics Department, Faculty of Science\ Assiut University, Assiut,71516, Egypt\ ahmed.abdelhakim@aun.edu.eg
--- abstract: 'The use of the binary tetrahedral group ($T^{''}$) as flavor symmetry is discussed. I emphasize the CKM quark and PMNS neutrino mixings. I present a novel formula for the Cabibbo angle.' address: \| Department of Physics and Astronomy, University of North Carolina, NC 27599\ $^$E-mail: frampton@physics.unc.edu author: - 'Paul H. Frampton' title: '$T^{''}$ and the Cabibbo Angle' --- Introduction on renormalizability ================================= In particle theory phenomenology, model building fashions vary with time and because the present lack of data (soon to be compensated by the Large Hadron Collider) does not allow discrimination between models some fashions develop a life of their own. In the present talk we take the apparently retrogressive step of imposing the requirement of renormalizability, as holds for quantum electrodynamics (QED), quantum chromodynamics (QCD) and the standard electroweak model, to show that non-abelian flavor symmetry becomes then much more restrictive and predictive. In a specific model we show that a normal neutrino mass hierarchy is strongly favored over an inverted hierarchy. For several years now there has been keen interest in the use of $A_4$ as a finite flavor symmetry in the lepton sector, especially neutrino mixing. In particular, the empirically approximate tribimaximal mixing of the three neutrinos can be predicted. It is usually stated that either normal or inverted neutrino mass spectrum can be predicted. We revisit these two questions in a minimal $A_4$ framework with only one $A_4$-[3]{} of Higgs doublets coupling to neutrinos and permitting only renormalizable couplings. For such a minimal model there is more predicitivity regarding neutrino masses. Although the standard model was originally discovered using the criterion of renormalizability, it is sometimes espoused that renormalizability is not prerequisite in an effective lagrangian. Nevertheless, imposing renormalizability in the present case is more sensible because it does render the model far more predictive by avoiding the many additional parameters associated with higher-order irrelevant operators. Our choice of Higgs sector also minimizes the number of free parameters. It is sufficiently important to emphasize the concept that every result mentioned in this talk would be impossible without imposing renormalizability. Although it has been fruitful in low-energy QCD, heavy-quark effective theory and technicolor this idea is inappropriate to fundamental model building in particle phenomenology. $A_4$ symmetry* ================ The group $A_4$ is the order g=12 symmetry of a regular tetrahedron $T$ and is a subgroup of the rotation group $SO(3)$. $A_4$ has irreducible representations which are three singlets $1_1, 1_2, 1_3$ and a triplet $3$. In the embedding $A_4 \subset SO(3)$ the [3]{} of $A_4$ is identified with the adjoint [3]{} of $SO(3)$. Since the only Higgs doublets coupling to neutrinos in our model are in a [3]{} of $A_4$, it is very useful to understand geometrically the three components of a [3]{}. A regular tetrahedron has four vertices, four faces and six edges. Straight lines joining the midpoints of opposite edges pass through the centroid and form a set of three orthogonal axes. Regarding the regular tetrahedron as the result of cutting off the four odd corners from a cube, these axes are parallel to the sides of the cube. With respect to the regular tetrahedron, a vacuum expectation value (VEV) of the [3]{} such as $<{\bf 3}> = v (1, 1, -2)$, as will be used,clearly breaks SO(3) to U(1) and correspondingly $A_4$ to $Z_2$, since it requires a rotation by $\pi$ about the 3-axis to restore the tetrahedron. At the same time, we can understand the appearance of tribimaximal mixing with matrix $$U_{TBM} = \left( \begin{array}{ccc} - \sqrt{\frac{1}{6}} & -\sqrt{\frac{1}{6}} & \sqrt{\frac{2}{3}} \\ \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} & 0 \end{array} \right), \label{TBM}$$ and our definitions are such that the ordering $\nu_{1, 2, 3}$ and $\nu_{\tau, \mu, e}$ satisfy $$\left( \begin{array}{c} \nu_1 \\ \nu_2 \\ \nu_3 \end{array} \right) = U_{TBM} \left( \begin{array}{c} \nu_{\tau} \\ \nu_{\mu} \\ \nu_e \end{array} \right)$$ Assuming no CP violation, the Majorana matrix $M_{\nu}$ is real and symmetric and therefore of the form $$M_{\nu} = \left( \begin{array}{ccc} A & B & C \\ B & D & F \\ C & F & E \end{array} \right) \label{ABCDEF}$$ and is related to the diagonalized form by $$M_{diag} = \left( \begin{array}{ccc} m_1 & 0 & 0 \\ 0 & m_2 & 0 \\ 0 & 0 & m_3 \end{array} \right) = U_{TBM} M_{\nu} U_{TBM}^{T}. \label{diag}$$ Substituting Eq.(\[TBM\]) into Eq.(\[diag\]) shows that $M_{\nu}$ must be of the general form in terms of real parameters $A, B, C$: $$M_{\nu} = \left( \begin{array}{ccc} A & ~~~ B & C \\ B & ~~~ A & C \\ C & ~~~ C & ~~~ A + B -C \end{array} \right), \label{ABC}$$ which has eigenvalues $$\begin{aligned} m_1 &=& (A + B - 2C) \nonumber \\ m_2 &=& (A + B + C) \nonumber \\ m_3 &=& (A - B). \label{masses}\end{aligned}$$ The observed mass spectrum corresponds approximately to $\|m_1\| = \|m_2\|$ which requires either $C=0$ or $C=2(A+B)$. For a normal hierarchy, $(A+B)=0$ and $C=0$. For an inverted hierarchy $A=B$ and $C=0$ or $C=4A$. Now we study our minimal $A_4$ model to examine the occurrence of the Majorana matrix Eq.(\[ABC\]) and the eigenvalues Eq.(\[masses\]). Minimal $A_4$ model =================== We assign the leptons to $(A_4, Z_2)$ irreps as follows $$\begin{array}{ccc} \left. \begin{array}{c} \left( \begin{array}{c} \nu_{\tau} \\ \tau^- \end{array} \right)_{L} \\ \left( \begin{array}{c} \nu_{\mu} \\ \mu^- \end{array} \right)_{L} \\ \left( \begin{array}{c} \nu_e \\ e^- \end{array} \right)_{L} \end{array} \right\} L_L (3, +1) & \begin{array}{c} ~ \tau^-_{R}~ (1_1, -1) \\ ~ \mu^-_{R} ~ (1_2, -1) \\ ~ e^-_{R} ~ (1_3, -1) \end{array} & \begin{array}{c} ~ N^{(1)}_{R} ~ (1_1, +1) \\ ~ N^{(2)}_R ~ (1_2, +1) \\ ~ N^{(3)}_{R} ~ (1_3, +1).\\ \end{array} \end{array}$$ The lepton lagrangian is $$\begin{aligned} {\cal L}^{(leptons)}_Y &=& \frac{1}{2} M_1 N_R^{(1)} N_R^{(1)} + M_{23} N_R^{(2)} N_R^{(3)} \nonumber \\ & & + \left[ Y_{1} \left( L_L N_R^{(1)} H_3 \right) + Y_{2} \left( L_L N_R^{(2)} H_3 \right) \right. \nonumber \\ & & \left. + Y_{3} \left( L_L N_R^{(3)} H_3 \right) \right. \nonumber \\ & & \left. + Y_\tau \left( L_L \tau_R H'_3 \right) + Y_\mu \left( L_L \mu_R H'_3 \right) \right. \nonumber \\ & & \left. + Y_e \left( L_L e_R H'_3 \right) \right] + {\rm h.c.}\end{aligned}$$ where $SU(2)$-doublet Higgs scalars are in $H_3(3, +1)$ and $H_3^{'}(3, -1)$. The charged lepton masses originate from $<H_3^{'}> = (\frac{m_{\tau}}{Y_{\tau}},\frac{m_{\mu}}{Y_{\mu}},\frac{m_{e}}{Y_{e}}) $ and are, to leading order, disconnected from the neutrino masses if we choose a flavor basis where the charged leptons are mass eigenstates. The $N_{R}^{i}$ masses break the $L_{\tau} \times L_{\mu} \times L_e$ symmetry but change the charged lepton masses only by very small amounts $\propto Y^2m_i/M_N$ at one-loop level. The right-handed neutrinos have mass matrix $$M_N = \left( \begin{array}{ccc} M_1 & 0 & 0 \\ 0 & 0 & M_{23} \\ 0 & M_{23} & 0 \end{array} \right). \label{MN}$$ We take the VEV of the scalar $H_3$ to be $$<H_3> = (V_1, V_2, V_3), \label{vev}$$ whereupon the Dirac matrix is $$M_D = \left( \begin{array}{ccc} Y_{1} V_1 & ~~~ Y_{2} V_3 & ~~~ Y_3 V_2 \\ Y_1 V_3 & ~~~ Y_2 V_2 & ~~~ Y_3 V_1 \\ Y_1 V_2 & ~~~ Y_2 V_1 & ~~~ Y_3 V_3 \end{array} \right). \label{MD}$$ The Majorana mass matrix $M_{\nu}$ is given by $$M_{\nu} = M_D M_N^{-1} M_D^{T}.$$ Technical details are provided in arXiv:0806.1707. Our conclusion is that the $A_4$ model in a minimal form does favor the normal hierarchy. We have considered a more restrictive model based on $A_4$ than previously considered. The theory has been required to be renormalizable and the Higgs scalar content is the minimum possible. We have required that the neutrino mixing matrix be of the tribimaximal form. We then find that the masses for the neutrinos are highly constrained and can be in a normal, not inverted hierarchy. Most, if not all, previous $A_4$ models in the literature permit higher-order irrelevant non-renormalizable operators and their concomitant proliferation of parameters and hence allow a wide variety if possiblities for the neutrino masses. We believe the renormalizabilty condition is sensible for these flavor symmetries because of the higher predictivity. The next step which is the subject of the rest of this talk is whether the present renornmalizable $A_4$ model can be extended to a renormalizable $T^{'}$ model. It is necessary but not sufficent condition for this that a successful renormalizable $A_4$ model, as presented here, exists. $T^{'}$ symmetry. ================= The first use of the binary tetrahedral group $T{'}$ in particle physics was [@Yang] by Case, Karplus and Yang in 1956 who were motivated to consider gauging a finite $T^{'}$ subgroup of $SU(2)$ in Yang-Mills theory. This led Fairbairn, Fulton and Klink (FFK) in 1964 to make an analysis[@FFK] of $T^{'}$ Clebsch-Gordan coefficients As a flavor symmetry, $T^{'}$ first appeared [@FK1]in 1994 motivated by the idea of representing the three quark families with the third treated differently from the first two. Since $T^{'}$ is the double cover of $A_4$, it was natural to suggest that $T^{'}$ be employed to accommodate quarks and simultaneously the established $A_4$ model building for tribimaximal neutrino mixing. We shall discuss such a $T^{'}$ model with simplifications to emphasize the largest quark mixing, the Cabibbo angle, for which we shall derive an entirely new formula [@FKM; @EFM] as an exact angle. Recall that charged lepton masses arise from the vacuum expectation value $$<H_3^{'}> = \left(\frac{m_{\tau}}{Y_{\tau}},\frac{m_{\mu}}{Y_{\mu}},\frac{m_{e}}{Y_{e}} \right) = ( M_{\tau}, M_{\mu}, M_e ) \label{Hprime}$$ where $M_i \equiv m_i/Y_i$ ($i = \tau, \mu, e$). Neutrino masses and mixings come from the see-saw mechanism and the VEV $$<H_3> = V( 1, -2, 1) \label{VEV}$$ We shall now promote $A_4$ to $T^{'}$ keeping renormalizability and including quarks. Minimal $T^{'}$ model ====================== The left-handed quark doublets $(t, b)_L, (c, d)_L, (u, d)_L$ are assigned under $(T^{'} \times Z_2)$ to $$\begin{array}{cc} \left( \begin{array}{c} t \\ b \end{array} \right)_{L} ~ {\cal Q}_L ~~~~~~~~~~~ ({\bf 1_1}, +1) \\ \left. \begin{array}{c} \left( \begin{array}{c} c \\ s \end{array} \right)_{L} \\ \left( \begin{array}{c} u \\ d \end{array} \right)_{L} \end{array} \right\} Q_L ~~~~~~~~ ({\bf 2_1}, +1) \end{array} \label{qL}$$ and the six right-handed quarks as $$\begin{array}{c} t_{R} ~~~~~~~~~~~~~~ ({\bf 1_1}, +1) \\ b_{R} ~~~~~~~~~~~~~~ ({\bf 1_2}, +1) \\ \left. \begin{array}{c} c_{R} \\ u_{R} \end{array} \right\} {\cal C}_R ~~~~~~~~ ({\bf 2_3}, -1)\\ \left. \begin{array}{c} s_{R} \\ d_{R} \end{array} \right\} {\cal S}_R ~~~~~~~~ ({\bf 2_2}, +1) \end{array} \label{qR}$$ We add only two new scalars $H_{1_1} (1_1, +1)$ and $H_{1_3} (1_3, +1)$ whose VEVs $$<H_{1_1}> = m_t/Y_t ~~~~ <H_{1_3}> = m_b/Y_b \label{H13VEV}$$ provide the $(t, b)$ masses. In particular, no $T^{'}$ doublet ($2_1, 2_2, 2_3$) scalars have been added. This allows a non-zero value only for $\Theta_{12}$. The other angles vanish making the third family stable [^1]. The allowed quark Yukawa and mass terms are $$\begin{aligned} {\cal L}_Y^{(quarks)} &=& Y_t ( \{{\cal Q}_L\}_{\bf 1_1} \{t_R\}_{\bf 1_1} H_{\bf 1_1}) \nonumber \\ && + Y_b (\{{\cal Q}_L\}_{\bf 1_1} \{b_R\}_{\bf 1_2} H_{\bf 1_3} ) \nonumber \\ && + Y_{{\cal C}} ( \{ Q_L \}_{\bf 2_1} \{ {\cal C}_R \}_{\bf 2_3} H^{'}_{\bf 3}) \nonumber \\ && + Y_{{\cal S}} ( \{ Q_L \}_{\bf 2_1} \{ {\cal S}_R \}_{\bf 2_2} H_{\bf 3}) \nonumber \\ && + {\rm h.c.} \label{Yquark}\end{aligned}$$ The use of $T^{'}$ singlets and doublets [^2] for quark families in Eqs.(\[qL\],\[qR\]) permits the third family to differ from the first two and thus make plausible the mass hierarchies $m_t \gg m_b$, $m_b > m_{c,u}$ and $m_b > m_{s,d}$. The Cabibbo angle ================= The nontrivial ($2 \times 2$) quark mass matrices $(c, u)$ and $(s, d)$ will be respectively denoted by $U^{'}$ and $D^{'}$ and calculated using the $T^{'}$ Clebsch-Gordan coefficients of Fairbairn, Fulton and Klink. Dividing out $Y_{{\cal C}}$ and $Y_{{\cal S}}$ in Eq.(\[Yquark\]) gives $U$ and $D$ matrices ($\omega = e^{i \pi/3}$) $$U \equiv \left( \frac{1}{Y_{{\cal C}}}\right) U^{'} = \left(\begin{array}{cc} \sqrt{\frac{2}{3}} \omega^2 M_{\tau} & \frac{1}{\sqrt{3}} M_e \\ - \frac{1}{\sqrt{3}} \omega^2 M_e & \sqrt{\frac{2}{3}} M_{\mu} \end{array} \right) \label{Umatrix}$$ $$D \equiv \left( \frac{1}{Y_{{\cal S}}} \right) D^{'} = \left( \begin{array}{cc} \frac{1}{\sqrt{3}} & - 2 \sqrt{\frac{2}{3}} \omega \\ \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \omega \end{array} \right) \label{Dmatrix}$$ Let us first consider $U$ of Eq.(\[Umatrix\]). Noting that $m_{\tau} > m_{\mu} \gg m_e$ we may simplify $U$ by setting the electron mass to zero, $M_e = 0$. This renders $U$ diagonal leaving free the c, u, $\tau$ and $\mu$ masses. This leaves only the matrix $D$ in Eq.(\[Dmatrix\]) which predicts both $\Theta_{12}$ and $(m_d^2/m_s^2)$. The hermitian square ${\bf {\cal D}} \equiv D D^{\dagger}$ is $${\bf {\cal D}} \equiv D D^{\dagger} = \left( \frac{1}{3} \right) \left( \begin{array}{cc} 9 & - \sqrt{2} \\ - \sqrt{2} & 3 \end{array} \right) \label{DDdagger}$$ which leads by diagonalization to a formula for the Cabibbo angle $$\tan 2\Theta_{12} = \left( \frac{\sqrt{2}}{3} \right) \label{Cabibbo}$$ or equivalently $\sin \Theta_{12} = 0.218..$ close to the experimental value footnote[Experimental results are from PDG2008; see references therein.]{} $\sin \Theta_{12} \simeq 0.227$. Our result of an exact angle for $\Theta_{12}$ can be regarded as on a footing with the tribimaximal values for neutrino angles $\theta_{ij}$. Note that the tribimaximal $\theta_{12}$ presently agrees with experiment within one standard deviation ($1 \sigma$). On the other hand, our analagous exact angle for $\Theta_{12}$ differs from experiment already by $9 \sigma$ which is probably a reflection of the fact that the experimental accuracy for $\Theta_{12}$ is $\sim 0.5\%$ while that for $\theta_{12}$ is $\sim 6\%$. It is thus very important to acquire better experimental data on $\theta_{12}$, $\theta_{23}$ and $\theta_{13}$ to detect their similar deviation from the exact angles predicted by TBM. Our result for $(m_d^2/m_s^2)$ from Eq.(\[DDdagger\]) is exactly $0.288..$ compared to the central experimental value $\simeq 0.003$ in a simplified model whose generalization to an extended scalar sector including $T^{'}$ doublets can avoid $\Theta_{23} = \Theta_{13} = 0$ and thereby change $(m_d^2/m_s^2)$ due to mixing of $(d, s)$ with the $b$ quark. This $T'\times Z_2$ extension of the standard model is an first step to tying the quark and lepton sectors together, providing calculability, and at the same time reducing the number of standard model parameters. The ultimate goal would be to understand the origin of this discrete symmetry. Since gauge symmetries can break to discrete symmetries, and gauge symmetries arise naturally from strings, perhaps there is a clever construction of our model with its fundamental origin in string theory. Summary ======= Renormalizability and simplification of $(A_4 \times Z_2)$ then $(T^{'} \times Z_2)$ models lead to: Cabibbo angle formula $$\tan 2\Theta_{12} = \left( \frac{\sqrt{2}}{3} \right)$$ [999]{} K.M. Case, R. Karplus and C.N. Yang, Phys. Rev. [101,]{} 874 (1956). W.M. Fairbairn, T. Fulton and W.H. Klink, J. Math. Phys. [5,]{} 1038 (1964). P.H. Frampton and T.W. Kephart, Int. J. Math. Phys. [A10,]{} 4689 (1995). [hep-ph/9409330]{}. P.H. Frampton, T.W. Kephart and S. Matsuzaki,\ Phys. Rev. [D78,]{} 073004 (2008). [arXiv:0807.4713 \[hep-ph\]]{}. D.A. Eby, P.H. Frampton and S. Matsuzaki, Phys. Lett. [B671,]{} 386 (2009). [arXiv:0810.4899 \[hep-ph\]]{}. [^1]: As we shall discuss non-vanishing $\Theta_{23}$ and $\Theta_{13}$ are related to $(d, s)$ masses. [^2]: It is discrete anomaly free. We thank the UF-Gainesville group for discussions.
--- abstract: \| We study Coulomb drag in double-layer graphene near the Dirac point. A particular emphasis is put on the case of clean graphene, with transport properties dominated by the electron-electron interaction. Using the quantum kinetic equation framework, we show that the drag becomes $T$-independent in the clean limit, $T\tau \to \infty$, where $T$ is temperature and $1/\tau$ impurity scattering rate. For stronger disorder (or lower temperature), $T\tau \ll 1/\alpha^2$, where $\alpha$ is the interaction strength, the kinetic equation agrees with the leading-order ($\alpha^2$) perturbative result. At still lower temperatures, $T\tau \ll 1$ (diffusive regime) this contribution gets suppressed, while the next-order ($\alpha^3$) contribution becomes important; it yields a peak centered at the Dirac point with a magnitude that grows with lowering $T\tau$. author: - 'M. Schütt' - 'P.M. Ostrovsky' - 'M. Titov' - 'I.V. Gornyi' - 'B.N. Narozhny' - 'A.D. Mirlin' title: Coulomb drag in graphene near the Dirac point --- Frictional drag in double-layer systems consisting of two closely spaced, but electronically isolated conductors is a well established experimental tool for studying the microscopic structure of solids [@roj; @ex1; @ex2; @ex3; @ex4; @ex5; @tut]. In such an experiment a current $I_1$ is passed through one of the conductors (the “active” layer) and the induced voltage drop $V_2$ is measured along the other (“passive”) layer. The ratio of this voltage to the driving current $\rho_D=-V_2/I_1$ (known as the drag coefficient or the transresistivity) is a measure of both the inter-layer interaction [@roj; @ex1] and the microscopic state [@ex2; @ex3; @ex4; @ex5] of the layers. At low temperatures the drag effect is dominated by direct Coulomb interaction between the carriers in the two layers. $ \begin{array}{ccc} \hspace{-0.6cm}\epsfig{file=fig_1_color_plot_1.eps,width=3.5cm} & \hspace{-0.6cm}\epsfig{file=fig_1_color_plot_2.eps,width=3.5cm} & \hspace{-0.5cm}\epsfig{file=fig_1_legend.eps,width=0.8cm} \\ \epsfig{file=fig_1_plot_3_m.eps,width=3.6cm} & \epsfig{file=fig_1_plot_4_m.eps,width=3.6cm} & \end{array}$ The physics of Coulomb drag is well understood if both layers are in the Fermi liquid state [@kor; @fl2]. The electric field in the passive layer is induced by exciting pairs of electron-like and hole-like excitations in a state with finite total momentum. The momentum is transferred from the current-carrying state in the active layer by the inter-layer Coulomb interaction. The inter-layer momentum transfer can be described by the effective relaxation rate $\tau_D^{-1}$. The most basic qualitative features of the drag measurement [@roj; @kor; @fl2] can already be inferred by estimating $\tau_D^{-1}$ with the help of Fermi’s golden rule, where it is crucial to take into account the energy dependence of the density of states (DoS) and/or diffusion coefficient $D$: indeed, the current-carrying states can be characterized by non-zero total momentum only in the case of electron-hole asymmetry. The drag coefficient $\rho_D$ and momentum relaxation rate $\tau_D^{-1}$ can be related using a simple Drude-like model. Consider the phenomenological equations of motion, assuming for simplicity that both layers are characterized by the same carrier density $n$ and effective mass $m$ $$\label{emp} \frac{d}{dt} \begin{pmatrix} {\boldsymbol{j}}_1 \cr {\boldsymbol{j}}_2 \end{pmatrix} = \frac{e^2n}{m} \begin{pmatrix} {\boldsymbol{E}}_1 \cr {\boldsymbol{E}}_2 \end{pmatrix} - \frac{1}{\tau_D} \begin{pmatrix} 1 & -1 \cr -1 & 1 \end{pmatrix} \begin{pmatrix} {\boldsymbol{j}}_1 \cr {\boldsymbol{j}}_2 \end{pmatrix} - \frac{1}{\tau} \begin{pmatrix} {\boldsymbol{j}}_1 \cr {\boldsymbol{j}}_2 \end{pmatrix} ,$$ where ${\boldsymbol{j}}_{1(2)}$ is the average current density in the active (passive) layer, ${\boldsymbol{E}}_{1(2)}$ is the electric field in the two layers, and $\tau$ is the impurity scattering time. Noting that in the drag measurement no net current is allowed to flow in the passive layer ${\boldsymbol{j}}_2=0$, we arrive at the Drude-like formula $$\label{dfd} \rho_D = -\rho_{12} = \left( e^2 n \tau_D / m \right)^{-1}.$$ Combining Eq. (\[dfd\]) with the Fermi’s golden rule estimate for $\tau_D^{-1}$ one can estimate the drag coefficient. More rigorous calculations based on either the diagrammatic perturbation theory [@kor] or the kinetic equation [@fl2] confirm the “Fermi-liquid” result $$\label{fld} \rho_D^{FL} = (\hbar/e^2) A_{12} T^2/(\mu_1\mu_2) ,$$ where $\mu_{1(2)}$ is the chemical potential of the active (passive) layer and $A_{12}$ is determined by the matrix elements of the inter-layer interaction (the precise form of $A_{12}$ as a function of the inter-layer spacing $d$ depends on whether transport in the two layers is ballistic or diffusive [@kor]). Even though the drag coefficient (\[fld\]) is apparently independent of the impurity scattering time $\tau$, transport properties of each individual layer are usually [@roj; @kor] assumed to be dominated by disorder, $\tau\ll\tau_D$. In particular, solving Eq. (\[emp\]) for the resistivity one finds the usual Drude formula. In contrast, the behavior of clean double-layer systems, i.e. with $\tau\gg\tau_D$, is less trivial. In this case, the last term in Eq. (\[emp\]) may be neglected leading to the non-zero result for the single-layer resistivity $$\label{rr} \rho_{11} = - \rho_{12} = \left( e^2 n\tau_D/m \right)^{-1} = \rho_D.$$ Note, that the system is still characterized by the infinite conductivity ($\hat\rho^{-1}=\infty$), as expected for disorder-free conductors on the grounds of Galilean invariance. The physical picture of the drag effect outlined so far is based on the following assumptions: (i) each of the layers is assumed to be in a Fermi-liquid state, which at the very least means $\mu_{1(2)}\gg T$; (ii) electron-electron interaction does not contribute to the transport scattering time; (iii) the inter-layer Coulomb interaction is assumed to be weak enough, $\alpha = e^2/(\hbar v_F)\ll 1$, such that $\rho_D$ is determined by the lowest-order perturbation theory [@kor]. Lifting one or more of the above assumptions leads to significant changes in the drag effect [@ex2; @ex3; @ex4; @tut; @tu2; @ge2]. In this Letter we focus on the system of two parallel graphene sheets [@tut; @tu2; @ge2; @me1; @dsa; @csn; @ds2; @tud; @us1; @pet; @per; @sch; @glz], which offers a great degree of control over the microscopic structure of the two layers. Indeed, using hexagonal boron nitride as a substrate [@ge2; @ge1], one can decrease disorder strength in the system and reach the regime, where transport properties of the two layers are dominated by electron-electron interaction, $\tau\gg\tau_{ee}$. Moreover, the carrier density can be electrostatically controlled allowing one to scan a wide range of chemical potentials from the Fermi liquid regime to the Dirac point. While inapplicable to massless fermions in graphene, the equations of motion (\[emp\]) provide an expectation of non-zero resistance in the case of the ultra-clean system. Below, we use the quantum kinetic equation (QKE) approach [@kas; @kin] to derive hydrodynamic equations [@ryz] that generalize Eq. (\[emp\]) for interacting Dirac fermions in graphene. Solving these equations (or equivalently, the QKE) we confirm that the system of two ultra-clean graphene sheets is indeed characterized by a non-zero, but degenerate resistance matrix whose elements satisfy Eq. (\[rr\]), with $\rho_D$ shown in Fig. \[fr1\]. [Kinetic equation. —]{} We now briefly outline the derivation of the QKE for double-layer graphene structures and its solution in the ballistic regime (see Supplemental Material [@sup]). Consider an infinite sample in an infinitesimal, homogeneous electric field ${\boldsymbol{E}}_1$ applied to the active layer. The response of the system to the field can be described by the small non-equilibrium corrections $h_{1(2)}$ to the Fermi distribution functions defined by $$n_i(\epsilon,\hat{\mathbf{v}})=n_F^{(i)}(\epsilon) + T \frac{\partial n_F^{(i)}(\epsilon)}{\partial \epsilon} h_i(\epsilon,\hat{\mathbf{v}}), \label{h}$$ where the eigenstates of the Dirac Hamiltonian $H = v {\boldsymbol{\sigma}} {\boldsymbol{p}}$ are labeled [@sup] by their energy $\epsilon$ and the velocity unit vector $\hat{\mathbf{v}}$; the momentum of the particle is ${\boldsymbol{p}} = \epsilon \hat{\mathbf{v}}/v$. Small corrections $h_{1(2)}$ can be found by linearizing the QKE [@fn2] $$\begin{aligned} && \frac{\partial h_1}{\partial t}+\frac{e{\boldsymbol{E}}_1{\boldsymbol{v}}}{T}= -\frac{h_1}{\tau} + I_{11}\{h_1\} + I_{12}\{h_1,h_2\}, \nonumber\\ && \nonumber\\ && \frac{\partial h_2}{\partial t}=-\frac{h_2}{\tau} + I_{22}\{h_2\} + I_{21}\{h_2,h_1\}, \label{ke1}\end{aligned}$$ where the linearized pair-collision integrals are given by $$\begin{aligned} \label{i} && I_{ij}= - \int d2\ d3\ d4\ W^{ij}(h_{i,1}-h_{i,2}+h_{j,3}-h_{j,4}), \nonumber\\ && \nonumber\\ && W^{ij}= \delta({\boldsymbol{p}}_1-{\boldsymbol{p}}_2+{\boldsymbol{p}}_{3}-{\boldsymbol{p}}_{4}) \ \delta(\epsilon_1-\epsilon_2+\epsilon_{3}-\epsilon_{4}) \nonumber\\ && \nonumber\\ && \quad \times \frac{ \cosh\frac{\epsilon_1-\mu_i}{2T}}{2 \cosh\frac{\epsilon_2-\mu_i}{2T} \cosh\frac{\epsilon_3-\mu_j}{2T}\cosh\frac{\epsilon_4-\mu_j}{2T}} K^{ij}_{1,2;3,4},\end{aligned}$$ and we have used short-hand notations $h_{i,a}=h(\epsilon_a,\hat{\mathbf{v}}_a)$, $da=\nu(\epsilon_a) d\hat{\mathbf{v}}_ad\epsilon_a$, with $a=1,2,3,4$. The kernel $$K^{ij}_{1,2;3,4}=\|U^{ij}({\boldsymbol{p}}_1-{\boldsymbol{p}}_2)\|^2 \frac{1+\hat{\mathbf{v}}_1\hat{\mathbf{v}}_2}{2} \frac{1+\hat{\mathbf{v}}_3\hat{\mathbf{v}}_4}{2}, \label{k}$$ contains the interaction matrix element describing the two-particle scattering $1\to 2 $ and $3\to 4$ and the corresponding Dirac factors. Here we take into account only the Hartree interaction term: there is no exchange interaction between the layers, whereas within the layers the Hartree term dominates in the large-$N$ limit ($N$ is the number of electron flavors; physically, $N=4$ due to spin and valley degeneracy). The peculiarity of the inelastic scattering in the Dirac spectrum is two-fold. First, since the velocity ${\boldsymbol{v}}=v^2{\boldsymbol{p}}/\epsilon$ is independent of the absolute value of the momentum, total momentum conservation does not prevent velocity (or current) relaxation. As a result, the intralayer collision integral $I_{ij}$ yields a non-zero transport relaxation rate due to electron-electron scattering. Second, the scattering of particles with almost collinear momenta is enhanced since the momentum and energy conservation laws coincide for collinear scattering. This restricts the kinematics [@kas; @kin; @po2] of the Dirac fermions leading to the singularity in the collision integral. This singularity leads to the fast thermalization of particles within a given direction, which justifies the Ansatz: $$\label{Ansatz} h_i(\epsilon,\hat{\bf{v}})=\left(\chi_v^{(i)}+\chi_p^{(i)} \; \epsilon/T\right) e{\boldsymbol{E}}{\boldsymbol{v}}/T^2.$$ The Ansatz (\[Ansatz\]) retains the only two modes for which the collision integral $I_{ij}$ is not singular: the “momentum mode” $\chi_p^{(i)}$, which nullifies the collision integral due to momentum conservation, and the “velocity mode” $\chi_v^{(i)}$, which nullifies $I_{ij}$ in the case of collinear scattering. The same kinematic restrictions lead to fast uni-directional thermalization between the layers. This allows us to set $\chi_p^{(1)}=\chi_p^{(2)}$, and hence reduce the QKE for the double-layer setup to a $3\times3$ matrix equation. Consider for simplicity the case of identical layers (for the more general case of $\mu_1\ne \mu_2$ see Supplemental Material [@sup]). Integrating the reduced QKE over the energies, we arrive at the set of steady-state hydrodynamic equations in terms of the particle currents $${\boldsymbol{J}}_i = - N T \int d\epsilon \nu(\epsilon) \frac{\partial n_F^{(i)}}{\partial \epsilon} \int d\hat{\mathbf{v}} {\boldsymbol{v}} h_i(\epsilon,\hat{\bf{v}}),$$ and the total momentum ${\boldsymbol{P}}=e\epsilon_0C_1^2({\boldsymbol{E}}_1+{\boldsymbol{E}}_2)\tau$: $$\label{heq} e\epsilon_0 \begin{pmatrix} {\boldsymbol{E}}_1 \cr {\boldsymbol{E}}_2 \end{pmatrix} = \left[ \frac{1}{\tau} + \widehat{\cal I}_{ee} - \widehat{\cal I}_D\right] \begin{pmatrix} {\boldsymbol{J}}_1 \cr {\boldsymbol{J}}_2 \end{pmatrix} +\left[\frac{1}{\tau_{D}}-\frac{1}{\tau_{ee}}\right] \begin{pmatrix} {\boldsymbol{P}} \cr {\boldsymbol{P}} \end{pmatrix},$$ where $\widehat{\cal I}_{ee(D)}=[(\hat\sigma_0+\hat\sigma_1)C_1^2+2\hat\sigma_{0(1)}C_2]/\tau_{ee(D)}$, the intra- and inter-layer electron-electron transport scattering rates are (${\cal W}^{ij} = W^{ij} \nu_1/\cosh^2[(\epsilon_1-\mu_i)/(2T)]$) $$\begin{aligned} && \frac{1}{\tau_{D}}=\frac{1}{4T\epsilon_0C_2} \int \prod_{a=1}^4 da {\cal W}^{12} \left({\boldsymbol{v}}_1-{\boldsymbol{v}}_2\right)\left({\boldsymbol{v}}_4-{\boldsymbol{v}}_3\right), \nonumber\\ && \nonumber\\ && \frac{1}{\tau_{ee}}=\frac{1}{8T\epsilon_0C_2} \int \prod_{a=1}^4 da \left[ {\cal W}^{11} \left({\boldsymbol{v}}_1-{\boldsymbol{v}}_2+{\boldsymbol{v}}_3-{\boldsymbol{v}}_4\right)^2 \right. \nonumber\\ && \nonumber\\ && \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \left. + 2 {\cal W}^{12}\left({\boldsymbol{v}}_1-{\boldsymbol{v}}_2\right)^2 \right],\end{aligned}$$ and $\sigma_k$ are the Pauli matrices in “layer space”. The coefficients $C_{1(2)}$ represent the average energy and energy variation, while $\epsilon_0= 2T {\cal J}\{1\}/N$ is a typical energy: $$\begin{aligned} && C_1 = \frac{\left\langle \epsilon \right\rangle_\epsilon}{T}\sim\frac{\mu}{T}, \quad C_2 = \frac{\left\langle\epsilon^2\right\rangle_\epsilon- \left\langle \epsilon \right\rangle^2_\epsilon}{T^2} \sim const, \nonumber\\ && \nonumber\\ && {\cal J}\{\dots\}= -\frac{v^2}{T}\int d\epsilon \nu(\epsilon) \frac{\partial n_F}{\partial \epsilon} \dots , \quad \left\langle\dots\right\rangle_\epsilon = \frac{{\cal J}\{\dots\}}{{\cal J}\{1\}}.\end{aligned}$$ The hydrodynamic equations (\[heq\]) generalize the equations of motion (\[emp\]) to the case of Dirac fermions in graphene. The kinematic peculiarity of Dirac fermions manifests itself in the appearance of the total momentum, which entangles the electric fields in the two layers. Solving the hydrodynamic equations (\[heq\]) we find $$\label{gr} \rho_D =\frac{\hbar}{e^2}\frac{C_2}{\epsilon_0} \frac{(\tau\tau_D)^{-1}+C_1^2\left[\tau^{-2}_{ee}-\tau^{-2}_{D}\right]} {\tau^{-1}+C_1^2\left[\tau_{ee}^{-1}-\tau_D^{-1}\right]}.$$ For a clean system, the resistivity matrix is degenerate and the drag coefficient is given by $$\rho_D (\tau\to\infty)=(\hbar/e^2) (C_2/\epsilon_0) \left(\tau_D^{-1}+\tau_{ee}^{-1}\right),$$ which remains non-zero $\rho_D\sim(\hbar/e^2)\alpha^2$ even at the Dirac point $\mu=0$, where it is determined by $\tau_{ee}^{-1}\sim\alpha^2T$ (to the second order in the inter-layer interaction $\tau_D^{-1}(\mu=0)=0$, while the third-order contribution $\tau_D^{-1}(\mu=0)\sim\alpha^3T$ is subleading; the latter is expected to dominate the effect for sufficiently strong disorder, see below). Equation (\[gr\]) gives the general expression for the drag coefficient in the ballistic regime based on the solution of the QKE (\[ke1\]). For arbitrary parameter values this expression is to be evaluated numerically (see Fig. \[fr1\] for the numerical results). Analytical expressions can be obtained for various limiting cases (summarized in Fig. \[pd\]). Below, we discuss the asymptotic behavior of $\rho_D$ for the case of two inequivalent layers [@sup] focusing on the experimentally relevant case [@tut; @tu2; @ge2] $Td/v<1$ and analyzing the evolution of $\rho_D$ with increasing disorder strength. [Ballistic regime. —]{} For weak disorder $\alpha^2T\tau\gg 1$ (or $\tau^{-1}\ll\tau_{ee}^{-1}$) and neglecting the third-order contribution to $\tau_D^{-1}$, we find for $\rho_D$ near the Dirac point $$\label{r1dp} \rho_D (\mu_i\ll T) \approx 2.87 \frac{h}{e^2} \; \alpha^2 \frac{\mu_1\mu_2}{\mu_1^2+\mu_2^2+0.49T/(\alpha^2\tau)},$$ where $\tau_D^{-1}\sim\alpha^2\mu_1\mu_2/T$, $\tau_{ee}^{-1}\sim\alpha^2T$, $\epsilon_0\sim T$, and $C_2\sim 1$. The value of $\rho_D$ precisely at the Dirac point depends on the experimental set-up. For clean samples, if one of the chemical potentials remains non-zero, while the other is scanned through the Dirac point [@tut], then $\rho_D(\mu_1=0, \mu_2\ne 0)=0$, similar to Ref. . On the contrary, if both chemical potentials are driven through the Dirac point simultaneously [@ge2], then Eq. (\[r1dp\]) predicts a non-vanishing value of $\rho_D(\mu_1=\pm\mu_2=0)\ne 0$, see Fig. \[fr1\]. For intermediate disorder strength $\alpha^2 T \ll \tau^{-1} \ll T$ the applicability region of the QKE overlaps with that of the conventional perturbation theory developed in Ref. and we recover perturbative results, see Fig. \[pd\]. For even stronger disorder (or at low temperatures) $T\tau\ll 1$ the electron motion becomes diffusive. In this case the kinematic restrictions are relaxed and the Ansatz (\[Ansatz\]) is no longer justified. However, in this regime, the perturbative approach is applicable and allows for a standard description of the diffusive transport. [Diffusive regime. —]{} The lowest-order perturbative calculation [@kor] amounts to evaluation of the Aslamasov-Larkin-type diagram for the drag conductivity given by $$\label{sd} \sigma^{\alpha\beta}_D = \frac{1}{16\pi T} \sum_{{\boldsymbol{q}}} \int \frac{d\omega}{\sinh^2\frac{\omega}{2T}} \Gamma_1^\beta(\omega, {\boldsymbol{q}}) \Gamma_2^\alpha(\omega, {\boldsymbol{q}}) \| {\cal D}^R_{12} \|^2,$$ where ${\cal D}^R_{12}$ is the retarded propagator of the inter-layer interaction and $\Gamma_a^\alpha(\omega, {\boldsymbol{q}})$ is the non-linear susceptibility \[in fact, all previous studies of the Coulomb drag in graphene [@me1; @dsa; @csn; @ds2; @us1; @tud; @pet; @per; @sch] focused on Eq. (\[sd\])\]. In the diffusive regime, $\Gamma_a^\alpha(\omega, {\boldsymbol{q}})$ can be found using the Ohm’s law and the continuity equation [@nas] ${\boldsymbol{\Gamma}} = e {\boldsymbol{q}} (\partial\sigma/\partial n) {\rm Im} \Pi^R$. All microscopic details are now encoded in the diffusion coefficient and the density dependence of the single-layer conductivity $\sigma$. Close to the Dirac point $\mu\ll T\ll \tau^{-1}$ the derivative $\partial\sigma/\partial n \sim n v^2\tau^2$ (independently of the precise nature of impurities). After this the evaluation of Eq. (\[sd\]) is rather standard (except that, in contrast to Ref. , the Thomas-Fermi screening length is much longer than the inter-layer spacing $\varkappa d\ll 1$) and yields $$\label{r3dp} \rho_D^{(2)}\left(\mu_i\ll T \ll\tau^{-1}\right) \sim (\hbar/e^2) \alpha^2 \mu_1\mu_2 T \tau^3.$$ This result vanishes at the Dirac point as a consequence of the electron-hole symmetry. The importance of the electron-hole asymmetry for the Coulomb drag follows from Eq. (\[sd\]): the non-linear susceptibility can be thought of as a measure of the asymmetry. However, Eq. (\[sd\]) is only the lowest-order contribution to $\sigma_D$. Under standard assumptions of the Fermi-liquid behavior in the two layers ($\mu\gg v/d \gg T$, $\mu\tau\gg 1$), this contribution indeed dominates the observable effect. On the contrary, in the vicinity of the Dirac point in graphene, the next-order contribution $\rho_D^{(3)}$ [@lak] becomes important since it is insensitive to the electron-hole symmetry and thus does not vanish at the Dirac point. The explicit results of Ref. were obtained in the usual limit $\varkappa d\gg 1$. Extending these calculations to the opposite case $\varkappa d\ll 1$ we find close to the Dirac point $$\label{rkl} \rho_D^{(3)}\left(\mu_i\ll T\ll\tau^{-1}\ll\alpha^{-2}T\right) \sim (\hbar/e^2)\alpha^3(T\tau)^{-3/2},$$ and $\rho_D^{(3)}\sim \hbar/e^2$ for $\tau^{-1}\gg\alpha^{-2}T$. Away from the Dirac point this contribution decays as a function of the chemical potential $\rho_D^{(3)}(\mu\tau\gg{\rm max}[1,\alpha^{-1}(T\tau)^{1/2}])\sim(\hbar/e^2)(\mu\tau)^{-3}$ and rapidly becomes subleading. As a result, $\rho_D^{(3)}$ is only detectable at low $T$ and $\mu$, see Fig. \[rdif\]. While estimating $\rho_D^{(3)}$ at the Dirac point, we assume the single-layer conductivity $\sigma\sim e^2/h$ discarding localization effects. Indeed, experiments on high-quality samples show $T$-independent $\sigma$ down to $T=30$ mK [@kim], that can be explained by the specific character of disorder in graphene [@ogm]. [Summary. —]{} We have studied Coulomb drag in double-layer graphene structures. By using the QKE formalism we have shown that for weak disorder (or high $T$; ballistic regime) $\rho_D$ near the Dirac point is given by Eq. (\[r1dp\]), see also Fig. \[fr1\], which is consistent with Ref. . For $\alpha^2 T\tau\ll 1$, the solution of the QKE agrees with the perturbative calculation of Ref. . For even stronger disorder (or lower $T$; diffusive regime) subleading third-order contribution dominates the effect in qualitative agreement with the experimentally observed peak at the Dirac point at low temperatures [@ge2]. A possible alternative origin of the low-$T$ peak at the Dirac point is a collective state of the double-layer system, either due to strong interaction [@vin] (not explored here) or correlated disorder [@gor; @lev] (discussed in Supplemental Material [@sup]). [Acknowledgements. —]{} We thank A.K. Geim, K.S. Novoselov, and L. Ponomarenko for communicating their experimental results prior to publication. We are grateful to J. Schmalian and L. Fritz for stimulating discussions. This research was supported by the Center for Functional Nanostructures and SPP 1459 “Graphene” of the Deutsche Forschungsgemeinschaft (DFG), and by BMBF. M.T. is grateful to KIT for hospitality. After this work was completed, we became aware of a closely related studies by J. Lux and L. Fritz [@lux]. Coulomb drag in graphene near the Dirac point: Supplemental Material ==================================================================== 1. Kinetic equation approach ============================ In this section we provide details of derivation of the resistivity tensor in double layer graphene from the kinetic equation approach. A. Kinetic equation ------------------- Eigenstates of the massless Dirac Hamiltonian $H = v \bm{\sigma} \mathbf{p}$ are characterized by the values of momentum $\mathbf{p}$ and the discrete variable $\chi = \pm 1$ indexing conduction and valence bands. In this representation, energy and velocity are $\epsilon = \alpha v \|\mathbf{p}\|$ and $\mathbf{v} = \chi v \mathbf{p}/p$. It is, however, more convenient to label the eigenstates by their energy $\epsilon$ and the unit velocity vector $\hat{\mathbf{v}}$. The momentum of the particle is then $\mathbf{p} = \epsilon \hat{\mathbf{v}}/v$ and the normalization of the states reads $$\int \frac{\|\epsilon\|\, d\epsilon\, d\hat{\mathbf{v}}}{(2\pi v)^2}\; \| \epsilon, \hat{\mathbf{v}} \rangle \langle \epsilon, \hat{\mathbf{v}} \| = 1.$$ We consider an infinite double-layer graphene sample (layers $1$ and $2$) in a homogeneous electric field $\mathbf{E}_1$ applied to the active layer $1$. Assuming weak electric field, we start with the linearized kinetic equation: $$\begin{aligned} &&\frac{\partial h_1}{\partial t}+\frac{e\mathbf{E}_1\mathbf{v}}{T}= -\frac{h_1}{\tau} + I_{11}\{h_1\} + I_{12}\{h_1,h_2\}, \nonumber\\ && \nonumber\\ && \frac{\partial h_2}{\partial t}=-\frac{h_2}{\tau} + I_{22}\{h_2\} + I_{21}\{h_2,h_1\}. \label{kineq1} \end{aligned}$$ Here the nonequilibrium correction $h_i$ to the Fermi distribution function is defined by $$n_i(\epsilon,\hat{\mathbf{v}})=n_F(\epsilon) + T \frac{\partial n_F(\epsilon)}{\partial \epsilon} h_i(\epsilon,\hat{\mathbf{v}}), \label{eq3}$$ and $I_{ij}$ is the linearized pair-collision integral: $$I_{ij} = - N \int d\epsilon_2d\epsilon_3d\epsilon_4 \int d\hat{\mathbf{v}}_2 d\hat{\mathbf{v}}_3 d\hat{\mathbf{v}}_4 \nu(\epsilon_2)\nu(\epsilon_3)\nu(\epsilon_4) W^{ij}(1,3;2,4)\ \left[(h_{i}(\epsilon_1,\mathbf{v}_1)-h_{i}(\epsilon_2,\mathbf{v}_2) +h_{j}(\epsilon_3,\mathbf{v}_3)-h_{j}(\epsilon_4,\mathbf{v}_4)\right], \label{eq2}$$ $$W^{ij}(1,2;3,4)=\delta(\mathbf{p}_1-\mathbf{p}_2+\mathbf{p}_{3}-\mathbf{p}_{4}) \ \delta(\epsilon_1-\epsilon_2+\epsilon_{3}-\epsilon_{4})\ \frac{\cosh\frac{\epsilon_1-\mu_i}{2T}} {2 \cosh\frac{\epsilon_2-\mu_i}{2T}\cosh\frac{\epsilon_3-\mu_j}{2T}\cosh\frac{\epsilon_4-\mu_j}{2T}} K^{ij}(1,2;3,4). \label{eq4}$$ Here $\nu(\epsilon)$ is the density of states for one of $N$ flavors (per spin and per valley in graphene, where $N=4$). We assume formally the large $N$ limit and neglect the intralayer exchange interaction. We further assume that the scattering does not mix flavors (i.e., we neglect the intervalley scattering due to Coulomb interaction): states $1(3)$ and $2(4)$ belong to the same flavor, which gives the overall factor $N$. The kernel $$K^{ij}(1,2;3,4)=\|M^{ij}\|^2 \frac{1+\hat{\mathbf{v}}_1\hat{\mathbf{v}}_2}{2}\frac{1+\hat{\mathbf{v}}_3\hat{\mathbf{v}}_4}{2} \label{Kernel}$$ contains the interaction matrix element $M_{ij}$ describing the collision of two particles $1\to 2 $ and $3\to 4$ and the corresponding Dirac factors. Within the Golden-rule approximation, this matrix element is given by the Fourier component of the interaction potential: $$M_{ij}^{(1)}=U_{ij}^{(0)}(\mathbf{p}_1-\mathbf{p}_2),$$ where $$\hat{U}^{(0)}(q)= V_0(q) \begin{pmatrix} 1 & e^{-q d}\\ e^{-q d} & 1 \end{pmatrix},$$ with $$V_0(q)=\frac{2\pi e^2}{q}.$$ Further, one can generalize the collision integral to the case of the RPA-screened interaction. Then $$M_{ij}=U_{ij}^{\text{RPA}}(\mathbf{p}_1-\mathbf{p}_2, vp_1-vp_2),$$ where $$\begin{aligned} \hat{U}^{\text{RPA}}(\mathbf{q}, \omega) &=&\frac{V_0(q)}{\left[1+V_0(q)\Pi_1(q,\omega)\right] \left[1+V_0(q)\Pi_2(q,\omega)\right]-e^{-2qd}V_0^2(q)\Pi_1(q,\omega)\Pi_2(q,\omega)} \nonumber\\ && \nonumber\\ &\times& \begin{pmatrix} 1+V_0(q)\Pi_2(q,\omega)\left(1-e^{-2qd}\right) & e^{-q d} \cr e^{-q d} & 1+V_0(q)\Pi_1(q,\omega)\left(1-e^{-2qd}\right) \end{pmatrix}, \label{RPA}\end{aligned}$$ where $\Pi_i(q,\omega)$ is the polarization operator in layer $i$. We will focus on the experimentally relevant case of closely located layers, $Td/v\ll 1$. Furthermore, here we will restrict our consideration to the case of relatively low concentrations, such that $\mu d/v\ll 1$ (the situation with large interlayer distance will be considered in detail elsewhere). Under these conditions we can set $d=0$ in the interaction matrix elements so that the intralayer and interlayer interactions are just the same. We further assume that the interaction coupling constant is small $$\alpha=\frac{e^2}{v}\ll 1.$$ For simplicity, we treat impurity scattering within the relaxation time approximation with an energy-independent transport time $\tau$. Generalization to the more realistic case of Coulomb impurities with an energy-dependent transport time will be discussed elsewhere. B. Collinear-scattering singularity ----------------------------------- The momentum and energy conservation establishes severe kinematic restrictions on the scattering in systems with linear spectrum [@kas]. This can be easily seen, when one rewrites the product of delta-functions in Eq. (\[eq4\]) as integrals over $\mathbf{q}=\mathbf{p}_1-\mathbf{p}_2$, $\omega=\epsilon_1-\epsilon_2$, and two angles $\phi_{1(3)}$ between $\mathbf{q}$ and $\mathbf{p}_{1(3)}$, respectively: $$\begin{aligned} \delta(\mathbf{v}_1\epsilon_1-\mathbf{v}_2\epsilon_2+\mathbf{v}_{3}\epsilon_{3}-\mathbf{v}_{4}\epsilon_{4}) &=&\int \frac{d^2 q}{(2\pi)^2} \delta(\mathbf{v}_1\epsilon_1-\mathbf{v}_2\epsilon_2-\mathbf{q}) \delta(\mathbf{v}_{3}\epsilon_{3}-\mathbf{v}_{4}\epsilon_{4}+\mathbf{q}),\nonumber \\ \delta(\epsilon_1-\epsilon_2+\epsilon_{3}-\epsilon_{4})&=&\int_\infty^\infty d\omega \delta(\epsilon_1-\epsilon_2-\omega)\delta(\epsilon_{3}-\epsilon_{4}+\omega),\nonumber \\ \label{qw} \end{aligned}$$ This allows one to integrate out $\mathbf{p}_2$ and $\mathbf{p}_4$ ($\mathbf{v}_{2,4}$ and $\epsilon_{2,4}$) in the collision integral, leading to the product $$\delta(\epsilon_1-\omega-\sqrt{\epsilon_1^2+q^2v^2-2\epsilon_1 qv \cos\phi_1}) \delta(\epsilon_3+\omega-\sqrt{\epsilon_3^2+q^2v^2+2\epsilon_3 qv \cos\phi_3}),$$ which sets $$\begin{aligned} \epsilon_1-\omega=v\|\mathbf{p}_1-\mathbf{q}\| \ &\Rightarrow & \ \cos\phi_1=\frac{q^2v^2-\omega^2+2\epsilon_1\omega}{2\epsilon_1 q v}, \\ \epsilon_3+\omega=\|\mathbf{p}_3+\mathbf{q}\| \ &\Rightarrow & \ \cos\phi_3=\frac{\omega^2-q^2v^2+2\epsilon_3\omega}{2\epsilon_3 q v}. \label{eq11} \end{aligned}$$ When calculating the scattering rates using the collision integral (\[eq2\]), the angular integration over $\phi_1$ and $\phi_3$ removes the delta-functions, producing the factor $$\begin{aligned} \frac{\epsilon_1-\omega}{\epsilon_1 q v \|\sin\phi_1\|} \frac{\epsilon_3+\omega}{\epsilon_3 q v \|\sin\phi_3\|} &=&\frac{4(\epsilon_1-\omega)(\epsilon_3+\omega)} {(q^2v^2-\omega^2)\sqrt{[(2\epsilon_1-\omega)^2-q^2v^2][(2\epsilon_3+\omega)^2-q^2v^2]}}. \label{eq16} \end{aligned}$$ In combination with the Dirac factors from Eq. (\[Kernel\]), $$\begin{aligned} (1+\mathbf{v}_1\mathbf{v}_2)(1+\mathbf{v}_3\mathbf{v}_4) =\frac{(2\epsilon_1-\omega)^2-q^2v^2}{2 \epsilon_1(\epsilon_1-\omega)}\ \frac{(2\epsilon_3+\omega)^2-q^2v^2}{2 \epsilon_3(\epsilon_3+\omega)}, \label{eq14} \end{aligned}$$ this yields $$\frac{1}{q^2v^2-\omega^2}\frac{\sqrt{(2\epsilon_1-\omega)^2-q^2v^2}}{\epsilon_1}\ \frac{\sqrt{(2\epsilon_3+\omega)^2-q^2v^2}}{\epsilon_3}.$$ Therefore, further integration over $q$ or $\omega$ generically produces a logarithmic divergence at $\omega=\pm qv$, which stems from the collinear scattering $\phi_3=\phi_1=0$ or $\pi$, see Eq. (\[eq11\]) at the light cone. This divergence reflects the fact that for a linear spectrum the momentum and energy conservation laws coincide in the “one-dimensional” collinear case. Note that this enhancement of the collinear scattering is not restricted to the case of undoped graphene. In order to regularize this divergence, one has to go beyond the Golden-rule level and take into account the screening of the interaction (which in the clean case is perfect exactly on the light cone) and renormalization of the spectrum due to interaction (leading to nonlinear corrections). These mechanisms lead to the appearance of a large factor $\|\ln(\alpha)\|\gg 1$ in generic relaxation rates in graphene. In disordered graphene, this singularity is also cut off by disorder-induced broadening of the momentum-conservation delta-function. C. Ansatz --------- The singularity in the collinear scattering in graphene leads to the fast thermalization of particles within given direction within each of the layers. Clearly, in an external electric field $\mathbf{E}$, the linearized nonequilibrium correction to the distribution function is proportional to the driving term $\mathbf{E v}$. Therefore, the nonequilibrium correction $$h_i(\epsilon,\mathbf{v})=\chi(\epsilon) \frac{e\mathbf{E v}}{T^2}$$ is characterized by some function of energy $\chi(\epsilon)$. One can formally expand this function in $\epsilon/T$: $$\chi(\epsilon)=\sum_{n=0}^\infty\chi_n (\epsilon/T)^n.$$ The action of the collision integral (which contains the combination $h_1-h_2+h_3-h_4$) on this function generates the equilibration rate in all terms except for $n=0,1$. Indeed, for $n=0$, the combination $$h^{(0)}_1-h^{(0)}_2+h^{(0)}_3-h^{(0)}_4 \propto \chi_0(\mathbf{v}_1-\mathbf{v}_2+\mathbf{v}_3-\mathbf{v}_4)$$ contains the differences of velocities. This cancels the kinematics-induced divergency $\propto \|\mathbf{v}_1-\mathbf{v}_2+\mathbf{v}_3-\mathbf{v}_4\|^{-1}$. For $n=1$ the combination $$\begin{aligned} h^{(1)}_1-h^{(1)}_2+h^{(1)}_3-h^{(1)}_4 \propto \chi_1(\epsilon_1\mathbf{v}_1-\epsilon_2\mathbf{v}_2+\epsilon_3\mathbf{v}_3-\epsilon_4\mathbf{v}_4) = \chi_1(\mathbf{p}_1-\mathbf{p}_2+\mathbf{p}_3-\mathbf{p}_4)\end{aligned}$$ contains the change of the total momentum of two colliding particles, which is exactly the argument of the momentum-conservation delta-function. All other contributions with $n>1$ produce a relaxation rate enhanced by the collinear scattering. The corresponding values of $\chi_n$ are therefore strongly suppressed compared to $\chi_0$ and $\chi_1$, which justifies the following Ansatz [@kas]: $$\label{EqAnsatz} h_i(\epsilon,\mathbf{v})=\left(\chi_\mu^{i}+\chi_T^{i}\frac{\epsilon-\mu}{T}\right) \frac{e\mathbf{E}\mathbf{v}}{T^2} = \left(\chi_0^{i}+\chi_1^{i}\frac{\epsilon}{T}\right) \frac{e\mathbf{E}\mathbf{v}}{T^2} \equiv \left(\chi_v^{i}+\chi_p^{i}\frac{\epsilon}{T}\right) \frac{e\mathbf{E}\mathbf{v}}{T^2}.$$ This correction to the distribution function contains only the two modes (proportional to velocity and momentum and characterized for each layer by the two constants $\chi_v=\chi_0$ and $\chi_p=\chi_1$, respectively), that nullify the collision integral in the case of collinear scattering. The notation $\chi_\mu$ and $\chi_T$ is chosen to emphasize that, after the linearization with respect to $\mathbf{E}$, these quantities reflect the angular-dependent corrections to the chemical potential and temperature, respectively, in the direction-equilibrated distribution function: $$\begin{aligned} n(\epsilon,\hat{\mathbf{v}})&=&\frac{1}{1+\exp\left[\dfrac{\epsilon-\mu(\hat{\mathbf{v}})}{2 T(\hat{\mathbf{v}})}\right]} \simeq n_F(\epsilon)-\frac{\partial n_F(\epsilon)}{\partial \epsilon} \left[\frac{\delta\mu(\hat{\mathbf{v}})}{2T}+(\epsilon-\mu)\frac{\delta T(\hat{\mathbf{v}})}{2T^2}\right] \\ &=&n_F(\epsilon)-\frac{1}{2T}\frac{\partial n_F(\epsilon)}{\partial \epsilon} \left\{\left[\delta\mu(\hat{\mathbf{v}})-\frac{\mu}{T}\delta T(\hat{\mathbf{v}})\right] +\frac{\epsilon}{T}\delta T(\hat{\mathbf{v}})\right\}.\end{aligned}$$ The Ansatz Eq. (\[EqAnsatz\]) greatly simplifies the solution of the kinetic equation, replacing the integral equation by a matrix one. Furthermore, the same kinematics-induced singularity in the collinear scattering as in $I_{ii}$ appears in also the intralayer collision integrals $I_{ij}$. This implies fast unidirectional thermalization between the layers. Therefore, we set $\chi_T^{(1)}=\chi_T^{(2)}$, and hence reduce the kinetic equation for the double-layer setup to a $3\times3$ matrix equation (“three-mode approximation”). D. Hydrodynamic equations ------------------------- In each of the two layers, we introduce the particle currents (the total velocities) $$\mathbf{J}_i = - N T \int d\epsilon\ \nu(\epsilon) \frac{\partial n_F^{(i)}}{\partial \epsilon} \int d\hat{\mathbf{v}}\ \mathbf{v} \, h_\alpha(\epsilon,\mathbf{v}),$$ and energy currents (or, equivalently, the total momenta) $$\mathbf{P}_i = - N \int d\epsilon\ \nu(\epsilon)\ \epsilon \ \frac{\partial n_F^{(i)}}{\partial \epsilon} \int d\hat{\mathbf{v}}\ \mathbf{v} \, h_i(\epsilon,\mathbf{v}),$$ and substitute the Ansatz, Eq. \[EqAnsatz\], into these expressions, which yields $$\begin{aligned} \mathbf{J}_i &=& \frac{N}{2} \left[ A_0^{(i)} \chi_0^{(i)} + A_1^{(i)} \chi_1^{(i)}\right] \mathbf{E} \\ \mathbf{P}_i &=& \frac{N}{2} \left[ A_1^{(i)} \chi_0^{(i)} + A_2^{(i)} \chi_1^{(i)}\right] \mathbf{E}.\end{aligned}$$ Here $$A_n^\alpha= -\frac{v^2}{T} \int d\epsilon\ \nu(\epsilon)\ \frac{\partial n_F^\alpha}{\partial \epsilon}\ \left(\frac{\epsilon}{T}\right)^n.$$ In terms of the currents, the fast interlayer thermalization ($\chi_T^{(1)}=\chi_T^{(2)}$) translates into the relation $$B_2^b(\mathbf{P}_a-B_1^a \mathbf{J}_a)=B_2^a(\mathbf{P}_b-B_1^b \mathbf{J}_b),$$ where $$\begin{aligned} \label{EqDefiningTheBs} B_0^\alpha = A_0^\alpha,\qquad B_1^\alpha=\frac{A_1^\alpha}{A_0^\alpha},\qquad B_2^\alpha = A_2^\alpha-\frac{(A_1^\alpha)^2}{A_0^\alpha}.\end{aligned}$$ The asymptotics of the functions $B^{(i)}_n(\mu_i/2T)$ read: $$\label{EqAssymptoticsOfBs} B^{(i)}_n(x\ll 1)=\left\{\begin{array}{cc} \ln2/\pi & n=0\\ 4x & n=1\\ 9\zeta(3)/2\pi & n=2 \end{array} \right. , \qquad B_n(x\gg 1)=\left\{\begin{array}{cc} \|x\|/\pi & n=0\\ 2x& n=1\\ \pi \|x\|/3 & n=2 \end{array} \right. .$$ It is also convenient to define $ B_2=B_2^{(1)}+B_2^{(2)}. $ Finally, we introduce the total momentum (total energy current) $$\mathbf{P}=\mathbf{P}_a+\mathbf{P}_b,$$ which is not affected by electron-electron collisions due to total momentum conservation in the e-e collision integral. These transformations allow us to rewrite the matrix kinetic equation in the “hydrodynamic form”. Here we can also introduce electric fields in both layers without doubling the number of relevant modes. Integrating the reduced matrix kinetic equation over the energy with $$\begin{aligned} &&-NT\int d\epsilon_1 \nu(\epsilon)\frac{\partial n_F(\epsilon_1)}{\partial \epsilon_1} \{\ldots\} =N\int d\epsilon_1 \frac{\nu(\epsilon)}{\cosh^2\frac{\epsilon_1-\mu_i}{2T}} \{\ldots\} \\ &&\text{and}\nonumber \\ &&-N\int d\epsilon_1 \epsilon_1 \nu(\epsilon)\frac{\partial n_F(\epsilon_1)}{\partial \epsilon_1} \{\ldots\} =N\int d\epsilon_1 \epsilon_1\frac{\nu(\epsilon)}{\cosh^2\frac{\epsilon_1-\mu_i}{2T}} \{\ldots\}\end{aligned}$$ yields the following steady-state equations (in order to avoid confusion in indices, from now on we label the layers by $a$ and $b$) $$\begin{aligned} \left\{\frac{1}{\tau} + \left[(B_1^a)^2+\frac{B_2}{B_0^a}\right] \frac{1}{\tau^a_{ee}} - B_1^a B_1^b \frac{1}{\tau_{D}} \right\}\mathbf{J}_a &+&\left\{ B_1^a B_1^b \frac{1}{\tau^a_{ee}}-\left[(B_1^b)^2+\frac{B_2}{B_0^b}\right] \frac{1}{\tau_{D}} \right\}\mathbf{J}_b\nonumber \\ &=&\frac{2}{N} T B_0^a e\mathbf{E}_a+\left( \frac{B_1^a}{\tau^a_{ee}} - \frac{B_1^b}{\tau_{D}}\right)\mathbf{P} \\ \left\{\frac{1}{\tau} + \left[(B_1^b)^2+\frac{B_2}{B_0^b}\right] \frac{1}{\tau^b_{ee}} - B_1^a B_1^b \frac{1}{\tau_{D}} \right\}\mathbf{J}_b &+&\left\{ B_1^a B_1^b \frac{1}{\tau^b_{ee}}- \left[(B_1^a)^2+\frac{B_2}{B_0^a}\right] \frac{1}{\tau_{D}} \right\}\mathbf{J}_a\nonumber \\ &=&\frac{2}{N} T B_0^b e\mathbf{E}_b+\left( \frac{B_1^b}{\tau^b_{ee}}- \frac{B_1^a}{\tau_{D}}\right)\mathbf{P} \\ \frac{\mathbf{P}}{\tau}&=& \frac{2}{N} T \left(B_0^a B_1^a \mathbf{E}_a+B_0^b B_1^b \mathbf{E}_b\right). \label{hydro-drag}\end{aligned}$$ Here we have introduced the following effective transport relaxation rates: $$\begin{aligned} \frac{1}{\tau^a_{ee}}&=&\frac{N}{8T^2B_2} \int\mathrm{d}\{\epsilon_i\}\mathrm{d}\{\hat{\mathbf{v}}_i\}\ \left[ \left(\mathbf{v}_1-\mathbf{v}_2+\mathbf{v}_3-\mathbf{v}_4\right)^2\ \mathcal{W}^{aa}\ + 2 \left(\mathbf{v}_1-\mathbf{v}_2\right)^2\ \mathcal{W}^{ab} \right], \\ \frac{1}{\tau^b_{ee}}&=&\frac{N}{8T^2B_2} \int\mathrm{d}\{\epsilon_i\}\mathrm{d}\{\hat{\mathbf{v}}_i\}\ \left[ \left(\mathbf{v}_1-\mathbf{v}_2+\mathbf{v}_3-\mathbf{v}_4\right)^2\ \mathcal{W}^{bb}\ + 2 \left(\mathbf{v}_1-\mathbf{v}_2\right)^2 \mathcal{W}^{ba} \right], \\ \frac{1}{\tau_D}&=&\frac{N}{4T^2B_2} \int\mathrm{d}\{\epsilon_i\}\mathrm{d}\{\hat{\mathbf{v}}_i\}\ \left(\mathbf{v}_1-\mathbf{v}_2\right)\left(\mathbf{v}_4-\mathbf{v}_3\right)\ \mathcal{W}^{ba}. \label{rates}\end{aligned}$$ Here $1/\tau^{a(b)}_{ee}$ are the intralayer transport relaxation rates describing the velocity relaxation within a layer due to inelastic scattering with electrons in the same layer (described by $\mathcal{W}^{aa}$) and in the other layer ($\mathcal{W}^{ab}$ term). The rate $1/\tau_D$ describes the interlayer velocity relaxation (velocity transfer from one layer to the other due to $\mathcal{W}^{ab}$): we call it the drag rate. The kernels $\mathcal{W}^{ij}$ here are related to the kernel of the collision integral (\[eq4\]) as follows: $$\mathcal{W}^{ij}(1,2;3,4)=\frac{\nu(\epsilon_1)}{\cosh^2\frac{\epsilon_1-\mu_i}{2T}}W^{ij}(1,2;3,4).$$ 2. Ballistic regime =================== A. Resistivity matrix --------------------- The hydrodynamic equations (\[hydro-drag\]) yield the following explicit expressions for the intralayer and interlayer resistivities: $$\begin{aligned} \rho_{aa}&=&\frac{\hbar}{e^2}\frac{2 B_2}{N \left(B_0^a\right)^2 T} \left[\frac{B_0^a}{\tau}+ \dfrac{\dfrac{1}{\tau\tau^a_{ee}}+\left(B_1^b\right)^2\left(\dfrac{1}{\tau^a_{ee}\tau^b_{ee}}-\dfrac{1}{\tau_D^2}\right)} {\dfrac{1}{\tau}+\dfrac{\left(B_1^a\right)^2}{\tau^a_{ee}} +\dfrac{\left(B_1^b\right)^2}{\tau^b_{ee}} -\dfrac{2B_1^aB_1^b}{\tau_D}}\right], \label{rhoaa} \\ \rho_{ab}&=&-\frac{\hbar}{e^2}\frac{2 B_2}{N B_0^a B_0^b T} \dfrac{\dfrac{1}{\tau\tau_D}+B_1^aB_1^b\left(\dfrac{1}{\tau^a_{ee}\tau^b_{ee}}-\dfrac{1}{\tau_D^2}\right)} {\dfrac{1}{\tau}+\dfrac{\left(B_1^a\right)^2}{\tau^a_{ee}} +\dfrac{\left(B_1^b\right)^2}{\tau^b_{ee}} -\dfrac{2B_1^aB_1^b}{\tau_D}} \label{rhoab}\end{aligned}$$ $\rho_{bb}=\rho_{aa}(a\leftrightarrow b)$, and $\rho_{ba}=\rho_{ab}$. The drag coefficient is defined as $$\rho_D=-\rho_{ab}.$$ It can also be rewritten in the following form, $$\rho_D=\frac{\hbar}{e^2}\frac{2 B_2}{N B_0^a B_0^b T \tau_D} \left[1-\dfrac{\left(\dfrac{B_1^a}{\tau^a_{ee}}+\dfrac{B_1^b}{\tau_D}\right) \left(\dfrac{B_1^b}{\tau^b_{ee}}+\dfrac{B_1^a}{\tau_D}\right)} {\dfrac{1}{\tau_D}\left(\dfrac{1}{\tau}+\dfrac{\left(B_1^a\right)^2}{\tau^a_{ee}} +\dfrac{\left(B_1^b\right)^2}{\tau^b_{ee}} -\dfrac{2B_1^aB_1^b}{\tau_D}\right)} \right],$$ which for the disorder-dominated case yields directly the conventional perturbative drag: $$\rho_D=\frac{\hbar}{e^2}\frac{2 B_2}{N B_0^a B_0^b T \tau_D}.$$ For equal layers, we denote $$\epsilon_0=2B_0 T/N, \qquad C_1=B_1, \qquad C_2=B_2/B_0.$$ Then the resistivity matrix reads $$\begin{aligned} {\hat \rho} = \frac{\hbar}{e^2}\frac{C_2}{\epsilon_0} \left\{ \frac{1}{\tau} \left[ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right.\right. &+& \left. \dfrac{C_2}{1/\tau+2 C_1^2\left(1/\tau_{ee}-1/\tau_D\right)} \begin{pmatrix} 1/\tau_{ee} & -1/\tau_D \\[0.5pt] -1/\tau_D & 1/\tau_{ee} \end{pmatrix}\right] \nonumber \\[1pt] &+& \left. \dfrac{C_2 C_1^2\left(1/\tau_{ee}^2-1/\tau_D^2\right)} {1/\tau+2 C_1^2\left(1/\tau_{ee}-1/\tau_D\right)} \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \right\}. \label{res-matrix}\end{aligned}$$ Here the first term is the intralayer resistivity determined by disorder, the second term describes the conventional Coulomb drag in combination with the intralayer inelastic transport relaxation, and the last term arises due to the fast unidirectional thermalization in graphene. In the clean limit $\tau=\infty$ the resistivity matrix has the form $${\hat \rho} = \frac{\hbar}{e^2}\frac{2 B_2}{N T} \dfrac{\dfrac{1}{\tau^a_{ee}\tau^b_{ee}}-\dfrac{1}{\tau_D^2}} {\dfrac{\left(B_1^a\right)^2}{\tau^a_{ee}} +\dfrac{\left(B_1^b\right)^2}{\tau^b_{ee}} -\dfrac{2B_1^aB_1^b}{\tau_D}} \begin{pmatrix} \dfrac{(B_1^b)^2}{(B_0^a)^2} & -\dfrac{B_1^a B_1^b }{B_0^a B_0^b} \\ -\dfrac{B_1^a B_1^b }{B_0^a B_0^b} & \dfrac{(B_1^a)^2}{(B_0^b)^2}, \end{pmatrix} \label{rho-clean}$$ which for equal layers simplifies to $${\hat \rho} = \frac{\hbar}{e^2} \frac{C_2}{\epsilon_0}\ \left(\frac{1}{\tau_{ee}}+\frac{1}{\tau_D}\right) \displaystyle\begin{pmatrix} 1 & - 1 \\ - 1 & 1 \end{pmatrix}. \label{rho-clean-equal}$$ The off-diagonal component of this matrix is given in Eq. (14) of the main text. B. Asymptotics of the drag coefficient -------------------------------------- The general condition separating the disorder-dominated and Coulomb-dominated transport regimes can be found from Eq. (\[rhoab\]), where one should compare the two terms in the numerator: $$\dfrac{1}{\tau} \sim B_1^aB_1^b \tau_D\left(\dfrac{1}{\tau^a_{ee}\tau^b_{ee}}-\dfrac{1}{\tau_D^2}\right). \label{condition}$$ In the vicinity of the Dirac point ($\mu_{a,b}\ll T$), the intralayer transport relaxation rates are $$\frac{1}{\tau^{a,b}_{ee}}\sim \alpha^2 N T,$$ whereas the drag rate was found in Ref. [@us1]: $$\frac{1}{\tau_D}\sim \alpha^2 N \frac{\mu_a\mu_b}{T},$$ so that $1/\tau_{ee}\gg 1/\tau_D$. Substituting these results into Eq. (\[condition\]), we find $$\dfrac{1}{\tau} \sim \frac{\mu_a\mu_b}{T^2} \frac{\tau_D}{\tau^a_{ee}\tau^b_{ee}} \sim \alpha^2 N T,$$ i.e. the crossover occurs at $1/\tau\sim 1/\tau_{ee}$. Away from the Dirac point, $T\ll \mu_{a,b}\ll T/\alpha$, the drag rate is (for simplicity we set $\mu_a\sim \mu_b\sim \mu$) $$\frac{1}{\tau_D}\sim \alpha^2 N \frac{T^2}{\mu} \ln\frac{\mu}{T},$$ and $$\frac{1}{\tau_{ee}}-\frac{1}{\tau_D}\sim \frac{1}{\tau_{ee}}\frac{T^2}{\mu^2}\ll \frac{1}{\tau_{ee}}.$$ It is worth mentioning that, for $\mu\gg T$, the intralayer scattering is much less efficient for the relaxation of velocity than the interlayer scattering. Indeed, the intralayer ($\propto \mathcal{W}^{ii}$) contribution to $1/\tau_{ee}$ in Eq. (\[rates\]) contains the combination of velocities $\mathbf{v}_1-\mathbf{v}_2+\mathbf{v}_3-\mathbf{v}_4$, which for $\mu\gg T$ is very close to $\mathbf{p}_1-\mathbf{p}_2+\mathbf{p}_3-\mathbf{p}_4$. Therefore, at high chemical potentials $1/\tau_{ee}$ is dominated by the interlayer contribution ($\propto \mathcal{W}^{ab}$). Thus the crossover between the two regimes occurs for $\mu\gg T$ at $$\dfrac{1}{\tau} \sim \frac{\mu^2}{T^2}\left(\dfrac{1}{\tau_{ee}}-\dfrac{1}{\tau_D}\right) \sim \alpha^2 N \frac{T^2}{\mu} \ln\frac{\mu}{T} \sim \frac{1}{\tau_{ee}}. \label{condition-high-mu}$$ The drag coefficient in the disorder-dominated regime coincides with the perturbative result, $$\begin{aligned} \rho_D &=& \frac{\hbar}{e^2} \frac{2 B_2}{N B_0^a B_0^b T \tau_D}\nonumber \\ &\sim& \frac{\hbar}{e^2}\ \alpha^2 \frac{T^2(\mu_a+\mu_b)}{(\mu_a\mu_b)^2} \ln\frac{\text{min}\{\mu_a,\mu_b\}}{T}, \qquad T\ll \mu \ll \frac{T}{\alpha},\ \frac{1}{\tau}\gg \alpha^2 N \frac{T^2}{\mu}.\end{aligned}$$ In the opposite, ultraclean case, assuming for simplicity equal layers, we get from Eq. (\[rho-clean-equal\]) $$\begin{aligned} \rho_D&=&\frac{\hbar}{e^2} \frac{C_2}{\epsilon_0}\ \left(\frac{1}{\tau_{ee}}+\frac{1}{\tau_D}\right) \simeq \frac{\hbar}{e^2} \frac{2C_2}{\epsilon_0 \tau_D}\nonumber \\ &\sim& \frac{\hbar}{e^2}\ \alpha^2 \frac{T^2}{\mu^2} \ln\frac{\mu}{T}, \qquad\qquad T\ll \mu \ll \frac{T}{\alpha},\ \frac{1}{\tau}\ll \alpha^2 N \frac{T^2}{\mu}.\end{aligned}$$ The difference between the disordered (perturbative) and ultraclean (equilibrated) results for the drag is in the presence of $1/\tau_{ee}$ in the latter case. In particular, for equal layers, this enhances the drag by a factor of 2. Since these results are qualitatively the same, we do not analyze the behavior of the drag at yet higher chemical potentials, referring the reader to Ref. [@us1]. C. Third-order drag rate ------------------------ It is important that the second-order (Golden-rule) drag rate vanishes at the Dirac point due to the particle-hole symmetry. However, the particle-hole symmetry does not affect the odd-order drag rates [@lak]. Near the Dirac point, such rates should not depend on $\mu$ and hence are proportional to $T$. Taking into account the second-order matrix element $M_{ab}^{(2)}\propto \alpha^2$, $$\left\|M_{ab}^{(1)}+M_{ab}^{(2)}\right\|^2\simeq \left\|M_{ab}^{(1)}\right\|^2 + 2 \text{Re} \left\{M_{ab}^{(1)} \left[M_{ab}^{(2)}\right]^\right\},$$ we estimate $$\frac{1}{\tau_D} \sim \alpha^2 N \frac{\mu^2}{T^2} + \alpha^3 N T,$$ where we skip the numerical prefactors in both terms. A similar correction arises in $1/\tau_{ee}$, but there it is always subleading for $\alpha\ll 1$. Substituting this correction into Eq. (\[res-matrix\]), we find $$\rho_D\sim \frac{\hbar}{e^2}\frac{1}{NT} \dfrac{\dfrac{N}{\tau}\left(\alpha^2 \dfrac{\mu^2}{T^2} + \alpha^3 T\right)+\dfrac{\mu^2 N^2}{T^2}\left[\alpha^4 T^2-\left(\alpha^2 \dfrac{\mu^2}{T^2} + \alpha^3 T\right)^2\right]} {\dfrac{1}{\tau}+\dfrac{\mu^2 N}{T^2} \left[\alpha^2 T + \left(\alpha^2 \dfrac{\mu^2}{T^2} + \alpha^3 T\right) \right]}$$ Clearly, for $\mu\gg \alpha^{1/2} T$ we can disregard the $\alpha^3$-terms. In the opposite limit, we neglect the conventional drag term: $$\rho_D \sim \frac{\hbar}{e^2} \dfrac{\alpha^3 T+\alpha^4 \mu^2 \tau N} {T+\alpha^2 \mu^2\tau N}, \qquad \mu\ll\alpha^{1/2} T, \quad \frac{1}{\tau}\ll T.$$ Exactly at the Dirac point this yields $$\rho_D\sim \frac{\hbar}{e^2} \alpha^3. \label{alpha3}$$ For finite chemical potential, this result is valid for $$\frac{1}{\tau}\gg \alpha N \frac{\mu^2}{T}.$$ In the opposite limit, one can neglect the $\alpha^3$-term, yielding $$\rho_D \sim \frac{\hbar}{e^2} \dfrac{\alpha^4 \mu^2 \tau N} {T+\alpha^2 \mu^2\tau N} = \frac{\hbar}{e^2}\begin{cases} \alpha^2, &\quad 1/\tau\ll \alpha^2 N \mu^2/T \\ \alpha^4 N \mu^2 \tau/T, &\quad \alpha^2 N \mu^2/T\ll 1/\tau\ll \alpha N \mu^2/T \end{cases}.$$ 3. Diffusive regime =================== A. Third-order contribution to the drag --------------------------------------- In this section we analyze the third-order drag [@lak] in the diffusive regime $T\tau\ll 1$ for the case of high dimensionless conductances (per spin and per valley), $g=\nu D \sim \mu\tau \gg 1$, where $D=v^2\tau/2$ is the diffusion coefficient. In this limit, one can calculate the prefactor analytically. In the vicinity of the Dirac point $\mu\tau\ll 1$, the conductance is of order unity. Indeed, experiments on high-quality samples show $T$-independent $\sigma$ down to $T=30$ mK [@kim], that can be explained by the specific character of disorder in graphene [@ogm]. Therefore, there we also assume the diffusive dynamics described by the diffusion propagators $$\mathcal{D}_i(q,\omega)=\frac{1}{\nu_{i}}\, \frac{1}{D_{i}q^{2}-i\omega}, \qquad qv,\omega \ll 1/\tau,$$ and polarization operators $$\Pi_i(q,\omega)=N\nu_i\dfrac{D_{i}q^{2}} {D_{i}q^{2}-i\omega},$$ where $\nu_{i}$ is the density of states of the layer (per spin and valley) $i=1,2$ and $D_i$ is its diffusion coefficient. The third-order drag resistivity was calculated in Ref. [@lak] for the case of large interlayer distance, $\kappa d\gg 1$. Here we generalize this result to the opposite case $\kappa d\ll 1$, which is experimentally relevant for graphene near the Dirac point. The analytic expression for the third-order drag resistivity is given by [@lak] $$\begin{aligned} \rho_{D}^{(3)}&=&\frac{\hbar}{e^2}\ 32 T g_{1}g_{2}\int\limits^{\infty}_{0} \mathrm{d}\omega\mathrm{d}\Omega\, \mathcal{F}_{1}(\omega,\Omega)\mathcal{F}_{2}(\omega,\Omega)\nonumber \\ &\times& \int \frac{d^2q}{(2\pi)^2} \int \frac{d^2Q}{(2\pi)^2} \, \mathrm{Im}\Big[\mathcal{D}_{1}(q,\omega)\mathcal{D}_{2}(q,\omega) \mathcal{V}_{12}(q,\omega) \mathcal{V}_{12}\left(\frac{\mathbf{q}}{2}-\mathbf{Q},{\frac{\omega}{2}}-\Omega\right) \mathcal{V}_{12}\left(\frac{\mathbf{q}}{2}+\mathbf{Q},{\frac{\omega}{2}}+\Omega\right)\Big]. \nonumber \\ \label{sigma-general}\end{aligned}$$ Here the thermal factors $\mathcal{F}_{i}(\omega,\Omega)$ are given by \[Spectral-F\] $$\mathcal{F}_{1}(\omega,\Omega)=T\frac{\partial}{\partial\Omega} \left[\mathcal{B}(\Omega+\omega/2)-\mathcal{B}(\Omega-\omega/2)\right],$$ $$\mathcal{F}_{2}(\omega,\Omega)=2-\mathcal{B}(\Omega+\omega/2) -\mathcal{B}(\Omega-\omega/2)+\mathcal{B}(\omega),$$ where $$\mathcal{B}(\omega)=\frac{\omega}{T}\coth\left(\frac{\omega}{2T}\right).$$ The propagators of longitudinal vector potentials $\mathcal{V}_{12}(q,\omega)$ in Eq. (\[sigma-general\]) include the dressing of the vertices by diffusons: $$\mathcal{V}_{12}(q,\omega)= \frac{q^{2}U^{\text{RPA}}_{12}(q,\omega)} {(D_{1}q^{2}-i\omega)(D_{2}q^{2}-i\omega)}, \label{calV}$$ where the retarded RPA-screened interlayer interaction $U^{\text{RPA}}_{12}(q,\omega)$ is defined in Eq. (\[RPA\]). For simplicity, we consider equal layers. For small interlayer distance $d\ll v\tau$ we have $qd\ll 1$. It is convenient to introduce the inverse screening length $$\kappa=2\pi e^2 \nu,$$ where $\nu$ is the thermodynamic density of states per one flavor of particles. Expanding $\exp(-qd)\simeq 1-qd$ we get $$\begin{aligned} U^{\text{RPA}}_{12}(q,\omega) &=& U_{12}^{(0)}(q) \left[\left(1+U_{11}^{(0)}(q) \Pi(q,\omega)\right)^2-\left( U_{12}^{(0)}(q) \Pi(q,\omega)\right)^2\right]^{-1} \nonumber \\ &=&\frac{1}{\nu} \frac{\kappa e^{-qd}}{q} \left[\left(1+\frac{N\kappa}{q}\frac{Dq^2}{Dq^2-i\omega}\right)^2-\left(\frac{N\kappa e^{-qd}}{q}\frac{Dq^2}{Dq^2-i\omega}\right)^2\right]^{-1} \nonumber \\ &\simeq&\frac{1}{\nu} \frac{\kappa}{q}\frac{(Dq^2-i\omega)^2}{[Dq(q+2N\kappa)-i\omega] [Dq^2(1+N\kappa d) -i\omega]},\end{aligned}$$ and hence $$\mathcal{V}_{12}(q,\omega)= \frac{1}{\nu} \frac{\kappa q}{[Dq(q+2N\kappa)-i\omega] [Dq^2(1+N\kappa d) -i\omega]}. \label{ourV12}$$ We first consider the case of large interlayer separation, $N\kappa d\gg 1$. For $N\kappa \gg \text{max}\{1/d,(T/D)^{1/2}\}$ we find $$\mathcal{V}_{12}(q,\omega)\simeq \frac{1}{2 N g} \frac{1}{Dq^2 N\kappa d -i\omega},$$ which reproduces the result of Ref. [@lak]: $$\rho_D^{(3)}\sim \frac{\hbar}{e^2}\, \frac{1}{N^3 g^3}\, \frac{1}{(N\kappa d)^2}.$$ For $1/d\ll N\kappa \ll (T/D)^{1/2}$, $$\mathcal{V}_{12}(q,\omega)\simeq \frac{1}{\nu}\frac{\kappa q}{Dq^2-i\omega} \frac{1}{Dq^2 N\kappa d -i\omega},$$ and we find $$\rho_D^{(3)}\sim \frac{\hbar}{e^2}\, \frac{1}{g^3}\, \frac{1}{(N\kappa d)^2} \left(\frac{D \kappa^2}{T}\right)^{3/2}.$$ In the opposite case $N\kappa d\ll 1$ (which is relevant to our problem), $$\mathcal{V}_{12}(q,\omega)\simeq \frac{1}{\nu} \frac{\kappa q}{[Dq(q+2N\kappa)-i\omega]\ [Dq^2-i\omega]}.$$ Substituting this into Eq. (\[sigma-general\]), we get $$\begin{aligned} \rho_{D}^{(3)}\!&=&\!\frac{\hbar}{e^2}\, 32T\frac{g^2}{\nu^5}\int\limits^{\infty}_{0} \mathrm{d}\omega\mathrm{d}\Omega\, \mathcal{F}_{1}(\omega,\Omega)\mathcal{F}_{2}(\omega,\Omega)\! \int \frac{d^2q}{(2\pi)^2} \int \frac{d^2Q}{(2\pi)^2}\ \mathrm{Im}\left\{\Big[\frac{1}{Dq^2-i\omega}\Big]^2\ \frac{\kappa q}{[Dq(q+2 N \kappa)-i\omega][Dq^2-i\omega]} \right. \nonumber \\ && \nonumber \\ &\times& \left. \frac{\kappa \|\mathbf{q}/2-\mathbf{Q}\|}{[D(\mathbf{q}/2-\mathbf{Q})^2+2 D\|\mathbf{q}/2-\mathbf{Q}\| N \kappa-i(\omega/2-\Omega)]\ [D(\mathbf{q}/2-\mathbf{Q})^2-i(\omega/2-\Omega)]} \right. \nonumber \\ && \nonumber \\ &\times& \left. \frac{\kappa \|\mathbf{q}/2-\mathbf{Q}\|}{[D(\mathbf{q}/2+\mathbf{Q})^2+2 D\|\mathbf{q}/2+\mathbf{Q}\| N \kappa-i(\omega/2+\Omega)]\ [D(\mathbf{q}/2+\mathbf{Q})^2-i(\omega/2+\Omega)]} \right\}. \label{sigma-small}\end{aligned}$$ The frequency integrals are dominated by $\omega\sim\Omega\sim T$, whereas the momentum integrals are dominated by $q\sim Q\sim q_T=\sqrt{T/D}$. Therefore, the drag conductivity in Eq. (\[sigma-small\]) can be estimated as $$\begin{aligned} \rho_D^{(3)} &\sim &\frac{\hbar}{e^2} \frac{T g^2}{\nu^5}\ \underbrace{T^2}_{d\omega d\Omega} \ \underbrace{q_T^4}_{d^2q d^2 Q} \ \frac{1}{(Dq_T^2+T)^5}\frac{\kappa^3 q_T^3}{(Dq_T^2+T+Dq_TN\kappa)^3} \nonumber \\ && \nonumber \\ &\sim &\frac{\hbar}{e^2} \frac{1}{g^3}\ \left(\frac{\kappa}{N\kappa+\sqrt{T/D}}\right)^3.\end{aligned}$$ Therefore, for $N \kappa\gg\sqrt{T/D}$ we get $$\rho_D^{(3)} \sim \frac{\hbar}{e^2}\ \frac{1}{N^3 g^3},$$ while for $N \kappa\ll\sqrt{T/D}$ we find $$\rho_D^{(3)} \sim \frac{\hbar}{e^2}\ \frac{1}{g^3} \ \left(\frac{D \kappa^2}{T}\right)^{3/2}.$$ In the first case the expression for the third-order drag coefficient is given by $$\begin{aligned} \rho_{D}^{(3)}\!&=&\!\frac{\hbar}{e^2}\ 32T\frac{g^2}{\nu^5}\int\limits^{\infty}_{0} \mathrm{d}\omega\mathrm{d}\Omega\, \mathcal{F}_{1}(\omega,\Omega)\mathcal{F}_{2}(\omega,\Omega)\! \int \frac{d^2q}{(2\pi)^2} \int \frac{d^2Q}{(2\pi)^2}\ \mathrm{Im}\left\{\Big[\frac{1}{Dq^2-i\omega}\Big]^2\ \right. \nonumber \\ && \nonumber \\ &\times& \left. \left(\frac{1}{2 N D}\right)^3 \frac{1}{Dq^2-i\omega} \frac{1}{D(\mathbf{q}/2-\mathbf{Q})^2-i(\omega/2-\Omega)} \ \frac{1}{D(\mathbf{q}/2+\mathbf{Q})^2-i(\omega/2+\Omega)} \right\}. \label{sigma-small-1}\end{aligned}$$ The integrals here are now dimensionless \[one measures momenta in units of $(T/D)^{1/2}$ and frequencies in units of $T$\]. In the second case the prefactor is again determined by a dimensionless integral: $$\begin{aligned} \rho_{D}^{(3)}\!&=&\!\frac{\hbar}{e^2}\ 32T\frac{g^2}{\nu^5} \ \int\limits^{\infty}_{0} \mathrm{d}\omega\mathrm{d}\Omega \, \mathcal{F}_{1}(\omega,\Omega)\mathcal{F}_{2}(\omega,\Omega)\! \int \frac{d^2q}{(2\pi)^2} \int \frac{d^2Q}{(2\pi)^2}\ \mathrm{Im}\left\{\Big[\frac{1}{Dq^2-i\omega}\Big]^2\ \right. \nonumber \\ && \nonumber \\ &\times& \left. \kappa^3 \frac{q}{[Dq^2-i\omega]^2} \frac{\|\mathbf{q}/2-\mathbf{Q}\|}{[D(\mathbf{q}/2-\mathbf{Q})^2-i(\omega/2-\Omega)]^2} \ \frac{\|\mathbf{q}/2+\mathbf{Q}\|}{[D(\mathbf{q}/2+\mathbf{Q})^2-i(\omega/2+\Omega)]^2} \right\}. \label{sigma-small-2}\end{aligned}$$ For $T\tau\ll 1$ and $\mu\tau\gg 1$, we have $$\kappa\sim \alpha \mu/v,$$ which implies $$N\kappa=\sqrt{\frac{T}{D}}\quad \leftrightarrow \quad \frac{1}{\tau}=\frac{\alpha^2N^2\mu^2}{T}.$$ For $\mu\tau \ll 1$, we have $$\kappa\sim \alpha/v\tau,$$ and hence $$N\kappa=\sqrt{\frac{T}{D}}\quad \leftrightarrow \quad \frac{1}{\tau}=\frac{T}{\alpha^2N^2}.$$ Thus, for $$\text{max}\{T,\alpha^2N^2\mu^2/T\}\ll 1/\tau \ll T/\alpha^2 N^2$$ the third-order drag resistivity reads: $$\rho_D^{(3)}\sim \frac{\alpha^3}{(T\tau)^{3/2}}.$$ At $T\tau\sim 1$ this result matches the ballistic third-order drag resistivity, Eq. (\[alpha3\]). For yet lower $T\ll \alpha^2N^2/\tau$ and $\mu\tau\ll 1$ the third-order drag saturates at $$\rho_D^{(3)}\sim \frac{\hbar}{e^2}\,\frac{1}{N^3}.$$ Finally, for $\mu\tau\gg \text{max}\{1,(T\tau)^{1/2}/\alpha N\}$, the third-order drag behaves as $$\rho_D^{(3)}\sim \frac{\hbar}{e^2}\,\frac{1}{(N \mu \tau)^3}.$$ 4. Correlated disorder ====================== In the original version of the paper we mentioned the correlations between the disorder potentials of the two layers [@gor] as an alternative mechanism leading to a low-$T$ peak in $\rho_D$ at the Dirac point. After the submission of the original version, we became aware of a preprint by Song and Levitov [@lev] that focused on such a mechanism. In this section we analyze the role of interlayer correlations of disorder potentials (both of short-range and long-range nature) within our general framework. This allows us to compare the effect of correlated disorder with the third-order drag considered in the main text and in Sections 2C and 3 of the Supplemental Material. As emphasized in Ref. [@lev], the correlations between the disorder potentials of the two layers might be especially important in drag experiments on graphene near the Dirac point for the two reasons: (i) similarly to the third-order drag, it does not require [@gor] the particle-hole symmetry and hence provides finite drag at the charge neutrality point [@lev]; (ii) in contrast to experiments on conventional semiconducting double wells, the interlayer distance in graphene experiments is rather small, which enhances the disorder correlations between the layers. In what follows, we analyze the two models of correlations: (A) Correlated scattering off common short-range impurities [@gor] and (B) correlations of large-scale inhomogeneities of the chemical potentials in the layers [@lev]. A. Short-range correlations: correlated impurity scattering ----------------------------------------------------------- Following Ref. [@gor], we introduce the matrix of disorder correlators $w^{(ij)}_{{\hat{\mathbf v}} {\hat{\mathbf v}}'}=\langle u^{(i)} u^{(j)} \rangle_{\text{imp}}$. The values of $w^{(ij)}$ at $i \neq j$ differ from zero due to correlations between the impurity potentials $u^{(i)}$ in different layers. The total scattering rates are defined by $$\begin{aligned} \frac{1}{\tau_{ij}}&=&\left<w^{ij}_{{\hat{\mathbf v}} {\hat{\mathbf v}}'}\frac{1+\hat{\mathbf{v}}\hat{\mathbf{v}}'}{2}\right>,\end{aligned}$$ where the symbol $\left<...\right>$ stands for the angular average. The disorder correlations between the layers are described by the characteristic rate $$\frac{1}{\tau_g}=\frac{\tau_{12}-\tau}{\tau^2}, \label{taug}$$ where $1/\tau = [1/ \tau_{11}+ 1/ \tau_{22}]/2$. The time scale $\tau_g$ is a characteristic scale on which carriers in the two layers start “feeling” the difference between the impurity potentials $u^{(1)}$ and $u^{(2)}$. The potentials in the two layers are strongly correlated when $\tau_g\gg \tau$. One might expect that for realistic systems the situation of moderately correlated potentials, $\tau_g\sim \tau \sim \tau_{12},$ is typically realized. Weakly correlated potentials ($\tau_{12}\gg \tau$) yield $\tau_g\ll\tau$. Below we assume that disorder is sufficiently short-ranged and do not distinguish between the total and transport scattering rates for the estimates. We start from the ballistic regime $T\tau\gg 1$. The correlated disorder affects the drag in a way similar to the third-order drag. With correlated disorder, one can include an interlayer disorder line $w_{12}$ into the inelastic scattering amplitude. In the ballistic $\rho^{(3)}_D$ drag we had one amplitude $M_2$ with two interaction lines and one with a single wave line ($M_1$). The corresponding drag rate contains $2Re[M_1(M_2)^] \propto \alpha^3$. Now one can form the second-order scattering amplitude $M_2$ using one interaction line ($\alpha$) and one interlayer-disorder line, which introduces a factor $(T\tau_{12})^{-1}$. This gives $$\frac{1}{\tau_D^{\text{corr}}} \sim \alpha^2 T (T \tau_{12})^{-1}=\alpha^2/\tau_{12},$$ and $$\rho_D^{\text{corr}}\sim \frac{\alpha^2}{T \tau_{12}}, \label{rhocorrball}$$ which overcomes the third-order drag $\rho_D^{(3)} \sim \alpha^3$ for $1/\tau_{12}>\alpha T$. This happens in the perturbative regime ($1/\tau>\alpha^2 T$, assuming correlated disorder, $\tau_{12}\sim \tau$), where the correlated-disorder contribution can be calculated diagrammatically. Similarly to $\sigma_D^{(3)}$, the corresponding diagram involves two four-leg vertices (hence finite drag at the Dirac point $\mu=0$), but now connected in all possible ways by two interaction lines and one disorder line $w_{12}$. The general expression for the drag resistivity in the ballistic regime, including both third-order and correlated-disorder drag rates for equal layers has the form: $$\rho_D\sim \frac{\hbar}{e^2}\frac{1}{NT} \dfrac{\dfrac{N}{\tau}\left(\alpha^2 \dfrac{\mu^2}{T^2} + \alpha^3 T + \dfrac{\alpha^2}{T\tau_{12}}\right)+\dfrac{\mu^2 N^2}{T^2}\left[\alpha^4 T^2-\left(\alpha^2 \dfrac{\mu^2}{T^2} + \alpha^3 T + \dfrac{\alpha^2}{T\tau_{12}}\right)^2 \right]} {\dfrac{1}{\tau}+\dfrac{\mu^2 N}{T^2} \left[\alpha^2 T + \left(\alpha^2 \dfrac{\mu^2}{T^2} + \alpha^3 T+ \dfrac{\alpha^2}{T\tau_{12}}\right) \right]}$$ Exactly at the Dirac point it reduces to: $$\rho_D(\mu=0)\sim \frac{\hbar}{e^2}\alpha^2\left(\frac{1}{T\tau_{12}}+\alpha\right).$$ Let us now analyze the role of correlated disorder in the diffusive regime $T\tau\ll 1$. Again, we assume the absence of localization at the Dirac point (see Section 3). The drag resistivity for the case of correlated disorder was calculated in the diffusive regime in Ref. [@gor]. It is dominated by the Maki-Thompson diagram with an interlayer Cooper propagator. It is worth noting that any difference in the disorder potentials (as well as in chemical potentials of the layers) leads to a finite gap in these propagators given by $1/\tau_g$. The main result of Ref. [@gor] is as follows: $$\rho^{\text{corr}}_D\sim \frac{\hbar}{e^2} \frac{1}{g^2 [\lambda^{-1}_{21} + \ln(\varepsilon_0 /T)]^2} \ln\frac{T\tau_{\varphi}\tau_{g}} {\tau_{\varphi}+\tau_{g}} \label{rhototal1}$$ at $\tau^{-1}_{g} \ll T \ll \tau^{-1}$, and $$\rho^{\text{corr}}_D\sim \frac{\hbar}{e^2} \frac{(T\tau_g)^2}{g^2 [\lambda^{-1}_{21} + \ln(\varepsilon_0 \tau_g)]^2}. \label{rhototal2}$$ at lower temperatures $T\ll \tau^{-1}_{g}$. In graphene near the Dirac point, for small interlayer distance $\kappa d\ll 1$ the interlayer interaction constant in the Cooper channel is $\lambda_{12}\sim \alpha$. The Cooper channel cutoff energy is $\epsilon_0=1/\tau$ (the logarithm in the Cooper channel appears only for a constant density of states; in graphene in the diffusive regime this happens only for energies below $1/\tau$), the dimensionless conductance $g \sim 1$, and $\tau_\phi \sim 1/T$. Substituting these values to Eqs. (\[rhototal1\]) and (\[rhototal2\]), we arrive at $$\begin{aligned} \rho^{\text{corr}}_D &\sim& \frac{\hbar}{e^2} \frac{\alpha^2 }{[1-\alpha \ln(T \tau)]^2},\qquad \tau^{-1}_{g} \ll T \ll \tau^{-1}, \label{rhocorr1} \\ && \nonumber \\ \rho^{\text{corr}}_D &\sim& \frac{\hbar}{e^2} \frac{\alpha^2 (T \tau_g)^2}{[1-\alpha \ln(T \tau)]^2}, \qquad T\ll \tau^{-1}_{g}. \label{rhocorr2}\end{aligned}$$ These results are $\propto \alpha^2$ for realistic temperatures, $T \tau \gg \exp(-1/\alpha)$. For a moderately correlated disorder $\tau_g\sim \tau$, Eqs. (\[rhocorrball\]) and (\[rhocorr2\]) then lead to $$\rho^{\text{corr}}_D \sim \frac{\hbar}{e^2} \alpha^2 \begin{cases} (T\tau)^{-1}, &\qquad T\tau\gg 1 \\[0.5pt] (T\tau)^2, &\qquad T\tau\ll 1 \end{cases},$$ which yields a maximum at $T\sim 1/\tau$ in the temperature dependence of the drag resistivity at the charge neutrality point. For strongly correlated disorder potentials ($\tau_g\gg \tau$), this maximum develops into a plateau between $\tau^{-1}_{g} \ll T \ll \tau^{-1}$. B. Long-range correlations: correlated macroscopic inhomogeneities ------------------------------------------------------------------ Let us now analyze within our kinetic-equation framework the model of correlated macroscopic spatial fluctuations $\delta \mu_i$ in chemical potentials of the two layers [@lev], characterized by the correlation function $$F_{ij}^{(\mu)}({\boldsymbol{r}}-{\boldsymbol{r}}')=\langle\delta\mu_i({\boldsymbol{r}})\delta\mu_j({\boldsymbol{r}}')\rangle\neq 0.$$ We restrict ourselves to the ballistic regime $T\tau\gg 1$. Assuming the spatial scale of the fluctuations to be much larger than all characteristic scales related to the particle scattering, $v\tau_{ee}$, $v\tau_D$, and $v\tau$, we solve the hydrodynamic equations locally, yielding Eq. (\[rhoab\]) with local values of the chemical potentials encoded in functions $B_1^{a(b)}\sim \mu_{a(b)}/T$, as well as in the local drag rate $$\frac{1}{\tau_D({\boldsymbol{r}})}\sim \alpha^2 N \frac{\mu_1({\boldsymbol{r}})\mu_2({\boldsymbol{r}})}{T}.$$ On the other hand, since the coefficients $B_0\sim 1$ and $B_2\sim 1$, as well as the transport electron-electron scattering rate $\tau_{ee}^{-1}\sim \alpha^2 T$ are finite at the neutrality point, we can neglect the fluctuations of $\mu_{i}$ in these quantities. Exactly at the Dirac point $\mu_{1,2}=0$, assuming that the fluctuations of chemical potentials are weak (the precise condition is established below), we can further neglect the $B_1$-terms in the denominator of Eq. (\[rhoab\]), yielding for the “local resistivity” $$\begin{aligned} \rho_D({\boldsymbol{r}})\simeq\frac{\hbar}{e^2}\frac{2 B_2 \tau}{N B_0^a B_0^b T} \left[\dfrac{1}{\tau\tau_D({\boldsymbol{r}})}+\dfrac{B_1^a({\boldsymbol{r}})B_1^b({\boldsymbol{r}})}{\tau^a_{ee}\tau^b_{ee}}\right] \nonumber \\ && \nonumber \\ \sim \frac{\hbar}{e^2}\frac{\tau}{N T}\, \delta \mu_1({\boldsymbol{r}}) \delta\mu_2({\boldsymbol{r}})\, \left(\dfrac{\alpha^2 N}{T \tau}+\alpha^4 N^2\right) . \label{rhoabcorr}\end{aligned}$$ Averaging this expression over the small fluctuations of the correlated chemical potentials [@lev], we arrive at the correction to the universal third-order result, $\rho_D^{(3)}(\mu=0)\sim (\hbar/e^2)\alpha^3$, $$\Delta \rho_D(\mu=0) \sim \frac{\hbar}{e^2}\frac{\alpha^2\, F^{(\mu)}_{12}(0)}{T^2}\, \left(1+\alpha^2 N T\tau\right) \sim \frac{\hbar}{e^2}\frac{F^{(\mu)}_{12}(0)}{T^2}\, \begin{cases} \alpha^4 N T\tau &\qquad \dfrac{1}{\tau}\ll \alpha^2 N T,\\[0.2pt] \alpha^2 &\qquad \alpha^2 N T\ll \dfrac{1}{\tau}\ll T. \end{cases}$$ We see that in the Coulomb-dominated transport regime, this correction is dominated by the fluctuations in $B_1$, whereas in the disorder-dominated (perturbative) regime, the main role is played by a locally finite drag rate. Finally, in the ultraclean limit $$\frac{1}{\tau}\ll \alpha^2 N F_{ii}^{(0)}/T, \label{cond-delta-mu}$$ we can neglect $1/\tau$ in the denominator of the local drag resistivity given by Eq. (\[rhoab\]), yielding a natural analog of Eq. (\[r1dp\]): $$\Delta \rho_D(\mu=0)(\mathbf{r}) \sim \frac{\hbar}{e^2}\alpha^2 \frac{\delta\mu_1\delta\mu_2}{\delta\mu_1\delta\mu_1+\delta\mu_2\delta\mu_2}. \label{DeltaRho}$$ In particular, for perfectly correlated chemical potentials, $\delta\mu_1(\mathbf{r})=\delta\mu_2(\mathbf{r})$, the fluctuations drops out from Eq. (\[DeltaRho\]) and the local resistivity turns out to be independent of $\mathbf{r}$. In a more general case, the averaging over fluctuations becomes nontrivial, but this can only affect the numerical prefactor in the final result. 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--- abstract: 'Our current understanding of the origin of barium and S stars is briefly reviewed, based on new orbital elements and binary frequencies.' author: - 'A. Jorissen' - 'S. Van Eck' title: 'Barium and Tc-poor S stars: Binary masqueraders among carbon stars' --- The relation of barium and S stars to carbon stars ================================================== Since the last conference (IAU Coll. 106, [Evolution of Peculiar Red Giants]{}, Johnson & Zuckermann eds., 1989) devoted to chemically-peculiar red giants (PRGs), much progress has been made in understanding how barium and S stars relate to the other PRGs. The discovery of the binary nature of barium stars (McClure et al. 1980; McClure 1983) suggested from the beginning that mass transfer was likely to play a key role in the formation of the barium syndrome. As far as S stars are concerned, it has become clear that Tc-rich and Tc-poor S stars form two separate families with similar chemical peculiarities albeit of very different origins (Iben & Renzini 1983; Little-Marenin et al. 1987; Jorissen & Mayor 1988; Smith & Lambert 1988; Brown et al. 1990; Johnson 1992; Jorissen & Mayor 1992; Groenewegen 1993; Johnson et al. 1993; Jorissen et al. 1993; Ake, this conference). Tc-rich (or ‘intrinsic’) S stars are genuine thermally-pulsing AGB stars where the s-process operates in relation with the thermal pulses, and where the third dredge-up brings the freshly synthesized s-elements (including Tc) to the surface (e.g. Iben & Renzini 1983; Sackmann & Boothroyd 1991). By contrast, Tc-poor (or ‘extrinsic’) S stars are believed to be the cool descendants of barium stars. Figure 1 summarizes our current understanding of the relationship between the different families of PRG stars. This general picture raises several questions, that will briefly be addressed in this paper:\ 1. Is binarity a necessary condition to produce a barium star? 2. What is the mass transfer mode (wind accretion or RLOF?) responsible for their formation? 3. Do barium stars form as dwarfs or as giants? 4. Do barium stars evolve into Tc-poor S stars? 5. What is the relative frequency of Tc-rich and Tc-poor S stars? 6. Are the abundances in the mass-loser star (i.e the AGB progenitor of the present white dwarf companion) compatible with those presently observed in the barium or extrinsic S star? We refer to Jorissen & Boffin (1992), Han et al. (1995) and Busso et al. (1995) for a detailed discussion of item 6. Is binarity a necessary condition to form a barium star? ======================================================== To answer that question, [all]{} 27 barium stars with [strong]{} anomalies (i.e. Ba3, Ba4 or Ba5 on the scale devised by Warner 1965) south of $\delta = -25^{\circ}$ from the list of Lü et al. (1983) have been monitored with the CORAVEL spectrovelocimeter (Baranne et al. 1979) since 1984. HD 19014 is the only star in that sample that does not show any sign of binary motion. No detailed abundance analysis to confirm the barium nature of that star is available, unfortunately. For a fictitious population of binaries observed with the same time sampling and the same internal errors as the real sample of barium stars, and having eccentricity and mass-function distributions matching the observed ones, a Monte-Carlo simulation yields a binary detection rate comprised between 96% (25.9/27) and 98% (26.5/27), depending on whether the observed period distribution is extrapolated or not towards periods as long as $2\times 10^4$ d \[see Jorissen et al. 1997 for more details\]. [Binarity is thus a necessary condition to produce strong barium stars]{}. In a comparison sample of 28 [mild]{} barium stars (i.e. with Ba1 and Ba2 indices) randomly selected from the list of Lü et al. (1983) and monitored in a similar way as the strong barium stars, 23 (82%) are definitely spectroscopic binaries, 2 (7%) are probably binaries, and 3 (11%; HD 50843, HD 95345, HD 119185) show no sign of radial velocity variations at the level 0.3 r.m.s. after more than 10 y of monitoring. Detailed spectroscopic abundance analyses performed on HD 95345 (Sneden et al. 1981) and HD 119185 (Začs et al. 1996) confirm the existence of mild heavy-element overabundances (\[s/Fe\] = 0.2 to 0.3 dex) for these stars with constant radial velocity. This frequency of constant stars is again consistent with the binary detection rate predicted for that sample by a Monte-Carlo simulation, provided that the period distribution of mild barium stars extends up to $2\times 10^4$ d. In these conditions, there is no need to invoke any formation mechanism other than mass transfer in a binary system to produce mild barium stars. On the contrary, an alternative formation scenario (like galactic fluctuations of the s/Fe ratio; Williams 1975, Sneden et al. 1981, Edvardsson et al. 1993) may be required to account for a population of non-binary stars found among [dwarf]{} mild barium stars (North et al., this conference). Is binarity a [sufficient]{} condition to produce a barium star? Probably not, since binary systems consisting of a [normal]{} red giant and a WD companion with Ba-like orbital parameters do exist (Jorissen & Boffin 1992). DR Dra (= HD 160538) is probably the best example, with $P = 904$ d, $e = 0.07$ (compare with Fig. \[Fig:elogP\]) and a hot WD companion detected by Fekel et al. (1993). Berdyugina (1994) finds a metallicity close to solar and normal Zr and La abundances in the giant. Začs et al. (1996) basically confirm that result. Metallicity may be the other key parameter, besides binarity, controlling the formation of barium stars. The s-process efficiency, expressed in terms of the neutron irradiation, seems to be larger in low-metallicity stars (Kovács 1985; Busso et al. 1995). Clayton (1988) provides a theoretical foundation for that empirical finding, provided that $^{13}\rm C(\alpha,n)^{16}O$ is the neutron source for the s-process. Barium stars would therefore be easier to produce in a low-metallicity population. Inferring the mass transfer mode from the orbital elements: Wind accretion and/or RLOF? ======================================================================================= Synthetic binary evolution models (Han et al. 1995; de Kool & Green 1995) suggest that the bimodal period distribution exhibited by strong barium stars (Fig. \[Fig:PBa\]) reflects the operation of two distinct mass-transfer modes, RLOF in the short-period mode (peaking around 500 d) and wind accretion in the long-period mode (around 3000 d). This general picture actually faces three major difficulties: first, the threshold period (about 1000 d) between the RLOF and wind-accretion modes is much too short to accomodate the large radii reached by AGB stars. Second, the period – eccentricity diagram (Fig. \[Fig:elogP\]) reveals that not all orbits in the short-period (i.e. post-RLOF) mode are circular, although tidal effects are expected to circularize the orbit in the phase of large radius just preceding RLOF (e.g. Zahn 1977). A similar problem exists for the orbits of dwarf barium stars (see North et al., this conference). Third, RLOF from AGB stars with a deep convective envelope is dynamically unstable (‘unstable case C RLOF’; e.g. Tout & Hall 1991), with the ensuing common envelope stage generally accompanied by dramatic orbital shrinkage leading to the formation of a cataclysmic binary with a period much shorter than that of barium stars (e.g. Meyer & Meyer-Hofmeister 1979). To solve these problems, Han et al. (1995), Livio (1996) and Jorissen et al. (1997) propose avenues to explore. One of these involves Peter Eggleton’s CRAP (Companion-Reinforced Attrition Process; Eggleton 1986) speculating that larger mass-loss rates for AGB stars in binary systems may reverse the mass ratio of the system prior to RLOF, thus stabilizing the mass transfer process (Tout & Eggleton 1988; Han et al. 1995). (6,6) (6,6) Do barium stars form as dwarfs or giants? ========================================= In Fig. 1, it is assumed that the mass transfer responsible for the barium syndrome occurred when the barium star was still on the main sequence. Because the stellar lifetime is longer on the main sequence than in the giant phase, that possibility indeed appears more probable than the formation of the barium star directly as a giant star. However, as pointed out by Iben & Tutukov (1985), the mismatch between the thermal time scale of the dwarf’s envelope and that of the mass-losing AGB star may prevent the formation of dwarf barium stars. A main-sequence star would indeed be driven out of thermal equilibrium in case of rapid mass accretion from its giant companion, and would swell to giant dimensions (e.g. Kippenhahn & Meyer-Hofmeister 1977), leading to a common envelope stage with possibly dramatic consequences on the fate of the binary system (see e.g. Meyer & Meyer-Hofmeister 1979 and Sect. 2). Dwarf barium stars long remained elusive, until Luck & Bond (1982, 1991) and North et al. (1994) recognized that some of the CH subgiants previously identified by Bond (1974), as well as some of the F dwarfs previously classified by Bidelman (1985) as having ’strong Sr $\lambda$ 4077’, had the proper abundance anomalies, gravities and galactic frequencies to be identified with the long-sought Ba dwarfs. A large fraction of binaries (about 90%) has been found among the stars with strong anomalies, as expected (McClure 1985; North & Duquennoy 1992; North et al., this conference). The very existence of binary dwarf Ba stars, in spite of Iben & Tutukov’s argument, is another indication that, if RLOF does indeed occur in these systems, it does not have the catastrophic consequences generally associated with unstable case C RLOF. The question of whether these dwarf barium stars will eventually evolve into giant barium stars is addressed by North et al. elsewhere in these Proceedings. The formation of a barium star directly as a giant, though probably less frequent, is by no means excluded. The barium star HD 165141 may be such a case. HD 165141 is unique in sharing properties of barium and RS CVn systems (Fekel et al. 1993; Jorissen et al. 1996). Its rapid rotation ($V \sin i = 14$ ) and X-ray flux (probably from a hot corona) are typical of RS CVn systems. However, the spin-up of that star (and the concomittant RS CVn properties) cannot be attributed to tidal effects synchronizing the stellar rotation with the orbit, as is the case for RS CVn systems, since the orbital period (about 5200 d) is much too long. That puzzle may be solved if the wind accretion episode responsible for the barium syndrome spun the star up, as suggested by detailed hydrodynamical simulations (Theuns & Jorissen 1993; Theuns et al. 1996). Since magnetic braking is generally faster than the stellar lifetime on the giant branch, wind accretion and concomittant spin-up must have occurred when HD 165141 was already a giant star. Strong support to that hypothesis comes from the fact that HD 165141 has a hot WD companion (Fekel et al. 1993) whose cooling time scale is shorter than the lifetime of HD 165141 on the red giant branch. Finally, note that Jeffries & Stevens (1996) have reported more cases of WIRRing (Wind-Induced Rapidly Rotating) stars among binary stars involving a hot WD. Do barium stars evolve into Tc-poor S stars? ============================================ Figure \[Fig:elogP\] shows that strong barium stars and Tc-poor S stars occupy the same region of the ($e, \log P$) diagram. The distributions of the mass function $f(M)$ presented in Fig. \[Fig:fMBaS\] \[where $f(M) = M_2^3 \sin^3 i /(M_1 + M_2)^2 \equiv Q \sin^3 i,\; M_1$ and $M_2$ being the masses of the giant and of the WD, respectively\] for the two families are compatible with the hypothesis that they are extracted from the same parent population. Following the usual analysis (Webbink 1986; McClure & Woodsworth 1990) of the mass function distribution in terms of a peaked distribution of mass ratios $Q$ convolved with randomly inclined orbits, an average ratio $Q = 0.045$ is found for the two classes, translating into a giant mass of 1.6 when adopting $M_2 = 0.6$ for the WD companion. These two results thus provide strong support to the hypothesis that barium and Tc-poor S stars represent successive stages in the evolutionary path sketched in Fig. 1. Note, however, that the above comparison of the mass functions does not include two Tc-poor S stars (HD 191589 and HDE 332077) with main sequence companions detected with the [International Ultraviolet Explorer]{} satellite (Ake & Johnson 1992; Ake et al. 1992). The evolutionary status of these stars is currently unknown. The relative frequency of intrinsic/extrinsic S stars ===================================================== The evaluation of the relative frequency of intrinsic/extrinsic S stars faces two difficulties: (i) one needs an efficient criterion for distinguishing extrinsic from intrinsic S stars, and (ii) the frequency evaluation must be corrected from the selection bias, since extrinsic and intrinsic S stars follow different galactic distributions (Jorissen et al. 1993). As far as (i) is concerned, the defining criterion of intrinsic/extrinsic S stars based on the presence/absence of Tc, respectively, may be difficult to apply to a complete sample of S stars like Henize’s (see below), since it involves many faint stars for which high-resolution spectroscopy is difficult to secure. Binarity may be an alternative, since the binary paradigm for S stars states that all Tc-poor S stars should be binaries (Brown et al. 1990; Johnson 1992). However, some binaries must be expected among Tc-rich S stars as well, like in any class of stars. Binary intrinsic S stars with main sequence companions (case 3 in Fig. 1) include the close visual binary $\pi^1$ Gru (Feast 1953) and stars with composite spectrum like T Sgr, W Aql, WY Cas (Herbig 1966; Culver & Ianna 1975), and possibly S Lyr (Merrill 1956). The situation is further confused by extrinsic S stars reaching the AGB phase and eventually becoming Tc-rich (case 8 in Fig. 1). $o^1$ Ori, a Tc-rich binary S star with a WD companion (Ake & Johnson 1988), may be such a case. The CORAVEL $Sb$ parameter, measuring the average line width (see Jorissen & Mayor 1988 for a more detailed definition), offers an interesting and efficient alternative to identify extrinsic/intrinsic S stars. In cool red giants where macroturbulence is the main line-broadening factor, the $Sb$ parameter may be expected to be a sensitive function of the luminosity, as is macroturbulence (e.g. Gray 1988). But at the same time, bright giants exhibit large velocity jitters probably caused by envelope pulsations (e.g. Mayor et al. 1984). A correlation between $Sb$ and the radial velocity jitter must thus be expected, as observed in Fig. \[Fig:Sbjitter\] for barium, intrinsic and extrinsic S stars (Jorissen & Mayor 1992; Jorissen et al. 1997). All Tc-poor S stars are binary stars, as expected, but moreover, they are restricted to $Sb < 5$ . That criterion has been used to identify extrinsic S stars among the Henize sample (Henize 1960). That sample covers the sky south of declination $-25^\circ$ uniformly to red magnitude 10.5, and 205 S stars were found from their ZrO $\lambda$6345 band on red-yellow spectra with a dispersion of 450 Å mm$^{-1}$ at H$\alpha$. The galactic distribution of the Henize sample is presented in Fig. \[Fig:Henize\]. Intrinsic S stars are clearly more concentrated towards the galactic plane than extrinsic S stars. Correcting for the uneven sampling of galactic latitudes, the frequency of intrinsic S stars (based on the $Sb > 5$ criterion) then amounts to at least $62\pm5$% (in a magnitude-limited sample). [ It is our pleasure to thank M. Mayor and the CORAVEL team at the Observatoire de Genève for making possible the long-term radial-velocity monitorings discussed here. A.J. is Research Associate, [Fonds National de la Recherche Scientifique]{} (Belgium); S.V.E. is [Boursier F.R.I.A.]{} (Belgium). We thank the [Fonds National de la Recherche Scientifique]{} (Belgium), the [ Communauté Française de Belgique]{} and the Organizing Committee for financial support. ]{} Ake, T.B. and Johnson, H.R. 1988, [Ap.J. 327]{}, 214 Ake, T.B. and Johnson, H.R. 1992, in: [Cool Stars, Stellar Systems and the Sun]{}, eds. M.S. Giampapa, J.A. Bookbinder (San Francisco: ASP), p. 579 Ake, T., Jorissen, A., Johnson, H.R., Mayor, M. and Bopp, B. 1992, [Bull. Am. Astron. 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J.]{} 402, 667 Jorissen, A. and Mayor, M. 1988, [A&A]{} 198, 187 Jorissen, A. and Mayor, M. 1992, [A&A]{} 260, 115 Jorissen, A., Frayer, D.T., Johnson, H.R., Mayor, M. and Smith, V.V. 1993, [A&A]{} 271, 463 Jorissen, A., Schmitt, J.H.M.M., Carquillat, J.M., Ginestet, N. and Bickert, K.F. 1996, [A&A]{} 306, 467 Jorissen, A., Van Eck, S., Mayor, M. and Udry, S. 1997, [A&A]{}, submitted Kippenhahn, R. and Meyer-Hofmeister, E. 1977, [A&A]{} 54, 539 Kovács, N. 1985, [A&A]{} 150, 232 Little, S.J., Little-Marenin, I.R. and Hagen-Bauer, W. 1987, [AJ]{} 94, 981 Livio, M. 1996, in: [Evolutionqry Processes in Binary Stars]{}, eds. R.A.M.J. Wijers, M.B. Davies and C.A. Tout (Dordrecht: Kluwer), p. 141 Lü, P.K., Dawson, D.W., Upgren, A.R. and Weis, E.W. 1983, [Ap. J. Supp.]{} 52, 169 Luck, R.E. and Bond, H.E. 1982, [Ap. J.]{} 259, 792 Luck, R.E. and Bond, H.E. 1991, [Ap. J. 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--- bibliography: - 'sample-bibliography.bib' title: 'JointDNN: An Efficient Training and Inference Engine for Intelligent Mobile Cloud Computing Services' ---
--- author: - \| [^1] for RIKEN-BNL-Columbia Collaboration\ RIKEN BNL Research Center, Brookhaven National Laboratory\ Upton, New York 11973, US\ E-mail: title: ' Calculation of $\Delta I = 3/2$ kaon weak matrix elements including two-pion interaction effects in finite volume' --- Introduction ============ It is difficult to directly calculate the $K \to \pi\pi$ weak matrix elements on lattice due to difficulties of calculation of the two-pion state in finite volume. Indirect method [@indirect_method] is one of the candidates to avoid the difficulty. In the indirect method $K\to\pi\pi$ process is reduced to $K\to\pi$ and $K\to 0$ processes through chiral perturbation theory (ChPT). The RBC [@RBC] and CP-PACS [@CP-PACS] Collaborations calculated full non-leptonic kaon decay processes with the method. Their final results of $\varepsilon^\prime / \varepsilon$, however, have the opposite sign to the experiment. In the calculations there are many systematic errors, [e.g.]{}, calculating at one finite lattice spacing with quenched approximation, and using the reduction with tree level ChPT. The indirect method might cause larger systematic errors than other sources, because the final state interaction of the two-pion is expected to play an important role in the decay process. Thus we have to treat the scattering effect of the two-pion state directly on lattice to eliminate this systematic error. There are two main difficulties for the direct method, where the two-pion state is calculated on lattice. The one is to extract the two-pion state contribution with non-zero relative momentum in the $K\to\pi\pi$ four-point function, which was pointed out by Maiani and Testa [@MT]. So far there are several ideas for solving the problem. A method is employed with a proper projection of the $K\to\pi\pi$ four-point functions [@NI]. In the method we need complicated calculations and analyses, [e.g.]{}, diagonalization of a matrix of the two-pion correlation functions [@LW], to treat the two-pion state with non-zero relative momentum on lattice. A simpler idea, where complicated analyses are not required, is to prohibit the zero momentum two-pion state. Recently Kim [@CK] reported an exploratory study of the idea with H-parity (anti-periodic) boundary conditions in the spatial direction. He succeeded to extract the two-pion state with non-zero relative momentum from the ground state contribution of correlation functions, because the zero momentum two-pion state is prohibited by the boundary condition. We can also forbid the zero momentum two-pion state in center-of-mass (CM) frame by performing the calculation in non-zero total momentum (Lab) frame, $\|\vec{P}\|\ne 0$. In the frame the ground state of the two-pion is $\|\pi(0)\pi(\vec{P})\rangle$, which is related to the two-pion state with the non-zero relative momentum in the CM frame. Thus, we can extract the two-pion state with non-zero momentum from the ground state contributions [@BGLLMPS] as well as in the H-parity boundary case. The other difficulty is related to the finite volume correction due to the two-pion interaction. We have to pay attention to the finite volume effect of the two-pion to obtain matrix elements in the infinite volume, because it is much larger than that of a one-particle state. Lellouch and Lüscher (LL) [@LL] suggested a solution which is a relation between the finite and infinite volume, center-of-mass frame decay amplitudes. However, this relation is valid only in the CM frame with periodic boundary condition in the spatial direction, so that we need a modified formula when we utilize H-parity boundary condition [@CK] or Lab frame calculation. Recently, two groups, Kim [et al.]{} [@KSS] and Christ [et al.]{} [@CKY], suggested a formula which is an extension of the LL formula for the Lab frame calculation. Here we attempt to apply these two methods, Lab frame calculation and the extended formula, to the calculation of the $\Delta I = 3/2$ kaon weak matrix elements. Our preliminary result was presented in Ref. [@LAT05]. The results presented here are systematically larger than the preliminary results reported in last year lattice conference. During the past year we discovered that larger time separations were needed to remove excited state contamination. Methods ======= Through the extended formula [@KSS; @CKY], the infinite volume decay amplitude $\|A\|$ in the CM frame is given by $$\| A \|^2 = 8\pi \gamma^2 \left(\frac{E_{\pi\pi}}{p}\right)^3 \left\{ p \frac{ \partial \delta(p) }{ \partial p } + q \frac{ \partial \phi_{\vec{P}}(q) }{ \partial q } \right\} \| M \|^2, \label{eq_Lab_Formula}$$ where $\|M\|$ is the finite volume, Lab frame decay amplitude, $\gamma$ is a boost factor, $E_{\pi\pi}$ is the CM two-pion energy, and $\delta$ is the scattering phase shift of the final state interaction. The relative momentum $p^2$ is defined by the two-pion energy, $p^2 = E_{\pi\pi}^2/4 - m_\pi^2$. The function $\phi_{\vec{P}}$ with $\vec{P}$ being the total momentum, derived by Rummukainen and Gottlieb [@RG], is defined by $$\tan \phi_{\vec{P}}(q) = - \frac{\gamma q\pi^{3/2}}{Z^{\vec{P}}_{00}(1;q^2;\gamma)},$$ where $q^2 = ( p L / 2\pi )^2$, and $$Z_{00}^{\vec{P}}(1;q^2;\gamma) = \frac{1}{\sqrt{4\pi}} \sum_{\vec{n}\in \mathbb{Z}^3}\frac{1}{n_1^2+n_2^2+\gamma^{-2}(n_3+1/2)^2-q^2}, \label{eq_phi}$$ in the $\vec{P} = (0, 0, 2\pi/L)$ case. The formula eq. (\[eq\_Lab\_Formula\]) is valid only for on-shell decay amplitude, [i.e.]{}, $E_{\pi\pi} = m_K$ as in LL formula [@LL]. We calculate the four-point function for $\Delta I = 3/2$ $K\to\pi\pi$ decay with total momentum $\vec{P} = \vec{0}$ and $(0, 0, 2\pi/L)$. The four-point function $G_i(\vec{P};t,t_\pi,t_K)$ is defined by $$G_i(\vec{P};t,t_\pi,t_K) = \langle 0 \| [ K^0(\vec{P};t_K) ]^\dagger O^{3/2}_i(t) \pi^+\pi^-(\vec{P};t_\pi) \| 0 \rangle,$$ where the operators $O^{3/2}_i$ are lattice operators entering $\Delta I = 3/2$ weak decays $$\begin{aligned} O^{3/2}_{27,88} &=& (\overline{s}^ad^a)_L \left[(\overline{u}^bu^b)_{L,R}-(\overline{d}^bd^b)_{L,R}\right] + (\overline{s}^au^a)_L(\overline{u}^bd^b)_{L,R}\ \ ,\\ O^{3/2}_{m88} &=& (\overline{s}^ad^b)_L \left[(\overline{u}^bu^a)_{R}-(\overline{d}^bd^a)_{R}\right] + (\overline{s}^au^b)_L(\overline{u}^bd^a)_{R}\ \ ,\end{aligned}$$ with $(\overline{q}q)_L = \overline{q}\gamma_\mu(1-\gamma_5)q$, $(\overline{q}q)_R = \overline{q}\gamma_\mu(1+\gamma_5)q$, and $a,b$ being color indices. $O^{3/2}_{27}$ and $O^{3/2}_{88}$ are the operators in the (27,1) and (8,8) representations of $SU(3)_L\otimes SU(3)_R$ with $I=3/2$, respectively. $O^{3/2}_{m88}$ is identical to $O^{3/2}_{88}$, except the color indices are mixed. We also calculate the four-point function of two pions $G_{\pi\pi}(\vec{P};t,t_\pi)$ and the two-point function of the kaon $G_K(\vec{P};t,t_K)$ with zero and non-zero total momenta, to obtain the needed energies and amplitudes. Simulation parameters ===================== We employ the domain wall fermion action with the domain wall height $M=1.8$, the fifth dimension length $L_s = 12$ and the DBW2 gauge action with $\beta = 0.87$ corresponding to $a^{-1} = 1.31(4)$ GeV and $m_{\mathrm{res}} = 1.25(3)\times 10^{-3}$. The lattice size is $L^3 \cdot T = 16^3 \cdot 32$, where the physical spatial size corresponds to about 2.4 fm. Our simulation is carried out at four $u,d$ quark masses, $m_u = 0.015, 0.03, 0.04$ and 0.05, for the chiral extrapolation of the decay amplitudes, using 252 gauge configurations, except for the lightest mass where we use 370 configurations. For interpolations of the amplitudes to the on-shell point, we calculate the decay amplitudes with six strange quark masses, $m_s = 0.12, 0.18, 0.24, 0.28, 0.35$ and 0.44, at the heavier three $m_u$, while we use the three lighter $m_s$ at the lightest $m_u$. It is enough for the on-shell interpolation with the three $m_s$ in the lightest $m_u$ case. We will see the point in a later section. We fix the two-pion operator at $t_\pi = 0$, and the kaon operator $t_K = 20$ to avoid contaminations from excited states as much as possible. A quark propagator is calculated by averaging quark propagators with periodic and anti-periodic boundary conditions for the time direction to obtain a propagator with $2T$ periodicity. We calculate the quark propagators with Coulomb gauge-fixed wall and momentum sources. \[0.27\][ ]{} \[0.27\][ ]{} Results ======= The final state of the $\Delta I = 3/2$ kaon decay is the S-wave isospin $I=2$ two-pion state. The scattering phase shift of the two-pion state can be obtained by the finite volume formulae for the CM [@FVF] and the Lab frames [@RG] with the two-pion energy in each frame. We define "scattering amplitude” $T(p) = \tan \delta(p) E_{\pi\pi} / 2 p$, where $E_{\pi\pi}$ is the two-pion energy in the CM frame. The scattering amplitude is used for the chiral extrapolation of the phase shift with a global fitting for $m_\pi^2$ and $p^2$ with a polynomial function $a_{10} m_\pi^2 + a_{20} m_\pi^4 + a_{01} p^2 + a_{11} m_\pi^2 p^2$. While $p^4$ and $m_\pi^4$ should both be treated as second-order in ChPT, for our calculation we have only two different relative momenta. For that reason we omit the additional $p^4$ term. The figure \[fig\_delta\] shows the result of $T(p)$ and the fit results at each pion mass in the left figure, and the measured values of $\delta(p)$ in the right figure. The phase shift at the physical pion mass $m_\pi = 0.14$ GeV, plotted by solid line with the error band in the right graph, is comparable with the prediction from ChPT with experiment. In order to utilize the extension eq.(\[eq\_Lab\_Formula\]) of the LL formula, we evaluate the derivative of the phase shift from the fit result, while the derivative of the function $\phi_{\vec{P}}$ is obtained by a numerical derivative. All the off-shell decay amplitudes are determined by the ratio of correlation functions with $i = 27, 88$ and $m88$, $\sqrt{3} G_i(\vec{P};t,t_\pi,t_K) Z_{\pi\pi} Z_K / G_{\pi\pi}(\vec{P};t,t_\pi) G_K(\vec{P};t,t_K) $, where $Z_{\pi\pi}$ and $Z_K$ are the overlap of the relevant operator with each state. We determine the off-shell amplitude in the flat region of the ratio as a function of $t$, because the ratio will be a constant for those values of $t$ when these correlation functions are dominated by each ground state. The figure \[fig\_ext\] shows the off-shell decay amplitude of the $27$ operator in finite volume and its interpolation to the on-shell $E_K = E_{\pi\pi}$ in the cases of both the frames. In the lightest pion mass case, we can linearly interpolate the data with the three kaon energies. On the other hand, the off-shell decay amplitudes at the heavier pion masses have large curvature for the kaon energy, so that we fit the data with a quadratic function. In all the pion mass cases, the interpolated, on-shell decay amplitude and the fit line are plotted in the figure. \[0.27\][ ]{} \[0.27\][ ]{} \[0.29\][ ]{} The decay amplitudes on finite volume in each frame are converted to those of the CM frame in the infinite volume through the LL formula [@LL] and its extended formula eq.(\[eq\_Lab\_Formula\]) using the derivatives of $\delta$ and $\phi_{\vec{P}}$ obtained in the above. In Fig. \[fig\_mom\] we plot the infinite volume decay amplitude of the 27 operator obtained from the different frames at the lightest pion mass as a function of $p^2$. The previous result obtained from H-parity boundary calculation is also plotted to compare with these results. The amplitude obtained from the Lab frame is consistent with the line determined from those of the CM frame and H-parity boundary calculations. \[0.29\][ ]{} \[0.27\][ ]{} \[0.32\][ ]{} \[0.32\][ ]{} The infinite volume decay amplitudes of all the operators are plotted in Fig. \[fig\_wme\]. The result of the 27 operator seems to vanish at the chiral limit with zero relative momentum, while the other elements remain a constant in the limit. These trends of the pion mass dependence are reasonably consistent with the prediction of ChPT at leading order [@BGLLMPS]. In order to investigate the $m_\pi^2$ and relative momentum $p^2$ dependences, we carry out a global fitting for each decay amplitude for $m_\pi^2$ and $p^2$ assuming a simple polynomial form as $$A_{00} + A_{10} m_\pi^2 + A_{01} p^2 + A_{11} m_\pi^2 p^2. \label{fit_func}$$ The $\chi^2$/d.o.f. of the 27 operator is larger than 5 with all the four pion mass data, so that we exclude the heaviest pion mass data in all the operators from the following analysis to make the $\chi^2$/d.o.f. reasonable. The constant term of the 27 operator is consistent with zero within the error, $A_{00}^{27} = -0.0020(29)$, as expected. In other operators, we employ the same fitting function as the 27 operator except $A_{11} = 0$. The fit results are plotted in the figures, and those at the both limits, $m_\pi^2 = p^2 = 0$, are represented by diamond symbols. It should be noted that while the linear fit to $\|A_{27}\|$ is the correct leading CHPT behavior, it is being used in a region of quite large mass. Likewise the linear fit to $\|A_{88}\|$ and $\|A_{m88}\|$ omits possible logarithm terms which are of the same order. The weak matrix elements are determined using non-perturbative matching factors previously calculated with the regularization independent (RI) scheme in Ref. [@CK]. The electroweak contributions of $\varepsilon^\prime / \varepsilon$, $\langle Q_7 \rangle_2$ and $\langle Q_8 \rangle_2$, are evaluated from $\|A_{88}\|$ and $\|A_{m88}\|$. At the physical pion mass $m_\pi = 0.14$ GeV and relative momentum $p = 0.206$ GeV, we obtain the matrix elements $\langle Q_7 \rangle_2 (\mu) = 0.2473(64)$ and $\langle Q_8 \rangle_2 (\mu) = 1.160(31)$ GeV$^3$ at the scale $\mu = 1.44$ GeV using the same fitting form as in the $\|A_{88}\|$ and $\|A_{m88}\|$ cases. We calculate $\mathrm{Re}A_2$ from the weak matrix elements in the RI scheme with the Wilson coefficients evaluated by NDR scheme calculated in Ref. [@CK]. The left figure in Fig. \[fig\_ReA2\] shows $\mathrm{Re}A_2$ obtained from the CM and Lab frame calculations. The measured values of $\mathrm{Re}A_2$ are almost same as those of the 27 decay amplitude, in Fig. \[fig\_wme\], apart from the overall constant. The reason is that the main contribution of $\mathrm{Re}A_2$ comes from the 27 amplitude. We carry out a global fit using the same polynomial assumption eq.(\[fit\_func\]) except the constant term $A_{00} = 0$ to evaluate the result at the physical pion mass $m_\pi = 0.14$ GeV and momentum $p=0.206$ GeV. In the fitting we omit the data at the heaviest pion mass because of the same reason mentioned in the above. We can also carry out a reasonable fitting, $\chi^2/$d.o.f. $=$ 1.2, without $A_{11}$. The fit results at each pion mass and the physical point are plotted in the figure. We plot the results of Re$A_2$ at the physical point in the right panel of Fig. \[fig\_ReA2\] together with those of the previous works using the indirect method [@RBC; @CP-PACS] and direct calculation with evaluating the finite volume, two-pion interaction effect through ChPT [@JLQCD]. We estimate $\mathrm{Re}A_2 = 2.26(41)$ and 1.61(24)$\times 10^{-8}$ GeV at the physical point with and without $A_{11}$ term, respectively. These results agree with each other and the experiment within two standard deviations. The result is encouraging, albeit it includes systematic errors due to quenched approximation, finite lattice spacing effects, and heavy pion and kaon masses. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Changhoan Kim for his previous study upon which the present work is based, and also thank RIKEN BNL Research Center, BNL and the U.S. DOE for providing the facilities essential for the completion of this work. [99]{} C. W. Bernard, T. Draper, A. 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Sharpe, [Nucl.Phys.]{} [B727]{} (2005) 218, \[[hep-lat/0507006](http://xxx.lanl.gov/abs/hep-lat/0507006)\]. N. H. Christ, C. Kim and T. Yamazaki, [Phys. Rev.]{} [D72]{} (2005) 114506, \[[hep-lat/0507009](http://xxx.lanl.gov/abs/hep-lat/0507009)\]. Collaboration, T. Yamazaki, [PoS]{} [LAT2005]{} (2006) 351, \[[hep-lat/0509135](http://xxx.lanl.gov/abs/hep-lat/0509135)\]. K. Rummukainen and S. Gottlieb, [Nucl. Phys.]{} [B450]{} (1995) 397, \[[hep-lat/9503028](http://xxx.lanl.gov/abs/hep-lat/9503028)\]. M. Lüscher, [Commun. Math. Phys.]{} [105]{} (1986) 153; [Nucl. Phys.]{} [B354]{} (1991) 531. Collaboration, S. Aoki [et al.]{}, [Phys. Rev.]{} [D58]{} (1998) 054503, \[[hep-lat/9711046](http://xxx.lanl.gov/abs/hep-lat/9711046)\]. [^1]: Present address: University of Connecticut, Physics Department, U-3046 2152 Hillside Road, Storrs, Connecticut 06269-3046, US
--- abstract: 'The dynamics of the expansion of the first order spatial coherence $g^{(1)}$ for a polariton system in a high-Q GaAs microcavity was investigated on the basis of Young’s double slit experiment under 3 ps pulse excitation at the conditions of polariton Bose-Einstein condensation. It was found that in the process of condensate formation the coherence expands with a constant velocity of about $10^8$ cm/s. The measured coherence is smaller than that in thermally equilibrium system during the growth of condensate density and well exceeds it at the end of condensate decay. The onset of spatial coherence is governed by polariton relaxation while condensate amplitude and phase fluctuations are not suppressed.' author: - 'V.V. Belykh' - 'N.N. Sibeldin' - 'V.D. Kulakovskii' - 'M.M. Glazov' - 'M.A. Semina' - 'C. Schneider' - 'S. Höfling' - 'M. Kamp' - 'A. Forchel' title: Coherence expansion and polariton condensate formation in a semiconductor microcavity --- One of the most important characteristics of Bose-Einstein condensate is the spatial coherence or the off-diagonal long range order, i.e. the property of the system to share the same wave function at different points separated by a distance larger than the thermal de Broglie wavelength. To understand the processes governing the Bose-Einstein condensation (BEC), it is important to know how fast the coherence is established throughout the system during the condensate formation. This question was addressed theoretically in Refs. [@Kagan1; @Kagan2], where it was shown that in the process of a condensation particles first relax to the low energy or so-called coherent region where the kinetic energy of the particle is of the order of its interaction energy with other particles. Second, the fluctuations of density are smoothed out and a “quasicondensate” is formed. Further, the phase fluctuations disappear resulting in the long range order formation within the system signifying the onset of the “true condensate”. The timescales of these processes are quite different: the relaxation to the low energy region is mainly determined by the stimulated scattering processes, whereas the quasicondensate formation is governed by the interparticle interactions. Experimentally, the dynamics of spatial coherence formation was first studied for a gas of ultracold atoms in Ref. [@Ritter1], where it was found that the coherence expands with a constant velocity of about 0.1 mm/s. In this Letter we discuss the expansion of the spatial coherence in a condensate of mixed exciton-photon states, polaritons, in a semiconductor microcavity (MC) with embedded quantum wells. The bosonic statistics of these particles and the extremely light effective mass $m$ ($\sim 10^{-4}$ of the free electron mass $m_e$) allow observing MC polariton BEC up to the room temperatures, which inspired a considerable attention to this system in the last decade. Up to now a number of bright phenomena in the MC polariton system have been observed and discussed: polariton BEC [@Kasp1], superfluidity [@Amo1], quantized vortices [@Lag1], spin-Meissner [@Lar1] and Josephson effect [@Lag2] (see [@Sanvitto1] for a review). Compared with atomic BEC, the MC polariton condensation is highly specific: the polariton system is strongly nonequilibrium due to the short lifetime [@Tassone1]. The first order spatial coherence function characterizes the ability of the polariton system to interfere [@Ons], and is related to off-diagonal elements of density matrix $\varrho(\bm \rho_1,\bm \rho_2)$ in the coordinate space: $$\label{def} g^{(1)}(\bm \rho_1 - \bm \rho_2) = \frac{\varrho(\bm \rho_1,\bm \rho_2)}{\sqrt{\varrho(\bm \rho_1,\bm \rho_1)\varrho(\bm \rho_2,\bm \rho_2)}}.$$ The $g^{(1)}$ can be probed in the MC polariton system by measuring the interference of the light emitted from different points on the sample, since the amplitude and phase of the electric field of the cavity emission is directly proportional to the amplitude and phase of the wavefunction of the polariton condensate [@Kasp1; @Deng1; @Szy]. Dynamics of $g^{(1)}$ in a process of MC polariton condensation was studied experimentally for the first time in a CdTe-based MC for the fixed separation between condensate regions [@Nar1] and very recently in a GaAs-based MC [@Ohadi1]. In the present work the spatial coherence dynamics of the polariton system in a high-Q GaAs-based MC is studied in detail for different separations $\Delta x$ between condensate regions. As a result it was found for the first time that the coherence expands with almost constant velocity and its value was measured. By tracing the dependencies $g^{(1)}(\Delta x)$ at different times we extract the dynamics of the coherence length $r_c$. We have found that in the time range of the condensate decay, the coherence is larger than that in the equilibrium system, which is highly unexpected. Our study indicates that in a polariton system under pulsed excitation of the MC the formation of the spatial coherence is governed by the processes of polariton relaxation towards the ground state, whereas amplitude and phase fluctuations of the quasicondensate are not suppressed. The sample is a half wavelength MC with Bragg reflectors made of 32 (for the top mirror) and 36 (for the bottom mirror) AlAs and Al$_{0.13}$Ga$_{0.87}$As pairs. It has a Q-factor of about 7000 and the Rabi splitting of 5 meV. The experiments were performed at $T=10$ K and photon-exciton detuning of $-9$ meV. The sample was excited by the radiation of a mode-locked Ti-sapphire laser generating a periodic ($f=76$ MHz) train of 2.5-ps-long pulses at the reflection minimum of the mirror 11 meV above the bare exciton energy. The beam was focused in a 20 $\mu$m spot on the sample surface. The spot was imaged with magnification of $\Gamma=6$ on the light-absorbing plate with two transparent parallel slits [@Deng1]. The interference pattern of the emission coming from the regions of the sample selected by the two slits was formed on the slit of the Hamamatsu streak camera, operating with time resolution of 3 ps. Spatial coherence $g^{(1)}$ was extracted as the visibility of the interference pattern $g^{(1)}=(I_{max}-I_{min})/(I_{max}+I_{min})$, where $I_{min}$ and $I_{max}$ are minimal and maximal intensities within one period of interference pattern, averaged over all the observed periods. The time-resolved MC emission spectra were recorded by a spectrometer coupled to the streak camera with spectral (temporal) resolution of 0.25 meV (20 ps). To convert the intensity $I(t)$, measured by the streak-camera to the number of polaritons $N$ at states with wavevectors $\|k\|<3$ $\mu$m$^{-1}$ (the collection aperture is $40^0$), the integrated intensity of the MC photoluminescence (PL) $I_{PL}$ was measured by the sensitive power meter. The number of polaritons was evaluated by the relation $N(t)= 2 I_{PL}\tau_{LP}I(t)/(f \hbar\omega \int I(t)dt)$, where factor 2 takes into account two directions of photon emission, $\hbar\omega$ is the energy of emitted photons, $\tau_{LP} \approx 3$ ps is the polariton lifetime at the bottom of the lower polariton (LP) branch. ![Dynamics of the energy (full squares, left axis) and FWHM (empty circles, right axis) of the LP emission line. $E_{MC}$ and $E_{LP}$ are marked by black arrows. Inset shows the number of LPs near $k=0$. Excitation power is $1.8 P_{thr}$.[]{data-label="Fig-IEW"}](Fig1.eps){width="0.8\columnwidth"} At low excitation density $P$ the PL dynamics of the LP branch is relatively slow, and the angular distribution of intensity indicates a bottleneck effect. As $P$ is increased above the BEC threshold $P_{thr}=0.7$ kW/cm$^2$ (this value corresponds to the time-averaged power of pulsed excitation), a fast and intense component in the PL dynamics corresponding to $k \approx 0$ appears. Onset of the fast component is accompanied by the blueshift of the spectral line and decrease of its width (Fig. \[Fig-IEW\]). The energy position of the spectral line is close the bare MC mode $E_{MC}$ just after the excitation pulse and relaxes to the energy of LP mode $E_{LP}$ with time. The maximum population (inset in Fig. \[Fig-IEW\]) is reached when the spectral line energy is between the MC and LP modes indicating that BEC is observed in the strong coupling regime despite the high particle density [@Keel1; @Kamide1; @Byrnes1]. Below $P_{thr}$ interference fringes in the double slit experiment are observed only for the smallest studied slits separation, and $g^{(1)}$($\Delta x =3$ $\mu$m)$<0.3$ at $0.9P_{thr}$. Above $P_{thr}$ the interference fringes are well resolved up to $\Delta x = 20$ $\mu$m (Fig. \[Fig-g1\](a)). The dynamics of $g^{(1)}$ for different $\Delta x$ at $P=1.8P_{thr}$ is presented in Fig. \[Fig-g1\](b) together with the dynamics of the polariton number $N$ at the bottom of the LP branch. Figure \[Fig-g1\](b) shows that the maximum value of $g^{(1)}$ decreases with increased $\Delta x$ and the decay of $g^{(1)}$ in the whole range of $\Delta x$ occurs much slower than that of the condensate density in agreement with [@Nar1]. Interestingly, Fig. \[Fig-g1\](b) shows that the coherence buildup time increases with an increase of $\Delta x$, indicating a finite velocity of the coherence expansion. We define the coherence buildup time $t_{0.5}(\Delta x)$ as the time when $g^{(1)}$ reaches half of its maximum value for a given $\Delta x$. These times are marked by arrows in Fig. \[Fig-g1\](b). Figure \[Fig-g1\](c) shows that the dependence of $t_{0.5}$ on $\Delta x$ is close to linear indicating that the coherence expands with almost constant velocity $v_c=0.6\cdot10^8$ cm/s. Figure \[Fig-rc\] shows the dynamics of the coherence length $r_c$, which is defined from $g^{(1)}(r_c)=1/e$, and number of polaritons at the bottom of the LP branch $N$ for $P>P_{thr}$. At the beginning of the BEC, $r_c$ grows almost linearly with time, reaches its maximum and decays afterwards. It follows from Fig. \[Fig-rc\] that $r_c$ and $N$ reach their maximal values almost at the same time, but the decay of $N$ at high $P$ occurs much faster than that of $r_c$: $N$ decays with the lifetime of 20-30 ps whereas $r_c$ decreases at most by $50 \%$ during the first 30-40 ps and then remains nearly constant during the next several tens of picoseconds where $N$ decreases by more than an order of magnitude. Furthermore, Fig. \[Fig-rc\] shows that the buildup of the coherence at $P=1.2P_{thr}$ begins when the particle number $N$ is more than one order of magnitude smaller than that at $P=4.2 P_{thr}$ and that the maximal value of $r_c$ decreases with $P$ at $P>2 P_{thr}$ in spite of a strong (about an two orders of magnitude) increase of the condensate density. These facts show that the length of coherence, as well as the rate of its increase, is not solely defined by the condensate occupation because of the nonequilibrium nature of the polariton BEC. In a 2D system of a finite size $L$ under the conditions of thermal equilibrium the number of particles in the ground state can be estimated as $$N_0=N-N'=N-\int_{1/L}^{\infty} \frac{k dk L^2}{\pi}(e^\frac{\hbar^2 k^2}{2m k_B T}-1)^{-1}, \label{N0}$$ where $N$ is the total number of LPs, $N'$ is the number of LPs in all states but the ground, $k_B$ is the Boltzmann constant. Here the chemical potential $\mu=0$. It follows from Figs. \[Fig-g1\](b) and \[Fig-rc\] that the onset of spatial coherence at $P < 2 P_{thr}$ starts at $N \sim 10^2$. This value is less than the estimated from Eq. (\[N0\]) $N' \approx 400$ at $T=10$ K in the investigated LP system with the lateral size $L\approx 17$ $\mu$m and $m=5 \cdot 10^{-5} m_e$. Thus, the formation of spatial coherence starts at negative chemical potential $\mu$ with respect to the bottom of the LP branch (e.g. $\mu \sim -0.05$ meV for $P=1.8P_{thr}$, $t=40$ ps) and hence governed by the relaxation process of polaritons to the low energy region, while interaction-induced suppression of amplitude and phase fluctuations plays no role [@Kagan1; @Kagan2]. We note also, that at $P<2P_{thr}$ the estimated value of the interaction energy for the condensed polaritons $\Delta E = \alpha N_0/L^2<2$ $\mu$eV $\ll k_B T \approx 1$ meV (where $\alpha=10^{-12}$ meVcm$^2$ is the polariton-polariton interaction constant). Hence, $g^{(1)}$ should be close to that for the classical noninteracting gas with the particle distribution function $N_k$ (see Eq. and [@Deng1]): $$g^{(1)}(\Delta x) =\frac{\sum_{\bm k} N_k \mathrm e^{\mathrm i k \Delta x}}{\sum_{\bm k} N_k}= \frac{\frac{L^2}{\pi}\int^{\infty}_{1/L}J_0(k \Delta x)N_k k dk+N_0}{\frac{L^2}{\pi}\int^{\infty}_{1/L}N_k k dk+N_0}, \label{g1:class}$$ where $J_0(x)$ is the zero order Bessel function. The detailed modeling of the polariton distribution similar to that in Refs. [@Kin] is beyond the scope of the present paper. Here we discuss the reason of the difference between the experimental dependences $g^{(1)}(\Delta x)$ and calculated ones for the thermal Bose distribution of LPs with the use of $T=10$ K and $\mu$ determined from the measured polariton number $N$ in the low energy region. The difference between the calculated and measured curves indicates how far the system is from the thermal equilibrium. The results of the calculation for three consecutive stages: condensation onset, maximum condensate density and its decay, are shown in Fig. \[Fig-g1\](d) for $P=1.8 P_{thr}$ and the condensate size $L=17$ $\mu$m determined from the experiment. It is seen that the experimental dependence $g^{(1)}(\Delta x)$ is slightly below the calculated thermally equilibrium one in the first stage ($t=45$ ps), the difference strongly increases in the second stage ($t=85$ ps) whereas in the third stage ($t=160$ ps) experimental values $g^{(1)}(\Delta x)$ turn out well above the calculated ones. This result is especially surprising as it indicates that the occupation of $k \approx 0$ states with respect to that of higher-energy states exceeds the thermally equilibrium one, i.e. the effective polariton temperature is lower than the bath temperature in the time range of condensate decay. Measurements of polariton distribution along the LP branch (not shown) indicate that the distribution function approaches the thermal one at the onset of condensation, but the occupation numbers of low $k \neq 0$ states are slightly increased compared to the thermal values, which explains a small discrepancy between the experimental and calculated $g^{(1)}(\Delta x)$ at $t=45$ ps, when $N \approx 350 < N'$. An increase in the discrepancy between the experimental and calculated $g^{(1)}(\Delta x)$ in the range of maximal $N$ at $t=85$ ps when $N \approx 7000 \gg N'$ indicates that the LP system becomes more nonequilibrium at high condensate density. The most probable reason for that is the runaway of condensed polaritons from the small bounded ($\sim 20$ $\mu$m) photoexcited region due to their repulsive interaction with a dense exciton reservoir [@Brichkin]. In addition, for the equilibrium system with $N_0 \gg N'$, the coherence is defined by the amplitude and phase fluctuations of the ground state wavefunction not taken into account in Eq. (\[g1:class\]). Indeed, the quasicondensate amplitude fluctuations are suppressed during the time $\tau_A \approx \hbar L^2 / (\alpha N)$ [@Kagan1; @Kagan2], determined by the interparticle interaction. For $N=7000$, $L=17$ $\mu$m we obtain value $\tau_A \sim 0.3$ ns that exceeds the lifetime of LP condensate at $P=1.8 P_{thr}$ (Fig. \[Fig-rc\]). It follows then, that the amplitude and, especially, phase fluctuations are not suppressed. Finally, let us discuss the reason for the unexpectedly high coherence in the decaying polariton system at $t \sim 160$ ps with $N \approx 200 <N'$ which exceeds markedly the coherence in the thermally equilibrium system. Note that the calculations underestimate experimental values of $g^{(1)}$ even for the unrealistically small condensate size $L=12$ $\mu$m (dashed line in Fig. \[Fig-g1\]d). At large $t > 100$ ps the reservoir becomes highly depleted, and the equilibrium in the system is established mainly via ineffective excitation of condensed polaritons by acoustic phonons with characteristic time exceeding the condensate decay time $\tau$. In this case the wave function at a given distance $\bm\rho$ decays as $\psi(\bm \rho)\propto \exp{(-\frac{t}{2\tau})}$. Thus, both diagonal and off-diagonal elements of the polariton density matrix decay approximately with the same rate $1/\tau$ resulting in a nearly constant ratio of the LP numbers in the ground state to that in the excited states well exceeding the equilibrium ratio at long times. As a result, $g^{(1)}$ weakly decays with time in agreement with the experiment. ![Dynamics of the coherence length (full squares, left axes, linear scales) and number of particles near the bottom of the LP branch (empty circles, right axes, logarithmic scales) at different excitation powers.[]{data-label="Fig-rc"}](Fig3.eps){width="\columnwidth"} It is worth to compare the measured coherence expansion velocity $v_c$ for the polariton condensate (about $10^8$ cm/s) and that for the atomic condensate (about $10^{-2}$ cm/s [@Ritter1]). The buildup of the spatial coherence is determined by the relaxation of the particles to the ground state and manifests itself by the condition $r_c > \lambda_{dB}$, where $\lambda_{dB}$ is thermal de Broglie wavelength. So one can estimate the coherence expansion velocity as $v_c \sim \lambda_{dB} / \tau_{rel}$. For polaritons $\lambda_{dB} \sim 2$ $\mu$m at $T=10$ K and for atoms $\sim 0.4$ $\mu$m at $T=0.2$ $\mu$K, so the large difference in $v_c$ is related to the difference in the relaxation times $\tau_{rel}$, which is $\sim 10$ ps for polaritons and $\sim 100$ ms for atoms [@Ritter1]. The relaxation for both polaritons and atoms is accomplished via interparticle scattering, while for polaritons scattering with phonons also plays role. The rate of the interparticle collisions depends on the scattering cross sections and on the average velocity of the particles. The latter is determined by the particle mass and temperature which differ in many orders of magnitude ensuring the necessary ratio of the relaxation times for polaritons and atoms. To conclude, we have studied the dynamics of the spatial coherence for a LP condensate under pulsed ps-long excitation for different excitation powers. It has been found that in the process of condensate formation, first order coherence expands with almost constant velocity of about $10^8$ cm/s. We have shown that the coherence is influenced by polariton relaxation from the reservoir. The onset of spatial coherence is determined by the narrowing of polariton distribution in $k$-space rather than formation of the condensate phase, and at high excitation density coherence is limited by condensate amplitude fluctuations. The true condensate i.e. macroscopic occupation of the ground state with suppressed phase fluctuation is not achieved under ps-long pulsed pumping. We are grateful to D.A. Mylnikov for help in the experiment and S.S. Gavrilov, N.N. Gippius, L.V. Keldysh, A.V. Sekretenko, S.G. Tikhodeev and V.B. Timofeev for valuable advice and useful discussions. This study was supported by the RFBR (projects no. 11-02-01310, 11-02-12261, 11-02-00573), RAS, Ministry of Education and Science of the Russian Federation (contract no. 8680), the State of Bavaria, and EU projects POLAPHEN and SPANGL4Q. MAS was partially supported by the RF President Grant NSh-2901.2012.2. [99]{} Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, JETP 75, 387 (1992). Yu. Kagan and B.V. Svistunov, JETP 78, 187 (1994). S. Ritter et al., Phys. Rev. Lett. 98, 090402 (2007). J. Kasprzak et al., Nature 443, 409 (2006). A. Amo et al., Nature Phys. 5, 805 (2009). K.G. Lagoudakis et al., Nature Phys. 4, 706 (2008). A.V. Larionov et al., Phys. Rev. Lett. 105, 256401 (2010). K.G. Lagoudakis, B. Pietka, M. Wouters, R. Andre, and B. Deveaud-Pledran, Phys. Rev. Lett. 105, 120403 (2010). D. Sanvitto and V. Timofeev (editors), Exciton Polaritons in Microcavities (Springer, New York, 2012), Springer Series in solid-state sciences Vol. 172. F. Tassone, C. Piermarocchi, V. Savona, A. 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--- abstract: 'We suggest that there are time-varying quanta of mass (gomidia) and of length (somia), thus pointing to a quantization of geometry and gravitation. The present numerical value of the gomidium and somium , are, $10^{-65}$ grams, and $10^{-91}$ centimeters. Gomidia may be responsible for dark matter in the Universe; Heisenberg’s principle, confirms the numerical estimates for gomidia and somia , either for the present Universe, or for Planck’s time.' author: - 'Marcelo Samuel Berman$^{1}$' date: '29 August, 2007.' title: '[ARE MASS AND LENGTH QUANTIZED?]{}' --- [ARE MASS AND LENGTH QUANTIZED?]{} [ MARCELO SAMUEL BERMAN ]{} We introduce the definitions of micromass and macromass, as well as those of microlength and macrolength, in the spirit of Wesson’s suggestions (Wesson, 2006). We show that by obtaining such quantities for Planck’s time, and the present Universe, both “micros” coincide with Planck’s mass and length, while for the present Universe, macrolength stands as the radius of the causal Universe, while macromass represents the mass of the Universe. We find a quantum of mass (“gomidium”) (Berman, 2007; 2007a), and a quantum of length (“somium”), to which we suggest interpretations. In the end of this paper, we discuss the novelties which appear here, in comparison with what has been already published (for instance, by Wesson, 2006). The definitions of macromass, micromass, macrolength, and microlength, given in this paper, are related with gauge parametrizations in penta-dimensional physics (Wesson, 2006). Dark matter in the Universe, responds for 27% of the total energy density, which is to be represented by the critical one, as far as we accept inflationary scenario (Güth, 1981). So we call the dark matter energy density, $\rho_{\nu}$ , and we write, $\rho_{\nu}=0.27\rho_{crit}$ . (1) Berman (2006 b) along with others (see Sabbata and Sivaram, 1994) have estimated that the Universe possess a magnetic field which, for Planck’s Universe, was as huge as $10^{55}$ Gauss. The relic magnetic field of the present Universe is estimated in $10^{-6}$ Gauss. We can then, suppose that some hypothetical particles with elementary spin, have been aligned with the magnetic field. On the other hand, the spin of the Universe is believed to have increased in accordance with a Machian relation. If we call $n$ the number of gomidia in the present Universe, and $n_{Pl}$ its value for Planck’s Universe, we may write, along with Berman (2007; 2007a), $\frac{n}{n_{Pl}}=\frac{L}{L_{Pl}}=10^{122}$ . (2) Then, $n=n_{Pl}\left[ \frac{R}{R_{Pl}}\right] ^{2}$ . (3) Thus, $n$ grows with $R^{2}$ . Now, we write the energy density of gomidia, $\rho_{\nu}\cong\frac{nm_{\nu}}{\frac{4}{3}\pi R^{3}}$ , (4) where $m_{\nu}$ is the rest mass of the individual gomidium. Berman’s suggestion, in the above citations, imply that all energy densities in the Universe decrease with $R^{-2}$ . Consider for instance, the inertial mass content. Its energy density is given by, $\rho_{i}=\frac{M}{\frac{4}{3}\pi R^{3}}$ . (5) For a Machian Universe, then, $M\propto R$ , from where the $R^{-2}$ inertial energy density appears in (5). If, then, $\rho_{\nu}\propto R^{-2}$ , we find: 1$^{st}.$) $\rho_{\nu}=0.27\rho_{Pl}\left[ \frac{R}{R_{Pl}}\right] ^{-2}$ . (6) 2$^{nd}.$) $m_{\nu}=\frac{\rho_{Pl}R_{Pl}^{4}}{R}$ . (7) We see now that while the number of gomidia in the Universe increases with $R^{2}$ , the rest mass decreases with $R^{-1}$ ; we may obtain, with $R\cong10^{28}$ cm, that the rest mass of * gomidia* should be, in the present Universe: $m_{\nu}\cong10^{-65}$ g . (8) ***A law of variation for the number of gomidia* in the Universe has been found. A law of variation for the rest mass of gomidia was also found. We remind the reader that Kaluza-Klein’s cosmology (Wesson, 1999; 2006; Berman and Som, 1993), considers time varying rest masses, in a penta-dimensional (“induced mass”) space-time-matter, of which the fifth coordinate is rest mass. The above results can not be rejected, for the time being, by any known data. We point out, that some of the features of the present calculation, resemble some points in a paper by Sabbata and Gasperini (1979). [II. Quantization of inertia and geometry]{} Wesson (2006), by citing Desloge(1984), comments that by means of the four fundamental “constants”, Planck’s ($h$), Newton’s ($G$), speed of light ($c$), and cosmological ($\Lambda$), one can obtain two different kind of mass, the micromass ($m$), and the macromass ($M$), given by: $m=\left( \frac{h}{c}\right) \Lambda^{1/2}$ , (9) and, $M=\frac{c^{2}}{G}\Lambda^{-1/2}$ . (10) Micromass involves Planck’s constant, hence its denomination; macromass is defined by means of $G$ , so its “macro” denomination. Notice that the above constant tetrad, is, of course, overlapping. With the present values for the cosmological “constant”, $\Lambda =\Lambda_{U}\approx10^{-56}$ cm$^{-2}$ , it is found, $m_{(U)}\approx10^{-65}$ g , (11) and, $M_{(U)}\approx10^{56}$ g . (12) The present Universe’s micromass ($m_{(U)}$), represents a present mass-quantum, i.e., the minimum mass in the present Universe. On the other hand, the present Universe’s macromass ($M_{(U)}$), is approximately the mass of the present Universe ($M_{U}$) . What Wesson overlooked, is that, when we apply the definitions (9) and (10), by plugging, $\Lambda=\Lambda_{PL}\approx L_{PL}^{-2}\approx10^{-66}$ cm$^{-2}$ , which stand for Planck’s time values, we find that micromass and macromass coincide approximately with Planck’s mass, $M_{PL}$ , $M_{(PL)}=m_{(PL)}=M_{PL}\approx10^{-5}$ g . (13) We are led to consider that, macromass, is always associated to the mass of the Universe ($M_{U}$), either in the very early Universe or in the present one. As to the micromass, we baptize this mass as the quantum mass value (gomidium, after F.M.Gomide): it is a time-varying mass, because $\Lambda$ is also so, because we expect its energy density $\frac{\Lambda}{\kappa}$ depend on $R^{-2}$ altogether. We now show, that associated with micromass and macromass, we have two distinct length values, which come associated to the present and Planck’s Universe. For each mass, we associate two kinds of lengths, namely, the macrolength, ($\lambda_{\nu}$), and the microlength ($l_{\nu}$); the first one, is Compton’s wavelength, given by, $\lambda_{\nu}=\frac{\bar{h}}{mc}$ . (14) It is a “macro”, because it is inversely proportional to micromass. The second length, that we call “microlength”, is a gravitationally associated length with mass, which we term the quantum of length, or somium , $l_{\nu}=\frac{Gm}{c^{2}}$ . (15) It is a “micro”, because it is proportional to micromass. We find, a microlength, $l_{\nu}$ ,for the present Universe, with $m=m_{\nu}=m_{(U)}$ , $l_{\nu(U)}\approx10^{-91}$ cm, (16) while, the macrolength is then found, $\lambda_{\nu(U)}\approx10^{28}$ cm . (17) On the other hand, for Planck’s Universe, the macrolength is found to be, $\lambda_{\nu(PL)}\approx10^{-33}$ cm , (18) and, the microlength has the same numerical value, $l_{\nu(PL)}\approx10^{-33}$ cm . (19) One can check, that the microlength represents a quantum, the somium. Macrolength is represented by the radius of the Universe. For Planck’s Universe, the somium, the macrolength, and the microlength then coincide with the Planck’s radius, $L_{PL}$ . According to Heisenberg’s uncertainty principle, any two conjugate quantities, in the sense of Hamilton’s canonical ones, carry uncertainties, $\Delta Q$ and $\Delta P$ , which obey the condition (Leighton, 1959), $\Delta Q\Delta P\approx h$ . (20) If we consider maxima $\Delta P$ , we obtain minima $\Delta Q$ . If $\Delta P$ stands for the uncertainty in linear momentum, given, say, by the product of mass and speed, then, its maximum value must be the product of the largest mass in the Universe by the largest speed, $\Delta P=M_{U}$ $c$ . (21) We thus, obtain a minimum length value, $\Delta Q\approx\frac{h}{c\text{ }M_{U}\text{ }}$ . (22) Now, let us think of the largest time numerical value in the Universe, $t_{U}\approx10^{10}$ years. (23) Its conjugate variable, will point out to a minimum inertial energy and a minimum inertial mass ( $\Delta m$ ) , $\bigskip$ $\Delta E=c^{2}\Delta m\approx\frac{h}{t_{U}}$ . (24) It turns out, that we have retrieved, from $\Delta m$ , and $\Delta Q$ , the minimum mass and length for the present Universe, with the same approximate values attached to gomidia and somia , in last Section, also for the present Universe. Analogously, we could repeat the calculation for Planck’s time and Planck’s mass, and we then would obtain numerically the same values attained by gomidia and somia in Planck’s Universe. We have, then, support for the quantization of mass and length, in a time-varying fashion, coinciding with the calculation in the last section. Wesson (2006), dealt classically with $D=5$ WEP (weak equivalence principle), as a symmetry in the “induced mass” pentadimensional Kaluza-Klein theory, or even the brane one. The new fifth forces and coordinate are then present. The geodesic equation adds and extra-acceleration. Microlength and macrolength were found by Wesson, to be good gauge parametrizations for mass, which allow a mass geometry consistent with the rest of Physics. The known laws of Mechanics and conservation of linear momentum, in limiting cases, were also preserved when $m$ is a representation of rest-mass. Because of the standard structure of $D=5$ Physics as an extension of the $D=4$ chapter, while introducing an extra coordinate, opens the possibility of quantization in the lower dimension Physics. Wesson even advanced that the Quantum domain would extend to the Cosmos, in the form of a broken symmetry for the angular momenta tied to the gravitational field. We have therefore, found in our present paper, that the micromass and microlength represent quanta of mass and length. We call them, respectively, gomidium and somium, but their numerical values are time-varying: present day’s gomidium is 10$^{-65}$ g, while somium is about 10$^{-91}$ cm. Planck’s values for gomidium and somium, coincide respectively with Planck’s mass and Planck’s length. We have thus hinted that mass is quantized, but geometry is altogether. As gravitation is associated with geometry, quantization of the latter, implies on the former: it seems that quantum gravity has been found. Much of what we have calculated here, like the Machian derivation, which led to time-varying quanta of mass and length, and also the interpretation, under which macromass and macrolength describe the Universe’s mass and radius, throughout its lifespan, (in particular, Planck’s and present times) are novelties in the literature. Though some of the topics dealt in our paper, were sparsely dealt in Wesson’s books (Wesson, 1999; 2006), and by other authors, we have here given a rational interpretation of otherwise disconnected elements. The numerical value for the present Universe’s microlength (somium), seems to have never appeared in the literature, in any context; our quantization ideas, as far as I know, are also novel; dark matter has been associated with gomidia. All the above, supported by Heisenberg’s uncertainty principle. The author expresses his recognition to his intellectual mentors, now friends and colleagues, M.M. Som and F.M. Gomide. He thanks the many other colleagues that collaborate with him, and the Editor of this Journal, for providing an important objection towards an earlier manuscript. The typing was made by Marcelo F. Guimarães, who I consider a friend and to whom my thanks are due for this and many other collaborations. [References]{} Berman,M.S. (2006) - Energy of Black-Holes and Hawking’s Universe in Trends in Black-Hole Research, Chapter 5. Edited by Paul Kreitler, Nova Science, New York. Berman,M.S. (2006 a) - Energy, Brief History of Black-Holes, and Hawking’s Universe in New Developments in Black-Hole Research, Chapter 5. Edited by Paul Kreitler, Nova Science, New York. Berman,M.S. (2006 b) - Los Alamos Archives, http://arxiv.org/abs/physics/0606208. Berman,M.S. (2007) - Introduction to General Relativity, and the Cosmological Constant Problem, Nova Science, New York.. Berman,M.S. (2007a) - Introduction to General Relativistic and Scalar-Tensor Cosmologies, Nova Science, New York.. Berman,M.S.; Som, M.M. (1993) - Astrophysics and Space Science, 207, 105. Brans, C.; Dicke, R.H. (1961) - Physical Review, 124, 925. Chen,W.;Wu,Y.- S. (1990) - Phys. Review D 41,695. Desloge, E.A. (1984) - Am. J. Phys. 52, 312. Gomide, F.M.(1963) - Nuovo Cimento, 30, 672. Guth, A. (1981) - Physical Review, D23, 347. Leighton, R.B. (1959) - Principles of Modern Physics, McGraw-Hill, N.Y. Sabbata, V. de; Gasperini, M. (1979) - Lettere al Nuovo Cimento. 25, 489. Sabbata, V. de; Sivaram, C. (1994) - Spin and Torsion in Gravitation, World Scientific, Singapore. Wesson, P.S. (1999) - Space-Time-Matter, World Scientific, Singapore. Wesson, P.S. (2006) - Five dimensional Physics, World Scientific, Singapore.
--- abstract: \| The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction-diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying several metabolic processes taking place in living cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes. The coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms. These nonlinearities, which model, at the microscopic scale, various volume or surface reaction processes, give rise in the macroscopic model to different effects, such as the appearance of additional source or sink terms or of a non-standard diffusion matrix.\ [Keywords]{}: homogenization; nonlinear flux conditions; reaction-diffusion equations. [AMS subject classifications]{}: 35B27; 35K57; 35Q92. author: - 'G. Cardone[^1], C. Perugia[^2], C. Timofte[^3]' title: 'Homogenization results for a coupled system of reaction-diffusion equations' --- Introduction\[sect1\] ===================== The goal of this paper is to analyze, via periodic homogenization techniques, the macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction-diffusion phenomena in periodic porous media. Our results can be useful for studying some metabolic processes occurring in biological cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the membranes of various organelles, which are present in the cytoplasm, such as chloroplasts, endoplasmic reticulum or mitochondria. From a mathematical point of view, we consider a perforated domain $\Omega^{\ast}_\varepsilon$, obtained by removing from a smooth domain $\Omega$ a set of periodically distributed inclusions, $\varepsilon$ being a positive small parameter related to the size and the periodicity of the structure. For applications in biology, the domain $\Omega$ can be seen, in a simplified setting, as the cytoplasm of a biological cell, consisting of the cytosol $\Omega^{\ast}_\varepsilon$ and of various organelles $S^\varepsilon$, periodically distributed, with period $\varepsilon$, in the cytoplasm (see Section \[Sec2\]). In such a geometry, we consider, at the microscale, a coupled system of three reaction-diffusion equations, governing the evolution of the concentration of three types of biochemical species (metabolites) in the perforated domain (cytosol), with suitable nonlinear boundary conditions at the surface of the inclusions (at the organellar membranes) and initial conditions. On the outer cellular membrane $\partial \Omega$, we impose Dirichlet conditions (see system and [@DV]). The model assumes that the coupling between the concentrations of the biochemical species is realized through nonlinear reaction terms, appearing in the perforated domain and at the boundary of the inclusions, as well. In particular, the coupling phenomenon at the boundary of the inclusions is described by imposing different nonlinear flux conditions for the three concentration fields. The nonlinear functions appearing in these flux conditions depend on the concentrations of the three species and are properly scaled, depending on the processes occurring at the membranes of the cellular organelles. The nonlinearities of our model were inspired by kinetics corresponding to multi-species enzyme catalyzed reactions, which are generalizations of the classical Michaelis-Menten kinetics to multi-species reactions. A rigorous mathematical model for such processes is analyzed in [@Gahn]$\div$[@Gahn3], but with a different scaling and taking into account more complex phenomena.\ The main mathematical difficulties behind our study come from the strong coupling between the equations governing the evolution of the concentrations of the biochemical species and, also, from the particular scaling of the nonlinear boundary terms. As a matter of fact, the novelty of our paper consists exactly in the special coupling and scaling of the boundary terms. The form of the nonlinearities arising at the microscale and this non-standard coupling lead, at the macroscale, to the appearance of additional source or sink terms and of a non-standard diffusion matrix, which is not constant (see Theorem \[teounf\] and Corollary \[teohom\]). Hence, we show that some fast reactions occurring, at the microscale, on the boundaries of $S^\varepsilon$ (see the second equation in system ) could lead, at the macroscale, to a cross - diffusion system (see for this terminology the recent papers [@JP] and [@J]).\ The metabolic pathway which our system better fits is the fatty acid activation on the outer mitochondrial membrane (see [@Enz]). Such a reactive system involves three metabolites, a fatty acid, the adenosine triphosphate (ATP) and the coenzyme A (CoA), which can diffuse in the cytosol and can also react on the mitochondrial membrane with different rate, with ATP being the slowest one. They can react in the cytosol thanks to the enzyme acyl coa - synthetase present on the outer mitochondrial membrane. Since this metabolic pathway is strongly ATP-dependent, the concentration of ATP itself limits the mithocondrial availability of fatty acid and CoA. This aspect could well explain why, in our limit problem, the effective diffusion matrices of the fastest metabolites depend on the concentration of the slowest one. Indeed, a shortage of ATP in the cytosol would imply the stop of the enzyme catalyzed reaction previously described limiting the diffusion of the fatty acid and the CoA in the whole cell.\ Our results might find some applicability also in the case of a porous medium in which various chemical substances are allowed to diffuse and to react inside and on the walls of the porous medium and in multiscale modeling of colloidal dynamics in porous media (see, for instance, [@Hornung] and [@KMK]).\ As already mentioned, the problem we consider here comprises nonlinear terms, properly (differently) scaled, asking, therefore, for different analytical techniques. For obtaining our macroscopic model, we use suitable extension operators (see [@Cio-Pau], [@Hopker], [@Bohm], and [@Meirmanov]) and the periodic unfolding method, adapted to time-dependent functions (see, for instance, [@Cio-Dam-Don-Gri-Zaki], [@Cio-Dam-Gri1], [@Cioranescu-Donato-Zaki], [@Amar1], and [@Amar2]). Related homogenization problems for reaction-diffusion problems in porous media were addressed, for instance, in [@Hornung], [@Hor-Jag], [@Jager], [@Neu-Jag], [@Muntean1], [@Malte], [@Fat-Mun-Pta], [@Gahn], [@Gahn2], [@Gahn3], [@Graf], [@Mar-Pta], [@Pop1], [@Pop2], [@Timofte1], and [@Timofte2]. Problems involving fast surface reaction terms can be encountered, for instance, in [@Amar], [@Allaire], [@Allaire1], and [@Iji-Mun]. As far as we know, there are only a few recent papers in which homogenized matrices depending on the solution itself appear. Non-constant homogenized matrices in which a nonlinear function depending on the solution of the macroscopic problem appears can be found, in different contexts from ours and for particular types of nonlinearities, in the recent papers [@Allaire], [@Iji-Mun], and [@Bun-Tim]. For the case in which the homogenized matrix depends on the solution of the limit problem, we also refer the reader to [@Allaire1], [@Allaire2], [@Allaire3], [@Timofte2], and [@DLN]. The rest of the paper is structured as follows: in Section \[Sec2\], we describe the geometry of the periodic porous medium and we set the microscopic problem. Existence and uniqueness results and [a priori]{} estimates for the solution of our microscopic problem are obtained in Section \[Sec3\]. The main homogenization results of this paper are stated and proven in Section \[Sec4\]. Setting of the problem\[Sec2\] ============================== We start this section by describing the geometry of the problem. We assume that the porous medium possesses a periodic microstructure. Let $Y =(0, 1)^n$ be the reference cell and $S$ be a closed strict subset of $\overline{Y}$ with the boundary $\Gamma:=\partial S$ being Lipschitz continuous. We restrict ourselves to the case in which $S$ is connected, but this assumption can be easily removed, since we can deal with the case in which $S$ has a finite number of connected components. Let $Y^:=Y\setminus S$. The sets $S$ and $Y^$ will be the reference inclusion and the perforated cell, respectively. Now, let us consider a bounded connected smooth open set $\Omega$ in $\Bbb R^n$, with a Lipschitz boundary $\partial \Omega$ (we shall be interested in the physically relevant cases $n=2$ or $n=3$). Let us notice that, for simplicity, we restrict ourselves here to a case in which the domain $\Omega$ is supposed to be representable by a finite union of axis-parallel cuboids with corner coordinates in $\Bbb Z^n$ (see [@Gra-Pet], [@Bohm], and [@Hopker]), but the results given in this paper still hold true for more general domains $\Omega$ (see [@Bohm] and [@Hopker]). Let $\varepsilon\in (0,1)$ be a small positive parameter, related to the characteristic dimensions of the structure and which takes values in a sequence of strictly positive numbers tending to zero, such that the stretched domain $\varepsilon ^{-1} \Omega$ can be represented as a finite union of axis-parallel cuboids having corner coordinates in ${\mathbb Z}^n$. For each ${\bf k}\in \Bbb Z^n$, let $Y_{\bf k}={\bf k}+Y^$, $\Gamma_{\bf k}:={\bf k}+\Gamma$, and $K_\varepsilon:=\{{\bf k}\in \Bbb Z^n \; \| \; \varepsilon Y_{\bf k} \subset \Omega\}$. Then, setting $S^\varepsilon:=\bigcup\limits_{{\bf k}\in K_\varepsilon}\varepsilon({\bf k}+S)$, the perforated domain $\Omega^{\ast}_{\varepsilon}$ is obtained by removing from $\Omega$ the set of inclusions $S^\varepsilon$, i.e. $\Omega^{\ast}_\varepsilon:=\Omega\setminus S^\varepsilon$. Moreover, if $\Gamma^\varepsilon:=\bigcup\limits_{{\bf k}\in K_\varepsilon}\varepsilon\Gamma_{\bf k}$ denotes the inner boundary of the porous medium, it holds that $\Gamma^\varepsilon\cap \partial \Omega=\emptyset$. In such a periodic microstructure, we shall consider a coupled system of nonlinear reaction-diffusion equations, with suitable boundary and initial conditions. The coupling phenomenon at the boundary of the inclusions is described by imposing suitably scaled nonlinear flux conditions for the concentrations. More precisely, if we denote by $[0,T]$, with $T>0$, the time interval of interest, we shall analyze the effective behavior, as the small parameter $\varepsilon \rightarrow 0$, of the solution $(c^{\varepsilon}_1, c^{\varepsilon}_2, c^\varepsilon_3)$ of the following system: $$\left\{ \begin{array} {ll} \partial_{t}c_{i}^{\varepsilon}-\hbox{div} (D_i^\varepsilon \nabla c_{i}^{\varepsilon })=F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon})&\text{ in }(0,T)\times\Omega^{\ast}_{\varepsilon},\,\, i\in \{1,2,3\}\\ \\ D_i^\varepsilon \nabla c_{i}^{\varepsilon}\cdot\nu^\varepsilon=\displaystyle \frac{1}{\varepsilon}\, G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon})&\text{ on }(0,T)\times \Gamma^{\varepsilon},\, \, i\in \{1,2\}\\ \\ D_3^\varepsilon \nabla c_{3}^{\varepsilon}\cdot\nu^\varepsilon=\varepsilon\, G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon})&\text{ on }(0,T)\times \Gamma^{\varepsilon}\\ \\ c_{i}^{\varepsilon}=0 & \text{ on }(0,T)\times\partial\Omega,\, \, i\in \{1,2,3\}\\ \\ c_{i}^{\varepsilon}(0)=c_{i}^0 & \text{ in }\Omega^{\ast}_{\varepsilon},\, \, i\in \{1,2,3\}, \end{array} \right. \label{eqmicro}$$ where $\nu^\varepsilon$ is the unit outward normal to $\Gamma^{\varepsilon}$.\ In , the concentrations $c_{i}^{\varepsilon}$ of the chemical species (metabolites), the diffusion matrix $D^\varepsilon$ and the nonlinear reactions rates $F_{i}^{\varepsilon}$, $i\in \{1,2,3\}$, $G_{i}$, $i\in \{1,2\}$, and $G_{3}^{\varepsilon}$ satisfy suitable conditions. More precisely, we make the following assumptions on the data (see also [@Gahn; @Conca-Diaz-Timofte; @DLN]):\ \ $(\mathbf{H}_1)$ For $i\in \{1,2,3\}$, the diffusion matrices are given by $D_i^\varepsilon(x)=D_i\left(\dfrac{x}{\varepsilon}\right)$, where - $D_i\in (L^\infty(Y))^{n\times n}$ is a $Y$-periodic and symmetric matrix-field; - for any $\xi\in\mathbb{R}^{n}$, $\alpha \|\xi\|^2 \leq (D_i(y)\xi, \xi) \leq \beta \|\xi\|^2$, almost everywhere in $Y$, for some positive real constants $\alpha$ and $\beta$. $(\mathbf{H}_2)$ For $i\in \{1,2,3\}$, the reaction rates are given by $F_{i}^{\varepsilon}(x,s_{1},s_{2},s_{3})=F_i(\frac {x}{\varepsilon},s_{1},s_{2},s_{3})$, where $F_i:\mathbb{R}^{n}\times\mathbb{R}^{3}\rightarrow\mathbb{R}$ satisfies the following hypotheses: - $F_i$ is continuous; - $F_i(\cdot,s)$ is a $Y$-periodic function for all $s\in \mathbb{R}^{3}$; - $F_i(y,\cdot)$ is Lipschitz continuous for all $y\in \mathbb{R}^{n}$ with constant independent of $y$; - $F_i(y, 0)=0$ for all $y\in \mathbb{R}^n$. $(\mathbf{H}_3)$ For $i\in \{1,2\}$, the nonlinear functions in the first flux condition are given by $$\label{formG} G_{i}(s_{1},s_{2},s_{3})=(-1)^i \left (s_1-s_2\right ) \, H \left (s_{3}\right),$$ where $H: \Bbb R \rightarrow \Bbb R^_+$ satisfies the following hypotheses: - there exists a positive constant $l$ such that $$0\leq H(s)\leq l<\infty \text{ for all } s\in\Bbb R;$$ - $H(0)=0$; - $H\in C^1(\mathbb{R})$; - there exists a positive constant $L$ such that $$\vert H^{'}(s)\vert \leq L<\infty \text{ for all } s\in\Bbb R.$$ $(\mathbf{H}_4)$ The nonlinear function in the second flux condition is given by $G_{3}^{\varepsilon}(x,s_{1},s_{2},s_{3})=G_3\left (\frac {x}{\varepsilon},s_{1},s_{2},s_{3}\right)$, where $G_3: \mathbb{R}^{n}\times\mathbb{R}^{3}\rightarrow\mathbb{R}$ satisfies the following hypotheses: - $G_3$ is continuous; - $G_3(\cdot,s)$ is a $Y$-periodic function for all $s\in \mathbb{R}^{3}$; - $G_3(y,\cdot)$ is Lipschitz continuous for all $y\in \mathbb{R}^{n}$ with constant independent of $y$; - $G_3(y, 0)=0$ for all $y\in \mathbb{R}^n$. $(\mathbf{H}_5)$ Set $\left( \cdot\right)_- =\min\left\{ \cdot,0\right\}$. We assume that for a.e. $(x,s)\in \mathbb{R}^n\times\mathbb{R}^3$ it holds $$\label{assF2} \begin{array}{l} F_{1}^{\varepsilon}(x,s_{1},s_{2},s_{3})\left( s_{1}\right) _{-} +F_{2}^{\varepsilon}(x,s_{1},s_{2},s_{3})\left( s_{2}\right) _{-} +F_{3}^{\varepsilon}(x,s_{1},s_{2},s_{3})\left( s_{3}\right) _{-}\\ \\ \leq C_1\left( \left\vert \left( s_{1}\right) _{-}\right\vert ^{2}+\left\vert \left( s_{2}\right) _{-}\right\vert ^{2}+\left\vert \left( s_{3}\right) _{-}\right\vert ^{2}\right) \end{array}$$ and $$\label{assG} G_{3}^{\varepsilon}(x,s_{1},s_{2},s_{3})\left( s_{3}\right) _{-} \leq C_2\left( \left\vert \left( s_{1}\right) _{-}\right\vert ^{2}+\left\vert \left( s_{2}\right) _{-}\right\vert ^{2}+\left\vert \left( s_{3}\right) _{-}\right\vert ^{2}\right),$$ with $C_1$ and $C_2$ positive constants independent of $\varepsilon$.\ \ $(\mathbf{H}_6)$ There exist $\Lambda>0$ and $A>0$ such that (see [@Gahn3]), for $i\in \{1,2,3\}$, $$\label{assFG_0} \left\{ \begin{array}{l} F_i(\cdot, s) \leq A s_i \quad \textrm{ for all } s\in \Bbb R^3 \textrm{ with } s_i \geq \Lambda,\\ \\ G_3(\cdot, s) \leq A s_3 \quad \textrm{ for all } s\in \Bbb R^3 \textrm{ with } s_3 \geq \Lambda.\\ \end{array} \right.$$ $(\mathbf{H}_7)$ The initial concentrations $c_{i}^0\in L^{2}(\Omega)$, $i\in \{1, 2, 3\}$, are assumed to be non-negative and bounded independently with respect to $\varepsilon$ by a constant $\Lambda$ (the same constant as the one in assumption $(\mathbf{H}_6)$, i.e. for $i\in \{1,2,3\}$ it holds $$\label{binitial} 0\leq c_i^0\leq \Lambda.$$ From the Lipschitz continuity, we obtain that for all $(x,s)\in \mathbb{R}^n\times \mathbb{R}^3$ the following growth conditions hold: $$\label{F2} \left\vert F_i^\varepsilon (x,s_{1},s_{2},s_{3})\right\vert \leq C_3(1+\|s_{1}\|+\|s_{2}\|+\|s_{3}\| ),\,\,i\in \{1,2,3\}$$ and $$\label{assG1} \left\vert G_3^\varepsilon (x,s_{1},s_{2},s_{3})\right\vert \leq C_4(1+\|s_{1}\|+\|s_{2}\|+\left\vert s_{3}\right\vert ),$$ with $C_3$ and $C_4$ positive constants independent of $\varepsilon$.\ Moreover by $(\mathbf{H}_2)_4$ and $(\mathbf{H}_4)_4$, and become $$\vert F_i^\varepsilon (x,s_{1},s_{2},s_{3})\vert \leq C_5(\|s_{1} \|+\|s_{2}\|+\|s_{3}\| ),\,\,i\in \{1,2,3\}$$ and $$\vert G_3^\varepsilon (x,s_{1},s_{2},s_{3})\vert \leq C_6(\|s_{1} \|+\|s_{2}\|+\|s_{3}\| ).$$ \[time\] For simplicity, we assume that all the parameters involved in our model are time independent, but the case in which they depend on time can be also treated. \[remF\] We point out that we don’t have too much freedom for our nonlinearities and for their scalings, since, when working with systems, it is difficult to ensure positivity, essential boundedness and, most important, strong convergence results. \[remG\] The adsorption/desorption processes and the reactions taking place at the surface of the inclusions depend on many factors, such as the nature of the molecules (various molecules can have very low or very high adsorption, some molecules can adsorb and react with other molecules, which do not adsorb), the value of the concentrations of the involved species, the nature of the solid surface, the total surface area of the adsorbent, etc. So, various surface kinetics corresponding to multi-species enzyme catalyzed reactions, such as Langmuir or Eley-Rideal mechanisms, can be considered in order to describe the nonlinear fluxes at the surface of the inclusions. The fact that the reactions of the biochemical species at the surface $\Gamma^\varepsilon$ are assumed to be fast or slow is reflected in the different scaling of the corresponding fluxes in equations $_{2,3}$. Such a specific scaling will lead, at the macroscale, to different effects, namely a non-standard homogenized matrix and an additional source or sink term, respectively (see Theorem \[teounf\] and Corollary \[teohom\]). Example 1. Let us give here some concrete examples of nonlinear functions $F_i$, for $i\in \{1,2,3\}$ (see, also, [@Gahn], Section 5 in [@Gahn2], and Chapter 4 in [@Gahn3]).\ If $a_0(y)$ is a positive bounded measurable $Y$ -periodic function, $c_k>0$ and $k\in \{0,1\}^3$, then $$F_i(y,s_1, s_2, s_3)= a_0(y)\displaystyle \frac{s_1 s_2 s_3}{\sum_{\vert k\vert \leq 3}c_k s^k}.$$ For simplicity, we can take all the coefficients in front of our variables to be equal to one. Thus, we consider functions $F_i$, for $1\leq i\leq 3$, of the following form: $$F_i(y,s_1, s_2, s_3)= a_0(y) \displaystyle \frac{s_1 s_2 s_3}{1+s_1+s_2+s_3+ s_1 s_2 +s_1s_3+s_2s_3+s_1 s_2 s_3}.$$ Other possible concrete examples could be: $$F_1(y, s_1, s_2, s_3)=a_0 (y) \, \displaystyle \frac{s_1 s_2}{1+s_1+s_2+s_1 s_2 +s_1 s_2 s_3}+s_1,$$ $$F_2(y, s_1, s_2, s_3)=\displaystyle \frac{s_2 s_3}{1+s_2+s_3+s_2s_3+s_1 s_2 s_3},$$ $$F_3(y, s_1, s_2, s_3)=\displaystyle \frac{s_1 s_3}{1+s_1+s_3+s_1s_3+s_1 s_2 s_3}+s_3.$$ A very simplified setting could be to assume that each $F_i$ depends only on the corresponding concentration $s_i$ and it is given by the Michaelis-Menten kinetics arising in biochemistry when dealing with enzyme-catalyzed reactions.\ Example 2. For $G_3$ we might take, for instance, kinetics similar to the ones used for $F_i$ or kinetics of the form $$G_3(s_1, s_2, s_3)= a(y) \dfrac{s_1^4}{1+s_1^4} \, s_3 +s_3,$$ where $a(y)$ is a positive bounded measurable $Y$ -periodic function.\ Example 3. As an example for a nonlinear function $H$ in , we can consider the Langmuir kinetics, i.e. $$\label{exampleH} H(s)=\displaystyle \frac{a s}{1+b s}, \quad a, b>0.$$ Thus, for a concrete relevant example of nonlinear functions $G_i$, for $i\in \{1,2\}$, we have $$\label{exGi} G_i(s_1, s_2, s_3)=(-1)^i a \dfrac{(s_1-s_2)s_3}{1+b s_3}.$$ We remark, that with this choice, the boundary term involving function $H$ is well-defined. Indeed, the function in satisfies assumption $(\mathbf{H}_3)$ since the concentration fields $c_i^\varepsilon$, for $i\in \{1,2,3\}$, are positive and essentially bounded, as we shall prove in Section \[Sec3\].\ \ For negative values of variables, in the above functions singularities may appear. To handle this type of nonlinearity, as in [@Gahn]$\div$[@Gahn3], we first consider the modified kinetics with the modulus appearing in all the terms in the denominator. After proving existence and uniqueness results for the problem with the modified kinetics, we prove that the solution is nonnegative, and, so, it is also a solution of the original problem (see, for instance, Example 1 in \[24\]).\ \[rem2\] Actually, having in mind , it would be enough to put $0<H'\leq L$ in order to consider a one to one and increasing function as done for instance in [@Allaire]. In this sense, our assumption $(\mathbf{H}_3)_4$ is more general from the mathematical point of view. The assumptions we made are natural structural assumptions. Most of them are mathematical requirements, which, however, have a strong physical justification, as underlined in the following remarks. \[remhyp1\] Hypotheses $(\mathbf{H}_1)$, $(\mathbf{H}_2)_{1,2,3}$, $(\mathbf{H}_3)$, and $(\mathbf{H}_4)_{1,2,3}$ are, on one hand, mathematical and are needed for proving existence and uniqueness results and estimates for the microscopic model. In general, apart from linear functions, one has to assume Lipschitz or monotonicity conditions to ensure well-posedness. $(\mathbf{H}_2)_{1,2,3}$ is also needed to prove strong convergence. However, $(\mathbf{H}_2)$ and $(\mathbf{H}_4)$ are standard hypotheses used for this kind of problems in the literature, describing chemical reactions which are relevant in many concrete applications. In biology and in chemistry, many important processes behave like having a monotone or a Lipschitz character and, so, our hypotheses on the nonlinear functions $F_i$ and $G_i$ are physically justified. Also, the regularity conditions imposed for our nonlinearities are technical requirements. This kind of regularity can probably be improved. The conditions asked in $(\mathbf{H}_3)$ for $G_i$ are not standard, but such nonlinearities are still relevant for describing several chemical reactions, such as those implied in the Eley-Rideal mechanisms. We notice that $(\mathbf{H}_2)_4$ and $(\mathbf{H}_4)_4$ are just physical assumptions: in absence of concentrations, we do expect no reaction to occur. These conditions are not relevant from a mathematical point of view. Technically, such conditions can be removed.\ \[remhyp2\] Hypotheses $(\mathbf{H}_5)$, $(\mathbf{H}_6)$, $(\mathbf{H}_7)$ and again $(\mathbf{H}_3)$ are needed for ensuring the non-negativity and the essential boundedness of the solutions, such properties being, in fact, reasonable from a biological point of view (see, for example, [@Gahn2] and [@Gahn3]). For instance, $(\mathbf{H}_5)$ is a technical requirement and it ensures the positivity of our concentrations. However, we mention that other choices are still possible. In fact, we could ask, as in [@HK], that $F_i(\cdot, s)\geq 0$, for all $s=(s_1, s_2, s_3)$ with $s_i\leq 0$. This means, in fact, that if a chemical species vanishes, then it can only be produced. The main difficulties behind our study come from the strong coupling between the equations governing the evolution of the concentration fields, from the nonlinearity of the model and from the interesting interface phenomena (which are far from being perfectly understood) appearing in the heterogeneous media under consideration. The microscopic model {#Sec3} ===================== Existence and uniqueness for the microscopic model -------------------------------------------------- The following lemma will be used several times throughout the paper (see [@Gahn Lemma 5]). \[lemtrace\] (i) Let $\Omega$ be an arbitrary Lipschitz domain and $\delta>0$. Then, $$\label{trace1} \Vert u\Vert^2_{L^2(\partial \Omega)}\leq C_{\delta} \Vert u\Vert^2_{L^2(\Omega)}+\delta \Vert \nabla u\Vert^2_{L^2(\Omega)},$$ for all $u\in H^1(\Omega)$ and with a constant $C_\delta>0$ depending on $\delta$.\ (ii) Let $p\in [1,+\infty)$ and $u_\varepsilon \in L^p(\Omega^{\ast}_{\varepsilon})$. Since the trace operator from $W^{1,p}(Y^)$ into $L^p(\Gamma)$ is continuous, one has $$\label{trace2} \varepsilon\Vert u^\varepsilon\Vert^p_{L^p(\Gamma^\varepsilon)}\leq C\left( \Vert u^\varepsilon\Vert^p_{L^p(\Omega^{\ast}_{\varepsilon})}+\varepsilon^p\Vert \nabla u^\varepsilon \Vert^p_{L^p(\Omega^{\ast}_{\varepsilon})}\right).$$ (iii) Under the same assumptions as in (ii) and for an arbitrary $\delta>0$, we have $$\label{trace3} \varepsilon\Vert u^\varepsilon\Vert^p_{L^p(\Gamma^\varepsilon)}\leq C_\delta \Vert u^\varepsilon\Vert^p_{L^p(\Omega^{\ast}_{\varepsilon})}+\delta\varepsilon^p \Vert \nabla u^\varepsilon \Vert^p_{L^p(\Omega^{\ast}_{\varepsilon})}.$$ For giving the definition of a weak solution of the system , following [@Gahn], for any Banach space $V$, we denote its dual by $V'$ and we introduce the space $$\mathcal{W}\left(0,T;V,V'\right):=\{u\in L^2\left(0,T;V\right):\partial_t u\in L^2\left(0,T;V'\right)\},$$ where the time derivative $\partial_t u$ is understood in the distributional sense. It is a Banach space if it is endowed with the norm of the graph $$\\|u\\|_{\mathcal{W}}:=\\|u\\|_{L^2(0,T;V)} + \\|\partial_t u\\|_{L^2(0,T;V')}.$$ We denote $$H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon})= \{ \, v\in H^{1}(\Omega^{\ast}_{\varepsilon} ) \, \mid \, v=0 \textrm{ on } \partial \Omega \}.$$ We say that $(c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \, c_{3}^{\varepsilon})\in \mathcal{W}\left(0,T;H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon}),(H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon}))'\right)^3$ is a weak solution of problem if for any $v\in H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon})$ and for a.e. $t\in(0,T)$ it holds that, for $i\in \{1,2\}$, $$\label{weakconcentration} \begin{array}{l} \displaystyle\langle\partial_{t}c_{i}^{\varepsilon},v\rangle_{\Omega^{\ast}_{\varepsilon}}+\int_{\Omega^{\ast}_{\varepsilon}}D_i^\varepsilon \nabla c_{i}^{\varepsilon}\cdot\nabla v\mathrm{d}x =\int_{\Omega^{\ast} _{\varepsilon}}F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})v\,\mathrm{d}x+\displaystyle \frac{1}{\varepsilon}\int_{\Gamma^{\varepsilon}}G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon},c_{3}^{\varepsilon})v\,\mathrm{d}\sigma _{x}\\ \\ \displaystyle\langle\partial_{t}c_{3}^{\varepsilon},v\rangle_{\Omega^{\ast}_{\varepsilon}}+\int_{\Omega^{\ast}_{\varepsilon}}D_3^\varepsilon \nabla c_{3}^{\varepsilon}\cdot\nabla v\mathrm{d}x =\int_{\Omega^{\ast} _{\varepsilon}}F_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})v\,\mathrm{d}x + \displaystyle \varepsilon\int_{\Gamma^{\varepsilon}}G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon},c_{3}^{\varepsilon})v\,\mathrm{d}\sigma _{x}, \end{array}$$ together with the initial conditions $$\label{weakinitial} c_{i}^{\varepsilon}(0)=c_{i}^0\quad\text{ in }\Omega^{\ast}_{\varepsilon}, \quad i\in \{1,2,3\}.$$ Here, we denoted by $\langle\cdot,\cdot\rangle_{\Omega^{\ast}_{\varepsilon}}$ the duality pairing $\langle\cdot,\cdot\rangle_{H_{\partial \Omega}^1(\Omega^{\ast}_{\varepsilon}),(H_{\partial \Omega}^1(\Omega^{\ast}_{\varepsilon}))'}$ of $H_{\partial \Omega}^1(\Omega^{\ast}_{\varepsilon})$ with its dual space $(H_{\partial \Omega}^1(\Omega^{\ast}_{\varepsilon}))'$. \[teoexun\] There exists a unique weak solution $(c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \,c_{3}^{\varepsilon})$ of problem . We can argue in a similar manner as in [@Gahn] (see, also [@Gahn3], [@Gra-Pet], and [@Graf]). Indeed, we can use Schaefer’s fixed point theorem. To this end, following [@Gahn], one can define the fixed point operator ${\cal F}: X \rightarrow X$, where $X =L^2\left(0,T;H^{\beta}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon})\right)^3$, for $\beta\in (\frac{1}{2},1)$. For $\widehat{c}^\varepsilon=(\widehat{c}_{1}^\varepsilon,\,\widehat{c}_{2}^\varepsilon, \, \widehat{c}_{3}^\varepsilon)\in X$, let ${\cal F}(\widehat{c}^\varepsilon) =c^{\varepsilon}= (c_1^{\varepsilon}, \, c_2^{\varepsilon}, \, c_3^{\varepsilon})$, where $c^{\varepsilon}\in \mathcal{W}\left(0,T;H^{1}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}),(H^{1}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}))'\right)^3$ is the unique solution of the linearization of the system (i.e. in we replace, for $i\in \{1,2,3\}$, $F_i^\varepsilon (x, c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \,c_{3}^{\varepsilon})$ by $F_i^\varepsilon (x, \widehat{c}_{1}^\varepsilon,\,\widehat{c}_{2}^\varepsilon, \, \widehat{c}_{3}^\varepsilon)$, $G_i(c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \,c_{3}^{\varepsilon})$ by $G_i(\widehat{c}_{1}^\varepsilon,\,\widehat{c}_{2}^\varepsilon, \, \widehat{c}_{3}^\varepsilon)$, for $i\in \{1,2\}$, and $G_3^\varepsilon(x, c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \,c_{3}^{\varepsilon})$ by $G_3^\varepsilon (x, \widehat{c}_{1}^\varepsilon,\,\widehat{c}_{2}^\varepsilon, \, \widehat{c}_{3}^\varepsilon)$, respectively). Due to the fact that the embedding $$\mathcal{W}\left(0,T;H^{1}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}),(H^{1}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}))'\right)^3 \hookrightarrow X$$ is compact, using estimates similar to those in Lemma \[lemest\] below, one gets that the operator ${\cal F}$ is continuous and compact and the set $\{c\in X \, \vert \, c=\lambda {\cal F}(c), \, \textrm{for some } \lambda \in [0,1]\}$ is bounded in $X$. Then, Schaefer’s theorem gives the existence of a weak solution of problem . As in [@Gahn], the uniqueness follows from the Lipschitz continuity of the functions $F_i^\varepsilon$ and $G_i^\varepsilon$ and Gronwall’s inequality. \ In what follows, we prove the nonnegativity and the uniform boundedness from above of the concentration fields $c_i^\varepsilon$, for $i\in \{1,2,3\}$, which is a reasonable condition from the point of view of applications in biology. In order to show nonnegativity, we have to take into account the fact that, for $i\in \{1,2,3\}$, the generalized time derivative $\partial_t c_i^\varepsilon$ is only an element of $L^2(0,T;((H^1_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}))')$ and we don’t know if the time derivative of $(c_i^\varepsilon)_{-}$ exists or not. So, as in [@Gahn], we regularize the solution $c_i^\varepsilon$ with the aid of Steklov average and we obtain an integral inequality for the concentrations $c_i^\varepsilon$. \[teopos\] Let $(c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \,c_{3}^{\varepsilon})$ be the weak solution of problem . Then, for a.e. $t\in(0,T)$ and for $i\in \{1,2,3\}$, we have $$\label{pos1} c_i^\varepsilon(\cdot,x)\geq 0\,\text{ a.e. in }\Omega^{\ast}_{\varepsilon}.$$ We start by proving that, for $i\in \{1,2\}$, for a.e. $t\in(0,T)$ we have $$\label{pos3} \begin{array}{l} \dfrac{1}{2}\Vert(c_i^\varepsilon)_{-}(t)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}+\alpha\left\Vert \nabla(c_{i}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))} \\ \\ \leq \displaystyle\int_0^{t}\int_{\Omega^{\ast}_{\varepsilon}}F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{i}^{\varepsilon })_{-}\mathrm{d}x\, \mathrm{d}s+ \displaystyle \frac{1}{\varepsilon} \int_0^{t}\int_{\Gamma^{\varepsilon}}G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{i}^{\varepsilon})_{-}\mathrm{d}\sigma_{x}\,\mathrm{d}s, \end{array}$$ while for $i=3$ we have $$\label{pos3-3} \begin{array}{l} \dfrac{1}{2}\Vert(c_3^\varepsilon)_{-}(t)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}+\alpha\left\Vert \nabla(c_{3}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))} \\ \\ \leq \displaystyle\int_0^{t}\int_{\Omega^{\ast}_{\varepsilon}}F_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{3}^{\varepsilon })_{-}\mathrm{d}x\, \mathrm{d}s+ \displaystyle \varepsilon \int_0^{t}\int_{\Gamma^{\varepsilon}}G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{3}^{\varepsilon})_{-}\mathrm{d}\sigma_{x}\,\mathrm{d}s. \end{array}$$ From the first equation in , after integration in time from $t$ to $t + h$, by integration by parts in the time derivative term and by multiplying the whole equation by $\dfrac{1}{h}$, we get for all $t, h\in (0,T)$ with $(t+h)\in (0,T)$ and for all $v\in H_{\partial \Omega}^1(\Omega^{\ast}_{\varepsilon})$ $$\label{pos1} \begin{array}{l} \dfrac{1}{h}\displaystyle\int_{\Omega^{\ast}_{\varepsilon}}\left(c_i^\varepsilon(t+h)-c_i^\varepsilon(t)\right)v\,\mathrm{d}x +\dfrac{1}{h}\int_t^{t+h}\int_{\Omega^{\ast}_{\varepsilon}} D_i^\varepsilon\nabla c_{i}^{\varepsilon} \cdot\nabla v\,\mathrm{d}x \,\mathrm{d}s\\ \\ =\displaystyle\dfrac{1}{h}\int_t^{t+h}\int_{\Omega^{\ast}_{\varepsilon}}F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon}) v\,\mathrm{d}x\, \mathrm{d}s+\dfrac{1}{h \varepsilon}\int_t^{t+h}\int_{\Gamma^{\varepsilon}}G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})v\,\mathrm{d}\sigma_{x}\,\mathrm{d}s,\,\,\,\,i\in \{1,2\}. \end{array}$$ Using the Steklov average, defined for a function $u\in L^2(0,T;L^2(D))$ with $D\subset \mathbb{R}^n$ open, $t, h\in (0, T )$ by $$\label{Stek} [u]_h(t):=\left\{ \begin{array}{ll} \dfrac{1}{h}\displaystyle\int_t^{t+h} u(s,\cdot)\,\mathrm{d}s & \text{for } t\in (0,T-h],\\ \\ 0 & \text{for } t>T-h, \end{array} \right.$$ we obtain $$\label{pos2} \begin{array}{l} \displaystyle\int_{\Omega^{\ast}_{\varepsilon}}\partial_t[c_i^\varepsilon]_{h} v \, \mathrm{d}x +\int_{\Omega^{\ast}_{\varepsilon}}D_i^\varepsilon \nabla [c_{i}^{\varepsilon}]_h \cdot\nabla v\mathrm{d}x \\ \\ =\displaystyle\int_{\Omega^{\ast}_{\varepsilon}}[F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})]_h v\,\mathrm{d}x+\displaystyle \frac{1}{\varepsilon} \int_{\Gamma^{\varepsilon}}[G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})]_h v\,\mathrm{d}\sigma_{x},\,\,\,i\in \{1,2\}. \end{array}$$ We have $\partial_t [c_i^\varepsilon]_h\in L^2(0,T; L^2(\Omega^{\ast}_{\varepsilon}))$ for all $0 < h < \delta$ and therefore $\partial_t ([c_i^\varepsilon]_h)_{-}\in L^2(0,T-\delta; L^2(\Omega^{\ast}_{\varepsilon}))$. For the function $[c_i^\varepsilon]_h$ it holds that $$\langle\partial_t [c_i^\varepsilon]_h(t), ([c_i^\varepsilon]_h(t))_{-}\rangle_{\Omega^{\ast}_{\varepsilon}}=\dfrac{1}{2}\dfrac{d}{dt}\Vert([c_i^\varepsilon]_h)_{-}\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}$$ and we obtain by integration from $0$ to $t \in (0, T )$ and testing the first equation in with $v=([c_i^\varepsilon ]_h)_{-}$, due to the definition of the cut off function $(c_i^\varepsilon)_{-}$ and assumption $(\mathbf{H}_1)_{2}$, for small $h$ and $i\in \{1,2\}$ $$\label{pos4} \begin{array}{l} \dfrac{1}{2}\Vert([c_i^\varepsilon]_h)_{-}(t)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}-\dfrac{1}{2}\Vert([c_i^\varepsilon]_h)_{-}(0)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}+\alpha \left\Vert \nabla([c_{i}^{\varepsilon}]_h)_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}\\\\ \leq\displaystyle\int_0^{t}\int_{\Omega^{\ast}_{\varepsilon}}[F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})]_h([c_{i}^{\varepsilon }]_h)_{-}\mathrm{d}x\,\mathrm{d}s+\displaystyle \frac{1}{\varepsilon}\int_0^{t}\int_{\Gamma^{\varepsilon}}[G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})]_h([c_{i}^{\varepsilon}]_h)_{-}\mathrm{d}\sigma_{x}\, \mathrm{d}s. \end{array}$$ Using the properties of the Steklov average, we can pass to the limit as $h\rightarrow 0$ in and by assumption $(\mathbf{H}_7)$ we use that the initial concentration fields are nonnegative, i.e. ($c_i^\varepsilon (0))_{-}=0$, to obtain . In a similar manner one can obtain . Summing up equations for $i\in \{1,2\}$ and , we get $$\begin{array}{l} \dfrac{1}{2}\left\Vert (c_{1}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+ \dfrac{1}{2}\left\Vert (c_{2}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+ \dfrac{1}{2}\left\Vert (c_{3}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})} +\\ \\ \alpha\left(\left\Vert \nabla(c_{1}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}+\left\Vert \nabla(c_{2}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}+\left\Vert \nabla(c_{3}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}\right)\\ \\ \leq \displaystyle\int_0^{t}\int_{\Omega^{\ast}_{\varepsilon}}\left[F_{1}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{1}^{\varepsilon})_{-}+F_{2}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{2}^{\varepsilon})_{-}+F_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{3}^{\varepsilon})_{-}\right]\mathrm{d}x\, \mathrm{d}s\\ \\+ \displaystyle \frac{1}{\varepsilon}\int_0^{t}\int_{\Gamma^{\varepsilon}}\left[G_{1}(c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{1}^{\varepsilon})_{-}+G_{2}(c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})(c_{2}^{\varepsilon})_{-}\right]\mathrm{d}\sigma_{x}\, \mathrm{d}s \\ \\ + \displaystyle \varepsilon\int_0^{t}\int_{\Gamma^{\varepsilon}} G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})(c_{3}^{\varepsilon})_{-}\mathrm{d}\sigma_{x}\, \mathrm{d}s. \end{array}$$ By taking into account , and , we obtain $$\begin{array}{l} \dfrac{1}{2}\left( \left\Vert (c_{1}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+\left\Vert (c_{2}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+\left\Vert (c_{3}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}\right) \\ \\ +\alpha\left(\left\Vert \nabla(c_{1}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}+\left\Vert \nabla(c_{2}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}+\left\Vert \nabla(c_{3}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}\right)\\ \\ \leq C_1\displaystyle\int_0^t\left(\Vert (c_1^\varepsilon)_{-}(s)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}+\Vert (c_2^\varepsilon)_{-}(s)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})} +\Vert (c_3^\varepsilon)_{-}(s)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}\right) \, \mathrm{d}s\\ \\ + \displaystyle C_2 \, \varepsilon \int_0^t\left(\Vert (c_1^\varepsilon)_{-}(s)\Vert^2_{L^2(\Gamma^\varepsilon)}+\Vert (c_2^\varepsilon)_{-}(s)\Vert^2_{L^2(\Gamma^\varepsilon)} +\Vert (c_3^\varepsilon)_{-}(s)\Vert^2_{L^2(\Gamma^\varepsilon)}\right) \, \mathrm{d}s\\ \\ +\displaystyle \frac{1}{\varepsilon}\int_0^{t}\int_{\Gamma^{\varepsilon}}\left(c_{1}^{\varepsilon}-c_{2}^{\varepsilon }\right)H\left(c_3^\varepsilon\right)\left[(c_2^\varepsilon)_{-}-(c_1^\varepsilon)_{-}\right]\mathrm{d}\sigma_{x}\, \mathrm{d}s. \end{array}$$ Since the function $H$ is positive, by the definition of $(c_i^\varepsilon)_-$, it is easy to show that $$\label{negG}\displaystyle \frac{1}{\varepsilon}\int_0^{t}\int_{\Gamma^{\varepsilon}}\left(c_{1}^{\varepsilon}-c_{2}^{\varepsilon }\right)H\left(c_3^\varepsilon\right)\left[(c_2^\varepsilon)_{-}-(c_1^\varepsilon)_{-}\right]\mathrm{d}\sigma_{x}\, \mathrm{d}s\leq 0,$$ and, hence, we get $$\begin{array}{l} \dfrac{1}{2}\left( \left\Vert (c_{1}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+\left\Vert (c_{2}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+\left\Vert (c_{3}^{\varepsilon})_{-}(t)\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}\right) \\ \\ +\alpha\left(\left\Vert \nabla(c_{1}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}+\left\Vert \nabla(c_{2}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}+\left\Vert \nabla(c_{3}^{\varepsilon})_{-}\right\Vert ^{2}_{L^2(0,t;L^2(\Omega^{\ast}_{\varepsilon}))}\right)\\ \\ \leq C_1\displaystyle\int_0^t\left(\Vert (c_1^\varepsilon)_{-}(s)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}+\Vert (c_2^\varepsilon)_{-}(s)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})} +\Vert (c_3^\varepsilon)_{-}(s)\Vert^2_{L^2(\Omega^{\ast}_{\varepsilon})}\right) \, \mathrm{d}s\\ \\ +\displaystyle C_2 \, \varepsilon \int_0^t\left(\Vert (c_1^\varepsilon)_{-}(s)\Vert^2_{L^2(\Gamma^\varepsilon)}+\Vert (c_2^\varepsilon)_{-}(s)\Vert^2_{L^2(\Gamma^\varepsilon)} +\Vert (c_3^\varepsilon)_{-}(s)\Vert^2_{L^2(\Gamma^\varepsilon)}\right) \, \mathrm{d}s. \end{array}$$ Using Gronwall’s inequality, we are led, as in [@Gahn], to . Following the same argument as in [@Gahn3], we prove that $c_{i}^{\varepsilon},\, i\in \{1,2, 3\}$, are essentially bounded. \[teobound\] Let $(c_{1}^{\varepsilon},\,c_{2}^{\varepsilon},\,c_{3}^{\varepsilon})$ be the weak solution of problem . Then, for a.e. $t\in(0,T)$ and for $i\in \{1,2, 3\}$, there exists a constant $C>0$ independent of $\varepsilon$ such that $$\label{b1} \left\Vert c_i^\varepsilon \right\Vert_{L^\infty((0,T)\times \Omega^{\ast}_{\varepsilon})} \leq C.$$ Set $\left( \cdot\right)_+ =\max\left\{\cdot,0\right\}$. Let $A$ and $\Lambda$ be as in assumptions $(\mathbf{H}_6)$ and $(\mathbf{H}_7)$ and let us consider, for $i\in \{1,2, 3\}$, $v=e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+$ as test function in . Then, we obtain, for $i\in \{1,2\}$, $$\begin{array}{l} \langle\partial_{t}c_{i}^{\varepsilon}, e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}}+\displaystyle\int_{\Omega^{\ast}_{\varepsilon}} D_i^\varepsilon \nabla c_{i}^{\varepsilon} \cdot\nabla e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+ \, \mathrm{d}x \\ \\ = \displaystyle \int_{\Omega^{\ast}_{\varepsilon}}F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon}) e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}x +\displaystyle \frac{1}{\varepsilon} \int_{\Gamma^{\varepsilon}}G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon}) e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}\sigma _{x} \end{array}$$ and, for $i=3$, $$\begin{array}{l} \langle\partial_{t}c_{3}^{\varepsilon}, e^{-tA}(c_3^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}}+\displaystyle\int_{\Omega^{\ast}_{\varepsilon}} D_3^\varepsilon \nabla c_{3}^{\varepsilon} \cdot\nabla e^{-tA}(c_3^\varepsilon \, e^{-tA}-\Lambda)_+ \, \mathrm{d}x \\ \\= \displaystyle \int_{\Omega^{\ast}_{\varepsilon}}F_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon}) e^{-tA}(c_{3}^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}x+\displaystyle \varepsilon \int_{\Gamma^{\varepsilon}}G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon}) e^{-tA}(c_3^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}\sigma _{x}. \end{array}$$ Using assumption $(\mathbf{H}_6)$ and , for $i\in \{1,2\}$, we have $$\label{est12} \begin{array}{l} \langle\partial_{t}c_{i}^{\varepsilon}, e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}}+\displaystyle\int_{\Omega^{\ast}_{\varepsilon}} D_i^\varepsilon \nabla c_{i}^{\varepsilon} \cdot\nabla e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+ \, \mathrm{d}x \\ \\ \leq \displaystyle \int_{\Omega^{\ast}_{\varepsilon}} A e^{-tA} c_i^\varepsilon (c_i^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}x +\displaystyle \frac{1}{\varepsilon} \int_{\Gamma^{\varepsilon}}(-1)^i e^{-tA}(c^\varepsilon_{1}-c^\varepsilon_{2})H\left(c_{3}\right)\left(c_i^\varepsilon e^{-tA} -\Lambda\right)_+\,\mathrm{d}\sigma_{x} \end{array}$$ and, for $i=3$, $$\label{est3} \begin{array}{l} \langle\partial_{t}c_{3}^{\varepsilon}, e^{-tA}(c_3^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}}+\displaystyle\int_{\Omega^{\ast}_{\varepsilon}} D_3^\varepsilon \nabla c_{3}^{\varepsilon} \cdot\nabla e^{-tA}(c_3^\varepsilon \, e^{-tA}-\Lambda)_+ \, \mathrm{d}x \\ \\ \leq \displaystyle \int_{\Omega^{\ast}_{\varepsilon}} A e^{-tA} c_3^\varepsilon (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}x+\displaystyle \varepsilon \int_{\Gamma^{\varepsilon}} A e^{-tA} c_3^\varepsilon (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}\sigma _{x}\\ \\= \displaystyle \int_{\Omega^{\ast}_{\varepsilon}} A e^{-tA} c_3^\varepsilon (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}x+ \displaystyle A \varepsilon \left\Vert (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\right \Vert ^{2}_{L^2(\Gamma^{\varepsilon})} \,\mathrm{d}\sigma _{x}+ \displaystyle A \varepsilon \int_{\Gamma^{\varepsilon}} \Lambda (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}\sigma _{x}. \end{array}$$ On the other hand, for $i\in \{1,2, 3\}$, one has\ $$\label{der} \begin{array}{l} \langle\partial_{t}c_{i}^{\varepsilon}, e^{-tA}(c_i^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}}\\ \\=\langle\partial_{t}(c_i^\varepsilon \, e^{-tA}-\Lambda), (c_i^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}} + \langle A e^{-tA}c_{i}^{\varepsilon}, (c_i^\varepsilon \, e^{-tA}-\Lambda)_+\rangle_{\Omega^{\ast}_{\varepsilon}}. \end{array}$$ Therefore, summing up for $i\in \{1,2\}$, by assumption $(\mathbf{H}_1)_{2}$ we obtain $$\begin{array}{l} \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\Vert (c_1^\varepsilon \, e^{-tA}-\Lambda)_+ \right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \left\Vert (c_2^\varepsilon \, e^{-tA}-\Lambda)_+\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}\\ \\ +\alpha\left(\left\Vert \nabla (c_1^\varepsilon \, e^{-tA}-\Lambda)_+ \right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})} +\left\Vert \nabla (c_2^\varepsilon \, e^{-tA}-\Lambda)_+ \right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}\right)\\ \\ \leq \displaystyle \frac{1}{\varepsilon} \int_{\Gamma^{\varepsilon}}\left[(c^\varepsilon_{1}e^{-tA}-\Lambda)-(c^\varepsilon_{2}e^{-tA}-\Lambda)\right]H\left(c_{3}\right)\left[(c_2^\varepsilon e^{-tA} -\Lambda)_+ - (c_1^\varepsilon e^{-tA} -\Lambda)_+ \right] \,\mathrm{d}\sigma_{x}. \end{array}$$ By arguing as to prove , we get that the second member of the previous inequality is non positive and then $$\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\Vert (c_1^\varepsilon \, e^{-tA}-\Lambda)_+ \right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+ \displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \left\Vert (c_2^\varepsilon \, e^{-tA}-\Lambda)_+\right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}\leq 0.$$ Integrating in time and taking into account that the initial data are bounded, we get the essential boundedness of $c_1^\varepsilon$ and $c_2^\varepsilon$. From , taking into account and assumption $(\mathbf{H}_1)_{2}$, we get $$\begin{array}{l} \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\Vert (c_3^\varepsilon \, e^{-tA}-\Lambda)_+ \right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})}+\alpha \left\Vert \nabla (c_3^\varepsilon \, e^{-tA}-\Lambda)_+ \right\Vert ^{2}_{L^2(\Omega^{\ast}_{\varepsilon})} \\ \\ \leq C \Big ( \displaystyle \varepsilon \left\Vert (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\right \Vert ^{2}_{L^2(\Gamma^{\varepsilon})} + \displaystyle \varepsilon \int_{\Gamma^{\varepsilon}} \Lambda (c_3^\varepsilon \, e^{-tA}-\Lambda)_+\,\mathrm{d}\sigma _{x} \Big ). \end{array}$$ Now, exactly as in [@Gahn3 Proposition 3.38], we are led to the essential boundedness of $c_3^\varepsilon$. Estimates for the microscopic model {#secest} ----------------------------------- Our goal is to obtain the effective behavior of the solutions of the microscopic system . To this aim, we need to pass to the limit, with $\varepsilon \rightarrow 0$, in the variational formulation of the microscopic model, by using compactness results with respect to suitable topologies. Thus, we need to prove [a priori]{} estimates for our solution. \[lemest\] For the solution $(c_{1}^{\varepsilon},\,c_{2}^{\varepsilon}, \,c_{3}^{\varepsilon})$ of problem , there exists a positive constant $C>0$, independent of $\varepsilon$, such that the following estimates hold true: $$\Vert c_i^{\varepsilon}\Vert_{L^{\infty}(0,T;L^{2}(\Omega^{\ast}_{\varepsilon}))}\leq C\,\,\,\,i\in \{1,2,3\}, \label{estimconcLinf}$$ $$\Vert c_i^{\varepsilon}\Vert_{L^{2}(0,T;H^{1}(\Omega^{\ast}_{\varepsilon}))}\leq C\,\,\,\,i\in \{1,2,3\}, \label{estimconcH1}$$ $$\Vert c_1^{\varepsilon}-c_2^{\varepsilon}\Vert_{L^{2}(0,T;L^{2}(\Gamma^{\varepsilon}))}\leq C \sqrt{\varepsilon}, \label{estimdifconc}$$ $$\sqrt{\varepsilon}\Vert c_3^{\varepsilon}\Vert_{L^{2}(0,T;L^{2}(\Gamma^{\varepsilon}))}\leq C, \label{estimconcb}$$ $$\Vert \partial_t c_i^{\varepsilon}\Vert_{L^{2}(0,T;(H_0^{1}(\Omega^{\ast}_{\varepsilon}))')}\leq C\,\,\,\,i\in \{1,2\}, \label{estimconcH1'}$$ $$\Vert \partial_t c_3^{\varepsilon}\Vert_{L^{2}(0,T;(H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon}))')}\leq C. \label{estimconc3H1'}$$ Let us prove . To this end, we test the first equation in with $c_{i}^{\varepsilon}$, for $i\in \{1,2\}$, and the second one with $c_{3}^{\varepsilon}$, respectively. We get $$\begin{array}{l} \displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{i}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\int_{\Omega^{\ast}_{\varepsilon}}D_i^\varepsilon \nabla c_{i}^{\varepsilon}\cdot\nabla c_{i}^{\varepsilon}\mathrm{d}x =\displaystyle\int_{\Omega^{\ast}_{\varepsilon}}F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})c_{i}^{\varepsilon}\mathrm{d}x+\displaystyle \frac{1}{\varepsilon}\int_{\Gamma ^{\varepsilon}}G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})c_{i}^{\varepsilon}\mathrm{d}\sigma_{x} \end{array}$$ and $$\begin{array}{l} \displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{3}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\int_{\Omega^{\ast}_{\varepsilon}}D_3^\varepsilon \nabla c_{3}^{\varepsilon}\cdot\nabla c_{3}^{\varepsilon}\mathrm{d}x =\displaystyle\int_{\Omega^{\ast}_{\varepsilon}}F_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})c_{3}^{\varepsilon}\mathrm{d}x+\displaystyle \varepsilon\int_{\Gamma ^{\varepsilon}}G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_{3}^{\varepsilon})c_{3}^{\varepsilon}\mathrm{d}\sigma_{x}. \end{array}$$ Then, by and assumption $(\mathbf{H}_1)_{2}$, we get $$\begin{array}{l} \displaystyle\frac{1}{2}\left (\frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{3}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2} \right )\\ \\ +\alpha\left(\Vert\nabla c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert\nabla c_{2}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert\nabla c_{3}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}\right)\\ \\ \leq \displaystyle\int_{\Omega^{\ast}_{\varepsilon}}F_{1}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})c_{1}^{\varepsilon}\mathrm{d}x+\displaystyle\int_{\Omega^{\ast}_{\varepsilon}}F_{2}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})c_{2}^{\varepsilon}\mathrm{d}x+\displaystyle\int_{\Omega^{\ast}_{\varepsilon}}F_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})c_{3}^{\varepsilon}\mathrm{d}x\\ \\ +\displaystyle \varepsilon\int_{\Gamma^{\varepsilon}}G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon },c_{3}^{\varepsilon})c_{3}^{\varepsilon}\mathrm{d}\sigma_{x} -\displaystyle \dfrac{1}{\varepsilon}\int_{\Gamma^{\varepsilon}}\left (c_{1}^{\varepsilon}-c_{2}^{\varepsilon}\right )^2 H \left (c_{3}\right )\mathrm{d}\sigma_{x}. \end{array}$$ Since $H$ is nonnegative, by using the growth conditions and for $F_i^\varepsilon$ and $G_3^\varepsilon$, respectively, we have $$\label{estc4} \begin{array}{l} \displaystyle\frac{1}{2}\left (\frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Vert c_{3}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2} \right )\\ \\ +\alpha\left(\Vert\nabla c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert\nabla c_{2}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert\nabla c_{3}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}\right)\\ \\ \leq C \displaystyle\int _{\Omega^{\ast}_{\varepsilon}}(1+\|c_{1}^\varepsilon\|+\|c_{2}^\varepsilon\|+\|c_{3}^{\varepsilon}\| )(\|c_{1}^{\varepsilon}\|+\|c_{2}^{\varepsilon}\|+\|c_{3}^{\varepsilon}\|)\mathrm{d}x+C\varepsilon\int_{\Gamma^{\varepsilon}}(1+\|c_{1}^\varepsilon \|+\|c_{2}^\varepsilon\|+\|c_{3}^{\varepsilon}\| )\|c_{3}^{\varepsilon}\|\mathrm{d}\sigma_{x}. \end{array}$$ Using the inequality $2ab\leq a^2+b^2$, the fact that $\|\Gamma^\varepsilon\|\leq C \varepsilon^{-1}$, $\|\Omega^{\ast}_{\varepsilon}\|\leq C$ and $\varepsilon<1$, as well as the modified trace inequality from Lemma \[lemtrace\]$ (iii)$, as in [@Gahn] or [@Gahn3], we are led to $$\label{est7} \begin{array}{ll} \displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left( \Vert c_{1}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert c_{2}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+ \Vert c_{3}^{\varepsilon}\Vert _{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}\right) + \alpha\left(\Vert\nabla c_{1}^{\varepsilon}\Vert_{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert\nabla c_{2}^{\varepsilon}\Vert_{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2} + \Vert\nabla c_{3}^{\varepsilon}\Vert_{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2} \right) \\ \\ \leq C_{1}+C_{2}\left( \Vert c_{1}^{\varepsilon}\Vert_{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}+\Vert c_{2}^{\varepsilon}\Vert_{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2} + \Vert c_{3}^{\varepsilon}\Vert_{L^{2}(\Omega^{\ast}_{\varepsilon})}^{2}\right) . \end{array}$$ Integrating with respect to time and using Gronwall’s inequality, we obtain the pointwise boundedness of $\Vert c_i^\varepsilon\Vert_{L^2(\Omega^{\ast}_{\varepsilon})}$, $i\in\{1,2,3\}$ in $(0,T)$, i.e. . Then, and estimate imply inequality . In order to prove estimate , let us test the first equation in the variational formulation , written for $i=1$ and $i=2$ with $c_1^\varepsilon$ and $c_2^\varepsilon$, respectively. Summing up these two equations, moving the boundary terms in the left-hand side and taking into account , and , we are led to . In order to prove the estimates for the time derivative of the concentration fields, i.e. inequality , we test the first equation in with $v\in H_0^1(\Omega^{\ast}_{\varepsilon})$, such that $\Vert v\Vert_{H_0^{1}(\Omega^{\ast}_{\varepsilon})}\leq 1$. Using similar arguments as before and the estimates , we get for $i\in {1,2}$.\ In fact, for $c_3^{\varepsilon}$ we get a slightly better estimate, as in [@Gahn]. It is enough to test the second equation in with $v\in H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon})$ such that $\Vert v\Vert_{H_{\partial \Omega}^{1}(\Omega^{\ast}_{\varepsilon})}\leq 1$ and using similar arguments as before. Estimate follows directly from and Lemma \[lemtrace\]$(ii)$. Homogenization results by the periodic unfolding method {#Sec4} ======================================================= In this section, we are interested in obtaining the effective behavior, as $\varepsilon\rightarrow0$, of the solution $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$ of problem -. To this aim, we shall use the [a priori]{} estimates given in Lemma \[lemest\] to derive convergence results for the sequences $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$. In order to pass to the limit in the nonlinear terms in the variational formulation of , we need to establish strong convergence results. For using classical compactness results, we shall extend the functions $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$ to the whole domain $\Omega$ and we shall use unfolding operators, which transform functions on varying domains to functions on fixed domains (see, for instance, [@Cio-Dam-Don-Gri-Zaki], [@Cio-Dam-Gri1], [@Cioranescu-Donato-Zaki], [@Donato], [@Amar1], and [@Amar2]). Through the unfolding method, which is more or less equivalent to the two-scale convergence (see [@Allaire4] and [@N]), we can handle easier the nonlinearities on $\Gamma^\varepsilon$. The time-dependent unfolding operator {#sub2.3} ------------------------------------- In this subsection, we start by briefly recalling the definition and the main properties of the unfolding operator $\mathcal{T}^\ast_{\varepsilon }$ introduced, for time-dependent functions, in [@Donato-Yang] (see also [@Donato-Yang]). Since time is a parameter, the results in [@Donato-Yang] are direct generalizations of the corresponding ones from [@Cio-Dam-Don-Gri-Zaki]. For a more general setting of unfolding operators with time, we refer to [@Amar1] (see also [@Amar2]). For the unfolding operator defined in fixed domains, we refer the reader to [@Cio-Dam-Gri1]. In the sequel, for $z\in \mathbb{R}^{n}$, we use $\left[ z\right] _{Y}$ to denote its integer part $k$, such that $z-\left[ z\right] _{Y}\in Y$, and we set $$\left\{ z\right\} _{Y}=z-\left[ z\right] _{Y}\in Y\text{ \ \ \ \ in } \mathbb{R}^{n}.$$Then, for $x\in \mathbb{R}^n$, one has$$x=\varepsilon \left( \left[ \frac{x}{\varepsilon }\right] _{Y}+\left\{ \frac{x}{\varepsilon }\right\} _{Y}\right).$$In order to define the periodic unfolding operators, let us introduce the following sets as in [@Cio-Dam-Don-Gri-Zaki; @Cio-Dam-Gri1]. Let $$\label{Omegahat} \widehat{\Omega}_\varepsilon=\text{interior} \left\{\bigcup_{{\bf k} \in K_\varepsilon}\varepsilon ( {\bf k} + \overline{Y})\right\},\quad\quad\Lambda_\varepsilon=\Omega \setminus \widehat{\Omega}_\varepsilon,$$ where $K_\varepsilon$ is the same as in Section \[Sec2\]. Set $$\label{Omegahat} \widehat{\Omega}^\ast_\varepsilon=\widehat{\Omega}_\varepsilon\setminus S_\varepsilon,\quad\quad\Lambda^\ast_\varepsilon=\Omega^\ast_\varepsilon \setminus \widehat{\Omega}^\ast_\varepsilon.$$ Moreover, throughout the paper we denote: - $\widetilde{u}$: the zero extension to the whole $\Omega $ of a function $u$ defined on $\Omega ^{\ast}_{\varepsilon }$, - $\mathcal{M}_{E }\left( f\right) :=\dfrac{1}{\left\vert E \right\vert }\displaystyle\int\nolimits_{E}f{\,\mathrm{d}}x$, the average on E of any function $f\in L^{1}\left(E\right)$. Let us first recall the unfolding operator $\mathcal{T}_\varepsilon$ for the fixed domain $\Omega\times(0,T)$ introduced in [@Cio-Dam-Gri1] (see also [@Gaveau] where the properties of $\mathcal{T}_\varepsilon$ are stated without proofs). Using the same notation as in [@Cio-Dam-Gri1], let us give the following definition: \[defunf\] Let $T>0$. For $p\in[1,+\infty)$ and $q\in [1+\infty]$, we define the operator $\mathcal{T}_{\varepsilon}:L^q(0,T;L^p(\Omega)) \rightarrow L^q(0,T;L^p(\Omega\times Y)) $ as follows: $$\mathcal{T}_{\varepsilon }\left( \varphi \right) \left( t,x,y\right)=\left\{ \begin{array}{ll} \varphi \left( t, \varepsilon \left[ \dfrac{x}{\varepsilon }\right]_{Y}+\varepsilon y\right) & \, \, {\rm a.e.\ for }\left( t,x,y\right) \in (0,T) \times \widehat{\Omega}_\varepsilon \times Y\\ \\ 0 & \, \, {\rm a.e.\ for }\left( t,x,y\right) \in (0,T) \times \Lambda_\varepsilon \times Y. \end{array} \right.$$ Concerning perforated domains, we have the definition below (see [@Donato-Yang]): \[defunf\\] Let $T>0$. For $p\in[1,+\infty)$ and $q\in [1+\infty]$, we define the operator $\mathcal{T}^\ast_{\varepsilon}:L^q(0,T;L^p(\Omega^\ast_\varepsilon)) \rightarrow L^q(0,T;L^p(\Omega\times Y^\ast)) $ as follows: $$\mathcal{T}^\ast_{\varepsilon }\left( \varphi \right) \left( t,x,y\right)=\left\{ \begin{array}{ll} \varphi \left( t, \varepsilon \left[ \dfrac{x}{\varepsilon }\right]_{Y}+\varepsilon y\right) & \, \, {\rm a.e.\ for }\left( t,x,y\right) \in (0,T) \times \widehat{\Omega}_\varepsilon \times Y^\ast\\ \\ 0 & \, \, {\rm a.e.\ for }\left( t,x,y\right) \in (0,T) \times \Lambda_\varepsilon \times Y^\ast. \end{array} \right.$$ Following the Remark 2.5 in [@Cio-Dam-Don-Gri-Zaki], since the time variable acts as a simple parameter, the relationship between $\mathcal{T}_\varepsilon$ and $\mathcal{T}^\ast_\varepsilon$ is given, for any $\varphi$ defined on $(0,T)\times\Omega^\ast_\varepsilon$, by $$\label{link} \mathcal{T}^\ast_\varepsilon(\varphi)=\mathcal{T}_\varepsilon(\widetilde{\varphi})_{\|\Omega\times Y^\ast}.$$ Actually, the previous equality still holds with every extension of $\varphi$ from $\Omega^\ast_\varepsilon$ into $\Omega$. In particular, for $\varphi$ defined on $\Omega$, we have $$\mathcal{T}^\ast_\varepsilon(\varphi_{\|\Omega^\ast_\varepsilon})=\mathcal{T}_\varepsilon(\varphi)_{\|\Omega\times Y^\ast}.$$ Hence, the operator $\mathcal{T}^\ast_\varepsilon$ inherits the properties of the operator $\mathcal{T}_\varepsilon$ (see [@Donato-Yang; @Donato-Yang]) and for the reader’s convenience they are recalled in the sequel.\ In particular, some immediate consequences of Definition \[defunf\\] are: 1. $\mathcal{T}^\ast_{\varepsilon }\left( \varphi \psi \right) =\mathcal{T}^\ast_{\varepsilon }\left( \varphi \right) \mathcal{T}^\ast_{\varepsilon }\left( \psi \right) $, for every $\varphi, \psi \in L^q(0,T;L^p(\Omega^\ast_\varepsilon))$; 2. $\mathcal{T}^\ast_{\varepsilon }\left( \varphi \psi \right) =\mathcal{T}^\ast_{\varepsilon }\left( \varphi \right)\psi$, for every $\varphi\in L^p(\Omega^\ast_\varepsilon)$ and $\psi\in L^q(0,T)$; 3. $\nabla _{y}\left[ \mathcal{T}^\ast_{\varepsilon }\left(\varphi \right) \right] =\varepsilon \mathcal{T}^\ast_{\varepsilon }\left(\nabla \varphi \right)$ for every $\varphi \in L^q(0,T;W^{1,p}\left( \Omega^\ast_\varepsilon\right) )$; 4. $\mathcal{T}^\ast_\varepsilon\left(\varphi\left(t,\frac{x}{\varepsilon}\right)\right)=\varphi(t,y)$ a.e. in $(0,T)\times \Omega \times Y^\ast$ for any $Y-$periodic function $\varphi \in L^q(0,T;L^p(Y^\ast))$; 5. for all $\varphi\in L^q(0,T;L^p(\Omega^\ast_\varepsilon))$, we get $$\label{derunf} \dfrac{\partial}{\partial t}(\mathcal{T}^\ast_{\varepsilon }(\varphi))(t,x,y)=\dfrac{\partial\varphi}{\partial t}\left(t, \varepsilon \left[ \dfrac{x}{\varepsilon }\right]_{Y}+\varepsilon y\right)=\mathcal{T}^\ast_{\varepsilon }\left(\dfrac{\partial\varphi}{\partial t}\right)(t,x,y)\,\text{ for a.e. }(t,x,y)\in (0,T)\times \Omega\times Y^\ast.$$ \[property\]Let $T>0$. For $p\in \lbrack 1,+\infty \lbrack $ and $q\in [1,+\infty]$, let $\varphi^\varepsilon \in L^q(0,T; L^{1}\left( \Omega^\ast_\varepsilon\right))$ satisfying $$\int_0^T\int_{\Lambda^\ast_\varepsilon}\|\varphi^\varepsilon\|{\,\mathrm{d}}x{\,\mathrm{d}}t \rightarrow 0.$$ Then, one has $$\int_0^T\int\nolimits_{\Omega^\ast_\varepsilon}\varphi^\varepsilon {\,\mathrm{d}}x {\,\mathrm{d}}t-\frac{1}{\left\vert Y\right\vert }\int_0^T\int\nolimits_{\Omega \times Y^\ast}\mathcal{T}^\ast_{\varepsilon }\left( \varphi^\varepsilon \right) {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\rightarrow 0.$$ As usual, this is denoted by $$\int_0^T\int\nolimits_{\Omega^\ast_\varepsilon}\varphi^\varepsilon {\,\mathrm{d}}x {\,\mathrm{d}}t \backsimeq \frac{1}{\left\vert Y\right\vert }\int_0^T\int\nolimits_{\Omega \times Y^\ast}\mathcal{T}^\ast_{\varepsilon }\left( \varphi^\varepsilon \right) {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t.$$ As a consequence, we have: 1. for every $\varphi \in L^q(0,T;L^{p}\left( \Omega^\ast_\varepsilon\right) )$, one gets $$\left\Vert \mathcal{T}^\ast_{\varepsilon }\left( \varphi \right) \right\Vert_{L^q(0,T;\, L^{p}\left( \Omega \times Y^\ast\right) )}\leq \left\vert Y\right\vert^{1/p}\left\Vert \varphi \right\Vert _{L^q(0,T;\, L^{p}\left( \Omega^\ast_\varepsilon\right)) },$$ which means that the operator $\mathcal{T}^\ast_{\varepsilon }$ is continuous from $L^q(0,T;L^p(\Omega^\ast_\varepsilon))$ to $L^q(0,T;L^p(\Omega\times Y^\ast)) $; 2. for every $\varphi \in L^q(0,T;W^{1,p}\left( \Omega^\ast_\varepsilon\right) ) $, one has $\mathcal{T}^\ast_{\varepsilon }\left(\varphi\right) \in L^q(0,T;L^{2}\left( \Omega,W^{1,p}\left( Y^\ast\right) )\right); $ 3. for $p, q\in (1,+\infty]$, let $\varphi^\varepsilon\in L^q(0,T; L^p(\Omega^\ast_\varepsilon))$ and $\psi\in L^{q'}(0,T; L^{p'}(\Omega^\ast_\varepsilon))$, with $\dfrac{1}{p}+\dfrac{1}{p'}=1$, $\dfrac{1}{q}+\dfrac{1}{q'}=1$ such that $$\\|\varphi_\varepsilon\\|_{L^q(0,T; L^p(\Omega^\ast_\varepsilon))}\leq C\quad\text{ and }\quad \\|\psi\\|_{L^{q'}(0,T; L^{p'}(\Omega^\ast_\varepsilon))}\leq C$$ with $C$ a positive constant independent of $\varepsilon$. Then, $$\int_0^T\int\nolimits_{\Omega^\ast_\varepsilon}\varphi^\varepsilon\psi {\,\mathrm{d}}x {\,\mathrm{d}}t \backsimeq \frac{1}{\left\vert Y\right\vert }\int_0^T\int\nolimits_{\Omega \times Y^\ast}\mathcal{T}^\ast_{\varepsilon }\left( \varphi_\varepsilon \right)\mathcal{T}^\ast_{\varepsilon }\left( \psi\right) {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t;$$ 4. for $p, q\in (1,+\infty]$, let $\varphi^\varepsilon\in L^q(0,T; L^p(\Omega^\ast_\varepsilon))$ and $\psi^\varepsilon\in L^{q'}(0,T; L^{p_0}(\Omega^\ast_\varepsilon))$, with $\dfrac{1}{p}+\dfrac{1}{p_0}<1$, $\dfrac{1}{q}+\dfrac{1}{q'}=1$ such that $$\\|\varphi^\varepsilon\\|_{L^q(0,T; L^p(\Omega^\ast_\varepsilon))}\leq C\quad\text{ and }\quad \\|\psi^\varepsilon\\|_{L^{q'}(0,T; L^{p_0}(\Omega^\ast_\varepsilon))}\leq C$$ with $C$ a positive constant independent of $\varepsilon$. Then $$\int_0^T\int\nolimits_{\Omega^\ast_\varepsilon}\varphi^\varepsilon\psi^\varepsilon {\,\mathrm{d}}x {\,\mathrm{d}}t \backsimeq \frac{1}{\left\vert Y\right\vert }\int_0^T\int\nolimits_{\Omega \times Y^\ast}\mathcal{T}^\ast_{\varepsilon }\left( \varphi^\varepsilon \right)\mathcal{T}^\ast_{\varepsilon }\left( \psi^\varepsilon\right) {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t.$$ Moreover, we have the following convergence properties. \[convergence\] 1. Let $\varphi \in L^q(0,T;L^{p}\left( \Omega \right) )$. Then, $$\mathcal{T}^\ast_{\varepsilon }\left( \varphi \right) \longrightarrow \varphi \text{ \ strongly in }L^q(0,T;L^{p}\left( \Omega \times Y^\ast\right)) \text{.}$$ 2. Let $\varphi ^{\varepsilon }\in L^q(0,T;L^{p}\left( \Omega \right)) $ such that $\varphi ^{\varepsilon}\longrightarrow \varphi $ strongly in $L^q(0,T;L^{p}\left( \Omega \right)) $. Then, $$\mathcal{T}^\ast_{\varepsilon }\left( \varphi ^{\varepsilon }\right)\longrightarrow \varphi \text{ \ strongly in }L^q(0,T;L^{p}\left( \Omega \times Y^\ast\right)) \text{.}$$ 3. Let $\varphi ^{\varepsilon }\in L^q(0,T;L^{p}\left( \Omega^\ast_\varepsilon\right)) $ satisfy $\left\Vert \varphi ^{\varepsilon}\right\Vert _{L^q(0,T;L^{p}\left( \Omega^\ast_\varepsilon\right))}\leq C$ and $$\mathcal{T}^\ast_{\varepsilon }\left( \varphi ^{\varepsilon }\right) \rightharpoonup \widehat{\varphi }\,\text{ weakly in } L^q(0,T;L^{p}\left( \Omega \times Y^\ast\right)).$$ Then, $$\widetilde{\varphi }^{\varepsilon }\rightharpoonup \dfrac{\vert Y^\ast\vert}{\vert Y\vert }\mathcal{M}_{Y^\ast}\left( \widehat{\varphi }\right) \text{ \ weakly in }L^q(0,T;L^{p}\left(\Omega \right) ).$$ Let us finally recall a known result about the convergences of the previously introduced unfolding operators $\mathcal{T}_\varepsilon$ and $\mathcal{T}^\ast_\varepsilon$, applied to bounded sequences in $H^1(\Omega)$ and $H^1(\Omega^\ast_\varepsilon)$, respectively. \[u1\] Let $v_\varepsilon$ be a sequence in $L^2(0,T;H^1(\Omega))$ such that $$\\|v_\varepsilon\\|_{L^2(0,T;H^1(\Omega))}\leq C,$$ with $C$ a positive constant independent of $\varepsilon$. Then, there exist $v\in L^2(0,T;H^1(\Omega))$ and $\widehat{v}\in L^2 \left (0,T;L^{2}( \Omega,H_{per}^1(Y)/\Bbb R)\right)$ such that, up to a subsequence, $$\left\{ \begin{array}{lll} \mathcal{T}_{\varepsilon }\left( v_{\varepsilon }\right) \rightharpoonup v & \text{weakly in} & L^2\left (0,T;L^{2}( \Omega,H^{1}\left( Y\right) )\right), \\[2 mm] \mathcal{T}_{\varepsilon }\left( \nabla v_{\varepsilon }\right) \rightharpoonup \nabla v+\nabla _{y}\widehat{v}& \text{weakly in} & L^2(0,T;L^{2}\left( \Omega \times Y\right)). \end{array}\right. \label{8}$$ \[u1\\] Let $v_\varepsilon$ be a sequence in $L^2(0,T;H^1(\Omega^\ast_\varepsilon))$ such that $$\\|v_\varepsilon\\|_{L^2(0,T;H^1(\Omega^\ast_\varepsilon))}\leq C,$$ with $C$ a positive constant independent of $\varepsilon$. Then, there exist $v\in L^2(0,T;H^1(\Omega))$ and $\widehat{v}\in L^2 \left (0,T;L^{2}( \Omega,H_{per}^1(Y^\ast)/\Bbb R)\right)$ such that, up to a subsequence, $$\left\{ \begin{array}{lll} \mathcal{T}^\ast_{\varepsilon }\left( v_{\varepsilon }\right) \rightharpoonup v & \text{weakly in} & L^2\left (0,T;L^{2}( \Omega,H^{1}\left( Y^\ast\right) )\right), \\[2 mm] \mathcal{T}^\ast_{\varepsilon }\left( \nabla v_{\varepsilon }\right) \rightharpoonup \nabla v+\nabla _{y}\widehat{v}& \text{weakly in} & L^2(0,T;L^{2}\left( \Omega \times Y^\ast\right)). \end{array}\right. \label{8}$$ Using the same notation as in [@Cio-Dam-Don-Gri-Zaki] (see, also, [@Cabarrubias-Donato] and [@Donato-Yang]), let us give the following definition: \[defunfbound\] Let $T>0$. For any function $\varphi$ which is Lebesgue measurable on $\Gamma_{\varepsilon}$, we define the boundary unfolding operator $\mathcal{T}^{b}_{\varepsilon}$ as follows: $$\mathcal{T}^{b}_{\varepsilon }\left( \varphi \right) \left(t, x,y\right)=\varphi \left(t, \varepsilon \left[ \dfrac{x}{\varepsilon }\right]_{Y}+\varepsilon y\right) \quad for \,\, a.e.\ \left(t,x,y\right) \in (0,T)\times \Omega \times \Gamma.$$ \[propertyb\]Let $p,q\in \lbrack 1,+\infty \lbrack $ and $T>0$. The operator $\mathcal{T}^b_{\varepsilon }$ is linear and continuous from $L^q(0,T;L^p(\Gamma^\varepsilon))$ to $L^q(0,T;L^p(\Omega \times \Gamma)) $. Moreover, 1. For every $\varphi\in L^q(0,T; L^{1}\left( \Gamma^\varepsilon)\right)$, one gets $$\label{intbound} \dfrac{1}{\varepsilon \|Y\|}\int_{\Omega\times\Gamma}T^b_\varepsilon(\varphi)(t,x,y) {\,\mathrm{d}}x {\,\mathrm{d}}\sigma_y=\int_{\Gamma_\varepsilon}\varphi(t,x) {\,\mathrm{d}}\sigma_x,$$ for a.e. $t\in (0,T)$. 2. For every $\varphi \in L^q(0,T;L^{p}\left( \Gamma^\varepsilon)\right) $, one gets $$\label{normbound} \\|T^b_\varepsilon(\varphi)\\|_{L^q(0,T;L^{p}\left( \Omega\times\Gamma\right) )}\leq \|Y\|^{1/p}\varepsilon^{1/p}\\|\varphi\\|_{L^q(0,T;L^{p}\left( \Gamma^\varepsilon\right) )}.$$ \[remgeo\] We shall be interested in working with these unfolding operators only for our particular form of the domain $\Omega$ (see Section \[Sec2\]). Hence, for such a geometry, it holds $\widehat{\Lambda}_\varepsilon=\widehat{\Lambda}^\ast_\varepsilon=\emptyset$. Weak and strong convergence results ----------------------------------- Under the assumptions we imposed on the geometry and on the data, we can use extensions for time-dependent functions to the whole of the domain $\Omega$. Following [@Bohm] and [@Hopker] (see, also, [@Acerbi], [@Cio-Pau], [@Gahn], and [@Meirmanov]), in our geometry, for $i\in \{1,2,3\}$, there exists a linear and bounded extension operator ${\cal L}_i^\varepsilon: L^2(0,T; H^1_{\partial \Omega}(\Omega^\ast_\varepsilon)) \rightarrow L^2(0,T; H_0^1(\Omega))$. We denote $$\label{extension} {\cal L}_i^\varepsilon (c_i^\varepsilon) =\overline{c}_i^\varepsilon.$$ We remark that the above linear and bounded extension operator to the whole of $\Omega$ preserves the non-negativity, the essential boundedness and the [a priori]{} estimates - obtained for the solution $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$. Moreover, as in [@Gra-Pet] and [@Graf], it follows that $$\overline{c}_i^\varepsilon \in L^2((0,T), H_0^1(\Omega))\cap H^1((0,T),(H^1_0(\Omega))^{'})\cap L^\infty((0,T)\times \Omega),$$ with bounds independent of $\varepsilon$, and there exists $c'_i\in L^2(0,T; H_0^1 (\Omega))$ such that, for $i\in \{1,2,3\}$, $$\label{convext} \overline{c}_i^\varepsilon \rightarrow c'_i\,\,\,\,\text{strongly in }L^2((0, T) \times \Omega).$$ Let us fix $i\in\{1,2,3\}$. For the function $c^\varepsilon_i\in \mathcal{W}\left(0,T;H^{1}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}),(H^{1}_{\partial \Omega}(\Omega^{\ast}_{\varepsilon}))'\right)$, we consider the time derivative $\partial_t \tilde{c}_i^\varepsilon \in L^2(0,T;(H_0^1(\Omega))')$ of the extension by zero of $c_i^\varepsilon$. It is obvious that the generalized time derivative of $\tilde{c}_i^\varepsilon$ exists and it holds $$\label{dergen} \langle\partial_{t}\tilde{c}_{i}^{\varepsilon}(t),v\rangle_{\Omega}=\langle\partial_{t}c_{i}^{\varepsilon}(t),v\|_{\Omega^{\ast}_{\varepsilon}}\rangle_{\Omega^{\ast}_{\varepsilon}}\,\,\forall v\in H_0^1(\Omega)\text{ and a.e. }t\in(0,T),$$ which implies $$\label{dergenbis} \\|\partial_t \tilde{c}_i^\varepsilon\\|_{L^2(0,T; (H_0^1(\Omega))')}\leq \\|\partial_t c_i^\varepsilon\\|_{L^2(0,T; (H_0^1(\Omega^{\ast}_\varepsilon))')} .$$ In the next lemma, we collect the main compactness results we have for the solution of our microscopic problem obtained by using the properties of the time-dependent unfolding operator $\mathcal{T}^\ast_{\varepsilon}$ for perforated domains recalled in Section \[sub2.3\] and the [a priori]{} estimates proved in Lemma \[lemest\]. \[lemma-conv\] Let $(c_1^\varepsilon, c_2^\varepsilon, c_3^\varepsilon)$ be the unique solution of problem -. Then, up to a subsequence, there exist $c$ and $c_3\in L^2( 0,T; H_0^{1}(\Omega))$, $\widehat{c}_i\in L^2((0,T) \times \Omega; H^1_{\textrm{per}}(Y^\ast)/\mathbb R)$, with $i\in \{1,2,3\}$, such that, for $\varepsilon \rightarrow 0$, we have $$\label{conv-genc12} \left\{ \begin{array}{lll} i)&\mathcal{T}^\ast_{\varepsilon } (c^{\varepsilon }_i) \rightharpoonup c& \textrm{weakly in } L^2((0,T) \times \Omega, H^1(Y^\ast)),\\ \\ ii)&\mathcal{T}^\ast_{\varepsilon } (\nabla c^{\varepsilon }_i) \rightharpoonup \nabla c+ \nabla_y \widehat{c}_i & \textrm{weakly in } L^2((0,T) \times \Omega \times Y^\ast),\\ \\ iii)&\mathcal{T}^\ast_{\varepsilon } (c_i^\varepsilon) \rightarrow c & \textrm{strongly in } L^2((0,T)\times \Omega\times Y^\ast),\\ \\ iv)&{\cal T}_\varepsilon^b (c^\varepsilon_i) \rightarrow c& \textrm{strongly in } L^2((0,T)\times \Omega \times \Gamma),\\ \\ v)&\partial_t \tilde{c}_i^\varepsilon \rightharpoonup \vert Y^* \vert \partial_t c & \textrm{weakly in } L^2(0,T; (H_0^1(\Omega))'), \end{array} \right.$$ for $i\in\{1,2\}$ and $$\label{conv-genc3} \left\{ \begin{array}{lll} i)&\mathcal{T}^\ast_{\varepsilon } (c^{\varepsilon }_3) \rightharpoonup c_3& \textrm{weakly in } L^2((0,T) \times \Omega, H^1(Y^\ast)),\\ \\ ii)&\mathcal{T}^\ast_{\varepsilon } (\nabla c^{\varepsilon }_3) \rightharpoonup \nabla c_3+ \nabla_y \widehat{c}_3 & \textrm{weakly in } L^2((0,T) \times \Omega \times Y^\ast),\\ \\ iii)&\mathcal{T}^\ast_{\varepsilon } (c_3^\varepsilon) \rightarrow c_3 & \textrm{strongly in } L^2((0,T)\times \Omega\times Y^\ast),\\ \\ iv)&{\cal T}_\varepsilon^b (c^\varepsilon_3) \rightarrow c_3& \textrm{strongly in } L^2((0,T)\times \Omega \times \Gamma),\\ \\ v)&\partial_t \tilde{c}_3^\varepsilon \rightharpoonup \vert Y^* \vert \partial_t c_3 & \textrm{weakly in } L^2(0,T; (H_0^1(\Omega))'). \end{array} \right.$$ Let us fix $i\in\{1,2,3\}$. By of Lemma \[lemest\] and Theorem \[u1\\], it follows that there exist $c_i\in L^2( 0,T; H_0^{1}(\Omega))$ and $\widehat{c}_i\in L^2((0,T) \times \Omega; H^1_{\textrm{per}}(Y^\ast)/\mathbb R)$ such that, up to a subsequence still denoted by $\varepsilon$, the following convergences hold $$\label{conv1} \left\{ \begin{array}{lll} i)&\mathcal{T}^\ast_{\varepsilon } (c^{\varepsilon }_i) \rightharpoonup c_i& \textrm{weakly in } L^2((0,T) \times \Omega, H^1(Y^\ast)),\\ \\ ii)&\mathcal{T}^\ast_{\varepsilon } (\nabla c^{\varepsilon }_i) \rightharpoonup \nabla c_i+ \nabla_y \widehat{c}_i & \textrm{weakly in } L^2((0,T) \times \Omega \times Y^\ast).\\ \end{array} \right.$$ On the other hand, by Theorem \[u1\] and , we get $$T_\varepsilon(\overline{c}_i^\varepsilon)\to c'_i \quad{ \rm in }\quad L^2((0,T)\times\Omega\times Y),$$ which implies $$T_\varepsilon(\overline{c}_i^\varepsilon)_{\|\Omega\times Y^\ast} \to (c'_i)_{\|\Omega\times Y^\ast}=c'_i\quad{ \rm in }\quad L^2((0,T)\times\Omega\times Y^\ast),$$ since $c'_i$ doesn’t depends on $y$. On the other hand, by it holds $$T_\varepsilon(\overline{c}_i^\varepsilon)_{\|\Omega\times Y^\ast}=T^{\ast}_\varepsilon(c_i^\varepsilon).$$ Hence, we can deduce that $$T^{\ast}_\varepsilon(c_i^\varepsilon) \to c'_i \quad{\rm in }\quad L^2((0,T)\times\Omega\times Y^\ast).$$ Due to i), by uniqueness $c'_i=c_i$ and we get indeed $$\label{strong-domain} \mathcal{T}^\ast_{\varepsilon } (c_i^\varepsilon) \rightarrow c_i \quad \textrm{strongly in } L^2((0,T)\times \Omega\times Y^\ast).$$ Moreover, by Lemma \[lemest\], acting as in [@Gahn Proposition 12], we get $$\label{strong-boundary} \mathcal{T}^{b}_{\varepsilon } (c_i^\varepsilon) \rightarrow c_i \quad \textrm{strongly in } L^2((0,T)\times \Omega \times \Gamma).$$ We remark that, for $i\in \{1,2,3\}$, $c_i\in \mathcal{W}\left(0,T;H_0^1(\Omega), (H_0^1(\Omega))'\right)$. Now it remains to prove that $c_1=c_2=c$ in $(0,T)\times \Omega$. To this, let us observe that, by and , we get $$\label{c1=c2} \begin{array}{l} \\|{\cal T}_\varepsilon^b (c^\varepsilon_2)-c_1\\|_{L^2((0,T)\times\Omega\times\Gamma)}\leq \\|{\cal T}_\varepsilon^b (c^\varepsilon_2)-{\cal T}_\varepsilon^b (c^\varepsilon_1)\\|_{L^2((0,T)\times\Omega\times\Gamma)}+\\|{\cal T}_\varepsilon^b (c^\varepsilon_1)-c_1\\|_{L^2((0,T)\times\Omega\times\Gamma)}\\ \\ \leq \sqrt\varepsilon\\|c_2^\varepsilon-c_1^\varepsilon\\|_{L^2((0,T)\times\Gamma_\varepsilon)}+\\|{\cal T}_\varepsilon^b (c^\varepsilon_1)-c_1\\|_{L^2((0,T)\times\Omega\times\Gamma)}\\ \\ \leq C\varepsilon+\\|{\cal T}_\varepsilon^b (c^\varepsilon_1)-c_1\\|_{L^2((0,T)\times\Omega\times\Gamma)}. \end{array}$$ Hence, when $\varepsilon$ tends to zero, by it holds $$\mathcal{T}^{b}_{\varepsilon } (c_2^\varepsilon) \rightarrow c_1\quad \textrm{strongly in } L^2((0,T)\times \Omega \times \Gamma),$$ which means that $c_1=c_2$ in $(0,T)\times \Omega \times \Gamma$. Since $c_1$ and $c_2$ are independent of $y$, we obtain $c_1=c_2$ in the whole $(0,T)\times \Omega$.\ By and , there exists $W_i \in L^2(0,T; (H_0^1(\Omega))')$ such that, up to a subsequence, we obtain $$\partial_t \tilde{c}_i^\varepsilon \rightharpoonup W_i \quad \textrm{weakly in } L^2(0,T; (H_0^1(\Omega))').$$ An easy integration by parts, Proposition \[convergence\](iii) and show that $W_i=\dfrac{\vert Y^\ast \vert}{\vert Y\vert} \partial_t c_i $. Hence, $$\label{derconv} \partial_t \tilde{c}_i^\varepsilon \rightharpoonup \dfrac{\vert Y^\ast \vert}{\vert Y\vert} \partial_t c_i \quad \textrm{weakly in } L^2(0,T; (H_0^1(\Omega))').$$ Finally, , , , and imply for $i\in\{1,2\}$ and for $i=3$. \[rem3\] In order to use classical compactness results and to obtain convergences of the microscopic solution, we are forced to extend, at first, the functions $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$ to the whole domain $\Omega$ by means of a suitable uniform extension operator before unfolding. Indeed, by using the a priori uniform estimate of Lemma \[lemest\] and Theorem \[u1\\] we could get only weak convergences of $c_i^\varepsilon$ and $T^\ast_\varepsilon(c_i^\varepsilon)$, $i\in\{1,2,3\}$. Moreover, due to the less regularity of the time derivative, these weak convergences can’t be improved unlike in [@Donato-Yang]. When handling nonlinear terms, as in our paper, weak convergence isn’t enough, but we need also strong convergence with respect to suitable topologies. More generally, we can deduce that when dealing with the homogenization by unfolding in a perforated domain, if there exists a classical uniform extension operator, it is like we could act directly with $T^\ast_\varepsilon$. On the other hand, if we cannot construct such a uniform extension operator, due to some particular reasons (for example, the lack of regularity of the boundary of the holes), we can homogenize as well (this is the main advantage of the unfolding), but, in the presence of nonlinear terms, we are forced to prove a convergence like . The macroscopic model --------------------- The main convergence result of this paper is stated in the next theorem, where we take into account that $\vert Y\vert=1$. \[teounf\] Let $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$ be the solution of system -. Then, under the assumptions $(\mathbf{H}_1)\div (\mathbf{H})_7$, there exist $c$ and $c_{3}\in L^{2}(0,T;H_0^{1}(\Omega))$ and $\widehat{c}_{i}\in L^{2}((0,T)\times\Omega;H_{per}^{1}(Y^\ast)/\Bbb R)$, $i\in \{1,2,3\}$, such that $$\label{unfold-convconc}\left\{ \begin{array}{lll} i)&\mathcal{T}^{\ast}_{\varepsilon}(c_{i}^{\varepsilon})\rightharpoonup c &\text{\rm weakly in }L^{2}((0,T)\times\Omega;H^{1}(Y^\ast)),\\[2mm] ii)&\mathcal{T}^{\ast}_{\varepsilon}(\nabla c_{i}^{\varepsilon})\rightharpoonup\nabla c+\nabla_{y}\widehat{c}_{i}&\text{\rm weakly in }L^{2}((0,T)\times \Omega\times Y^\ast),\\[2mm] iii)&\mathcal{T}^\ast_{\varepsilon } (c_i^\varepsilon) \rightarrow c & \text{\rm strongly in } L^2((0,T)\times \Omega\times Y^\ast), \end{array} \right.$$ for $i\in\{1,2\}$ and $$\label{unfold-convconc3}\left\{ \begin{array}{lll} i)&\mathcal{T}^{\ast}_{\varepsilon}(c_{3}^{\varepsilon})\rightharpoonup c_{3}&\text{\rm weakly in }L^{2}((0,T)\times\Omega;H^{1}(Y^\ast)),\\[2mm] ii)&\mathcal{T}^{\ast}_{\varepsilon}(\nabla c_{3}^{\varepsilon})\rightharpoonup\nabla c_{3}+\nabla_{y}\widehat{c}_{3}&\text{\rm weakly in }L^{2}((0,T)\times \Omega\times Y^\ast),\\[2mm] iii)&\mathcal{T}^\ast_{\varepsilon } (c_3^\varepsilon) \rightarrow c_3 & \text{\rm strongly in } L^2((0,T)\times \Omega\times Y^\ast), \end{array} \right.$$ where $(c, \widehat{c_1}, \widehat{c}_{2}, c_3, \widehat{c}_{3})$ is the unique solution of the following problem $$\label{probunf} \left\{ \begin{array}{l} \text{\rm Find } (c, c_3)\in (L^{2}(0,T;H_0^{1}(\Omega)))^2 \ {\text{\rm and }} \widehat{c}_{i}\in L^{2}((0,T)\times\Omega;H_{per}^{1}(Y^\ast)/\Bbb R),\,i\in \{1,2,3\}, \,\text{\rm such that } \\ \\ 2\displaystyle \vert Y^\ast \vert \, \langle\partial_t c, \varphi\rangle_{\Omega}+ \displaystyle \int_{\Omega\times Y^{\ast}}D_1(y)(\nabla c+\nabla_{y}\widehat{c_{1}})(\nabla\varphi+\nabla_{y}\Psi_1) {\,\mathrm{d}}x {\,\mathrm{d}}y\\ \\ +\displaystyle \int_{\Omega\times Y^{\ast}}D_2(y)(\nabla c+\nabla_{y}\widehat{c_{2}})(\nabla\varphi+\nabla_{y}\Psi_2)\,\mathrm{d}x\mathrm{d}y +\displaystyle \int_{\Omega\times\Gamma} (\widehat{c_{1}}-\widehat{c_{2}}) H(c_3) (\Psi_1-\Psi_2) {\,\mathrm{d}}x {\,\mathrm{d}}\sigma_{y}\\ \\ \displaystyle = \int_{\Omega\times Y^{\ast}}F_{1}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y+\int_{\Omega\times Y^{\ast}}F_{2}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y,\\ \\ \displaystyle \vert Y^\ast \vert \, \langle\partial_t c_3, \varphi\rangle_{\Omega}+ \displaystyle \int_{\Omega\times Y^{\ast}}D_3(y)(\nabla c_{3}+\nabla_{y}\widehat{c_{3}})(\nabla\varphi+\nabla_{y}\Psi_3) {\,\mathrm{d}}x {\,\mathrm{d}}y\\ \\ \displaystyle=\int_{\Omega\times\Gamma}G_{3}(y,c,c,c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}\sigma_{y}+ \int_{\Omega\times Y^{\ast}}F_{3}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y,\\ \\ \text{\rm in }\mathcal{D}'(0,T)\,\text{\rm and for all } \varphi\in H_0^{1}(\Omega),\ \Psi_i\in L^2(\Omega ;H_{per}^{1}(Y^\ast)),\\ \\ c(x,0)=(c_1^0+c_2^0)/2 \,\,\,\,\,\,\text {\rm in } \Omega,\\ \\ c_3(x,0)=c_3^0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ \rm in } \Omega. \end{array} \right.$$ By Lemma \[lemma-conv\], we get convergences and , up to a subsequence, still denoted by $\varepsilon$. It remains to prove that $(c, \widehat{c}_{1}, \widehat{c}_{2}, c_3, \widehat{c}_{3})$ is solution of the limit problem . To this aim, let $$\label{test} v_i^\varepsilon(x)=\varphi(x) +\varepsilon \, \omega_i(x) \, \psi_i^\varepsilon(x),$$ where $\varphi, \omega_i \in \mathcal{D}(\Omega)$, $\psi_i^\varepsilon(x)=\psi_i\left(\dfrac{x}{\varepsilon}\right)$, and $\psi_i\in H_{\textrm{per}}^{1}(Y^\ast)$.\ Since $\mathcal{T}^{\ast}_{\varepsilon}\left(\nabla v_i^\varepsilon \right)=\mathcal{T}^{\ast}_{\varepsilon}\left(\varphi\right) +\varepsilon\mathcal{T}^{\ast}_{\varepsilon}\left(\nabla\omega\right) \mathcal{T}^{\ast}_{\varepsilon}\left(\psi_i^\varepsilon\right)+ \mathcal{T}^{\ast}_{\varepsilon}\left(\omega\right)\mathcal{T}^{\ast}_{\varepsilon}\left(\nabla_y\psi_i^\varepsilon\right)$, we have $$\label{convtest} \begin{array}{ll} \mathcal{T}^{\ast}_{\varepsilon}\left(v_i^\varepsilon \right)\rightarrow \varphi&\text{strongly in }L^2(\Omega\times Y^\ast),\\ \\ \mathcal{T}^{\ast}_{\varepsilon}\left(\nabla v_i^\varepsilon \right)\rightarrow \nabla\varphi+\omega_i\nabla_y \psi_i&\text{strongly in }L^2(\Omega\times Y^\ast). \end{array}$$ In order to get the first equation in , let us take $v=v_1^\varepsilon$ and $v=v_2^\varepsilon$ as test functions in the first equation in , written for $i=1$ and $i=2$, respectively. Multiplying these equations by $w\in \mathcal{D}(0,T)$, integrating by parts and summing them up, we get $$\begin{array}{l} -\displaystyle\int_0^T\int_{\Omega^{\ast}_{\varepsilon}}c_1^\varepsilon \, v_1^\varepsilon w\, ' \mathrm{d}x\,\mathrm{d}t-\displaystyle\int_0^T\int_{\Omega^{\ast}_{\varepsilon}}c_2^\varepsilon \, v_2^\varepsilon w\,' {\,\mathrm{d}}x {\,\mathrm{d}}t\\ \\ + \displaystyle \int_0^T\int_{\Omega^{\ast}_{\varepsilon}}D_1^\varepsilon \nabla c_{1}^{\varepsilon}\cdot\nabla v_1^{\varepsilon}\, w {\,\mathrm{d}}x {\,\mathrm{d}}t + \displaystyle \int_0^T\int_{\Omega^{\ast}_{\varepsilon}}D_2^\varepsilon \nabla c_{2}^{\varepsilon}\cdot\nabla v_2^{\varepsilon}\, w {\,\mathrm{d}}x {\,\mathrm{d}}t\\ \\ =\displaystyle \frac{1}{\varepsilon}\int_0^T\int_{\Gamma^{\varepsilon}}(G_{1}(c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_3^{\varepsilon})v_1^{\varepsilon} +G_{2}(c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_3^{\varepsilon})v_2^{\varepsilon}) \, w {\,\mathrm{d}}\sigma _{x} {\,\mathrm{d}}t\\ \\ +\displaystyle \int_0^T\int_{\Omega ^{\varepsilon}}F_{1}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_3^{\varepsilon})v_1^{\varepsilon} \, w {\,\mathrm{d}}x {\,\mathrm{d}}t +\displaystyle \int_0^T\int_{\Omega ^{\varepsilon}}F_{2}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_3^{\varepsilon})v_2^{\varepsilon} \, w {\,\mathrm{d}}x {\,\mathrm{d}}t. \end{array}$$ By we obtain $$\label{unf3} \begin{array}{l} -\displaystyle\int_0^T\int_{\Omega^{\ast}_{\varepsilon}}c_1^\varepsilon \, v_1^\varepsilon w\, ' \mathrm{d}x\,\mathrm{d}t-\displaystyle\int_0^T\int_{\Omega^{\ast}_{\varepsilon}}c_2^\varepsilon \, v_2^\varepsilon w\,' {\,\mathrm{d}}x {\,\mathrm{d}}t + \displaystyle \int_0^T\int_{\Omega^{\ast}_{\varepsilon}}D_1^\varepsilon \nabla c_{1}^{\varepsilon}\cdot\nabla v_1^{\varepsilon}\, w {\,\mathrm{d}}x {\,\mathrm{d}}t \\ \\ + \displaystyle \int_0^T\int_{\Omega^{\ast}_{\varepsilon}}D_2^\varepsilon \nabla c_{2}^{\varepsilon}\cdot\nabla v_2^{\varepsilon}\, w {\,\mathrm{d}}x {\,\mathrm{d}}t +\displaystyle\int_0^T\int_{\Gamma^{\varepsilon}}(c_{1} ^{\varepsilon}-c_{2}^{\varepsilon})H\left (c_3^{\varepsilon}\right )(\omega_1 \, \psi_1^\varepsilon- \omega_2 \, \psi_2^\varepsilon) \, w {\,\mathrm{d}}\sigma _{x} {\,\mathrm{d}}t\\ \\ =\displaystyle \int_0^T\int_{\Omega^\ast _{\varepsilon}}F_{1}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_3^{\varepsilon})v_1^{\varepsilon} \, w {\,\mathrm{d}}x {\,\mathrm{d}}t +\displaystyle \int_0^T\int_{\Omega^\ast _{\varepsilon}}F_{2}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon }, c_3^{\varepsilon})v_2^{\varepsilon} \, w {\,\mathrm{d}}x {\,\mathrm{d}}t. \end{array}$$ By Proposition \[property\] and Proposition \[propertyb\], we unfold by means of the operators $\mathcal{T}^{\ast}_{\varepsilon}$ and $\mathcal{T}^{b}_{\varepsilon}$ and by assumption $(\mathbf{H}_1)_{1}$ we have $$\label{unf3bis} \begin{array}{l} -\displaystyle\int_0^T\int_{\Omega\times Y^\ast}\mathcal{T}^{\ast}_{\varepsilon}(c_1^\varepsilon) \, \mathcal{T}^{\ast}_{\varepsilon}(v_1^\varepsilon) w\, ' \mathrm{d}x\,{\,\mathrm{d}}y\,\mathrm{d}t-\displaystyle\int_0^T\int_{\Omega\times Y^\ast}\mathcal{T}^{\ast}_{\varepsilon}(c_2^\varepsilon) \, \mathcal{T}^{\ast}_{\varepsilon}(v_2^\varepsilon) w\,' {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ + \displaystyle \int_0^T\int_{\Omega\times Y^\ast}D_1(y)\mathcal{T}^{\ast}_{\varepsilon}(\nabla c_{1}^{\varepsilon})\cdot\mathcal{T}^{\ast}_{\varepsilon}(\nabla v_1^{\varepsilon})\, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t + \displaystyle \int_0^T\int_{\Omega\times Y^\ast}D_2(y)\mathcal{T}^{\ast}_{\varepsilon}(\nabla c_{2}^{\varepsilon})\cdot\mathcal{T}^{\ast}_{\varepsilon}(\nabla v_2^{\varepsilon})\, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ +\displaystyle\int_0^T\int_{\Omega\times \Gamma}\mathcal{T}^{b}_{\varepsilon}\left(\dfrac{c_{1} ^{\varepsilon}-c_{2}^{\varepsilon}}{\varepsilon}H(c_3^{\varepsilon})\right)(\omega_1 \, \mathcal{T}^{\ast}_{\varepsilon}(\psi_1^\varepsilon)- \omega_2 \, \mathcal{T}^{\ast}_{\varepsilon}(\psi_2^\varepsilon)) \, w {\,\mathrm{d}}\sigma _{x} {\,\mathrm{d}}y{\,\mathrm{d}}t\\ \\ =\displaystyle \int_0^T\int_{\Omega\times Y^\ast}\mathcal{T}^{\ast}_{\varepsilon}(F_{1}^{\varepsilon}(x, c_{1}^{\varepsilon}, c_{2}^{\varepsilon }, c_3^{\varepsilon}))\mathcal{T}^{\ast}_{\varepsilon}(v_1^{\varepsilon}) \, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t \\ \\ +\displaystyle \int_0^T\int_{\Omega\times Y^\ast}\mathcal{T}^{\ast}_{\varepsilon}(F_{2}^{\varepsilon}(x, c_{1}^{\varepsilon}, c_{2}^{\varepsilon }, c_3^{\varepsilon}))\mathcal{T}^{\ast}_{\varepsilon}(v_2^{\varepsilon}) \, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t. \end{array}$$ In order to obtain the macroscopic problem, we want to pass to the limit with $\varepsilon\rightarrow0$ in , by using the convergence results for the microscopic solutions proved in the previous section. To this aim, let us analyze at first the nonlinear terms.\ By assumption $(\mathbf{H}_2)$, due to the strong convergences $_{iii}$ and $_{iii}$ , for $i\in \{1,2,3\}$, one has $$\label{convFG} \mathcal{T}^\ast_{\varepsilon}(F_i^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_3^{\varepsilon})) \rightarrow F_i(y,c_{1},c_{2}, c_{3})\text{ strongly in }L^{2}((0,T)\times\Omega\times Y^\ast).$$ On the other hand, assumptions $(\mathbf{H}_3)_{2,4}$ imply that the function $H$ is globally Lipschitz-continuous and it holds $\vert H(s)\vert \leq L\vert s\vert$. Consequently $H(v)\in L^2(0,T;L^2(\Omega\times \Gamma))$ if $v\in L^2(0,T;L^2(\Omega\times\Gamma))$. Hence, the boundary term involving the function $H$ is well-defined, since the concentration fields are essentially bounded. Moreover, by and , we get $$\label{boundest} \begin{array}{l} \left \\| \mathcal{T}^{b}_{\varepsilon }\left((c_{1} ^{\varepsilon}-c_{2}^{\varepsilon})H(c_3^{\varepsilon})\right)\right \\|_ {L^2((0,T) \times \Omega \times \Gamma)} = \\ \\ = \\| (\mathcal{T}^{b}_{\varepsilon } (c_1^\varepsilon) - \mathcal{T}^{b}_{\varepsilon } (c_2^\varepsilon)) \\|_{L^2((0,T) \times \Omega \times \Gamma)}\\| H(\mathcal{T}^{b}_{\varepsilon } (c_3^\varepsilon) ) \\|_{L^\infty((0,T) \times \Omega \times \Gamma)} \leq C \, \varepsilon. \end{array}$$ Therefore, there exists a function $U\in L^2((0,T) \times \Omega \times \Gamma)$ such that $$\label{convbound} \mathcal{T}^{b}_{\varepsilon } \left( \frac{c_1^\varepsilon - c_2^\varepsilon}{\varepsilon}H(c_3^\varepsilon)\right) \rightharpoonup U \quad \textrm{weakly in } L^2((0,T)\times \Omega \times \Gamma).$$ In order to identify the function $U$, let us define in $(0,T)\times\Omega^\ast_\varepsilon$ the function $$U^\varepsilon:=\dfrac{c_1^\varepsilon-c_2^\varepsilon}{\varepsilon}H(c_3^\varepsilon).$$ Clearly, $U^\varepsilon \in L^2(0,T;H^1(\Omega^\ast_\varepsilon))$. Moreover, the [a priori]{} estimate yields $$\label{estU} \sqrt{\varepsilon}\|\|U^\varepsilon\|\|_{L^2((0,T)\times\Gamma_\varepsilon)}\leq C$$ and since the chain rule applies according to [@Stampacchia], Lemma 1.1 and $H'$ is bounded due to assumption $\mathcal{H}_3$, by we get $$\label{estgradU} \begin{array}{l} \varepsilon \|\|\nabla U^\varepsilon \|\|_{L^2((0,T)\times\Omega^\ast_\varepsilon)}\leq \|\|\nabla c_1^\varepsilon-\nabla c_2^\varepsilon\|\|_{L^2((0,T)\times\Omega^\ast_\varepsilon)}\|\|H(c_3^\varepsilon)\|\|_{L^\infty((0,T)\times\Omega^\ast_\varepsilon)}\\ \\ +\|\|c_1^\varepsilon-c_2^\varepsilon\|\|_{L^\infty((0,T)\times\Omega^\ast_\varepsilon)}\|\|H'(c_3^\varepsilon)\|\|_{L^\infty((0,T)\times\Omega^\ast_\varepsilon)}\|\|\nabla c_3^\varepsilon\|\|_{L^2((0,T)\times\Omega^\ast_\varepsilon)}\leq C. \end{array}$$ Estimates and , by using Lemma 6.1 in [@Conca], imply $$\|\|U^\varepsilon\|\|_{L^2((0,T)\times\Omega^\ast_\varepsilon)}\leq C\left( \varepsilon \|\|\nabla U^\varepsilon \|\|_{L^2((0,T)\times\Omega^\ast_\varepsilon)}+ \sqrt{\varepsilon}\|\|U^\varepsilon\|\|_{L^2((0,T)\times\Gamma_\varepsilon)}\right)\leq C.$$ Now, we can apply Proposition 2.8 in [@Donato-Yang] and get the existence of $\widehat{U}\in L^2(0,T;L^2(\Omega;H^1(Y^\ast)))$ such that $$\label{convunfU} \mathcal{T}^\ast_\varepsilon(U^\varepsilon)\rightharpoonup \widehat{U} \quad \textrm{weakly in } L^2((0,T)\times \Omega; H^1(Y^\ast))$$ and $$\label{convunfgradU} \mathcal{T}^\ast_\varepsilon(\varepsilon\nabla U^\varepsilon)\rightharpoonup \nabla_y\widehat{U} \quad \textrm{weakly in } L^2((0,T)\times \Omega\times Y^\ast).$$ By the definition of the boundary unfolding operator $T^b_\varepsilon$ as the trace on $\Gamma$ of $T^\ast_\varepsilon$, due to the continuity of the trace operator, by and , we deduce that $U$ is indeed the trace on $\Gamma$ of $\widehat{U}$. The equality $U=\widehat{U}$ in $(0,T)\times\Omega\times\Gamma$ shows that it is enough to identify the function $\widehat{U}$. We are able to calculate the limit . Indeed, for $\Psi\in \mathcal{D}((0,T)\times\Omega\times Y^\ast)^n$, by using the properties of the unfolding operator for perforated domains, we get $$\label{U} \begin{array}{c} \displaystyle\int_0^T\int_{\Omega\times Y^\ast} \mathcal{T}^\ast_\varepsilon(\varepsilon \nabla U^\varepsilon)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t=\int_0^T\int_{\Omega\times Y^\ast} \mathcal{T}^\ast_\varepsilon \left((\nabla c_1^\varepsilon-\nabla c_2^{\varepsilon})H(c_3^\varepsilon)\right)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ \displaystyle+ \int_0^T\int_{\Omega\times Y^\ast} \mathcal{T}^\ast_\varepsilon \left((c_1^\varepsilon-c_2^{\varepsilon})H^{'}(c_3^\varepsilon)\nabla c_3^\varepsilon \right)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ =\displaystyle \int_0^T\int_{\Omega\times Y^\ast} \left((\mathcal{T}^\ast_\varepsilon (\nabla c_1^\varepsilon)-T^\ast_\varepsilon (\nabla c_2^{\varepsilon}))H(\mathcal{T}^\ast_\varepsilon(c_3^\varepsilon)\right)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t \\ \\ \displaystyle+\int_0^T\int_{\Omega\times Y^\ast} \left((\mathcal{T}^\ast_\varepsilon (c_1^\varepsilon)-T^\ast_\varepsilon (c_2^{\varepsilon}))T^\ast_\varepsilon (H^{'}(c_3^\varepsilon))T^\ast_\varepsilon (\nabla c_3^\varepsilon) \right)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t. \end{array}$$ We remark now that, according to , and assumption $(\mathbf{H}_3)_3$, we have that $$\vert \mathcal{T}^{\ast}_\varepsilon(H^{'}(c_3^\varepsilon))\vert=\vert H^{'}(\mathcal{T}^{\ast}_\varepsilon(c_3^\varepsilon))\vert\leq L \quad \textrm{in } L^\infty((0, T )\times \Omega\times Y^\ast),$$ hence by iii) and ii), we get $$\label{boundLinf} \int_0^T\int_{\Omega\times Y^\ast} \left((\mathcal{T}^\ast_\varepsilon (c_1^\varepsilon)-T^\ast_\varepsilon (c_2^{\varepsilon}))T^\ast_\varepsilon (H^{'}(c_3^\varepsilon))T^\ast_\varepsilon (\nabla c_3^\varepsilon) \right)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\rightarrow 0.$$ Using now convergence iii) and assumption $(\mathbf{H}_3)$ we obtain $$\label{convhh'} \mathcal{T}^{\ast}_\varepsilon (H(c_3^\varepsilon))=H(\mathcal{T}^{\ast}_\varepsilon(c_3^\varepsilon)) \rightarrow H(c_3) \quad \textrm{strongly in } L^2((0,T)\times \Omega\times Y^\ast).$$ Hence, by ii), and , we obtain $$\begin{array}{c} \displaystyle\int_0^T\int_{\Omega\times Y^{\ast}} \mathcal{T}^\ast_\varepsilon(\varepsilon \nabla U^\varepsilon)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\rightarrow \displaystyle\int_0^T\int_{\Omega\times Y} \left(\nabla_{y}\widehat{c}_{1}-\nabla_{y}\widehat{c}_{2})H(c_3\right)\Psi(x,y,t)\,{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t. \end{array}$$ for any $\Psi\in \mathcal{D}((0,T)\times\Omega\times Y^\ast)^n$.\ Comparing with and taking into account the fact that $H$ does not depend on the variable y and the functions $\widehat{c}_{1}$ and $\widehat{c}_{2}$ are defined up to an additive function depending on $t$ and $x$ only, we obtain $\widehat{U}=(\widehat{c}_{1}-\widehat{c}_{2})H(c_3)$. So, finally $$\label{Ubound} U=\widehat{U}=(\widehat{c}_{1}-\widehat{c}_{2})H(c_3).$$ Thus, by , , and we can pass to the limit as $\varepsilon\rightarrow0$ in and we obtain: $$\begin{array}{l} \displaystyle -2 \vert Y^\ast \vert \int_{0}^{T}\int_{\Omega}c \varphi w'\,\mathrm{d}x \mathrm{d}t+\int_{0}^{T}\int_{\Omega\times Y^{\ast}}D_1(y)(\nabla c+\nabla_{y}\widehat{c_{1}})(\nabla\varphi+\omega_1\nabla_{y}\psi_1)\, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ \displaystyle +\int_{0}^{T}\int_{\Omega\times Y^{\ast}}D_2(y)(\nabla c+\nabla_{y}\widehat{c_{2}})(\nabla\varphi+\omega_2\nabla_{y}\psi_2)\, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ +\displaystyle \int_{0}^{T}\int_{\Omega\times\Gamma} (\widehat{c_{1}}-\widehat{c_{2}}) H(c_3) (\omega_1\psi_1-\omega_2\psi_2) w {\,\mathrm{d}}x {\,\mathrm{d}}\sigma_{y} {\,\mathrm{d}}t\\ \\ =\displaystyle \int_{0}^{T} \int_{\Omega\times Y^{\ast}}F_{1}(y,c,c, c_{3})\varphi w{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t+\int_{0}^{T} \int_{\Omega\times Y^{\ast}}F_{2}(y,c,c, c_{3})\varphi w{\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t, \end{array}$$ for every $\varphi, \omega_i \in \mathcal{D}(\Omega)$, $\psi_i\in H_{\textrm{per}}^{1}(Y^\ast)$ and $w\in\mathcal{D}(0,T)$ which, by density, leads to the first equation in .\ In a similar way, by and since due to assumption $(\mathbf{H}_4)$ and the strong convergences $_{iii}$ we get $$\mathcal{T}_{\varepsilon}(G_3^{\varepsilon}(x, c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_3^{\varepsilon})) \rightarrow G_3(y, c,c, c_3)\text{ strongly in }L^2((0,T)\times\Omega \times\Gamma),$$ we obtain the equation governing the evolution of the concentration $c_3^\varepsilon$, as $\varepsilon \rightarrow 0$, which reads: $$\begin{array}{l} \displaystyle -\vert Y^\ast \vert \int_{0}^{T}\int_{\Omega}c_{3}\varphi\partial_{t}w\,\mathrm{d}x \mathrm{d}t+\displaystyle \int_{0}^{T}\int_{\Omega\times Y^{\ast}}D_3(y)(\nabla c_{3}+\nabla_{y}\widehat{c_{3}})(\nabla\varphi+\omega_3\nabla_{y}\psi_3)\, w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t\\ \\ =\displaystyle \int_{0}^{T}\int_{\Omega\times\Gamma}G_{3}(y,c,c, c_3)\varphi w\mathrm{d}x\mathrm{d}\sigma_{y}\mathrm{d}t+\int_{0}^{T}\int_{\Omega\times Y^{\ast}}F_{3}(y,c,c,c_{3})\varphi w {\,\mathrm{d}}x {\,\mathrm{d}}y {\,\mathrm{d}}t \end{array}$$ for every $\varphi, \omega_i \in \mathcal{D}(\Omega)$, $\psi_i\in H_{\textrm{per}}^{1}(Y^\ast)$ and $w\in\mathcal{D}(0,T)$ which, by density, gives the second equation in .\ The initial condition is obtained in a standard way. \ \[teohom\] Let $(c_{1}^{\varepsilon}, c_{2}^{\varepsilon}, c_{3}^{\varepsilon})$ be the solution of system - . Then, if the assumptions $(\mathbf{H}_1) \div (\mathbf{H}_5)$ hold, there exist $c$ and $c_{3}$ in $L^{2}(0,T;H_0^{1}(\Omega))$ such that convergences and hold and the pair $(c, c_3)$ is the unique solution of the coupled system $$\label{homsystem} \left\{ \begin{array}{ll} 2 \dfrac{\partial c }{\partial t}-\operatorname{div} (B(c_3)\nabla c)= \mathcal{M}_{Y^{\ast}} (F_{1}(\cdot,c,c,c_{3}))+\mathcal{M}_{Y^{\ast}} (F_{2}(\cdot,c,c,c_{3}))& \text{\rm in }(0,T)\times\Omega\\ \\ \dfrac{\partial c_3 }{\partial t}-\operatorname{div} (D^{0}\nabla c_3)= \mathcal{M}_{Y^{\ast}} (F_{3}(\cdot,c,c,c_{3}))+\dfrac{\vert \Gamma \vert }{\vert Y^{\ast}\vert}\mathcal{M}_{\Gamma} (G_{3}(\cdot,c,c,c_{3}))& \text{\rm in }(0,T)\times\Omega\\ \\ c(x,0)=(c_1^0+c_2^0)/2 & \text{\rm in } \Omega\\ \\ c_3(x,0)=c_3^0 & \text{\rm in } \Omega. \end{array} \right.$$ The positive definite constant homogenized diffusion matrix $D^0$ is given, for $i,j\in \{1,\dots,n\}$, by its entries $$\label{matrixA} D^0_{ij}=\mathcal{M}_{Y^\ast}\left((D_3(y))_{ij}-\sum_{k=1}^{n}(D_3(y))_{ik}\dfrac{\partial\chi^j}{\partial y_k}(y)\right),$$ where, for $j\in \{1,\dots,n\}$, $\chi^j \in H^1_{\textrm{per}}(Y^{\ast})/{\Bbb R}$ verifies the local problem $$\label{cellprobl1} \left\{ \begin{array}{ll} -\displaystyle {\rm div}_y \left(D_3(y)(\nabla_y \chi^{j}-\mathbf{e}_{j})\right)=0 & \text{\rm in }Y^{\ast},\\ \\ D_3(y)(\nabla_y\chi^{j}-\mathbf{e}_{j})\cdot\nu=0 & \text{\rm on }\Gamma,\\ \\ \chi^{j} \text{ \rm Y-periodic }\,\text{\rm such that }\mathcal{M}_{Y^\ast}(\chi^j)=0, \end{array} \right.$$ where $\nu$ is the outward unit normal to the boundary $\Gamma$.\ For every $s\in\mathbb{R}$, the non-constant matrix $B(s)$ is defined, for $i,j\in \{1,\dots,n\}$, by its entries $$\label{matrixA0} (B(s))_{ij}=\mathcal{M}_{Y^\ast}\left((D_1(y))_{ij}-\sum_{k=1}^{n}(D_1(y))_{ik}\dfrac{\partial\chi_1^j}{\partial y_k}(y,s)\right)+\mathcal{M}_{Y^\ast}\left((D_2(y))_{ij}-\sum_{k=1}^{n}(D_2(y))_{ik}\dfrac{\partial\chi_2^j}{\partial y_k}(y,s)\right),$$ where the pair $(\chi_1^j(\cdot, s), \, \chi_2^j(\cdot, s))\in H^1_{\textrm{per}}(Y^{\ast})/{\Bbb R}\times H^1_{\textrm{per}}(Y^{\ast})/{\Bbb R}$ is, up to the addition of the same constant to $\chi_1^j(\cdot, s)$ and $\chi_2^j(\cdot, s)$, the unique solution of the local problem $$\label{cellprobl2} \left\{ \begin{array}{ll} - \displaystyle{\rm div}_{y} \left(D_1(y)(\nabla_{y} \chi_1^j-\mathbf{e}_{j})\right)=0 & \text{\rm in } Y^\ast, \\ \\ - \displaystyle{\rm div}_{y} \left(D_2(y)(\nabla_{y} \chi_2^j-\mathbf{e}_{j})\right)=0 & \text{\rm in } Y^\ast, \\ \\ D_1(y)(\nabla_{y} \chi_1^j-\mathbf{e}_{j})\cdot \nu=-H(s)(\chi_1^j -\chi_2^j) & \text{\rm on } \Gamma, \\ \\ D_2(y)(\nabla_{y} \chi_2^j-\mathbf{e}_{j}) \cdot \nu= H(s)(\chi_1^j -\chi_2^j) & \text{\rm on } \Gamma,\\ \\ \chi_i^{j} \text{ \rm Y-periodic },\, i\in\{1,2 \}\,\text{\rm and such that }\mathcal{M}_{Y^\ast}(\chi_1^j)=0, \end{array} \right.$$ where $\nu$ is the outward unit normal to the boundary $\Gamma$. Taking $\varphi=0$ in the second equation in , we get $$\displaystyle\int_{\Omega\times Y^{\ast}}D_3(y)(\nabla c_3+\nabla_{y}\widehat{c_{3}})\nabla_{y}\Psi_3 {\,\mathrm{d}}x {\,\mathrm{d}}y =0\,\text{ in }\mathcal{D}'(0,T).$$ for all $\Psi_3\in L^2(\Omega;H_{per}^{1}(Y^\ast))$. Hence, for a.e. $t\in (0,T)$, we get $$\label{probconc} \left\{ \begin{array}{ll} -\hbox{div}_{y}\left(D_3(y)\nabla_y\widehat{c_3}\right)=\hbox{div}_{y}(D_3(y)\nabla c_3) &\text{ in }\Omega\times Y^{\ast},\\[2mm] D_3(y)\nabla_{y}\widehat{c_3}\cdot\nu=-D_3(y)\nabla c_3\cdot\nu&\text{ on }\Omega\times\Gamma,\\[2mm] \widehat{c_3}\quad\text{periodic in }y. \end{array} \right.$$ By linearity, we get, for a.e. $t\in (0,T)$ and $(x,y)\in \Omega\times Y^\ast$, $$\label{cihat} \widehat{c_{3}} (t,x,y)=-\sum\limits_{j=1}^{n}\chi^{j}(y)\frac{\partial c_{3}}{\partial x_{j}}(t,x),$$ where $\chi^{j},\,j\in \{1,\dots,n\}$, are the solutions of the local problems . Taking $\Psi_3=0$ in the second equation of , we get $$\begin{array}{l} \vert Y^\ast \vert \displaystyle \langle\partial_t c_3, \varphi\rangle_{\Omega}+ \int_{\Omega\times Y^{\ast}}D_3(y)(\nabla c_{3}+\nabla_{y}\widehat{c_{3}})\nabla\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y\\ \\ =\displaystyle \int_{\Omega\times\Gamma}G_{3}(y,c,c, c_{3})\varphi\mathrm{d}x\mathrm{d}\sigma_{y}+ \int_{\Omega\times Y^{\ast}}F_{3}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y ,\text{ in }\mathcal{D}'(0,T). \end{array}$$ Replacing $\widehat{c}_3$ given by in the previous equality, we obtain $$\begin{array}{l} \vert Y^\ast \vert \displaystyle \langle\partial_t c_3, \varphi\rangle_{\Omega}+\int_{\Omega}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n} \left( \int_{Y^{\ast}}\left((D_3(y))_{ij}-\sum_{k=1}^{n}(D_3(y))_{ik}\dfrac{\partial \chi^{j}}{ \partial y_{k}}\left( y\right) \right) {\,\mathrm{d}}y\right)\dfrac{\partial c_3}{\partial x_{j}}\dfrac{\partial \varphi}{\partial x_{i}} \,{\,\mathrm{d}}x \\ \\ =\displaystyle\int_{\Omega\times\Gamma}G_{3}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}\sigma_{y}+ \int_{\Omega\times Y^{\ast}}F_{3}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y\text{ in }\mathcal{D}'(0,T) \end{array}$$ for all $\varphi \in H_0^{1}\left( \Omega \right)$, which means that $(c, c_3)$ satisfies the following problem $$\left\{ \begin{array}{ll} \displaystyle \dfrac{\partial c_{3}}{\partial t}-\sum\limits_{i=1}^{n}\dfrac{\partial }{\partial x_{i}}\sum\limits_{j=1}^{n} \left(\dfrac{1}{\|Y^\ast\|} \int_{Y^{\ast}}\left((D_3(y))_{ij}-\sum_{k=1}^{n}(D_3(y))_{ik}\dfrac{\partial \chi^{j}}{ \partial y_{i}}\left( y\right) \right) {\,\mathrm{d}}y\right)\dfrac{\partial c_3}{\partial x_{j}}\\ \\ \\= \dfrac{\|\Gamma\|}{\|Y^\ast\|}\mathcal{M}_{\Gamma} (G_{3}(\cdot,c,c, c_{3}))+\mathcal{M}_{Y^{\ast}} (F_{3}(\cdot,c,c, c_{3}))& \text{\rm in }(0,T)\times\Omega,\\ \\ c_3=0& \text{\rm on }(0,T)\times\partial\Omega. \end{array}\right.$$ Thus, we are led to the homogenized equation for $c_3$ in , where the constant matrix $D^0$ is defined through .\ Let us take now $\varphi=0$ in the first equation in . We have: $$\begin{array}{ll} \displaystyle \int_{\Omega\times Y^{\ast}}D_1(y)(\nabla c+\nabla_{y}\widehat{c_{1}})\nabla_{y}\Psi_1\,{\,\mathrm{d}}x {\,\mathrm{d}}y+\displaystyle \int_{\Omega\times Y^{\ast}}D_2(y)(\nabla c+\nabla_{y}\widehat{c_{2}})\nabla_{y}\Psi_2\,{\,\mathrm{d}}x {\,\mathrm{d}}y \\ \\ +\displaystyle \int_{\Omega\times\Gamma} (\widehat{c_{1}}-\widehat{c_{2}}) H(c_3) (\Psi_1-\Psi_2) {\,\mathrm{d}}x {\,\mathrm{d}}\sigma_{y}=0\text{ in }\mathcal{D}'(0,T), \end{array}$$ for all $\Psi_i\in L^2(\Omega;H_{per}^{1}(Y^\ast))$, $i\in\{1,2\}$. Hence, for a.e. $t\in (0,T)$, we have $$\label{probconc} \left\{ \begin{array}{ll} -\hbox{div}_{y}\left(D_1(y)\nabla_y\widehat{c_1}\right)=\hbox{div}_{y}(D_1(y)\nabla c) &\text{ in }\Omega\times Y^{\ast},\\ \\ -\hbox{div}_{y}\left(D_2(y) \nabla_y \widehat{c_2}\right)=\hbox{div}_{y}(D_2(y)\nabla c)&\text{ in }\Omega\times Y^{\ast},\\ \\ D_1(y)\nabla_{y}\widehat{c_1}\cdot\nu=-D_1(y)\nabla c\cdot \nu -H(c_3)(\widehat{c_1}-\widehat{c_2})&\text{ on }\Omega\times\Gamma,\\ \\ D_2(y)\nabla_{y}\widehat{c_2}\cdot\nu=-D_2(y)\nabla c\cdot \nu + H(c_3)(\widehat{c_1}-\widehat{c_2})&\text{ on }\Omega\times\Gamma,\\ \\ \widehat{c_i}\ \text{periodic in }y,\,i\in\{1,2\}. \end{array} \right.$$ Then, acting as previously, for a.e. $t\in (0,T)$ and $(x,y)\in \Omega\times Y^\ast$, we get $$\label{uhat1} \widehat{c}_1(t,x,y)=-\displaystyle \sum\limits_{j=1}^n \chi_1^j (y, c_3(t,x)) \displaystyle \frac{\partial c}{\partial x_j}(t,x) \quad \textrm{in } (0,T) \times \Omega \times Y^{\ast},$$ $$\label{uhat2} \widehat{c}_2(t,x,y)=-\displaystyle \sum\limits_{j=1}^n \chi_2^j (y, c_3(t,x)) \displaystyle \frac{\partial c}{\partial x_j}(t,x) \quad \textrm{in } (0,T) \times \Omega \times Y^{\ast},$$ where $\chi_k^j(c_3)$, $k\in \{1,2\}$ and $j\in \{1,\dots,n\}$, up to the addition of the same constant, are the unique solutions of the local problems . Taking $\Psi_1=\Psi_2=0$ in the first equation of , we get $$\begin{array}{l} 2\displaystyle \vert Y^\ast \vert \, \langle\partial_t c, \varphi\rangle_{\Omega}+ \displaystyle \int_{\Omega\times Y^{\ast}}D_1(y)(\nabla c+\nabla_{y}\widehat{c_{1}})\nabla\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y+\displaystyle \int_{\Omega\times Y^{\ast}}D_2(y)(\nabla c+\nabla_{y}\widehat{c_{2}})\nabla\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y \\ \\ \displaystyle = \int_{\Omega\times Y^{\ast}}F_{1}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y+\int_{\Omega\times Y^{\ast}}F_{2}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y,\text{ in }\mathcal{D}'(0,T). \end{array}$$ Replacing $\widehat{c}_i$, $i\in\{1,2\}$, given by and in the previous equality, we obtain $$\begin{array}{l} 2\vert Y^\ast \vert \displaystyle \langle\partial_t c, \varphi\rangle_{\Omega}+\int_{\Omega}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n} \left( \int_{Y^{\ast}}\left((D_1(y))_{ij}-\sum_{k=1}^{n}(D_1(y))_{ik}\dfrac{\partial \chi_1^{j}}{ \partial y_{k}}\left( y\right) \right) {\,\mathrm{d}}y\right)\dfrac{\partial c}{\partial x_{j}}\dfrac{\partial \varphi}{\partial x_{i}} \,{\,\mathrm{d}}x \\ \\ + \displaystyle\int_{\Omega}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n} \left( \int_{Y^{\ast}}\left((D_2(y))_{ij}-\sum_{k=1}^{n}(D_2(y))_{ik}\dfrac{\partial \chi_2^{j}}{ \partial y_{k}}\left( y\right) \right) {\,\mathrm{d}}y\right)\dfrac{\partial c}{\partial x_{j}}\dfrac{\partial \varphi}{\partial x_{i}} \,{\,\mathrm{d}}x\\ \\ =\displaystyle\int_{\Omega\times Y^{\ast}}F_{1}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y+ \int_{\Omega\times Y^{\ast}}F_{2}(y,c,c, c_{3})\varphi {\,\mathrm{d}}x {\,\mathrm{d}}y\text{ in }\mathcal{D}'(0,T) \end{array}$$ for all $\varphi \in H_0^{1}\left( \Omega \right)$, which means that $(c, c_3)$ satisfies the following problem $$\left\{ \begin{array}{ll} 2\displaystyle \dfrac{\partial c}{\partial t}-\sum\limits_{i=1}^{n}\dfrac{\partial }{\partial x_{i}}\sum\limits_{j=1}^{n} \sum\limits_{l=1}^{2}\left(\dfrac{1}{\|Y^\ast\|} \int_{Y^{\ast}}\left((D_l(y))_{ij}-\sum_{k=1}^{n}(D_l(y))_{ik}\dfrac{\partial \chi_l^{j}}{ \partial y_{i}}\left( y\right) \right) {\,\mathrm{d}}y\right)\dfrac{\partial c}{\partial x_{j}}\\ \\ \\= \mathcal{M}_{Y^{\ast}} (F_{1}(\cdot,c,c, c_{3}))+\mathcal{M}_{Y^{\ast}} (F_{2}(\cdot,c,c, c_{3}))& \text{\rm in }(0,T)\times\Omega,\\ \\ c=0& \text{\rm on }(0,T)\times\partial\Omega. \end{array}\right.$$ Thus, we are led to the homogenized equation for $c$ in , where the non-constant matrix $B(c_3)$ is defined through .\ We remark that, for any given $s \in \mathbb{R}$, under the assumptions $(\mathbf{H}_1)$ and $(\mathbf{H}_3)$, one can prove, by using Lax-Milgram theorem, the existence and the uniqueness, up to a constant, of a solution $(\chi_1^j(\cdot,s), \chi_2^j(\cdot, s)) \in H^1_{\textrm{per}}(Y^{\ast})/{\mathbb R} \times H^1_{\textrm{per}}(Y^{\ast})/{\mathbb R} $ for problem . The value of the constant needs to be the same. The constant homogenized matrix $D^0$ is positive definite. By assumption $(\mathbf{H}_3)$, the non-constant homogenized matrix $B$ is uniformly coercive and uniformly bounded from above. Therefore, as in [@Allaire], this implies the uniqueness of the solution of the limit problem and, thus, all the above convergence results hold for the whole sequences. So, the couple $(c, c_3)$ is the unique solution of problem , where the matrices $D^0$ and $B$ are defined by and , respectively. \ \[rem5\] We point out that in system we have two homogenized matrices, a standard one $D^0$ and a non-constant one $B$, generated by the special coupling and scalings of the boundary terms in the microscopic system . Moreover, it is easy to prove that the positive definite constant homogenized diffusion matrix $D^0$ can be written also as $$\label{matrixAbis} D^0_{ij}=\dfrac{1}{\vert Y^\ast \vert}\displaystyle \int_{Y^{\ast}} D_3(y)\left(e_i-\nabla_y \chi^i \right) \left(e_j-\nabla_y \chi^j\right){\,\mathrm{d}}y\,\,\,\text{\rm for }i,j\in \{1,\dots,n\}$$ and, for every $s\in \mathbb{R}$, the non-constant dispersion matrix $B(s)$ can also be written as $$\label{matrixA0bis} \begin{array}{c} (B(s))_{ij}=\dfrac{1}{\vert Y^\ast \vert}\displaystyle \int_{Y^{\ast}} D_1(y)\left(e_i-\nabla_y \chi_1^i\right) \left(e_j-\nabla_y \chi_1^j\right){\,\mathrm{d}}y+\dfrac{1}{\vert Y^\ast \vert}\displaystyle \int_{Y^{\ast}} D_2(y)\left(e_i-\nabla_y \chi_2^i\right) \left(e_j-\nabla_y \chi_2^j\right){\,\mathrm{d}}y\\ \\ +\dfrac{1}{\vert Y^\ast \vert}\displaystyle \int_{\Gamma} H(s) \left(\chi_1^i - \chi_2^i\right)\left(\chi_1^j -\chi_2^j\right) {\,\mathrm{d}}\sigma_y,\,\,\,\text{\rm for }i,j\in \{1,\dots,n\}. \end{array}$$ Here, for $i\in \{1,\dots,n\}$, $\chi^i \in H^1_{\textrm{per}}(Y^{\ast})/{\Bbb R}$ verifies the local problem , while for every $s\in\mathbb{R}$, $\chi_1^i(\cdot,s), \, \chi_2^i (\cdot, s)\in H^1_{\textrm{per}}(Y^{\ast})/{\Bbb R}$ are, up to the addition of the same constant, the unique solutions of the local problem . \[remA0\] The fact that the non-linearity with respect to $c_3^\varepsilon$ passes from the reaction term, at the microscopic level, to the diffusion term, at the macroscopic one (see ), is a manifestation of the different scaling of reaction terms on the boundary (see ). In other words, the diffusion of concentration $c_3^\varepsilon$ on the boundary is slower than the ones of $c_1^\varepsilon$ and $c_2^\varepsilon$. In particular, we observe that for small values of the concentration $c_3$, due to the Lipschitz continuity property of the function $H$ and the fact that $H(0)=0$, the cell problems become decoupled and the dispersion tensor $B$ is the sum of two matrices of the same type of $D^0$ carrying at the macroscopic level the contributions of each concentration $c^\varepsilon_i$, $i\in \{1,2\}$. On the other hand, for large values of $c_3$, if in particular $H$ is of Langmuir type (see ), as in [@Allaire], we have the saturation effect of the Langmuir isotherm. In this case, since by the limit of $H(s)$ for $s \to \infty$ is the constant $\dfrac{a}{b}$, if we assume that $c_3$ goes to infinity, then the cell problem corresponding to such an infinite reaction limit is no longer dependent on the homogenized solution $c_3$, but it remains coupled. Of course, other mathematical settings could be considered in place of for getting such an effect and they will be dealt with in a forecoming paper. If in the microscopic system all the reaction terms on $\Gamma^{\varepsilon}$ are scaled with $\varepsilon$, as, for instance, in [@Gahn], i.e. if we consider the system $$\left\{ \begin{array} {ll} \partial_{t}c_{i}^{\varepsilon}-\operatorname{div}(D_i^\varepsilon \nabla c_{i}^{\varepsilon })=F_{i}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon})&\text{ in }(0,T)\times\Omega^{\ast}_{\varepsilon},\,\, i\in \{1,2,3\},\\ \\ D_i^\varepsilon \nabla c_{i}^{\varepsilon}\cdot\nu^\varepsilon=\varepsilon\, G_{i}(c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon})&\text{ on }(0,T)\times \Gamma^{\varepsilon},\, \, i\in \{1,2\},\\ \\ D_3^\varepsilon \nabla c_{3}^{\varepsilon}\cdot\nu^\varepsilon=\varepsilon\, G_{3}^{\varepsilon}(x,c_{1}^{\varepsilon},c_{2}^{\varepsilon}, c_{3}^{\varepsilon})&\text{ on }(0,T)\times \Gamma^{\varepsilon},\\ \\ c_{i}^{\varepsilon}=0 & \text{ on }(0,T)\times\partial\Omega,\, \, i\in \{1,2,3\},\\ \\ c_{i}^{\varepsilon}(0)=c_{i}^0 & \text{ in }\Omega^{\ast}_{\varepsilon},\, \, i\in \{1,2,3\}, \end{array} \right. \label{eqmicro-v}$$ then it is not difficult to see that there exist $c_{i}\in L^{2}(0,T;H_0^{1}(\Omega))$ and $\widehat{c}_{i}\in L^{2}((0,T)\times\Omega;H_{per}^{1}(Y^\ast)/\Bbb R)$, $i\in \{1,2,3\}$, such that $$\label{conv-v} \left\{ \begin{array}{ll} \mathcal{T}^{\ast}_{\varepsilon}(c_{i}^{\varepsilon})\rightharpoonup c_{i}&\text{\rm weakly in }L^{2}((0,T)\times\Omega;H^{1}(Y^\ast)),\\[2mm] \mathcal{T}^{\ast}_{\varepsilon}(\nabla c_{i}^{\varepsilon})\rightharpoonup\nabla c_{i}+\nabla_{y}\widehat{c}_{i}&\text{\rm weakly in }L^{2}((0,T)\times \Omega\times Y^\ast),\\ [2mm] \mathcal{T}^\ast_{\varepsilon } (c_i^\varepsilon) \rightarrow c_i & \text{\rm strongly in } L^2((0,T)\times \Omega\times Y^\ast). \end{array} \right.$$ In this case, the limit function $(c_1, c_2, c_3)$ in is the unique solution of the following system: $$\left\{ \begin{array}{ll} \dfrac{\partial c_1 }{\partial t}-\operatorname{div} (D_1^0\nabla c_1)= \mathcal{M}_{Y^{\ast}} (F_{1}(\cdot,c_{1},c_{2},c_{3}))-(c_1-c_2)\, \dfrac{\vert \Gamma \vert}{\vert Y^\ast \vert}\, H(c_{3})& \text{\rm in }(0,T)\times\Omega,\\ \\ \dfrac{\partial c_2 }{\partial t}-\operatorname{div} (D_2^0\nabla c_2)= \, \mathcal{M}_{Y^{\ast}} (F_{2}(\cdot,c_{1},c_{2},c_{3}))+(c_1-c_2) \, \dfrac{\vert \Gamma \vert}{\vert Y^\ast \vert} \, H(c_{3})& \text{\rm in }(0,T)\times\Omega,\\ \\ \dfrac{\partial c_3 }{\partial t}-\operatorname{div} (D_3^0\nabla c_3)= \mathcal{M}_{Y^{\ast}} (F_{3}(\cdot,c_{1},c_{2},c_{3}))+\dfrac{\vert \Gamma \vert}{\vert Y^\ast \vert} \mathcal{M}_{\Gamma} (G_{3}(\cdot,c_{1},c_{2},c_{3}))& \text{\rm in }(0,T)\times\Omega, \\ \\ c_i=0 & \text{\rm on } (0,T)\times\partial\Omega,\\ \\ c_i(x,0)=c_i^0 & \text{\rm in } \Omega, \end{array} \right.$$ where the entries of the positive definite constant homogenized diffusion matrices $D_i^0$, $i\in\{1, 2, 3\}$, are of the same type of . 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--- abstract: 'We show that, as $n$ goes to infinity, the free group on $n$ generators, modulo $n+u$ random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen-Lenstra heuristics. For each $n$, these random groups belong to the few relator model in the Gromov model of random groups.' address: \| Department of Mathematics\ University of Wisconsin-Madison\ 480 Lincoln Drive\ Madison, WI 53705 USA\ and American Institute of Mathematics\ 600 East Brokaw Road\ San Jose, CA 95112 USA author: - Yuan Liu and Melanie Matchett Wood title: The free group on $n$ generators modulo $n+u$ random relations as $n$ goes to infinity --- Introduction ============ For an integer $u$ and positive integers $n$, we study the random group given by the free group $F_n$ on $n$ generators modulo $n+u$ random relations. In particular we find that these random groups have a nice limiting behavior as $n{\rightarrow}\infty$ and we explicitly describe the limiting random group. There are two ways to take relations in a “uniform” way: 1)complete $F_n$ to the profinite complete group ${\hat{F}}_n$ on $n$ generators and take relations with respect for Haar measure, or 2)take relations from $F_n$ uniformly among words up to length $\ell$ and then let $\ell{\rightarrow}\infty$. In Proposition \[P:same\], we show that the random groups obtained from the second method weakly converge, as $\ell{\rightarrow}\infty$, to the random groups obtained from the first method. For a positive integer $n$, let ${\hat{F}}_n$ be the profinite free group on $n$ generators. For an integer $u$, we define the random group $X_{u,n}$ by taking the quotient of ${\hat{F}}_n$ by (the closed, normal subgroup generated by) $n+u$ independent random generators, taken from Haar measure on ${\hat{F}}_n$. We need to define a topology to make precise the convergence of $X_{u,n}$ as $n{\rightarrow}\infty$. Let $S$ be a set of (isomorphism classes of) finite groups. Let $\bar{S}$ be the smallest set of groups containing $S$ that is closed under taking quotients, subgroups, and finite direct products. For a profinite group $G$, we write $G^{\bar{S}}$ for its pro-$\bar{S}$ completion. We consider the set $\mathcal{P}$ of isomorphism classes of profinite groups $G$ such that $G^{\bar{S}}$ is finite for all finite sets $S$ of finite groups. All finitely generated profinite groups are in $\mathcal{P}$ and all groups in $\mathcal{P}$ are small in the sense of [@Fried2008 Section 16.10]. We define a topology on $\mathcal{P}$ in which the basic opens are, for each finite set $S$ of finite groups and finite group $H$, the sets $U_{S,H}:=\{G \mid G^{\bar{S}} {\simeq}H\}$. \[T:Main\] Let $u$ be an integer. Then there is a probability measure $\mu_u$ on $\mathcal{P}$ for the $\sigma$-algebra of Borel sets such that as $n{\rightarrow}\infty$, the distributions of $X_{u,n}$ weakly converge to $\mu_u$. We give these $\mu_u$ explicitly in fact. See Equation for a formula for $\mu_u$ on each basic open, and see Section \[S:examples\] for several other interesting examples of the values of these measures. In fact, we prove in Theorem \[T:calcinf\] a stronger form of convergence than weak convergence, which, in particular, tell us the measure of any finite group. In particular, we have $$\label{E:trivialgp} \mu_u(\textrm{trivial group})=\prod_{\substack{G \textrm{ finite simple}\\\textrm{abelian group}}} \prod_{i=u+1}^{\infty} (1-\|G\|^{-i}) \prod_{\substack{G \textrm{ finite simple}\\\textrm{non-abelian group}}} e^{-\|{\operatorname{Aut}}(G)\|^{-1}\|G\|^{-u}},$$ which is $\approx.4357$ when $u=1$. The abelian group version of this problem has been well-studied, as the limiting groups when $u=0,1$ are the random groups of the Cohen-Lenstra heuristics. The first factor above, as a product over primes, is very familiar from the random groups of the Cohen-Lenstra heuristics, but here it naturally appears as part of a product over all finite simple groups. Cohen and Lenstra [@Cohen1984] defined certain random abelian groups that they predicted gave the distribution of class groups of random quadratic fields. Friedman and Washington [@Friedman1989] later realized that these random abelian groups arose as the limits of cokernels of random matrices, which is just a rewording of the abelianization of our construction above. These random abelian groups are universal, in the sense that, as $n{\rightarrow}\infty$, taking ${\ensuremath{{\mathbb{Z}}}\xspace}^n$ modulo almost any collection of $n+u$ independent relations will give these same random abelian groups, even if the relations are taken from strange and lopsided distributions [@Wood2015a]. One motivation for our work is to develop a non-abelian version of the random abelian groups of Cohen and Lenstra, in order to eventually be able to model non-abelian versions of class groups of random number fields. Boston, Bush, and Hajir [@Boston2016a] have defined random pro-$p$ groups that they conjecture model the pro-$p$ generalizations of class groups of random imaginary quadratic fields. In their definition, they were able to use special properties of $p$-groups to give a definition that avoids the limit as $n{\rightarrow}\infty$ that we study above (or rather, reduces the question of the limit as $n{\rightarrow}\infty$ to the abelian case, which was already understood). There is a large body of work on the Gromov, or density, model of random groups (see [@Ollivier2005] for an excellent introduction). In this model, one takes $F_n$ modulo $r(\ell)$ random relations uniform among words of length $\ell$, and studies the behavior as $\ell{\rightarrow}\infty$. When $r(\ell)$ grows like $(2n-1)^{d\ell}$ this is called the density $d$ model. There has been a great amount of work to understand, as $\ell{\rightarrow}\infty$, what properties hold asymptotically almost surely for these groups (e.g see [@Ollivier2005; @Ollivier2010] for an overview and [@Calegari2015; @Kotowski2013; @Mackay2016; @Ollivier2011] for some more recent examples). Our $X_{u,n}$ are limits as $\ell{\rightarrow}\infty$ of density $0$ models of these random groups. However, the emphasis of our work is different from much of the previous work on the Gromov model of random groups. That work has often emphasized a random group with given generators, and we consider only the isomorphism class of the group and focus on the convergence to a limiting random variable. Also, from the point of view of our topology, any Gromov random group with $r(\ell){\rightarrow}\infty$ weakly converges to the trivial group (see Proposition \[P:alltrivial\]). Our topology is aimed at understanding finite quotients of groups, and is rather different than the topology due to Chabauty [@Chabauty1950] and Grigorchuk [@Grigorchuk1984] on the space of marked groups that emphasizes the geometry of the Cayley graphs but isn’t well behaved on isomorphism classes of groups. The closest previous work to ours is that of Dunfield and Thurston [@Dunfield2006]. They studied $F_n$ modulo $r$ random relations (with both methods described above of taking relations) in order to contrast those random groups with random $3$-manifold groups. Their main consideration was the probability that these random groups (for fixed $n$ and $r$) had a quotient map to a fixed finite group. They do observe [@Dunfield2006 Theorem 3.10] that for a fixed non-abelian finite simple group $G$, the distribution of the number of quotient maps to $G$ has a Poisson limiting behavior as $n{\rightarrow}\infty$; this is the first glimpse of the nice limiting behavior as $n{\rightarrow}\infty$ that we study in this paper. Jarden and Lubotzky [@Jarden2006] studied the normal subgroup of ${\hat{F}}_n$ generated by a fixed number of random elements, in particular proving that when it is infinite index that it is almost always the free profinite group on countably many generators. Our work here complements theirs, as they have determined the structure of the random normal subgroup and we determine the structure of the quotient by this random normal subgroup. The bulk of this paper is devoted to showing the existence of the measure $\mu_u$ of Theorem \[T:Main\]. Let $\mu_{u,n}$ be the distribution of our random group $X_{u,n}$. Since the basic opens in our topology are also closed, it is clear that if $\mu_u$ exists, then for any basic open $U$ we have $\mu_u(U)=\lim_{n{\rightarrow}\infty} \mu_{u,n} (U).$ The argument for the existence of $\mu_u$ breaks into two major parts. The first part is to show the limit $\lim_{n{\rightarrow}\infty} \mu_{u,n} (U)$ exists. The second part is to show that these measures on basic opens define a countably additive measure. After giving some notation and basic definitions in Section \[S:Notations\], we will give the values of $\mu_u$ on basic opens in Section \[S:definemu\] for easy reference. Then in Section \[S:setup\] we set up the strategy for proving $\lim_{n{\rightarrow}\infty} \mu_{u,n} (U)$ exists, which is entirely group theoretical. This argument will take us through Section \[S:basics\]. It is easy to express $\mu_{u,n}$ in group theory terms involving $\hat{F}_n$. However, such expressions do not allow one to take a limit as $n{\rightarrow}\infty$, and so the main challenge is to extract $\hat{F}_n$ from the description of the probabilities so that they only involve the number $n$ and group theoretical quantities that do not depend on $n$. This requires several steps. In Section \[S:genprob\], we express the probabilities in terms of multiplicities of certain groups appearing in $\hat{F}_n$. In Section \[S:factors\], we bound what possible groups can have positive multiplicities. In Section \[S:countqt\], we relate the multiplicities to a count of certain surjections, and finally in Section \[S:basics\] we count these surjections in another way that eliminates $\hat{F}_n$ from our description of the probabilities. The next challenge is to show the countable additivity of the $\mu_u$ that we have then defined on basic opens. It follows from Fatou’s lemma that for a finite set $S$ of finite groups, $$\sum_{H \text{ is finite}} \lim_{n{\rightarrow}\infty} \mu_{u,n} (U_{S, H}) \leq 1.$$ However, a priori, this inequality may be strict. In the limit as $n$ goes to infinity there could be escape of mass. To show that this does not occur, we require bounds on the $\mu_{u,n} (U_{S, H})$ that are sufficiently uniform in $n$. The difficultly is that our group theoretical expressions do not easily lend themselves to the kind of bounds useful for an analytic argument. We obtain the necessary uniformity by considering a notion of chief factor pairs, which generalizes the notion of a chief factor of a group to also include the conjugation action on the chief factor. We are able to bound the size of the outer action of conjugation on chief factors for a given $S$ in Section \[S:factors\], which then, combined with an induction on $S$, gives us the uniformity necessary to show in Section \[S:cntadd\] that the above inequality is actually an equality. That is the heart of the proof of countable additivity, which we show in Section \[S:cntadd\]. Once we have established the existence of the measure $\mu_u$ with the desired measure on basic opens, Theorem \[T:Main\] follows immediately in Section \[S:mainproof\]. In Section \[S:arbitraryS\], we give the measures of sets of the form $\{X \in\mathcal{P} \mid X^{{\bar{S}}}\simeq H^{{\bar{S}}}\}$ for arbitrary sets $S$ of finite groups, and see that $\mu_u$ and $\lim_{n{\rightarrow}\infty} \mu_{u,n}$ agree there, giving a stronger convergence than the weak convergence of Theorem \[T:Main\]. There, one inequality is automatic, and we then we argue that we either have the necessary uniformity to get equality, or that the larger probability is $0$, which also gives equality. The result of Section \[S:arbitraryS\] then allows us to compute measures of many different Borel sets, and in Section \[S:examples\], we give many examples including the trivial group, infinite groups, and distributions of the abelianization and pro-nilpotent quotient. In Section \[S:prob0\], we see that the measures $\mu_u$ give positive measure to any basic open where groups can be generated by $u$ more relations than generators. Finally, in Section \[S:non-profinite\], we compare the profinite model used in this paper to the discrete group model described above. This is the beginning of investigation into these random groups, and there are many further questions we would like to understand. Are these measures universal in the sense of [@Wood2015a], i.e. would we still get $\mu_u$ as $n{\rightarrow}\infty$ even if we took our relations from a different measure? Are these measures determined by their moments, which in [@Heath-Brown1994-1 Lemma 17], [@Fouvry2006 Section 4.2], [@EVW09 Lemma 8.1], [@Wood2017 Theorem 8.3], and [@Boston2017 Theorem 1.4] has been an important tool to identify analogous random groups? What is the measure of the set of all infinite groups when $u\geq 0$ (see Example \[Ex:inf\], and note by [@Jarden2006] this implies the normal subgroup generated by the relations is free on countably many generators with probability $1$)? What is the measure of the set of finitely generated groups, and of finitely presented groups? Do the $\mu_{u,n}$ converge strongly to $\mu_u$? Besides their inherent interest, many of these questions have implications for the possible connections to number theory described above. Notation and basic group theoretical definitions {#S:Notations} ================================================ Notation -------- Whenever we take a quotient by relations, we always mean by the closed, normal subgroup generated by those relations. For elements $x_1,\dots$ of a group $G$, we write $[x_1,\dots]_G$ for the closed normal subgroup of $G$ generated by $x_1,\dots$. We write $G{\simeq}H$ to mean that $G$ and $H$ are isomorphic. For profinite groups, we always mean isomorphic as profinite groups. For two groups $G$ and $H$, we write $G=H$ when there is an obvious map from one of $G$ or $H$ to the other (e.g. when $H$ is defined as a quotient or subgroup of $G$) and that map is an isomorphism. For a group $G$, we write $G^j$ for the direct product of $j$ copies of $G$. If $H$ is a subgroup of $G$, then we denote the centralizer of $H$ by ${\mathbf{C}}_G(H)$. When we say a set of finite groups, we always mean a set of isomorphism classes of finite groups. $F$-groups ---------- If $F$ is a group, an $F$-group is a group $G$ with an action of $F$. A morphism of $F$-groups is a group homomorphism that respects the $F$-action. An $F$-subgroup is a subgroup $G$ such that $f(G)=G$ for all $f\in F$, and an $F$-quotient is a group quotient homomorphism that respects the $F$-action. An irreducible $F$-group is an $F$-group with no normal $F$-subgroups except the trivial subgroup and the group itself. We write ${\operatorname{Hom}}_F(G_1,G_2)$ for the $F$-group morphisms from $G_1$ to $G_2$ and $h_F(G):=\|{\operatorname{Hom}}_F(G,G)\|$. We write ${\operatorname{Sur}}_F(G_1, G_2)$ for the $F$-group surjections from $G_1$ to $G_2$, and ${\operatorname{Aut}}_F(G)$ for the $F$-group automorphisms of $G$. For a sequence $x_k$ in an $F$-group $G$, let $[x_1,\dots ]_F$ be the closed normal $F$-subgroup of $G$ generated by the $x_k$. $H$-extensions -------------- For a group $H$, an $H$-extension is a group $E$ with a surjective morphism $\pi: E{\rightarrow}H.$ If $\pi:E{\rightarrow}H$ and $\pi':E'{\rightarrow}H$ are $H$-extensions, a morphism from $(E,\pi)$ to $(E',\pi')$ is a group homomorphism $f:E{\rightarrow}E'$ such that $\pi=\pi' \circ f$. If $\pi:E{\rightarrow}H$ and $\pi':E'{\rightarrow}H$ are $H$-extensions, we write ${\operatorname{Sur}}_H(\pi,\pi')$ for the set of surjective morphisms from $(G,\pi)$ to $(G',\pi')$. For an $H$-extension $E$, we write ${\operatorname{Aut}}_H(E,\pi)$ for the automorphisms of $(E,\pi)$ as an $H$-extension. If $(E,\pi)$ is an $H$-extension, a sub-$H$-extension is a subgroup $E'$ of $E$ with $\pi\|_{E'}$, such that $\pi\|_{E'}$ is surjective. Note that when $\ker \pi$ is abelian, it is an $H$-group under conjugation in $E$. Pro-${\bar{S}}$ completions and level $S$ groups {#SS:proSdef} ------------------------------------------------ Given a set $S$ of finite groups, we let $\bar{S}$ denote the smallest set of groups containing $S$ that is closed under taking quotients, subgroups and finite direct products. (This is called the variety of groups generated by $S$.) Given a profinite group $G$, we write $G^{{\bar{S}}}$ for its pro-${\bar{S}}$ completion, which is defined as $$G^{{\bar{S}}} = \varprojlim_{M} G/M,$$ where the inverse limit is taken over all closed normal subgroups $M$ of $G$ such that $G/M\in {\bar{S}}$. For a set $S$ of finite groups, we say that a profinite group $G$ is level $S$ if $G \in {\bar{S}}$. Also, for a positive integer $\ell$, let $S_\ell$ be the set consisting of all groups whose order is less than or equal to $\ell$. Then we say $G$ is level $\ell$ if $G\in {\bar{S}}_\ell$. Note that for $G\in\mathcal{P}$ we have that $G$ is level $S$ if and only if $G=G^{{\bar{S}}}$. Definition of $\mu_u$ {#S:definemu} ===================== For finite set $S$ of finite groups and finite group $H$, let $U_{S,H}:=\{X\in \mathcal{P} \,\|\, X^{{\bar{S}}}{\simeq}H \}.$ For integers $n\geq 1$ and $u > -n$, we have a measure $\mu_{u,n}$ on the $\sigma$-algebra of Borel sets of $\mathcal{P}$ such that $\mu_{u,n}(A)={\operatorname{Prob}}(X_{u,n}\in A)$. We will define a measure $\mu_u$, for each integer $u$, at first as a measure on the algebra $\mathcal{A}$ of sets generated by the $U_{S,H}$. For $A\in \mathcal{A}$, we define $$\label{E:defmu} \mu_u(A):=\lim_{n{\rightarrow}\infty} \mu_{u,n}(A).$$ We will below establish that 1) this limit exists when $A=U_{S,H}$ (see Theorem \[T:calc\], and Equation just below, in which we give the value of the limit), and hence for any $A\in \mathcal{A}$ since the limit is compatible with finite sums and subtraction from $1$; and 2) $\mu_u$ is countably additive on $\mathcal{A}$ (see Theorem \[T:countadd\]). These two results represent the bulk of the work of the paper. Then by Carathéodory’s extension theorem, it follows that $\mu_u$ extends uniquely to a probability measure on $\mathcal{P}$. Value of $\mu_u$ on basic open sets {#SS:basicq} ----------------------------------- Given a finite group $H$, let $\mathcal{A}_H$ be the set of isomorphism classes of non-trivial finite abelian irreducible $H$-groups. Let $\mathcal{N}$ be the set of isomorphism classes of finite groups that are isomorphic to $G^j$ for some finite simple non-abelian group $G$ and a positive integer $j$. Let $S$ be a set of finite groups, and $H$ a finite level $S$ group. For $G\in \mathcal{A}_H$, we define the quantity $$\lambda(S,H,G):=(h_H(G)-1) \sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{such that $\ker \pi$ isom. $G$ as $H$-groups,} \\ \textrm{and $E$ is level $S$} }} \frac{1}{\|{\operatorname{Aut}}_H(E,\pi)\|}.$$ We will see in Remark \[R:lamint\] that for $G\in \mathcal{A}_H$, the number $\lambda(S,H,G)$ is an integer power of $h_H(G)$. If $G\in \mathcal{N}$, we define $$\lambda(S,H,G):=\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{such that $\ker \pi$ isom. $G^j$ as groups, }\\ \textrm{$\ker \pi$ irred. $E$-group,} \\ \textrm{and $E$ is level $S$} }} \frac{1}{\|{\operatorname{Aut}}_H(E,\pi)\|}.$$ The definitions are not quite parallel in the abelian and non-abelian cases, but this is unavoidable given the different behavior of abelian and non-abelian simple groups. It will follow from Theorem \[T:calc\] below that for a finite set $S$ of finite groups and a finite level $S$ group $H$, we have $$\begin{aligned} \label{E:limval} \mu_u(U_{S,H})=&\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=0}^{\infty} (1-\lambda(S,H,G) \frac{h_H(G)^{-i-1}}{ \|G\|^{{u}}}) & \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(S,H,G)}.\end{aligned}$$ Theorem \[T:calcinf\] gives the analogous result for an infinite set $S$. We will see in Section \[S:factors\] that for finite $S$ only finitely many elements of $\mathcal{A}_H$ and $\mathcal{N}$ contribute non-trivially to this product. Setup and organization of the proofs {#S:setup} ==================================== The proof of Equation will be established from Section \[S:genprob\] to Section \[S:basics\], which are dominated by group theoretical methods. Here we outline the proof for the reader’s convenience. Suppose $n$ is a positive integer, $S$ is a finite set of finite groups, and $H$ is a finite level $S$ group. Then $(\hat{F}_n)^{{\bar{S}}}$ is a finite group [@Neumann1967 Cor. 15.72] and $(X_{u,n})^{{\bar{S}}}$ has the same distribution as the quotient of $(\hat{F}_n)^{{\bar{S}}}$ by $n+u$ independent, uniform random relations $r_1, \cdots r_{n+u}$ from $(\hat{F}_n)^{{\bar{S}}}$. By the definition of $\mu_u$, we have that $$\mu_u(U_{S,H})= \lim_{n\to \infty} {\operatorname{Prob}}((\hat{F}_n)^{{\bar{S}}}/[r_1,\cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}\simeq H).$$ We consider a normal subgroup $N$ of $(\hat{F}_n)^{{\bar{S}}}$ with an isomorphism $(\hat{F}_n)^{{\bar{S}}}/N \simeq H$. Let $M$ be the intersection of all maximal proper $(\hat{F}_n)^{{\bar{S}}}$-normal subgroups of $N$. We denote $F=(\hat{F}_n)^{{\bar{S}}}/M$ and $R=N/M$. Then for independent, uniform random elements $r_1, \cdots, r_{n+u}$ of $(\hat{F}_n)^{{\bar{S}}}$, we have that $[r_1, \cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}=N$ if and only if $R$ is the normal subgroup generated by the images of $r_1, \cdots, r_{n+u}$ in $F$. Indeed, the “only if” direction is clear; and if $[r_1, \cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}/M = R$, then $[r_1, \cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}=N$ since $[r_1, \cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}$ being contained in a proper maximal $(\hat{F}_n)^{{\bar{S}}}$-normal subgroup of $N$ would imply that its image is contained in a proper maximal $F$-normal subgroup of $R$. Any two surjections from $(\hat{F}_n)^{{\bar{S}}}$ to $H$ are isomorphic as $H$-extensions [@Lubotzky2001 Proposition 2.2]. Thus, the short exact sequence $$\label{E:fundSES} 1 {\rightarrow}R {\rightarrow}F {\rightarrow}H {\rightarrow}1$$ does not depend (up to isomorphisms of $F$ as an $H$-extension) on the choice of the normal subgroup $N$. Given a finite set $S$ of finite groups, a positive integer $n$ and a finite level $S$ group $H$, the short exact sequence defined in Equation is called the fundamental short exact sequence associated to $S$, $n$ and $H$. By the above arguments, ${\operatorname{Prob}}((\hat{F}_n)^{{\bar{S}}}/[r_1, \cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}\simeq H)$ equals the number of normal subgroups $N$ of $(\hat{F}_n)^{{\bar{S}}}$ with $(\hat{F}_n)^{{\bar{S}}}/N\simeq H$ times the probability that independent, uniform random elements $x_1, \cdots, x_{n+u}\in F$ normally generate $R$. Note that the number of such normal subgroups $N$ is $\|{\operatorname{Sur}}((\hat{F}_n)^{{\bar{S}}}, H)\|/\|{\operatorname{Aut}}(H)\|$, and there is a one-to-one correspondence between ${\operatorname{Sur}}((\hat{F}_n)^{{\bar{S}}}, H)$ and ${\operatorname{Sur}}(\hat{F}_n, H)$. It follows that $$\label{E:setup} {\operatorname{Prob}}((\hat{F}_n)^{{\bar{S}}}/[r_1, \cdots, r_{n+u}]_{(\hat{F}_n)^{{\bar{S}}}}\simeq H) = \frac{\|{\operatorname{Sur}}(\hat{F}_n, H)\|}{\|{\operatorname{Aut}}(H)\|} {\operatorname{Prob}}([x_1, \cdots, x_{n+u}]_F = R).$$ It therefore suffices to compute ${\operatorname{Prob}}([x_1, \cdots, x_{n+u}]_F=R)$. Note that $R$ is an $F$-group under the conjugation action. It will follow from Lemma \[L:irred\] that $R$ is a direct product of irreducible $F$-groups. Theorem \[T:probfrommult\] will prove the formula for ${\operatorname{Prob}}([x_1, \cdots, x_{n+u}]_F=R)$ for $F$ and $R$ where $R$ is a direct product of irreducible $F$-groups, in terms of the multiplicities of the various irreducible $F$-group factors of $R$. In Section \[S:factors\], we will give some criteria for which irreducible $F$-groups can appear in $R$. Then in Section \[S:countqt\], we will relate the multiplicities of irreducible factors in $R$ to the number of normal subgroups of $R$ with specified quotients. In Section \[S:basics\], we will count these normal subgroups of $R$ in another way in order to finally give an explicit formula for $\mu_{u,n}(U_{S,H})$. This formula will be explicit enough that we can easily take the limit as $n{\rightarrow}\infty$, giving Equation . Generating probabilities for products of irreducible $F$-groups {#S:genprob} =============================================================== Throughout this section, we let $n\geq 1$ and $u>-n$ be integers, $F$ a group, and $R$ a finite product of finite irreducible $F$-groups. (We don’t require $R$ to be a subgroup of $F$.) The goal of this section is to prove the following theorem which gives the probability that the normal $F$-subgroup generated by $n+u$ random elements of $R$ is the whole group. \[T:probfrommult\] Let $F$ be a group and $G_i$ be finite irreducible $F$-groups for $i=1,\dots, k$ such that for $i\ne j$, we have that $G_i$ and $G_j$ are not isomorphic $F$-groups, and let $m_i$ be non-negative integers. Let $R=\prod_{i=1}^k G_i^{m_i}$. Then $${\operatorname{Prob}}( [x_1,\dots, x_{n+u} ]_F=R)=\prod_{\substack{1\leq i\leq k\\ G_i \textrm{ abelian}}} \prod_{j=0}^{m_i-1} (1-h_F(G_i)^j \|G_i\|^{-n-u}) \prod_{\substack{1\leq i\leq k\\ G_i \textrm{ non-abelian}}} (1-\|G_i\|^{-n-u})^{m_i}$$ where the $x_i$ are independent, uniform random elements of $R$. \[R:Gpow\] Given a finite abelian irreducible $F$-group $G$, if we let $\mathfrak{m}$ be maximal such that $G^\mathfrak{m}$ can be generated by one element as an $F$-group, then we have $h_F(G)^\mathfrak{m}=\|G\|$. This follows from Theorem \[T:probfrommult\] because if we take $m_i=\mathfrak{m}$, the probability that one element generates $G^{m_i}$ is positive, but if we take $m_i=\mathfrak{m}+1$ the probability is $0$. We will build up to Theorem \[T:probfrommult\] through several lemmas. First, we determine the structure of normal $F$-subgroups of products of irreducible $F$-groups. \[L:subprod\] If $G_i$ are irreducible $F$-groups and $N$ is an $F$-subgroup of $\prod_{i=1}^m G_i$ that projects to $1$ or $G_i$ in each factor, then there exists a subset $J{\subset}\{1,\dots,m\}$ such that the projection of $N$ to $\prod_{i\in J} G_i$ is an isomorphism. We prove this by induction on $m$. Let $\pi_m$ be the projection map from $\prod_{i=1}^m G_i$ to $G_m$, and $\pi$ the projection map from $N$ to $\prod_{i=1}^{m-1} G_i$. Since $\pi_m(N)$ is $1$ or $G_m$, and $\pi_m(\ker \pi)$ is a normal $F$-subgroup of $\pi_m(N)$, we have $\pi_m(\ker \pi)$ is $1$ or $G_m$. If $\pi_m(\ker \pi)=1$, then since $\ker \pi\cap \ker \pi_m=1$, we have $\ker \pi=1$ and $N$ is isomorphic to $\pi(N)$. If $\pi_m(\ker \pi)=G_m$, then $N$ is isomorphic to $\pi(N) \times G_m$. In either case, we apply the inductive hypothesis to $\pi(N)$ and conclude the lemma. \[L:preSchur\] Let $G_1$ and $G_2$ be irreducible $F$-groups. Then any homomorphism of $F$-groups $\phi: G_1{\rightarrow}G_2$ with normal image is either trivial or an isomorphism. If it is not trivial, then $\ker(\phi)$ is a normal $F$-subgroup and so must be trivial, and ${\operatorname{im}}(\phi)$ is a normal $F$-subgroup and must be $G_2$, so it is a bijection. \[L:subsplit\] Let $G_i$ be irreducible $F$-groups for $i=1,\dots, k$ such that for $i\ne j$, we have that $G_i$ and $G_j$ are not isomorphic as $F$-groups. Let $N$ be a normal $F$-subgroup of $\prod_{i=1}^k G_i^{m_i}$, then $N=\prod_{i=1}^k N_i,$ where $N_i$ is a normal $F$-subgroup of $G_i^{m_i}$. Since $N$ is a normal $F$-subgroup of $\prod_{i=1}^k G_i^{m_i}$, its projection to each factor $G_i$ is normal $F$-subgroup of $G_i$, hence it’s either 1 or $G_i$. By Lemma \[L:subprod\], we can write $N$ abstractly as $\prod_{i=1}^k G_i^{n_i}$ and $N_i=G_i^{n_i}$. From Lemma \[L:preSchur\], we see that for $i\neq j$ the projection $N_i\to G_j^{m_j}$ is trivial, and it follows that $N_i$ is the subgroup of elements of $N$ that are trivial in the projections to $G_j^{m_j}$ for all $j\ne i$. Finally, if $n\in N_i$, then we can see that any $\prod_{i=1}^k G_i^{m_i}$ conjugate of $n$ is trivial in the projections to $G_j^{m_j}$ for all $j\ne i$ and in is $N$. Hence $N_i$ is a normal $F$-subgroup of $G_i^{m_i}$. The followings are two corollaries of Lemma \[L:subsplit\]. \[C:subwhole\] Let $G_i$ be irreducible $F$-groups for $i=1,\dots, k$ such that for $i\ne j$, we have that $G_i$ and $G_j$ are not isomorphic as $F$-groups. Let $N$ be a normal $F$-subgroup of $\prod_{i=1}^k G_i^{m_i}$. Then $N=\prod_{i=1}^k G_i^{m_i}$ if and only if $\pi_i(N)=G_i^{m_i}$ for each projection $\pi_i : N {\rightarrow}G_i^{m_i}$. \[C:probsplit\] Let $G_i$ be finite irreducible $F$-groups for $i=1,\dots, k$ such that for $i\ne j$, we have that $G_i$ and $G_j$ are not isomorphic as $F$-groups, and let $m_i$ be non-negative integers. Let $R=\prod_{i=1}^k G_i^{m_i}$. Then $${\operatorname{Prob}}( [x_1,\dots, x_{n+u} ]_F=R)=\prod_{i=1}^m {\operatorname{Prob}}( [y_{i,1},\dots, y_{i,n+u} ]_F=G_i^{m_i}),$$ where the $x_k$ are independent, uniform random elements of $G$, and the $y_{i,k}$ are independent, uniform random elements of $G_i^{m_i}.$ The next lemma will help us determine when $ [y_{i,1},\dots, y_{i,n+u} ]_F=G_i^{m_i}$. \[L:critgen\] Let $G$ be an irreducible $F$-group. If $G$ is non-abelian, then a normal $F$-subgroup $N$ of $G^m$ is all of $G^m$ if and only if it is non-trivial in each of the $m$ projections to $G$. If $G$ is abelian, then a normal $F$-subgroup $N$ of $G^m$ is all of $G^m$ if and only if the projection onto the product of the first $m-1$ factors is surjective and the projection of $N$ onto the $m$th factor does not factor through the projection onto the product of the first $m-1$ factors. The only if direction is clear. We let $\pi$ be the projection of $N$ onto the first $m-1$ factors of $G^m$ and $\pi^m$ the projection onto the last factor. For the other direction, for non-abelian $G$ we induct and so we have by the inductive hypothesis $\pi(N)=G^{m-1}$. For $G$ abelian we have $\pi(N)=G^{m-1}$ as a hypothesis. We consider $\pi_m (\ker \pi)$, which must be $1$ or $G$. If $\pi_m (\ker \pi)$ is $G$, then we see $N=G^m$, as it includes element with every possible first $m-1$ coordinates, and then an element with trivial first $m-1$ coordinates and every possible $m$th coordinate. Now we show that we cannot have $\pi_m (\ker \pi)=1.$ Suppose for the sake of contradiction that $\pi_m (\ker \pi)=1.$ Then since $\ker \pi_m\cap \ker \pi =1$, we have $\ker \pi=1,$ and $\pi$ is an isomorphism on $N$, and in particular $\pi_m$ factors through $\pi$. So given our hypotheses, this can only happen when $G$ is non-abelian. We write elements $(a,b) \in G^{m-1} \times G$. Since $\pi_m(N)$ is non-trivial, but must be $G$ be the irreducibility of $G$. For every $b\in G$, we have some $a\in G^{m-1}$ such that $(a,b)\in N$. However, since $N$ is normal, that means $(a,gbg^{-1})\in N$ for every $g\in G$. Since $\pi_m$ factors through $\pi$, we have that $b=gbg^{-1}$ for every $b,g\in G$, which is a contradiction, since above we saw we can only be in this case if $G$ is non-abelian. Lemma \[L:critgen\] lets us compute the probabilities appearing in the right-hand side of Corollary \[C:probsplit\] in the following two corollaries. \[C:nonabprob\] If $G$ is a finite non-abelian irreducible $F$-group, and $y_{k}$ for $k=1,\dots,n+u$ are independent, uniform random elements of $G^{m}$, then $${\operatorname{Prob}}( [y_{1},\dots, y_{n+u} ]_F=G^{m})=(1-\|G\|^{-n-u})^m.$$ \[C:abprob\] If $G$ is a finite abelian irreducible $F$-group, and $y_{k}$ for $k=1,\dots,n+u$ are independent, uniform random elements of $G^{m}$, then $${\operatorname{Prob}}( [y_{1},\dots, y_{n+u} ]_F=G^{m}) =\prod_{k=0}^{m-1} (1-h_F(G)^k\| G\|^{-n-u}).$$ Let $\pi_k$ be the projection of $G^m$ onto the $k$th factor, and $\Pi_k$ the projection of $G^m$ to the first $k$ factors. We have $$\begin{aligned} &{\operatorname{Prob}}( [y_{1},\dots, y_{n+u} ]_F=G^{m})\\ =&\prod_{k=0}^{m-1} {\operatorname{Prob}}( \Pi_{k+1}([y_{1},\dots, y_{n+u} ]_F)=G^{k+1} \,\|\, \Pi_{k}([y_{1},\dots, y_{n+u} ]_F)=G^{k} ).\end{aligned}$$ We condition on the values of $\Pi_{k}(y_i)$, and we still have, with this conditioning, that the $\pi_{k+1}(y_i)$ are uniform, independent random in $G$. By Lemma \[L:critgen\], given $\Pi_{k}([y_{1},\dots, y_{n+u} ]_F)=G^{k}$, we will have $\Pi_{k+1}([y_{1},\dots, y_{n+u} ]_F)=G^{k+1},$ exactly if the map $\pi_{k+1}\|_{[y_{1},\dots, y_{n+u} ]_F}$ does not factor through $\Pi_{k}\|_{[y_{1},\dots, y_{n+u} ]_F}$. We have a total of $\|G\|^{n+u}$ choices for the $(n+u)$-tuple $(\pi_{k+1}(y_1), \cdots, \pi_{k+1}(y_{n+u}))$. Call choice for $(\pi_{k+1}(y_1),\dots, \pi_{k+1}(y_{n+u}))$ bad if $\pi_{k+1}\|_{[y_{1},\dots, y_{n+u} ]_F}$ factors through $\Pi_{k}\|_{[y_{1},\dots, y_{n+u} ]_F}$. Since $\Pi_{k}([y_{1},\dots, y_{n+u} ]_F)=G^{k}$, there are $\|{\operatorname{Hom}}_F(G^k,G)\|$ choices for maps from $G^k$ to $G$, each of which gives a bad choice for $(\pi_{k+1}(y_1),\dots, \pi_{k+1}(y_{n+u}))$ (and all bad choices arise this way). For two maps in ${\operatorname{Hom}}_F(G^k,G) $ to give the same bad choice, they would have to agree on $\Pi_k(y_i)$ for all $i$, and since $\Pi_{k}([y_{1},\dots, y_{n+u} ]_F)=G^{k}$, this would imply the two maps in ${\operatorname{Hom}}_F(G^k,G) $ would be the same. Thus there are $\|{\operatorname{Hom}}_F(G^k,G)\|$ bad choices in $\|G\|^{n+u}$ for the $\pi_{k+1}(y_i)$, and as $\|{\operatorname{Hom}}_F(G^k,G)\|=h_F(G)^k$, the corollary follows. Theorem \[T:probfrommult\] now follows from Corollaries \[C:probsplit\], \[C:nonabprob\], and \[C:abprob\]. Also, we can now prove the following lemma which is key for our general approach in Section \[S:setup\]. \[L:irred\] Let $G$ be a finite group, and let $N$ be a normal subgroup of $G$. Let $M$ be the intersection of all maximal proper, $G$-normal subgroups of $N$. Then $N/M$ is a $G/M$-group under the action of conjugation. We have that $N/M$ is isomorphic, as an $G/M$-group, to a direct product of irreducible $G/M$-groups. Moreover, among these irreducible $G/M$-groups, the abelian ones all have the action of $G/M$ factor through $G/N$, so are also irreducible $G/N$-groups. We consider $N$ as a $G$-group under conjugation. A subgroup of $N$ is a normal subgroup of $G$ if and only if it is a $G$-subgroup of $N$. Taking the quotient modulo $M$ gives us a containment respecting bijection between the $G$-subgroups of $N$ containing $M$ and the $G/M$-subgroups of $N/M$. Since all maximal proper $G$-subgroups of $N$ contain $M$, the quotient map gives us a bijection between the maximal proper $G$-subgroups of $N$ and the maximal proper $G/M$-subgroups of $N/M$, and in particular the quotient $M/M=1$ is the quotient of all the maximal proper $G/M$-subgroups of $N/M$. Let $M_i$ be the maximal proper $G/M$-subgroups of $N/M$. Each $(N/M)/M_i$ is an irreducible $G/M$-group. We have that $N/M$ is a subgroup of $\prod_i (N/M)/M_i$ that surjects into each factor, $N/M$ is isomorphic to a direct product of irreducible $G/M$-groups by Lemma \[L:subprod\]. On an abelian irreducible $G/M$-group factor, conjugation by any element in $N/M$ gives the trivial group action, so we have the last statement of the lemma. Determining factors appearing in $R$ {#S:factors} ==================================== Throughout this section, we assume $S$ is a set of finite groups, $n$ is a positive integer, and $H$ is a finite level $S$ group. If $S$ is finite, then we let $$1{\rightarrow}R {\rightarrow}F{\rightarrow}H {\rightarrow}1$$ be the fundamental short exact sequence associated to $S$, $n$ and $H$ (see Section \[S:setup\]). In this section, we will bound which irreducible $F$-groups are possible factors in $R$. A finite irreducible $F$-group is characteristically simple and thus, as a group, a direct product $\Gamma^m$ of isomorphic simple groups. First, when the group $H$ is fixed, Lemma \[L:HfactR\] will bound the possible power $m$ for factors in $R$. For fixed $S$, Corollary \[C:nonewsimple\] will then bound the possible simple group $\Gamma$ for factors in $R$. We take a slightly longer than necessary route to Corollary \[C:nonewsimple\] because along the way we will develop the technology to prove Corollary \[C:CfpFin\], which will later be critical in Section \[S:cntadd\] for our proof of countable additivity of $\mu_u$. \[L:HfactR\] Let $(E, \pi)$ be an $H$-extension such that $G=\ker \pi$ is a finite irreducible $E$-group. Then $G$ is isomorphic to $\Gamma^m$ for some finite simple group $\Gamma$ and $m\leq \|H\|$. Let $G{\simeq}\Gamma^m$, where $\Gamma$ is a finite simple group. If $\Gamma={\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$, then $G \subset {\mathbf{C}}_E(G)$ and we have that the map $H{\rightarrow}{\operatorname{Aut}}(G)={\operatorname{GL}}_m({\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})$ defined by conjugation action is an irreducible representation of $H$. Since for any non-zero vector $v \in {\ensuremath{{\mathbb{F}}}}_p^m$, the vectors $hv$, for $h\in H$, span a subrepresentation of $H$, we have the dimension $m$ is at most $\|H\|$. If $\Gamma$ is non-abelian, then consider an embedding $ \iota: \Gamma \hookrightarrow \Gamma^m$ such that the image is a normal subgroup. There is an element $a=(a_1, \cdots, a_m) \in \iota(\Gamma)$ such that $a_i$ is not the identity element for some $i$. Let $b\in \Gamma^m$ have $j$th coordinate $1$ for $j\neq i$ and $i$th coordinate $\gamma\in \Gamma$. Then since $\iota(\Gamma)$ is normal, we have that the commutator $[a,b]\in \iota(\Gamma)$. The element $[a,b]$ is trivial in all but the $i$th coordinate, where it is $[a_i,\gamma]$. So the intersection of $\iota(\Gamma)$ and the $i$th factor (which is a normal subgroup of $\Gamma$) contains $[a_i,\gamma]$ for some non-trivial $a_i\in \Gamma$ and all $\gamma\in \Gamma$. Since $\Gamma$ is a non-abelian simple group, this means the intersection of $\iota(\Gamma)$ and the $i$th factor is non-trivial, and hence all of the $i$th factor. So $\iota(\Gamma)$ is exactly the $i$th factor of $\Gamma^m$. We have thus showed that a normal subgroup of $\Gamma^m$ that is isomorphic to $\Gamma$ must be one of the $m$ factors. So we have a well-defined map ${\operatorname{Aut}}(\Gamma^m) \to S_m$ (the symmetric group on $m$ elements), and note that ${\operatorname{Inn}}(\Gamma^m)$ is in the kernel of this map. If $\Gamma^m$ is an irreducible $E$-group, the action of $H$ on the factors must be transitive, which proves $m\leq \|H\|$. Recall that a chief series of a finite group $G$ is a chain of normal subgroups $$\label{eq:chs} 1=G_0 \lhd G_1 \lhd \cdots \lhd G_r =G$$ such that for each $0\leq i \leq r-1$, $G_i$ is normal in $G$ and the quotient group $G_{i+1}/G_i$ is a minimal normal subgroup of $G/G_i$. If $M$ is a minimal normal subgroup of $G$, then define $\rho_M$ to be the homomorphism $$\begin{aligned} \rho_M : G & \to & {\operatorname{Aut}}(M)\\ g & \mapsto & (x \mapsto g x g^{-1})_{x\in M}.\end{aligned}$$ The kernel of $\rho_M$ is the centralizer ${\mathbf{C}}_G(M)$ of $M$ in $G$. So $\rho_M$ gives an isomorphism from $G/{\mathbf{C}}_G(M)$ to the subgroup $\rho_M(G)$ of ${\operatorname{Aut}}(M)$. In fact, since $M$ is a minimal normal subgroup of $G$, it is a direct product of isomorphic simple groups. If $M$ is a direct product of isomorphic abelian simple groups, i.e. an elementary abelian $p$-group, then $\rho_M(M)={\operatorname{Inn}}(M)=1$; otherwise, $\rho_M(M)={\operatorname{Inn}}(M)\simeq M$. Thus, ${\operatorname{Inn}}(M)$ is always a normal subgroup of $\rho_M(G)$ and $\rho_M(G) /{\operatorname{Inn}}(M) \simeq G/(M\cdot {\mathbf{C}}_G(M))$. A chief factor pair is a pair of finite groups $(M,A)$ such that $M$ is an irreducible $A$-group and the $A$-action on $M$ is faithful (hence $A$ is naturally a subgroup of ${\operatorname{Aut}}(M)$). In particular, the chief series (\[eq:chs\]) gives a sequence of chief factor pairs $(G_{i+1}/G_i, \rho_{G_{i+1}/G_i}(G/G_i))$, and we call them chief factor pairs of the series (\[eq:chs\]). Two chief factor pairs $(M_1, A_1)$ and $(M_2, A_2)$ are isomorphic if there exists an isomorphism $\alpha: M_1 \to M_2$ such that the induced isomorphism $\alpha^: {\operatorname{Aut}}(M_1)\to {\operatorname{Aut}}(M_2)$ maps $A_1$ to $A_2$. The following is an analog of the Jordan-Hölder Theorem. \[L:J-H\] Let $G$ be a finite group. Suppose there are two chief series of $G$: $$\begin{aligned} && 1= G_0 \lhd G_1 \lhd \cdots \lhd G_r =G \label{eq:cs1} \\ &\text{and}& 1= I_0 \lhd I_1 \lhd \cdots \lhd I_s=G. \label{eq:cs2}\end{aligned}$$ Then 1. $r=s$; 2. the list of isomorphism classes of chief factor pairs $\Big\{\Big(G_{i+1}/G_i , \rho_{G_{i+1}/G_i}(G/G_i) \Big)_{i=0}^{r-1}\Big\}$ is a rearrangement of the list $\Big\{\Big(I_{i+1}/I_i , \rho_{I_{i+1}/I_i}(G/I_i) \Big)_{i=0}^{s-1}\Big\}$. We prove this by induction on $\|G\|$. The case that $\|G\|=1$ is trivial. Assume the lemma is true for all groups of order less than $k$ and $G$ is a group of order $k$. If $G_1=I_1$ then $$\begin{aligned} && 1 \lhd G_2/G_1 \lhd \cdots \lhd G_r/G_1 =G/G_1\\ &\text{and }& 1 \lhd I_2/I_1 \lhd \cdots I_s/I_1 = G/I_1\end{aligned}$$ are two chief series of $G/G_1$. So the lemma is proved for $G$ by the induction hypothesis. Assume $G_1\neq I_1$. Since they are minimal normal subgroups, $G_1\cap I_1 = 1$ and $G_1I_1=G_1\times I_1$. Define $J_2$ to be the product $G_1I_1$. Then $J_2/G_1\simeq I_1$ is a minimal normal subgroup of $G/G_1$ and we can construct a chief series of $G$ passing through $G_1$ and $J_2$ $$\label{eq:cs3} 1\lhd G_1 \lhd J_2 \lhd J_3 \lhd \cdots \lhd J_t=G.$$ Comparing chief series (\[eq:cs1\]) and (\[eq:cs3\]), it follows that $r=t$ and $$\begin{aligned} \left\{ \Big( G_{i+1}/G_i , \rho_{G_{i+1}/G_i} (G/G_i)\right)_{i=0}^{r-1}\Big\} &\sim& \Big\{ \Big( G_1, \rho_{G_1}(G)\Big), \Big(J_2/G_1, \rho_{J_2/G_1} (G/G_1)\Big), \label{eq:list} \\ & &\Big(J_{i+1}/J_i, \rho_{J_{i+1}/J_i}(G/J_i)\Big)_{i=2}^{t-1} \Big\} \nonumber \end{aligned}$$ where the symbol $\sim$ means “is a rearrangement of”. Let $\pi$ be the quotient map $G\to G/G_1$. As $G_1\unlhd {\mathbf{C}}_G(I_1)$, if an element in $G$ centralizes $I_1$, then its image under $\pi$ centralizes $\pi(I_1)=J_2/G_1$. It follows that $\pi({\mathbf{C}}_G(I_1))\subseteq {\mathbf{C}}_{G/G_1}(J_2/G_1)$. Conversely, if $a$ is an element in $G$ such that $\pi(a)\in {\mathbf{C}}_{G/G_1}(J_2/G_1)$, then for every $h \in I_1$, we have $\pi(aha^{-1})=\pi(h)$, which indicates that there exists $g\in G_1$ such that $aha^{-1}=hg$. But $I_1\unlhd G$, so $aha^{-1}\in I_1$. It follows from $I_1\cap G_1=1$ that $g=1$, and hence $a\in {\mathbf{C}}_G(I_1)$, which proves $\pi({\mathbf{C}}_G(I_1))={\mathbf{C}}_{G/G_1}(J_2/G_1)$. Thus we have $$\begin{aligned} \faktor{G}{{\mathbf{C}}_G(I_1)}&\cong& \faktor{G/G_1}{{\mathbf{C}}_G(I_1)/G_1}\\ &\cong& \faktor{G/G_1}{{\mathbf{C}}_{G/G_1}(J_2/G_1)}. \end{aligned}$$ Therefore the chief factor pairs $\Big(I_1, \rho_{I_1}(G)\Big)$ and $\Big(J_2/G_1, \rho_{J_2/G_1}(G/G_1)\Big)$ are isomorphic. So the list (\[eq:list\]) is $$\label{list1} \sim \Big\{\Big(G_1, \rho_{G_1}(G)\Big), \Big(I_1, \rho_{I_1}(G) \Big), \Big(J_{i+1}/J_i, \rho_{J_{i+1}/J_i}(G/J_i)\Big)_{i=2}^{t-1}\Big\}.$$ Similarly, by comparing the following chief series of $G$ $$\label{cs4} 1\lhd I_1 \lhd J_2 \lhd J_3 \lhd \cdots \lhd J_t=G.$$ with (\[eq:cs2\]), we finish the proof of the lemma. If $G$ is a finite group, then define ${\mathcal{CF}}(G)$ to be the set consisting of all isomorphism classes of chief factor pairs of a chief series of $G$ (${\mathcal{CF}}(G)$ does not depend on the choice of chief factor series by Lemma \[L:J-H\]). If $T$ is a set of finite groups, then $${\mathcal{CF}}(T):=\bigcup\limits_{G\in T} {\mathcal{CF}}(G).$$ The following lemma shows that every factor in $R$ comes from ${\mathcal{CF}}({\bar{S}})$. \[L:R-cfp\] Let $S$ be a finite set of finite groups, and $R$, $F$ and $H$ as defined at the beginning of this section. If $G$ is an irreducible $F$-subgroup of $R$, then $(G, \rho_G(F))\in {\mathcal{CF}}({\bar{S}})$ and $\rho_G(F)/{\operatorname{Inn}}(G)$ is isomorphic to a quotient of $H$. Since $F$ is a finite level $S$ group and $G$ is a minimal normal subgroup of $F$, we have $(G, F/{\mathbf{C}}_F(G)) \in {\mathcal{CF}}({\bar{S}})$. Further, $R$ is a direct product of irreducible $F$-groups, so $R$ is contained in ${\mathbf{C}}_F(G) \cdot G$ and it follows that $(F/{\mathbf{C}}_F(G))/{\operatorname{Inn}}(G) = F/({\mathbf{C}}_F(G)\cdot G)$ is a quotient of $H$. In the rest of this section, we will bound the size of chief factor pairs. \[L:CFbarS\] If $S$ is a set of finite groups that is closed under taking subgroups and quotients, then ${\mathcal{CF}}({\bar{S}})={\mathcal{CF}}(S)$. Since ${\bar{S}}$ is the closure of $S$ under taking finite direct products, quotients and subgroups, it suffices to show that none of these three actions creates new chief factor pairs not belonging to ${\mathcal{CF}}(S)$. First, taking direct products and quotients does not create new chief factor pairs. If $G$ and $J$ are finite groups with chief series $1\lhd G_1 \lhd \cdots \lhd G_r = G$ and $1\lhd J_1 \lhd \cdots \lhd J_s = J$. Then the following chief series of $G\times J$ $$1 \lhd G_1 \times 1 \lhd \cdots G\times 1 \lhd G\times J_1 \lhd \cdots \lhd G\times J$$ implies that ${\mathcal{CF}}(G\times J)={\mathcal{CF}}(G)\cup {\mathcal{CF}}(J)$. If $N$ is a normal subgroup of $G$, then ${\mathcal{CF}}(G/N)\subseteq {\mathcal{CF}}(G)$ since we can always find a chief series of $G$ passing through $G/N$. Finally, assume $J$ is a subgroup of $G$ for $G\in {\bar{S}}$ such that ${\mathcal{CF}}(G)\subseteq {\mathcal{CF}}(S)$. We want to prove ${\mathcal{CF}}(J)\subseteq {\mathcal{CF}}(S)$. Let $1 \lhd G_1 \lhd \cdots \lhd G_r = G$ be a chief series of $G$. We can construct a chief series of $J$ that passes through $G_i \cap J$ for every $i=1, \cdots, r$. The chief factor pairs achieved from the elements between $G_i \cap J$ and $G_{i+1}\cap J$ are achieved from the group $J/(G_i\cap J)\simeq (J\cdot G_i)/ G_i$, which is a subgroup of $G/G_i$. Thus it’s enough to consider the positions between $1$ and $G_1\cap J$. Since $(G_1, \rho_{G_1}(G))\in {\mathcal{CF}}(G) \subseteq {\mathcal{CF}}(S)$, there is a group $G'\in S$ and a minimal subgroup $G'_1$ of $G'$ such that the chief factors $(G_1, \rho_{G_1}(G))$ and $(G'_1, \rho_{G'_1}(G'))$ are isomorphic, i.e. $\exists$ $\alpha: G_1\overset{\sim}{\to}G'_1$ such that $\alpha^: {\operatorname{Aut}}(G_1)\overset{\sim}{\to} {\operatorname{Aut}}(G'_1)$ maps $\rho_{G_1}(G)$ to $\rho_{G'_1}(G')$. Define $A:=\rho_{G_1}(J)=(J\cdot {\mathbf{C}}_G(G_1))/{\mathbf{C}}_G(G_1)$ that is a subgroup of $\rho_{G_1}(G)$. Note that the action of $A$ on $G_1$ actually stabilizes $G_1\cap J$. Let $J':=\rho_{G'_1} ^{-1}(\alpha^(A))$ and $J'_1=\alpha(G_1\cap J)$. So $J'$ is a subgroup of $G'$ satisfying the following short exact sequence $$1 \to {\mathbf{C}}_{G'}(G'_1) \to J' \to \alpha^(A)\to 1.$$ and $J'_1$ is a subgroup of $G'_1\cap J'$. Since ${\mathbf{C}}_{G'}(G_1')\leq {\mathbf{C}}_{J'}(G'_1\cap J')$, the action of $J'$ via conjugation on $G'_1\cap J'$ factors through $\alpha^(A)$. Also, since the $\alpha^(A)$ action on $G'_1$ stabilizes $J'_1$, we have that $J'_1$ is a normal subgroup of $J'$. Because $G_1\cap J$ with the action of $A$ is isomorphic to $J'_1$ with the action of $\alpha^(A)$, every chief factor pair of $G$ achieved from positions between 1 and $G_1\cap J$ is also a chief factor pair of $J'$ achieved via a series passing through $J'_1$. Finally, $J'$ as a subgroup of $G'$ belongs to $S$, so ${\mathcal{CF}}(J)\subseteq {\mathcal{CF}}(S)$ and we prove the lemma. \[C:nonewsimple\] If $S$ is a set of finite groups, and $\Gamma \in \bar{S}$ is a simple group, then $\Gamma$ is in the closure of $S$ under taking subgroups and quotients. If $\Gamma\in {\bar{S}}$ is a simple group, then $(\Gamma, {\operatorname{Inn}}(\Gamma))\in {\mathcal{CF}}({\bar{S}})$. By Lemma \[L:CFbarS\], $\Gamma$ is in the closure of $S$ under taking subgroups and quotients. \[C:CfpFin\] Let $S$ be a finite set of finite groups. Then ${\mathcal{CF}}({\bar{S}})$ is a finite set. Moreover, if $\ell$ is the upper bound of the orders of groups in $S$, then for any pair $(M,A)\in {\mathcal{CF}}({\bar{S}})$, the quotient $A/{\operatorname{Inn}}(M)$ is of level $\ell-1$. Without lost of generality, let’s assume $S$ is closed under taking subgroups and quotients. By Lemma \[L:CFbarS\], ${\mathcal{CF}}({\bar{S}})={\mathcal{CF}}(S)$ is finite, and for any chief factor pair $(M,A)\in {\mathcal{CF}}({\bar{S}})$, there is a group $G\in S$ such that $(M,A)\in {\mathcal{CF}}(G)$. If $M$ is abelian, then $\|M\|\|A\|\leq \|G\| \leq \ell$; otherwise, $M$ is non-abelian and $\|A\|=\|M\|\|A/{\operatorname{Inn}}(M)\|\leq \|G\| \leq \ell$. In either case, we have $\|A/{\operatorname{Inn}}(M)\|\leq \ell-1$. Counting maximal quotients of irreducible $F$-groups {#S:countqt} ==================================================== In order to apply Theorem \[T:probfrommult\] to a group that we know, abstractly, to be a product of irreducible $F$-groups, we need to know the multiplicities of the various irreducible $F$-groups in the product. In this section, we relate those multiplicities to a count of surjections. \[T:countRsub\] Let $G_i$ be finite irreducible $F$-groups for $i=1,\dots, k$ such that $G_i$ and $G_j$ are not isomorphic for $i\ne j$. Then if $G_j$ is abelian $$\# {\operatorname{Sur}}_F\left(\prod_{i=1}^k G_i^{m_i} , G_j \right) =h_F(G_j)^{m_j}-1$$ and if $G_j$ is non-abelian $$\# {\operatorname{Sur}}_F\left(\prod_{i=1}^k G_i^{m_i} , G_j \right) = m_j\|{\operatorname{Aut}}_F(G_j)\|.$$ This theorem follows immediately from Lemma \[L:preSchur\] and the following lemmas. \[L:justj\] Let $G_i$ be finite irreducible $F$-groups for $i=1,\dots, k$ such that for $i\ne j$, we have that $G_i$ and $G_j$ are not isomorphic. The restriction map $${\operatorname{Sur}}_F\left(\prod_{i=1}^k G_i^{m_i} , G_j \right) {\rightarrow}{\operatorname{Hom}}_F\left( G_j^{m_j} , G_j \right)$$ is a bijection to ${\operatorname{Sur}}_F\left( G_j^{m_j} , G_j \right){\subset}{\operatorname{Hom}}_F\left( G_j^{m_j} , G_j \right)$. Note that in a surjection, each $G_i$ must go to a normal subgroup of $G_j$, and so by Lemma \[L:preSchur\] the restriction to every $G_i$ factor for $i\ne j$ is trivial. So that proves the above restriction map is injective. The restriction map is surjective to ${\operatorname{Sur}}_F\left( G_j^{m_j} , G_j \right)$ since $G_j^{m_j}$ is a quotient of $\prod_{i=1}^k G_i^{m_i}$. \[L:morto1\] Let $G$ be a finite irreducible $F$-group and $m$ a positive integer. We have $${\operatorname{Hom}}_F(G^m, G) {\subset}{\operatorname{Hom}}_F(G,G)^m$$ by restriction to each factor. If $G$ is abelian, then this inclusion is an equality. If $G$ is non-abelian, then we have that ${\operatorname{Hom}}_F(G^m, G) $ is the subset of the $m$-tuples ${\operatorname{Hom}}_F(G,G)^m$ where at most $1$ coordinate is a non-trivial morphism in ${\operatorname{Hom}}_F(G,G)$. The only homomorphism that is not surjective among those above is the trivial morphism. If $G$ is abelian, then for $\phi_i\in {\operatorname{Hom}}_F(G,G)$, we have a morphism $\phi: G^m {\rightarrow}G$ such that $\phi(a_1,\dots,a_m)=\prod_{i=1}^m \phi_i(a_i)$. Note for $\phi\in {\operatorname{Hom}}_F(G^m, G)$, with restrictions $\phi_i$ to the factors, we have that $a\in\phi_i(G)$ and $b\in\phi_j(G)$ commute for $i\neq j$. Since $\phi_i(G)$ is $1$ or $G$, if $G$ is non-abelian we see that at most one $\phi_i$ can be non-trivial. Moreover, clearly the $m$-tuples ${\operatorname{Hom}}_F(G,G)^m$ where at most $1$ coordinate is a non-trivial morphism in ${\operatorname{Hom}}_F(G,G)$ give elements of ${\operatorname{Hom}}_F(G^m,G)$. For an $F$-morphism $G{\rightarrow}G$, if it is non-trivial, it must be injective (since its kernel is a normal $F$-subgroup), and thus surjective. Determination of $\mu_{u,n}$ on basic open sets {#S:basics} =============================================== The goal of this section is to prove Theorem \[T:calc\], in which we will give ${\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)$ for every finite set $S$ and finite level $S$ group $H$, i.e. determine the measures of the basic open sets in the distributions coming from our random groups. Throughout this section, we assume $n\geq 1$, $S$ is a finite set of finite groups, $H$ is a finite level $S$ group and $$1{\rightarrow}R {\rightarrow}F {\rightarrow}H {\rightarrow}1$$ is the fundamental short exact sequence associated to $S$, $n$ and $H$. For any abelian irreducible $H$-group $G$, we define $m(S,n,H,G)$ to be the multiplicity of $G$ in $R$ as an $H$-group under conjugation (see Lemma \[L:irred\]). Let $G$ be a non-abelian finite group. Let $G_i$ be the irreducible $F$-group structures one can put on $G$. Then we define $m(S,n,H,G)$ to be the sum (over $i$) of the multiplicity of the $G_i$ in $R$ as an $F$-group under conjugation. Equation and Theorem \[T:probfrommult\] allow us to express ${\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)$ in terms of the multiplicities $m(S,n,H,G)$. The work of this section will be to find explicit formulas for these $m(S,n,H,G)$ (given in Corollaries \[C:finalmab\] and \[C:finalmnon\]). \[T:calc\] Let $S$ be a finite set of finite groups and $H$ a finite level $S$ group. Let $n\geq 1$ and $u> -n$ be integers. Then $$\begin{aligned} \label{E:fixedn} &&{\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\\ &=&\frac{\|{\operatorname{Sur}}(\hat{F}_n,H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{n+u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{k=0}^{m(S,n,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{n+u}}}) \prod_{\substack{ G\in \mathcal{N}}} (1-\|G\|^{-{n-u}})^{m(S,n,H,G)},\nonumber\end{aligned}$$ and we have $$\begin{aligned} \label{E:lim} && \lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\\ &=&\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(S,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(S,H,G)}.\nonumber\end{aligned}$$ Further, if $G\in \mathcal{A}_H\cup \mathcal{N}$ is isomorphic as a group to $\Gamma^j$ for some simple group $\Gamma$, and either 1) $\Gamma$ is not in the closure of $S$ under taking subgroups and quotients, or 2) $j>\|H\|$, then $m(S,n,H,G)=\lambda(S,H,G)=0$. \[R:finprod\] The products over $\mathcal{A}_H$ and $\mathcal{N}$ appearing in Theorem \[T:calc\] are actually finite products (except for trivial terms), because of the last statement in the theorem. We will show in Section \[S:arbitraryS\] that statement of Theorem \[T:calc\] also works for an arbitrary set $S$ of finite groups. First, we need to define the Möbius function on a poset of $H$-extensions. Given a finite group $H$, there is a poset $\mathcal{E}_H$ of $H$-extensions (not* isomorphism classes of $H$-extensions) where $(E,\pi) \leq (E',\pi')$ if $(E,\pi)$ is a sub-$H$-extension of $(E',\pi')$. (This relation is defined for literal sub-$H$-extensions and not $H$-extensions just isomorphic to a subextension.) We let $\nu(D,E)$ be the Möbius function of this poset (we drop the maps to $H$ in the notation but they are implicit) so that for two $H$-extensions $D$ and $E$ we have $$\begin{aligned} \nu(E,E)&=1\\ \nu(D,E)&=-\sum_{\substack{D'\in \mathcal{E}_H \\D< D' \leq E }} \nu(D',E) \quad\quad \textrm{ if $D\ne E$}\end{aligned}$$ so that in particular $$\begin{aligned} \nu(D,E)&=0 \textrm{ if $D$ is not a sub-$H$-extension of $E$}\\ \sum_{\substack{D'\in \mathcal{E}_H \\D\leq D' \leq E }} \nu(D',E)&= \begin{cases} 1 &\textrm{if $D=E$}\\ 0 & \textrm{otherwise}. \end{cases}\end{aligned}$$ Theorem \[T:countRsub\] relates our key multiplicities $m(S,n,H,G)$ to the number of $F$-surjections from $R$ to $G$. An $F$-surjection $R{\rightarrow}G$ has a kernel $K$, and we have a surjection from our $H$-extension $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}H$ to the $H$-extension $F/K {\rightarrow}H$. The next proposition will count such surjections of $H$-extensions. \[P:countfromF\] Let $n\geq1$ be an integer, $S$ a finite set of finite groups, and $H$ a finite level $S$ group. Let $({\hat{F}}_n)^{{\bar{S}}}\stackrel{\rho}{{\rightarrow}}H$ be an $H$-extension structure on $({\hat{F}}_n)^{{\bar{S}}}$. Let $E \stackrel{\pi}{{\rightarrow}} H $ be a finite $H$ extension. We have $$\|{\operatorname{Sur}}(\rho,\pi)\|=\begin{cases} \sum_{\substack{D\in\mathcal{E}_H, D\leq E}} \nu(D,E) \left( \frac{\|D\|}{\|H\|} \right)^n &\textrm{ if $E$ is level $S$} \\ 0 &\textrm{ otherwise}. \end{cases}$$ If $({\hat{F}}_n)^{{\bar{S}}} {\rightarrow}E$ is a surjection, then $E$ is level $S$. If $E$ is level $S$ and $(D,\psi)\leq (E,\pi)$, surjections $({\hat{F}}_n)^{{\bar{S}}} {\rightarrow}D$ exactly correspond to surjections ${\hat{F}}_n {\rightarrow}D$, i.e. choices of image for each generator $x_1,\dots,x_n$ of $\hat{F}_n$ such that their images generate $D$. For each generator $x_i$ of ${\hat{F}}_n$, we have a fixed coset of $\ker(\pi)$ in $D$ it can land in to actually obtain a surjection compatible with the maps to $H$. We have that the number of homomorphisms ${\hat{F}}_n {\rightarrow}D$ where the generators go to the appropriate cosets is $(\|D\|/\|H\|)^n$. Let $E'$ be a subgroup of $D$ that could be generated by some $y_1,\dots,y_n$ with each $y_i$ in the required cosets of $\ker(\pi)$. Since $\rho$ is a surjection, it follows that $\pi(E')=H$. So we have $$\left(\frac{\|D\|}{\|H\|}\right)^n=\sum_{\substack{(E',\phi)\in\mathcal{E}_H\\ (E',\phi)\leq (D,\psi)}} \|{\operatorname{Sur}}_H(\rho,\phi)\|.$$ Using Möbius inversion, we obtain the result. We can sum the above as follows. Given a finite $H$-extension $(E,\pi)$ of level $S$, we have $$\begin{aligned} \sum_{(D,\psi) \leq (E,\pi)} \nu(D,E) \left(\frac{\|D\|}{\|H\|}\right)^n&=\sum_{(D,\psi) \leq (E,\pi)} \nu(D,E) \sum_{\substack{ (E',\phi)\leq (D,\psi)}} \|{\operatorname{Sur}}_H(\rho,\phi)\|\\ &=\sum_{(E',\phi) \leq (E,\pi)} \|{\operatorname{Sur}}_H(\rho,\phi)\| \sum_{ (E',\phi)\leq (D,\psi) \leq (E,\pi)} \nu(D,E) \\ &= \|{\operatorname{Sur}}_H(\rho,\pi)\|,\end{aligned}$$ as desired. Now we will build on Proposition \[P:countfromF\] to find $\|{\operatorname{Sur}}_F(R,G)\|$, after which we can then use Theorem \[T:countRsub\] to find the $m(S,n,H,G)$. We will first do the case of abelian $G$, and then non-abelian $G$. \[P:countNother\] Let $H$, $F$, and $R$ be defined as at the beginning of this section. Let $G$ be an abelian irreducible $F$-group. Then $$\begin{aligned} \|{\operatorname{Sur}}_F(R,G)\|=\|{\operatorname{Aut}}_F(G)\|\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ as an $H$-group} \\ \textrm{$E$ is level $S$} }} \frac{\sum_{\substack{D\in\mathcal{E}_H, D\leq E}} \nu(D,E) \left( \frac{\|D\|}{\|H\|} \right)^n}{\|{\operatorname{Aut}}_H(E,\pi)\|}.\end{aligned}$$ if the action of $F$ on $G$ factors through $F{\rightarrow}H$ (i.e. elements of $R$ act trivially on $G$) and $\|{\operatorname{Sur}}_F(R,G)\|=0$ otherwise. We have that $\|{\operatorname{Sur}}_F(R,G)\|$ is $\|{\operatorname{Aut}}_F(G)\|$ times the number of $F$-subgroups $M$ of $R$ such that $R/M$ under $F$-conjugation is isomorphic to $G$ as an $F$-group. If $M$ is an $F$-subgroup of $R$ such that $R/M$ is abelian, then the action of $F$ via conjugation on $R/M$ factors through $H$ (because conjugation by elements from $R$ is trivial in $R/M$ as $R/M$ is abelian). So suppose that the action of $F$ on $G$ factors through $H$. We have the number of $F$-subgroups $M$ of $R$ such that $R/M$ is isomorphic to $G$ as an $F$-group is $$\begin{aligned} &&\sum_{\substack{\textrm{isom. classes of} \\ \textrm{$H$-extensions $(E,\pi)$}}} \#\left\{ F\textrm{-subgroups $M$ of $R$}\, \Bigg\|\, \begin{aligned} & (F/M{\rightarrow}H){\simeq}(E,\pi) \textrm{ as $H$-exts,}\\ & \textrm{$R/M{\simeq}G$ as $F$-groups } \end{aligned} \right\}\\ &=&\sum_{\substack{\textrm{isom. classes of} \\ \textrm{$H$-extensions $(E,\pi)$}\\ \ker \pi{\simeq}G \textrm{ as groups} }} \#\left\{ F\textrm{-subgroups $M$ of $R$}\, \Bigg\|\, \begin{aligned} & (F/M{\rightarrow}H){\simeq}(E,\pi) \textrm{ as $H$-exts,}\\ & \textrm{$R/M{\simeq}G$ as $F$-groups } \end{aligned} \right\}.\end{aligned}$$ Given that $R/M$ is abelian (which is guaranteed by the group isomorphism $\ker \pi{\simeq}G$ and the $H$-extension isomorphism $(F/M{\rightarrow}H){\simeq}(E,\pi)$), since the action of $F$ on $R/M$ factors through $H$, we have that $R/M$ is isomorphic to $G$ as an $F$-group if and only if it is isomorphic to $G$ as an $H$-group. Given $(F/M{\rightarrow}H){\simeq}(E,\pi)$, this is the same as requiring $\ker \pi{\simeq}G$ as an $H$-group. Thus the above sum is equal to $$\begin{aligned} && \sum_{\substack{\text{isom. classes of $H$-extension }(E,\pi) \\ \ker \pi \text{ isom. $G$ as an $H$-group}}} \#\left\{ F\text{-subgroups $M$ of $R$}\mid (F/M {\rightarrow}H) \simeq (E,\pi) \text{ as $H$-exts} \right\}\\ &=& \sum_{\substack{\text{isom. classes of $H$-extension }(E,\pi) \\ \ker \pi \text{ isom. $G$ as an $H$-group}}} \frac{ \# \left\{ (M, \phi)\,\Bigg\|\, \begin{aligned} & M \text{ an $F$-subgroup of }R\\ & \phi: (F/M {\rightarrow}H) \simeq (E, \pi) \end{aligned} \right\}}{\|{\operatorname{Aut}}_H(E,\pi)\|}.\end{aligned}$$ Note that the data $(M,\phi)$ above is exactly the same as the data of a surjection of $H$-extensions from $F{\rightarrow}H$ to $E{\rightarrow}H$. Now let $(E,\pi)$ be an $H$-extension with $\ker \pi$ (via conjugation) an abelian irreducible $H$-group. Consider a surjection of $H$-extensions from $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}H$ to $E{\rightarrow}H$, in which the map $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}E$ has kernel $K$. Let $N$ denote the kernel of $(\hat{F}_n)^{{\bar{S}}}\to H$. Then $N/K$ is an irreducible $(\hat{F}_n)^{{\bar{S}}}$-group, and so $K$ is a maximal proper $(\hat{F}_n)^{{\bar{S}}}$-subgroup of $N$. So the map $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}E$ factors through $F$. On the other hand, any surjection of $H$-extensions from $F{\rightarrow}H$ to $E{\rightarrow}H$ clearly extends to a surjection of $H$-extensions from $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}H$ to $E{\rightarrow}H$. Thus the above sum is equal to $$\begin{aligned} \sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ as an $H$-group}}} \frac{\|{\operatorname{Sur}}_H((\hat{F}_n)^{{\bar{S}}}{\rightarrow}H,\pi)\|}{\|{\operatorname{Aut}}_H(E,\pi)\|}.\end{aligned}$$ The result now follows from applying Proposition \[P:countfromF\] above, after dividing out by the number of choices of isomorphism to $(E,\pi)$. We now can determine the multiplicities of the abelian irreducible $F$-groups in $R$ by combining Theorem \[T:countRsub\] and Proposition \[P:countNother\]. \[C:finalmab\] Let $H$, $F$ and $R$ be as above. Let $G$ be an abelian irreducible $H$-group. Then $$\frac{h_H(G)^{m(S,n,H,G)}-1}{h_H(G)-1} =\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ as an $H$-group} \\ \textrm{$E$ is level $S$} }} \frac{\sum_{\substack{D\in\mathcal{E}_H, D\leq E}} \nu(D,E) \left( \frac{\|D\|}{\|H\|} \right)^n}{\|{\operatorname{Aut}}_H(E,\pi)\|}.$$ Next, we will apply a similar plan to obtain the multiplicities of the non-abelian $G$, but there is an important difference from the abelian case. When $\ker ( E \stackrel{\pi}{{\rightarrow}} H)$ is non-abelian, a surjection of $H$-extensions $F{\rightarrow}E$ still gives an $F$-group structure on $\ker \pi$ by conjugation in $E$, but, unlike in the case when $\ker\pi$ is abelian, that $F$-group structure is not necessarily determined by the isomorphism type of the $H$-extension $(E,\pi).$ So in this case it is most convenient to add together, for each possible underlying group $G$ of a non-abelian irreducible $F$-group, all surjections $F{\rightarrow}E$ over all $G$ extensions $E$ of $H$. \[P:mulnonab\] Let $H$, $F$ and $R$ be as above. Let $G$ be a finite non-abelian group. Let $G_i$ be the pairwise non-isomorphic irreducible $F$-group structures on $G$ for $1\leq i \leq k$ ($k$ may be $0$). Then $$\begin{aligned} \sum_{i=1}^k \frac{\|{\operatorname{Sur}}_F(R,G_i)\|}{\|{\operatorname{Aut}}_F(G_i)\|}= &\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} \\ \textrm{$E$ is level $S$} }} \frac{\sum_{\substack{D\in\mathcal{E}_H, D\leq E}} \nu(D,E) \left( \frac{\|D\|}{\|H\|} \right)^n}{\|{\operatorname{Aut}}_H(E,\pi)\|}.\end{aligned}$$ We note that ${\|{\operatorname{Sur}}_F(R,G_i)\|}/{\|{\operatorname{Aut}}_F(G_i)\|}$ is the number of $F$-subgroups of $R$ whose corresponding quotient is isomorphic to $G_i$ as an $F$-group. We have $$\begin{aligned} &&\sum_{i=1}^k \frac{\|{\operatorname{Sur}}_F(R,G_i)\|}{\|{\operatorname{Aut}}_F(G_i)\|}\\ &=&\sum_{i=1}^{k} \sum_{\substack{\textrm{isom. classes of}\\ \textrm{ $H$-extensions $(E,\pi)$} }} \#\left\{\textrm{$F$-subgroups $M$ of $R$} \,\Bigg\|\, \begin{aligned} &(F/M{\rightarrow}H){\simeq}(E,\pi) \text{ as $H$-exts}\\ & R/M{\simeq}G_i \text{ as $F$-groups } \end{aligned}\right\}\\ &=&\sum_{i=1}^{k} \sum_{\substack{\textrm{isom. classes of}\\\textrm{ $H$-extensions $(E,\pi)$}\\\textrm{$\ker \pi$ isom. $G$ } }} \#\left\{\textrm{$F$-subgroups $M$ of $R$} \,\Bigg\|\, \begin{aligned} &(F/M{\rightarrow}H){\simeq}(E,\pi) \text{ as $H$-exts}\\ & R/M{\simeq}G_i \text{ as $F$-groups } \end{aligned}\right\}\\ &=& \sum_{\substack{\textrm{isom. classes of}\\\textrm{ $H$-extensions $(E,\pi)$}\\\textrm{$\ker \pi$ isom. $G$ } }} \sum_{i=1}^{k} \#\left\{\textrm{$F$-subgroups $M$ of $R$} \,\Bigg\|\, \begin{aligned} &(F/M{\rightarrow}H){\simeq}(E,\pi) \text{ as $H$-exts}\\ & R/M{\simeq}G_i \text{ as $F$-groups } \end{aligned}\right\} \\ &=& \sum_{\substack{\textrm{isom. classes of}\\\textrm{ $H$-extensions $(E,\pi)$}\\\textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} }} \sum_{i=1}^{k} \#\left\{\textrm{$F$-subgroups $M$ of $R$} \,\Bigg\|\, \begin{aligned} &(F/M{\rightarrow}H){\simeq}(E,\pi) \text{ as $H$-exts}\\ & R/M{\simeq}G_i \text{ as $F$-groups } \end{aligned}\right\} \\ &=&\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} }} \#\{\textrm{$F$-subgroups $M$ of $R \,\|\,(F/M{\rightarrow}H){\simeq}(E,\pi)$ as $H$-exts }\}\\ &=&\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} }}\frac{ \#\left\{(M,\phi)\,\Bigg\|\, \begin{aligned} &\textrm{$M$ an $F$-subgroup of $R$} \\ &\textrm{$\phi:(F/M{\rightarrow}H){\simeq}(E,\pi)$ as $H$-exts} \end{aligned} \right\}}{\|{\operatorname{Aut}}_H(E,\pi)\|}.\end{aligned}$$ The second equality follows because $(F/M{\rightarrow}H) {\simeq}(E,\pi)$ and $R/M{\simeq}G_i$ imply that $\ker \pi {\simeq}G$. The fourth equality follows because $(F/M{\rightarrow}H) {\simeq}(E,\pi)$ and $R/M$ and irreducible $F$-group implies that $\ker \pi$ is an irreducible $E$-group. The final equality follows because $\ker \pi{\simeq}G$ and $\ker\pi$ and irreducible $E$-group and $(F/M{\rightarrow}H) {\simeq}(E,\pi)$ implies that $R/M$ is isomorphic to some $G_i$ as an $F$-group. Note that the data $(M,\phi)$ in the final equation above is exactly the same as the data of a surjection of $H$-extensions from $F{\rightarrow}H$ to $E{\rightarrow}H$. Now let $(E,\pi)$ be an $H$-extension with $\ker \pi$ (via conjugation) an irreducible $E$-group. Consider a surjection of $H$-extensions from $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}H$ to $E{\rightarrow}H$, in which the map $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}E$ has kernel $K$. Again, we let $N$ denote the kernel of $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}H$. Then $N/K$ is an irreducible $(\hat{F}_n)^{{\bar{S}}}$-group, and so $K$ is a maximal proper $(\hat{F}_n)^{{\bar{S}}}$-subgroup of $N$. So the map $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}E$ factors through $F$. On the other hand, any surjection of $H$-extensions from $F{\rightarrow}H$ to $E{\rightarrow}H$ clearly extends to a surjection of $H$-extensions from $(\hat{F}_n)^{{\bar{S}}}{\rightarrow}H$ to $E{\rightarrow}H$. Thus, the above sum is equal to $$\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} }} \frac{\|{\operatorname{Sur}}((\hat{F}_n)^{{\bar{S}}}{\rightarrow}H,\pi)\| }{\|{\operatorname{Aut}}_H(E,\pi)\|}.$$ Then by applying Proposition \[P:countfromF\] we obtain the result. We now can determine the multiplicities of the non-abelian irreducible $F$-groups in $R$ by combining Theorem \[T:countRsub\] and Proposition \[P:mulnonab\]. \[C:finalmnon\] Let $H$, $F$ and $R$ be as above. Let $G$ be an non-abelian finite group. Then $$m(S,n,H,G)=\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} \\ \textrm{$E$ level $S$} }} \frac{\sum_{\substack{D\in\mathcal{E}_H, D\leq E}} \nu(D,E) \left( \frac{\|D\|}{\|H\|} \right)^n}{\|{\operatorname{Aut}}_H(E,\pi)\|}.$$ Finally, before we prove Theorem \[T:calc\], we need the following lemma, whose proof is straightforward. \[L:seq\] Suppose $x_1,x_2,\dots$ is a sequence of real numbers with limit $x$, and $y_1,\dots$ is a sequence of real numbers with limit $\infty$. Let $a>1$ be a real number. Then $f(x)=\prod_{i=1}^\infty (1-xa^{-i})$ is continuous in $x$ and $$\lim_{n{\rightarrow}\infty} \prod_{i=1}^{y_n} (1-x_na^{-i})=\prod_{i=1}^{\infty} (1-xa^{-i}).$$ Equation and Theorem \[T:probfrommult\], combined with the multiplicities given by Corollaries \[C:finalmab\] and \[C:finalmnon\], establish Equation of Theorem \[T:calc\]. By the definition of $\lambda(S,H,G)$ defined in Section \[SS:basicq\] and Corollaries \[C:finalmab\] and \[C:finalmnon\], we see that $\lambda(S,H,G)$ is related to $m(S,n,H,G)$ as follows: $$\begin{aligned} \label{E:relmandlam} \lim_{n{\rightarrow}\infty} \frac{h_H(G)^{m(S,n,H,G)}}{\|G\|^n}&= \lambda(S,H,G) \quad\quad &\textrm{for $G\in \mathcal{A}_H$}\\ \lim_{n{\rightarrow}\infty} \frac{m(S,n,H,G)}{\|G\|^n} &= \lambda(S,H,G) \quad\quad &\textrm{for $G\in \mathcal{N}$}. \notag\end{aligned}$$ \[R:lamint\] Note that since by Remark \[R:Gpow\], for $G\in\mathcal{A}_H$, we have that $\|G\|$ is a power of $h_H(G)$, it follows that $\lambda(S,H,G)$ is an integral power of $h_H(G)$, and that the limit above stabilizes for sufficiently large $n$. We next establish the final statement of Theorem \[T:calc\]. Since any irreducible $F$-group factor of $R$ is in $\bar{S}$, Corollary \[C:nonewsimple\] shows that it is a power of a simple group in the closure of $S$ under taking subgroups and quotients. Lemma \[L:HfactR\] shows that the power is bounded by $\|H\|,$ showing the final statement of Theorem \[T:calc\]. To establish Equation , it will suffice to take the limit of a factor in Equation corresponding to a single $G$ (since there are only finitely many $G$ with non-trivial factors, independent of $n$, by Lemma \[L:R-cfp\] and Corollary \[C:CfpFin\]). The factor in Equation for a $G\in\mathcal{A}_H$ is $$\prod_{k=0}^{m(S,n,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{n+u}}})=\prod_{i=1}^{m(S,n,H,G)} (1-\frac{h_H(G)^{m(S,n,H,G)} h_H(G)^{-i}}{ \|G\|^{{n+u}}}).$$ If there are no extensions $E$ in the sum in Corollary \[C:finalmab\], then $m(S,n,H,G)$ and $\lambda(S,H,G)$ are $0$. Otherwise $\lambda(S,H,G)>0$, and thus it follows from Equation that $m(S,n,H,G){\rightarrow}\infty$ as $n{\rightarrow}\infty$. So using Lemma \[L:seq\] and Equation , we obtain the limit in Equation for a single factor $G\in\mathcal{A}_H$. In a similar but simpler fashion, from Equation , we obtain the limit in Equation for a single factor $G\in\mathcal{N}$. This completes the proof of Theorem \[T:calc\]. Countably additivity of $\mu_u$ {#S:cntadd} =============================== The goal of this section is to prove Theorem \[T:countadd\] which states that $\mu_u$ defined in Equation is countably additive on the algebra $\mathcal{A}$. It then follows from Carathéodory’s extension theorem that $\mu_u$ can be uniquely extended to a measure on the Borel sets of $\mathcal{P}$. The heart the proof of Theorem \[T:countadd\] is Theorem \[T:totall\]. We will first prove Theorem \[T:totall\] in Section \[SS:proof-totall\], and then prove Theorem \[T:countadd\] in Section \[SS:countadd\]. \[T:countadd\] Let $u$ be an integer. Then $\mu_u$ is countably additive on the algebra $\mathcal{A}$ generated by the $U_{S,H}$ for $S$ a finite set of finite groups and $H$ a finite group. \[T:totall\] Let $\ell$ be a positive integer. Recall that $S_{\ell}$ is defined to be the set consisting of all groups of order less than or equal to $\ell$. For a non-negative integer $u$, we have $$\sum_{H \text{ is finite and level }\ell} \mu_u (U_{S_{\ell}, H}) = 1.$$ Proof of Theorem \[T:totall\] {#SS:proof-totall} ----------------------------- Assume $\ell$ is a positive integer, $H$ is a finite level $\ell$ group and $\tilde{H}=H^{{\bar{S}}_{\ell-1}}$ . In Lemmas \[L:ab-con\] and \[L:nab-con\], we will first give upper bounds for the number of irreducible factors $G$ with non-zero $m(S_{\ell}, n, H, G)$ for some $n$ that are isomorphic to a given underlying group $M$. \[L:ab-con\] Suppose $M$ is a direct product of isomorphic abelian simple groups. Then $$\#\left\{ G\in \mathcal{A}_H \,\Bigg\|\, \begin{aligned} & G\simeq M \text{ and }\\ & m(S_\ell, n, H, G)\neq 0\text{ for some }n \end{aligned} \right\} \leq \sum_{(M,A)\in {\mathcal{CF}}({\bar{S}}_{\ell})} \|{\operatorname{Sur}}(\widetilde{H}, A) \|,$$ where the notation on the right-hand side above means that the sum is taken over all chief factor pairs in ${\mathcal{CF}}({\bar{S}}_\ell)$ whose first components are isomorphic to $M$ as groups. We give an injection $$\left\{ G\in \mathcal{A}_H, \phi \,\Bigg\|\, \begin{aligned} & \phi: G\simeq M \text{ and }\\ & m(S_\ell, n, H, G)\neq 0\text{ for some }n \end{aligned} \right\} {\rightarrow}\{(M,A)\in {\mathcal{CF}}({\bar{S}}_{\ell}),\pi \,\|\, \pi\in{\operatorname{Sur}}(H, A)\}.$$ Consider $G\in \mathcal{A}_H$ and $\phi:G\simeq M$ such that $m(S_\ell,n,H,G)\neq 0$ for some $n$. Assume $$1 {\rightarrow}R {\rightarrow}F {\rightarrow}H {\rightarrow}1$$ is the fundamental short exact sequence associated to $S_\ell,$ $n,$ and $H$. Then $G$ appears as a factor in $R$ and $(G, \rho_{G}(F))\in {\mathcal{CF}}({\bar{S}}_{\ell})$. Using $\phi:G\simeq M$, we have that the quotient $\rho_{G}(F)$ of $H$ acts on $M$, and so $(M, \rho_{G}(F))\in {\mathcal{CF}}({\bar{S}}_{\ell}).$ We let $\pi$ be the quotient map from $H$ to $\rho_{G}(F)$. Given $(M,A)\in {\mathcal{CF}}({\bar{S}}_{\ell})$ and $\pi\in{\operatorname{Sur}}(H, A)$, we can use $\pi$ to give $M$ the structure of an irreducible $H$-group, and let $\phi$ be the identity. This recovers $G$ and $\phi$, though possibly without $m(S_\ell, n, H, G)\neq 0$. By Corollary \[C:CfpFin\], if $(M,A)\in {\mathcal{CF}}({\bar{S}}_\ell)$, then $A/{\operatorname{Inn}}(M)\simeq A$ is a group of level $\ell-1$. Then by the definition of pro-${\bar{S}}$ completion, ${\operatorname{Sur}}(H, A)$ is one-to-one corresponding to ${\operatorname{Sur}}(\widetilde{H}, A)$ and we finish the proof. Similarly, for non-abelian irreducible factors, we have the following lemma. \[L:nab-con\] Suppose $M$ is a direct product of isomorphic non-abelian simple groups. Then $$\begin{aligned} && \# \left\{ \text{isom. classes of $H$-extensions } (E,\pi)\, \Bigg\|\, \begin{aligned} & \ker\pi \simeq M \text{ is irred. $E$-group}\\ & E \text{ is level }\ell \end{aligned} \right\} \\ &\leq& \sum_{(M,A)\in {\mathcal{CF}}({\bar{S}}_\ell)} \|{\operatorname{Sur}}(\widetilde{H}, A/{\operatorname{Inn}}(M))\|.\end{aligned}$$ We give an injection $$\begin{aligned} && \left\{ \text{isom. classes of $H$-extensions } (E,\pi)\, \Bigg\|\, \begin{aligned} & \ker\pi \simeq M \text{ is irred. $E$-group}\\ & E \text{ is level }\ell \end{aligned} \right\} \\ &\to& \{(M,A)\in {\mathcal{CF}}({\bar{S}}_{\ell}), \phi \mid \phi \in {\operatorname{Sur}}(H,A/{\operatorname{Inn}}(M))\}.\end{aligned}$$ Consider an isomorphism class of $H$-extension $(E,\pi)$ such that $\ker\pi\simeq M$ is an irreducible $E$-group and $E$ is level $\ell$. Then $(\ker \pi, \rho_{\ker\pi}(E)) \in {\mathcal{CF}}({\bar{S}}_{\ell})$, and $\rho_{\ker\pi}$ induces a surjection $\phi: H\to \rho_{\ker \pi}(E)/{\operatorname{Inn}}(M)$ since $\rho_{\ker \pi}$ is an isomorphism when restricted on $\ker\pi$ that maps $\ker\pi$ to ${\operatorname{Inn}}(M)$. Suppose $(M,A)\in {\mathcal{CF}}({\bar{S}}_{\ell})$ and $\phi\in {\operatorname{Sur}}(H, A/{\operatorname{Inn}}(M))$. If two $H$-extensions $(E_1, \pi_1)$ and $(E_2, \pi_2)$ both map to $(M,A)$ and $\phi$, then from the diagram below we see that $E_1$ and $E_2$ are both the fiber product of $\phi$ and $A\to A/{\operatorname{Inn}}(M)$, so $(E_1,\pi_1)$ and $(E_2, \pi_2)$ are isomorphic as $H$-extensions. Therefore, the map defined at the begin of the proof is an injection. Then the lemma follows as ${\operatorname{Sur}}(H,A)$ is one-to-one corresponding to ${\operatorname{Sur}}(\widetilde{H},A)$. E\_2 & &\ & E\_1 & H\ & A & A/(M) Let $P_{u,n}(U_{S_\ell, H})$ denote the product in Equation , i.e. $$P_{u,n}(U_{S_\ell,H}) = \prod_{G\in \mathcal{A}_H} \prod_{k=0}^{m(S_\ell, n, H, G)-1} (1-\frac{h_H(G)^k}{\|G\|^{n+u}}) \prod_{G\in \mathcal{N}}(1-\|G\|^{-n-u})^{m(S_\ell, n, H,G)}.$$ \[L:P-Const\] Suppose $\ell>1$ ,$n\geq 1$ and $u> -n$ are integers and $\widetilde{H}$ is a finite level $\ell-1$ group. Then there exists a non-zero constant $c(u,\ell,\widetilde{H})$ depending on $u, \ell$ and $\widetilde{H}$ such that, for every finite level $\ell$ group $H$ with $H^{{\bar{S}}_{\ell-1}}=\widetilde{H}$, either $P_{u,n}(U_{S_\ell,H})\geq c(u,\ell,\widetilde{H})$ or $P_{u,n}(U_{S_\ell,H})=0$ . For each $G\in \mathcal{A}_H$, $G$ is a direct product of isomorphic abelian simple groups, i.e. $G$ is a direct product of ${\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$ for some prime $p$. By Remark \[R:Gpow\], $h_H(G)$ is a power of $p$. Note that both of the trivial map (every element maps to 1) and the identity map of $G$ respect the $H$-action, therefore $h_H(G)>1$ and the product $\prod_{k=0}^{m(S_\ell, n, H, G)-1}(1-\frac{h_H(G)^k}{\|G\|^{n+u}})$ is either 0 or greater than $\prod_{k=1}^{\infty}(1-p^{-k}) \geq \prod_{k=1}^{\infty}(1-2^{-k})$. If $P_{u,n}(U_{S_\ell,H})\neq 0$, then $$\begin{aligned} \prod_{G \in \mathcal{A}_H} \prod_{k=0}^{m(S_\ell, n, H, G)-1}(1-\frac{h_H(G)^k}{\|G\|^{n+u}}) &=& \prod_{\substack{G\in \mathcal{A}_H \text{ and }\\ m(S_\ell, n, H, G)\neq 0}} \prod_{k=0}^{m(S_\ell, n, H, G)-1}(1-\frac{h_H(G)^k}{\|G\|^{n+u}})\\ &\geq& \prod_{\substack{G\in \mathcal{A}_H \text{ and }\\ m(S_\ell, n, H, G)\neq 0 \text{ for some }n}} \prod_{k=1}^{\infty} (1-2^{-k})\\ &=& \left[\prod_{k=1}^{\infty}(1-2^{-k})\right]^{\#\{G\in\mathcal{A}_H \mid\, m(S_\ell, n, H, G)\neq 0 \text{ for some }n\}}\\ &\geq& \left[\prod_{k=1}^{\infty}(1-2^{-k})\right]^{\sum\limits_{\substack{(M,A)\in {\mathcal{CF}}({\bar{S}}_\ell)\\M \text{ is abelian}}} \|{\operatorname{Sur}}(\widetilde{H}, A)\|},\end{aligned}$$ where the last inequality follows from Lemma \[L:ab-con\]. Therefore, if $P_{u,n}(U_{S_\ell,H})$ is non-zero, then its abelian part has a lower bound depending only on $\ell$ and $\widetilde{H}$. Similarly, for the non-abelian part, we consider $$\begin{aligned} \prod_{G \in \mathcal{N}} (1-\|G\|^{-n-u})^{m(S_\ell,n,H,G)} &\geq& \prod_{G\in \mathcal{N}} \left[(1-\frac{1}{2})^2\right]^{\frac{m(S_\ell,n,H,G)}{\|G\|^{n+u}}}\\ &=& \left[(1-\frac{1}{2})^2\right]^{\sum\limits_{G\in \mathcal{N}}\frac{m(S_\ell,n,H,G)}{\|G\|^{n+u}}}.\end{aligned}$$ By Corollary \[C:finalmnon\], we have $$\begin{aligned} &&\sum_{G\in \mathcal{N}} \frac{m(S_\ell,n,H,G)}{\|G\|^{n+u}}\\ &=& \sum_{G\in \mathcal{N}} \|G\|^{-u}\left(\sum_{\substack{\text{isom. classes of }H\text{-extensions }(E,\pi) \\ \ker\pi \text{ isom. $G$ }\\ \ker \pi \text{ irred. $E$-group}\\ E \text{ is level }\ell}} \frac{\|G\|^{-n}\sum_{D\in \mathcal{E}_H, D\leq E} \nu(D,E)\frac{\|D\|}{\|H\|}^n}{\|{\operatorname{Aut}}_H(E,\pi)\|} \right) \\ &\leq& \sum_{G \in \mathcal{N}} \|G\|^{-u} \# \left\{ \text{isom. classes of $H$-extensions } (E,\pi) \,\Bigg\|\, \begin{aligned} & \ker\pi \simeq G \text{ is irred. $E$-group}\\ & E \text{ is level }\ell \end{aligned} \right\} \\ &\leq& \sum_{G \in \mathcal{N}} \|G\|^{-u} \left(\sum_{(G,A)\in {\mathcal{CF}}({\bar{S}}_\ell)} \|{\operatorname{Sur}}(\widetilde{H}, A/{\operatorname{Inn}}(G))\| \right)\\ &=& \sum_{\substack{(G,A)\in {\mathcal{CF}}({\bar{S}}_\ell)\\ G \text{ non-abelian}}} \|G\|^{-u} \|{\operatorname{Sur}}(\widetilde{H}, A/{\operatorname{Inn}}(G))\|.\end{aligned}$$ The first inequality above follows from the fact that $\|{\operatorname{Sur}}(\rho, \pi)\|$ in Proposition \[P:countfromF\] is less than or equal to $\|G\|^n$. The second inequality follows by Lemma \[L:nab-con\]. It shows that the non-abelian part also has a lower bound depending on $u, \ell$ and $\widetilde{H}$. By Corollary \[C:CfpFin\], ${\mathcal{CF}}({\bar{S}}_\ell)$ is a finite set, so these lower bounds for abelian part and non-abelian parts are both non-zero. Then we proved the theorem. Now, we establish the inductive step that is crucial in the proof of Theorem \[T:totall\]. \[L:l-1tol\] Let $\ell>1$, $n\geq 1$, $u> -n$ be integers, and $\widetilde{H}$ be a finite level $\ell-1$ group. Then $$\lim_{n\to \infty} \sum_{\substack{H \text{ is finite level }\ell\\ \text{s.t. }\widetilde{H}=H^{{\bar{S}}_{\ell-1}}}} \mu_{u,n}(U_{S_\ell,H})=\sum_{\substack{H \text{ is finite level }\ell\\ \text{s.t. }\widetilde{H}=H^{{\bar{S}}_{\ell-1}}}}\mu_u (U_{S_\ell,H}).$$ Assume $H$ is finite and level $\ell$ such that $\widetilde{H}=H^{{\bar{S}}_{\ell-1}}$. Then either $\mu_{u,n}(U_{S_\ell,H})=0$ or $$\begin{aligned} \mu_{u,n}(U_{S_\ell,H}) &=& \frac{\|{\operatorname{Sur}}(\hat{F}_n,H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{n+u}} P_{u,n}(U_{S_\ell,H})\\ &\geq& \frac{1}{2}c(u, \ell, \widetilde{H}) \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u}\end{aligned}$$ for $n>i(H)$, where $i(H)$ is the smallest integer such that $\|{\operatorname{Sur}}(\hat{F}_n, H)\|/\|H\|^n\geq \frac{1}{2}$ for all $n>i(H)$ (note that $i(H)$ is finite since $\lim_{n\to \infty}\|{\operatorname{Sur}}(\hat{F}_n, H)\|/\|H\|^n =1$). Let’s call $H$ achievable if it is finite level $\ell$ and there exists $n$ such that $\mu_{u,n}(U_{S_\ell, H})\neq 0$ (we will give an equivalent definition in Section \[S:prob0\]). The function $\mu_{u,n}(U_{S_\ell,H})$ of $H$ is dominated by the function of $H$ that is $\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u}$ when $H$ is achievable and $0$ otherwise. We have $$\begin{aligned} \sum_{\substack{H \text{ is achievable} \\ \text{ s.t. } \widetilde{H}\simeq H^{{\bar{S}}_{\ell-1}}}} \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u} &=& \lim_{n\to\infty}\sum_{\substack{H \text{ is achievable} \\ \text{ s.t. } \widetilde{H}\simeq H^{{\bar{S}}_{\ell-1}}\\ \text{and } i(H)<n}} \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u}\\ &\leq& \lim_{n\to \infty}\frac{2}{c(u,\ell, \widetilde{H})} \sum_{\substack{H \text{ is achievable} \\ \text{ s.t. } \widetilde{H}\simeq H^{{\bar{S}}_{\ell-1}}\\ \text{and } i(H)<n}} \mu_{u,n}(U_{S_\ell,H})\\ &\leq& \frac{2}{c(u,\ell, \widetilde{H})}.\end{aligned}$$ Thus by Lebesgue’s Dominated Convergence Theorem, we have $$\begin{aligned} \lim_{n\to \infty} \sum_{\substack{H \text{ is finite level }\ell \\ \text{ s.t. } \widetilde{H}\simeq H^{{\bar{S}}_{\ell-1}}}} \mu_{u,n}(U_{S_\ell,H}) &=& \sum_{\substack{H \text{ is finite level }\ell \\ \text{ s.t. } \widetilde{H}\simeq H^{{\bar{S}}_{\ell-1}}}} \lim_{n\to \infty} \mu_{u,n}(U_{S_\ell,H}),\end{aligned}$$ which completes the lemma. We proceed by induction on $\ell$. When $\ell=1$, note that the trivial group is the only group that is finite level 1 and it’s obvious that $\mu_u(U_{S_1,1})=1$. Assume the theorem is true for $\ell-1$, i.e. $$\sum\limits_{\widetilde{H} \text{ is finite level }\ell-1} \mu_u(U_{S_{\ell-1},\widetilde{H}})=1.$$ We see that for any finite level $\ell-1$ group $\widetilde{H}$ $$\begin{aligned} \mu_u(U_{S_{\ell-1}, \widetilde{H}}) &=& \lim_{n\to \infty} \mu_{u,n}(U_{S_{\ell-1},\widetilde{H}}) \\ &=& \lim_{n\to \infty} \sum_{\substack{H \text{ is finite level }\ell \\ \text{s.t. }\widetilde{H}=H^{{\bar{S}}_{\ell-1}}}} \mu_{u,n}(U_{S_\ell,H})\\ &=& \sum_{\substack{H \text{ is finite level }\ell \\ \text{s.t. }\widetilde{H}=H^{{\bar{S}}_{\ell-1}}}} \mu_u(U_{S_\ell,H}), \end{aligned}$$ where the second equality above follows from the definition of $\mu_{u,n}$ on basic open sets and the last step follows from Lemma \[L:l-1tol\]. Therefore, we finish the proof by $$\begin{aligned} \sum_{H \text{ is finite level }\ell} \mu_u(U_{S_\ell,H}) &=& \sum_{\widetilde{H} \text{ is finite level }\ell-1} \sum_{\substack{H \text{ is finite level }\ell \\ \text{s.t. }\widetilde{H}=H^{{\bar{S}}_{\ell-1}}}} \mu_u(U_{S_\ell,H})\\ &=& \sum_{\widetilde{H} \text{ is finite level }\ell-1} \mu_u(U_{S_{\ell-1},\widetilde{H}})\\ &=& 1. \end{aligned}$$ Proof of Theorem \[T:countadd\] {#SS:countadd} ------------------------------- We will use the following corollary of Theorem \[T:totall\]. \[C:addB\] Let $\ell$ be a positive integer, and $B=\cup_{j=1}^{\infty} U_{S_\ell,H_j}$ for some finite groups $H_j$ such that $U_{S_\ell,H_j}\ne U_{S_\ell,H_{j'}}$ for $j\ne j'$. Suppose that $B\in \mathcal{A}$, the algebra of sets generated by the basic open sets $U_{S,G}$ for a finite set $S$ of finite groups and a finite level $S$ group $G$. Let $u$ be an integer. Then $\mu_u(B)=\sum_{j=1}^\infty \mu_u(U_{S_\ell,H_j})$. Let $G_j$ be the level $\ell$ finite groups not among the $H_j$. Then for every positive integer $M$, we have $$\sum_{j=1}^M \mu_u(U_{S_\ell,H_j}) \leq \mu_u(B) \leq 1-\sum_{j=1}^M \mu_u(U_{S_{\ell},G_j}).$$ Taking limits as $M{\rightarrow}\infty$ gives $$\sum_{j=1}^\infty \mu_u(U_{S_\ell,H_j}) \leq \mu_u(B) \leq 1-\sum_{j=1}^\infty \mu_u(U_{S_\ell,G_j})=\sum_{j=1}^\infty \mu_u(U_{S_\ell,H_j}),$$ where the last equality is by Theorem \[T:totall\]. Since $\mu_u(A)$ is defined as a limit of measures $\mu_{u,n}(A)$, it is immediate that $\mu_u$ is finitely additive because finite sums can be exchanged with the limit. If we have disjoint sets $A_n \in\mathcal{A}$ with $A=\cup_{n\geq 1 } A_n \in \mathcal{A}$, by taking $B_n=A\setminus \cup_{j=1}^n A_j$, it suffices to show that for $B_1 \supset B_2 \supset \dots$ (with $B_n\in \mathcal{A}$) with $\cap_{n\geq 1} B_n=\emptyset$ we have $\lim_{n{\rightarrow}\infty} \mu_u(B_n)=0$. We can assume, without loss of generality, that for each $\ell\geq 1$, we have $B_\ell=\cup_{j} U_{S_\ell,G_{\ell,j}}$ (i.e. $B_\ell$ is defined at level $\ell$). (Note that when all groups in $S$ have order at most $m$ that $U_{S,H}$ is a union of sets of the form $U_{S_m,G}$ for varying $G$. We can always insert redundant $B_i$’s if the level required to define the $B_\ell$ increase quickly.) We will show by contradiction that $\lim_{\ell {\rightarrow}\infty} \mu_u(B_\ell)=0$. Suppose, instead that there is an $\epsilon>0$ such that for all $\ell$, we have $\mu_u(B_\ell)\geq \epsilon$. It follows from Corollary \[C:addB\] that for each $\ell$ we have a subset $K_\ell{\subset}B_\ell$ such that $\mu_u(B_\ell\setminus K_\ell)<\epsilon/2^{\ell+1}$ and $K_\ell$ is a finite union of $U_{S_\ell,G_{\ell,j}}$. Next, let $C_\ell=\cap_{j=1}^\ell K_j$. Then $\mu_u(B_\ell\setminus C_\ell)<\epsilon/2$, since $$\begin{aligned} \mu_u(B_\ell\setminus C_\ell)=& \mu_u(B_\ell\setminus K_\ell)+\mu_u(K_\ell\setminus K_\ell \cap K_{\ell-1})+\cdots +\mu_u(K_\ell \cap \cdots \cap K_2 \setminus K_\ell \cap \cdots \cap K_1) \\ < &\epsilon/2^{\ell+1} + \mu_u(B_{\ell-1} \setminus K_{\ell-1}) + \cdots +\mu_u(B_1\setminus K_{1}) \\ < &\epsilon/2^{\ell+1} + \epsilon/2^{\ell} + \cdots +\epsilon/2^{2}.\end{aligned}$$ So $\mu_u(C_\ell)\geq \epsilon/2$ for each $\ell$ and in particular it is non-empty. Note $C_{\ell+1}{\subset}C_\ell$ for all $\ell$. Pick $x_\ell\in C_\ell$ for all $\ell$. Note $C_\ell$ is defined at level $\ell$ and a finite union of the basic open sets $U_{S_\ell,G_{\ell,j}}$. Pick an $H_1$ so that infinitely many of the $x_\ell$ are in $U_{S_1,H_1}$ (this is possible since all $x_\ell$ are in $C_1$ and there are only finitely many $U_{S_1,H}$ that make up $C_1$), and then disregard the $x_\ell$ that are not in $U_{S_1,H_1}$. In particular note $U_{S_1,H_1}\subset C_1$. Then pick $H_2$ so that infinitely many of the remaining $x_\ell$ are in $U_{S_2,H_2}$, and disregard the $x_\ell$ that are not. Since all of the remaining $x_\ell$ are in $U_{S_1,H_1}$, we have $U_{S_2,H_2}{\subset}U_{S_1,H_1}$. Also note $U_{S_2,H_2}{\subset}C_2$. We continue this process and then consider the profinite group $H$ that is the inverse limit of the $H_i$’s. Since $H\in U_{S_\ell,H_\ell} {\subset}C_\ell {\subset}B_\ell$ for all $\ell$, we have a point $H\in \cap_{\ell\geq 1} B_\ell $ which is a contradiction. Proof of Theorem \[T:Main\] {#S:mainproof} =========================== The last section established the existence of the probability measure $\mu_u$ on Borel sets of $\mathcal{P}$. Now we are able to give the proof of Theorem \[T:Main\], the weak convergence of the $\mu_{u,n}$ to $\mu_u$. Note that the weak convergence $\mu_{u,n} \Rightarrow \mu_u$ is equivalent to that $$\liminf_{n\to\infty} \mu_{u,n} (U) \geq \mu_u(U)$$ for all open sets $U$. In the topological space $\mathcal{P}$, every open set is a countable disjoint union of basic open sets. Assume $U=\cup_{i\geq 1} U_i$ is an open set, where $U_i$ are disjoint basic open sets. By Fatou’s lemma, we have $$\mu_u(U) = \sum_{i\geq 1} \mu_u(U_i) = \sum_{i\geq 1} \lim_{n\to \infty} \mu_{u,n} (U_i) \leq \liminf_{n\to \infty }\sum_{i\geq 1} \mu_{u,n}(U_i) = \liminf_{n\to \infty} \mu_{u,n}(U).$$ For arbitrary set $S$ {#S:arbitraryS} ===================== In this section, we let $S$ be an arbitrary (not necessarily finite) set of finite groups and consider the value of $\mu_u$ on the specific type of Borel sets $$V_{S,H}:=\{X \in\mathcal{P} \mid X^{{\bar{S}}}\simeq H\}$$ for a finite level $S$ group $H$. We will first prove an analogue of Theorem \[T:calc\] for an arbitrary set $S$ (see Theorem \[T:calcinf\]), the proof of which shows that Equation gives the value $\mu_{u}(V_{S,H})$. Note that $V_{S,H}$ is not a basic open set, but is the intersection of a sequence of basic open sets. Since we will approximate $S$ by increasing finite subsets, we need the following lemma. \[L:mincinS\] Consider two sets $T{\subset}T'$ of finite groups. For any positive integer $n$, finite group $H$ of level $T$, and $G\in \mathcal{A}_H\cup \mathcal{N}$, we have $m(T,n,H,G)\leq m(T',n,H,G).$ Also if $T_1{\subset}T_2{\subset}\cdots$ are finite sets of finite groups, then $m(T_m,n,H,G)$ eventually stabilizes as $m{\rightarrow}\infty$. Consider the case when $G$ is abelian. Let $\rho: {\hat{F}}^{\bar{T}_m} {\rightarrow}H$ be a surjection. Corollary \[C:finalmab\] and Proposition \[P:countfromF\] give $$\frac{h_H(G)^{m(T_m,n,H,G)}-1}{h_H(G)-1} =\sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ as an $H$-group} \\ \textrm{$E$ is level $T_m$} }} \frac{\|{\operatorname{Sur}}(\rho,\pi)\|}{\|{\operatorname{Aut}}_H(E,\pi)\|}.$$ The right-hand side is clearly non-decreasing in $m$. There are only finitely many isomorphism classes of $H$-extensions whose kernel is isomorphic to $G$, which proves the stabilization. The case of non-abelian $G$ is similar. Let $S$ be a set of finite groups, $n$ a positive integer, and $H$ a finite level $S$ group. Let $T_1\subset T_2 \subset \cdots$ be finite sets of finite groups such that $\cup_{m\geq 1}T_m=S$. For any $G\in \mathcal{A}_H \cup \mathcal{N}$, we define $m(S,n,H,G)= \lim_{m\to \infty} m(T_m, n, H, G).$ It’s clear that $m(S,n,H,G)$ does not depend on the choice of the increasing sequence $T_i$, and $m(S,n,H,G)$ is always a non-negative integer. It is actually easier to determine $\mu_u(V_{S,H})$, as we will in the next lemma, than to find $\lim_{n{\rightarrow}\infty} \mu_{u,n}(V_{S,H}),$ which we will do in Theorem \[T:calcinf\]. \[L:wrongorder\] Let $S$ be a set of finite groups. Let $T_1{\subset}T_2{\subset}\cdots$ be finite sets of finite groups such that $\cup_{m\geq 1} T_m=S$. Let $H$ be a finite group of level $S$. Let $u$ be an integer. Then $$\begin{aligned} \mu_u(V_{S,H}) &=&\lim_{m{\rightarrow}\infty} \lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H)\\ &=&\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(S,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(S,H,G)}.\end{aligned}$$ First of all, since $\mu_u$ is a measure, we have $$\mu_u(V_{S,H})=\mu_u( \cap_{m\geq 1} U_{T_m,H^{{\bar{T}}_m}})=\lim_{m\to \infty} \mu_u (U_{T_m, H^{{\bar{T}}_m}}) = \lim_{m\to \infty} \lim_{n \to \infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m} \simeq H).$$ By definition, we have that $\lambda(T,H,G)$ is non-decreasing in $T$, i.e. if $T{\subset}T'$ then $\lambda(T,H,G)\leq \lambda(T',H,G)$. Further, again by definition, we have $$\lambda(S,H,G) =\lim_{m{\rightarrow}\infty} \lambda(T_m,H,G).$$ When $m$ is sufficiently large such that $H$ is level $T_m$, we have $$\begin{aligned} &&\lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H)\\ &=& \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(T_m,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(T_m,H,G)}\end{aligned}$$ by Equation since $T_m$ is finite. For each $G\in \mathcal{A}_H$, the factor $$\prod_{i=1}^{\infty} (1-\lambda(T_m,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}})$$ is a limit of terms $$\prod_{i=1}^{m(T_m,n,H,G)} (1-\frac{h_H(G)^{m(T_m,n,H,G)} h_H(G)^{-i}}{ \|G\|^{{n+u}}}),$$ by Lemma \[L:seq\], each of which is a probability and in the interval $[0,1]$. Since the factors $$\prod_{i=1}^{\infty} (1-\lambda(T_m,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \quad \quad \textrm{and} \quad \quad e^{-\|G\|^{-u}\lambda(T_m,H,G)}$$ are all in $[0,1]$ and are non-increasing in $m$, we have the second equality in the following $$\begin{aligned} & &\lim_{m{\rightarrow}\infty}\lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H)\\ &=& \lim_{m{\rightarrow}\infty} \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(T_m,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(T_m,H,G)}\\ &=& \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \lim_{m{\rightarrow}\infty} \prod_{i=1}^{\infty} (1-\lambda(T_m,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} \lim_{m{\rightarrow}\infty} e^{-\|G\|^{-u}\lambda(T_m,H,G)}\\ &=& \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(S,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(S,H,G)}.\end{aligned}$$ The last equality uses the continuity from Lemma \[L:seq\]. \[T:calcinf\] The statement in Theorem \[T:calc\] also works for an arbitrary set $S$ of finite groups. Let $T_m$ be the subset of $S$ of all groups of order at most $m$ in $S$. Since $H$ is level $S$, for large enough $m$ we have that $ H^{{\bar{T}}_m}=H$, and from now on we only consider $m$ this large. We can show that $G^{{\bar{S}}}{\simeq}H$ if and only if for every $m\geq 1$ we have $G^{{\bar{T}}_m}{\simeq}H^{{\bar{T}}_m}.$ Since $G^{\bar{T}_m}$ is a quotient of $G^{\bar{S}}$, the “only if” direction is clear. If we take the inverse limit of the sets $\operatorname{Isom}(G^{\bar{T}_m},H^{\bar{T}_m})$, with the natural maps, we have an inverse limit of non-empty finite sets, which is non-empty. An element of this inverse limit gives us an isomorphism $G^{\bar{S}}{\simeq}H^{\bar{S}}$. From this, and the basic properties of a measure, and Equation for finite $S$, we have that $$\begin{aligned} &&{\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\\ &=& \lim_{m{\rightarrow}\infty} \frac{\|{\operatorname{Sur}}(\hat{F}_n,H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{n+u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{k=0}^{m(T_m,n,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{n+u}}}) \prod_{\substack{ G\in \mathcal{N}}} (1-\|G\|^{-{n-u}})^{m(T_m,n,H,G)}.\end{aligned}$$ From Lemma \[L:mincinS\], we have that $$\prod_{k=0}^{m(T_m,n,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{n+u}}}) \quad \quad \textrm{and} \quad \quad (1-\|G\|^{-{n-u}})^{m(T_m,n,H,G)}$$ are non-increasing in $m$, and as they are probabilities they are in the interval $[0,1]$. Thus it follow from basic analysis that $$\begin{aligned} & {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\\= & \frac{\|{\operatorname{Sur}}(\hat{F}_n,H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{n+u}}\prod_{\substack{G \in \mathcal{A}_H}} \lim_{m{\rightarrow}\infty} \prod_{k=0}^{m(T_m,n,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{n+u}}}) \prod_{\substack{ G\in \mathcal{N}}} \lim_{m{\rightarrow}\infty} (1-\|G\|^{-{n-u}})^{m(T_m,n,H,G)}\end{aligned}$$ By definition of $m(S,n,H,G)$, we have that $\lim_{m{\rightarrow}\infty} m(T_m,n,H,G)=m(S,n,H,G)$ (and the latter is finite). Thus, we have $$\begin{aligned} & {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\\= & \frac{\|{\operatorname{Sur}}(\hat{F}_n,H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{n+u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{k=0}^{m(S,n,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{n+u}}}) \prod_{\substack{ G\in \mathcal{N}}} (1-\|G\|^{-{n-u}})^{m(S,n,H,G)},\end{aligned}$$ which is Equation for arbitrary $S$. Next, towards Equation for arbitrary $S$, we will show that the order of the limits in Lemma \[L:wrongorder\] could be exchanged. For every $m$, we have ${\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\leq {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H)$ and so $$\begin{aligned} &&\limsup_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)\notag\\ &\leq& \lim_{m{\rightarrow}\infty} \lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H)\notag\\ &=& \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(S,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(S,H,G)}.\label{E:doublim}\end{aligned}$$ From here we consider two cases. Case 1 will be the following: $$\sum_{G\in \mathcal{A}_H } \frac{\lambda(S,H,G)}{h_H(G)\|G\|^u} +\sum_{G\in \mathcal{N} } \frac{\lambda(S,H,G)}{\|G\|^u} \quad \textrm{diverges}.$$ In case $1$, the product in Equation is $0$, and we have proven $\lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H)=0$, establishing Equation . Case 2 will be the following: $$\sum_{G\in \mathcal{A}_H } \frac{\lambda(S,H,G)}{h_H(G)\|G\|^u} +\sum_{G\in \mathcal{N} } \frac{\lambda(S,H,G)}{\|G\|^u} \quad \textrm{converges}.$$ We define a minimal non-trivial $H$-extension $(E,\pi)$ to be an $H$-extension whose only quotient $H$-extensions are itself and the trivial one. These are exactly the $H$-extensions with $\ker \pi$ an irreducible $E$-group (under conjugation). Also, these are exactly the $H$-extensions $(E,\pi)$ such that $\ker \pi$ is an abelian irreducible $H$-group or $\ker \pi$ is a power of a non-abelian simple group and an irreducible $E$-group. Since $\|{\operatorname{Aut}}(E)\|\geq \|{\operatorname{Aut}}_H(E,\pi)\|$ and $h_H(G)\geq 2$, we have $$\sum_{G\in \mathcal{A}_H } \frac{\lambda(S,H,G)}{h_H(G)\|G\|^u} +\sum_{G\in \mathcal{N} } \frac{\lambda(S,H,G)}{\|G\|^u} \geq \frac{1}{2}\sum_{\substack{(E,\pi) \textrm{ min. non-triv. $H$-extension } \\ \textrm{$E$ level $S$}}}\|{\operatorname{Aut}}(E)\|^{-1} \|G\|^{-u}.$$ Since we are in case 2, the sum on the right converges, and $$\begin{aligned} &&\lim_{m{\rightarrow}\infty} \sum_{\substack{(E,\pi) \textrm{ min. non-triv. $H$-extension }\\ \textrm{$E$ level $S$, but not level $T_m$}}}\|{\operatorname{Aut}}(E)\|^{-1} \|E\|^{-u}\\ &=&\|H\|^{-u}\lim_{m{\rightarrow}\infty} \sum_{\substack{(E,\pi) \textrm{ min. non-triv. $H$-extension }\\ \textrm{$E$ level $S$, but not level $T_m$}}}\|{\operatorname{Aut}}(E)\|^{-1} \|G\|^{-u} \\ &=&0.\end{aligned}$$ If $(X_{u,n})^{{\bar{S}}}\not{\simeq}H,$ but $(X_{u,n})^{{\bar{T}}_m}{\simeq}H$ for some $m$, then $X_{u,n}$ has a surjection to $H$ and thus $X_{u,n}$ has a surjection to some minimal non-trivial $H$-extension $(E,\pi)$ of level $S$ but not level $T_m$. Note that $$\begin{aligned} {\ensuremath{{\mathbb{P}}}}(X_{u,n} \textrm{ has a surjection to $E$})&\leq {\ensuremath{{\mathbb{E}}}}(\textrm{quotients of $X_{u,n}$ isom. to $E$})\\ &= \|{\operatorname{Aut}}(E)\|^{-1} {\ensuremath{{\mathbb{E}}}}( \|{\operatorname{Sur}}(X_{u,n},E)\|)\\ &= \|{\operatorname{Aut}}(E)\|^{-1} \frac{\|{\operatorname{Sur}}({\hat{F}}_n,E)\|}{\|E\|^{n+u}}\\ &\leq \|{\operatorname{Aut}}(E)\|^{-1} \|E\|^{-u}.\end{aligned}$$ Thus, $$\begin{aligned} {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H) \geq {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H) -\sum_{\substack{(E,\pi) \textrm{ min. non-triv. $H$-extension }\\ \textrm{$E$ level $S$, but not level $T_m$}}}\|{\operatorname{Aut}}(E)\|^{-1} \|E\|^{-u} \end{aligned}$$ and $$\begin{aligned} &&\liminf_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{S}}}{\simeq}H) \\ &\geq& \lim_{n{\rightarrow}\infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}{\simeq}H) -\sum_{\substack{(E,\pi) \textrm{ min. non-triv. $H$-extension }\\ \textrm{$E$ level $S$, but not level $T_m$}}}\|{\operatorname{Aut}}(E)\|^{-1} \|E\|^{-u} \end{aligned}$$ Now we take a $\lim_{m{\rightarrow}\infty}$ of both sides and conclude Equation for arbitrary $S$. Finally, note that if $m(S,n,H,G)\ne 0$, then $m(T_m,n,H,G)\ne 0$ for some $m$, and so the last statement of Theorem \[T:calc\] for infinite $S$ follows from the same statement for finite $S$. Though this doesn’t follow from weak convergence (see Proposition \[P:alltrivial\] and Remark \[R:alltrivial\], for example), we see here that $\mu_u$ and $\lim_{n{\rightarrow}\infty} \mu_{u,n}$ agree on the $V_{S,H}$. \[C:coninfS\] Let $S$ be a set of finite groups and $H$ a finite level $S$ group. Then we have $$\lim_{n\to \infty}\mu_{u,n} (V_{S,H}) = \mu_u(V_{S,H}).$$ In the proof of Theorem \[T:calcinf\], we showed that $$\lim_{n\to \infty} \lim_{m\to \infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m} \simeq H) = \lim_{m \to \infty} \lim_{n \to \infty} {\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}\simeq H).$$ By Lemma \[L:wrongorder\], the right-hand side in the above equation is $\mu_u(V_{S,H})$. Also, since $\mu_{u,n}$ are measures on $\mathcal{P}$, we have $\lim_{m\to \infty}{\operatorname{Prob}}((X_{u,n})^{{\bar{T}}_m}\simeq H)= \mu_{u,n}(V_{S,H})$. Examples of the values of $\mu_u$ {#S:examples} ================================= In this section, we will apply Theorem \[T:calcinf\] to compute $\mu_u(A)$ for some interesting Borel sets $A$. Let $S$ contain every finite group. Then the trivial group is the only element in $V_{S,1}$. By Lemma \[L:HfactR\], if $(E,\pi)$ is an extension of the trivial group such that $\ker \pi$ is irreducible $E$-group, then $E$ is a finite simple group. Then it follows from the definition of $\lambda(S,H,G)$ that $$\lambda(S,1,G)=\begin{cases} 1 & G \text{ is an abelian simple group}\\ \|{\operatorname{Aut}}(G)\|^{-1} & G \text{ is a non-abelian simple group}\\ 0 & otherwise \end{cases}$$ By Theorem \[T:calcinf\], we have $$\mu_u(\text{trivial group})=\prod_{p\text{ prime}}\prod_{i=u+1}^{\infty}(1-p^{-i}) \prod_{\substack{G \text{ finite simple}\\ \text{non-abelian group}}} e^{-\|G\|^{-u}\|{\operatorname{Aut}}(G)\|^{-1}}.$$ The above product over prime integers is zero if and only if $u\leq 0$. When $u\geq 1$, by the classification of finite simple groups, the number of finite simple groups of given order is at most 2. Note that $\|{\operatorname{Aut}}(G)\|\geq \|{\operatorname{Inn}}(G)\|=\|G\|$ for every non-abelian simple group $G$. We have $$\prod_{\substack{G \text{ finite simple}\\ \text{non-abelian group}}} e^{-\|G\|^{-u}\|{\operatorname{Aut}}(G)\|^{-1}} \geq \exp (-\sum_{\substack{G \text{ finite simple}\\ \text{non-abelian group}}}\|G\|^{-u-1}) > 0,$$ which shows that $\mu_u(\text{trivial group})>0$ if and only if $u\geq 1$. By using the classification of finite simple groups, we are able to give the following approximations $$\mu_u(\text{trivial group})\approx\begin{cases} 0.4357 & \text{when }u=1\\ 0.7168 & \text{when }u=2\\ 0.8616 & \text{when }u=3. \end{cases}$$ We observe that the product over non-abelian factors is very close to 1 and cannot be seen in this many digits. Again let $S$ contain all finite groups. Let $H$ be an infinite profinite group in $\mathcal{P}$, and $H_{\ell}$ denote the pro-${\bar{S}}_{\ell}$ completion of $H$. Since $U_{S_\ell, H_{\ell}}$ is a sequence of basic opens that is decreasing in $\ell$ and $\cap_{\ell} U_{S_{\ell}, H_{\ell}} = \{H\}$, we obtain $$\begin{aligned} \mu_u(\{H\})&=& \lim_{\ell \to \infty} \mu_u(U_{S_\ell, H_\ell})\\ &\leq& \lim_{\ell\to \infty} \frac{1}{\|{\operatorname{Aut}}(H_\ell)\| \|H_\ell\|^u}.\end{aligned}$$ Note that $H$ is the inverse limit of $H_{\ell}$, so $\lim_{\ell\to \infty} \|H_\ell\| =\infty$. It follows that $\mu_u(\{H\})=0$ when $u\geq 1$. When $u=0$, since $\{H\}$ is contained in the Borel set $A:=V_{\{\text{all abelian groups}\}, H^{ab}}$ and $\mu_0(A)=0$ (see Example \[E:abelian\]), we have $\mu_0(\{H\})=0$. \[ex:abelian\] Let $p$ be a prime integer and $S$ the set consisting of all finite abelian $p$-groups. Then ${\bar{S}}=S$ and $({\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}, 1)$ is the only element in ${\mathcal{CF}}({\bar{S}})$. Let $H$ be a finite abelian $p$-group of generator rank $d$. Then for any $G\in\mathcal{A}_H\cup \mathcal{N}$, the factor in Theorem \[T:calc\] associated to $G$ is $1$, unless $G={\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$ with the trivial $H$-action. We consider the Borel set $V_{S,H}$ that is the set of all profinite groups whose maximal abelian pro-$p$ quotient is $H$. For any integer $n\geq d$ , there is a normal subgroup $N$ of $(\hat{F}_n)^{{\bar{S}}}=({\ensuremath{{\mathbb{Z}}}\xspace}_p)^n$ such that the corresponding quotient is $H$. Since $H$ is finite, $N$ is isomorphic to $({\ensuremath{{\mathbb{Z}}}\xspace}_p)^n$ with the trivial $(\hat{F}_n)^{{\bar{S}}}$-action, which shows that $m(S,n,H,{\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})=n$ and $\lambda(S,H,{\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})=1$. By Lemma \[L:wrongorder\], we have $$\mu_u(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u} \prod_{i=1}^{\infty}(1-p^{-i-u}).$$ When $u<0$, this probability is 0, which is as expected since we can never get a finite quotient of $({\ensuremath{{\mathbb{Z}}}\xspace}_p)^n$ with fewer than $n$ relators. When $u\geq 0$, we get a finite group with probability 1. When $u=0$ or $1$, these are the measures used in the Cohen-Lenstra heuristics for class groups of quadratic number fields. More generally, let’s consider an infinite abelian pro-$p$ group $H$ in $\mathcal{P}$. Since $H\in \mathcal{P}$, the pro-$\overline{\{{\ensuremath{{\mathbb{Z}}}\xspace}/2{\ensuremath{{\mathbb{Z}}}\xspace}\}}$ completion of $H$ is finite, so $H$ is finitely generated, i.e. $H=H_1 \times ({\ensuremath{{\mathbb{Z}}}\xspace}_p)^r$ for a finite abelian $p$-group $H_1$ and a positive integer $r$. Let $T_j:=\{{\ensuremath{{\mathbb{Z}}}\xspace}/p^j{\ensuremath{{\mathbb{Z}}}\xspace}\}$. So ${\bar{T}}_j$ is an increasing sequence and $\cup {\bar{T}}_j=S$. Assume $n\geq d$ and $j$ is greater than the exponent of $H_1$. Then we have $$(\hat{F}_n)^{{\bar{T}}_j}=({\ensuremath{{\mathbb{Z}}}\xspace}/p^j{\ensuremath{{\mathbb{Z}}}\xspace})^n \text{ and } H^{{\bar{T}}_j}=H_1 \times ({\ensuremath{{\mathbb{Z}}}\xspace}/p^j{\ensuremath{{\mathbb{Z}}}\xspace})^r.$$ So $m(T_j, n,H, {\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})=n-r$, $\lambda(T_j, H, {\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})=p^{-r}$ and $$\begin{aligned} \mu_u(V_{T_j,H}) &=& \frac{1}{\|{\operatorname{Aut}}(H^{{\bar{T}}_j})\|\|H_1\|^u p^{jru}} \prod_{i=1+u+r}^{\infty}(1-p^{-i})\\ &=&\frac{1}{\|{\operatorname{Aut}}(H_1)\|\|H_1\|^{2r+u}p^{jr(r+u)}} \prod_{i=1}^r(1-p^{-i})^{-1}\prod_{i=1+u+r}^{\infty}(1-p^{-i}),\end{aligned}$$ since $$\|{\operatorname{Aut}}(H^{{\bar{T}}_j})\| = \|{\operatorname{Aut}}(H_1)\|\|H_1\|^{2r} p^{jr^2} \prod_{i=1}^r (1- p^{-i}).$$ It follows that $\mu_u(V_{S,H})=\lim_{j\to \infty}(V_{T_j,H})> 0$ if and only if $u+r=0$, in which case $$\mu_u(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H_1)\|\|H_1\|^{-u}} \prod_{i=1-u}^{\infty} (1-p^{-i}).$$ So we see that when $u<0$, $\mu_u(V_{S, H})>0$ if and only if the (torsion-free) rank of $H$ is $-u$ and we get the groups in such form with probability 1. \[E:abelian\] Similar to the example above, when $S$ is the set of all finite abelian groups and $H$ is a finite abelian group, we have $$\mu_u(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u} \prod_{p \textrm{ prime}}\prod_{i=1}^{\infty}(1-p^{-i-u}),$$ which is $0$ if $u\leq 0$ and is positive if $u\geq 1$. If $H=H_1 \times (\hat{{\ensuremath{{\mathbb{Z}}}\xspace}})^r$, then $$\mu_u(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H_1)\|\|H_1\|^{-u}} \prod_{p \textrm{ prime}} \prod_{i=1-u}^{\infty} (1-p^{-i})>0$$ if $u=-r<0$ and $\mu_u(V_{S,H})=0$ otherwise. In order to consider the pro-$p$ quotients of our random groups, we will first need to recall the definitions of some $p$-group invariants. Let $H$ be a finite $p$-group of generator rank $d$. The relation rank of $H$ is defined to be $r(H):=\dim_{{\ensuremath{{\mathbb{F}}}}_p}\operatorname{H}^2(H, {\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})$. On the other hand, there is a normal subgroup $N$ of $\hat{F}_d$ such that $\hat{F}_d/N\simeq H$. Define $N^:=[N,\hat{F}_d]\cdot N^p$. Then $N^$ is the minimal $\hat{F}_d$-normal subgroup of $N$ such that $N/N^$ is a finite elementary abelian $p$-group with trivial $\hat{F}_d/N^$-action. Then $\hat{F}_d/N^$ is called the $p$-covering group* of $H$, and $N/N^$ is called the $p$-multiplicator* of $H$, and $\dim _{{\ensuremath{{\mathbb{F}}}}_p}(N/N^)$ is called the $p$-multiplicator rank* of $H$. One can show that $\operatorname{H}^2(H, {\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace})\simeq N/N^$, so $r(H)$ equals to the $p$-multiplicator rank of $H$. \[L:p-multi\] Let $H$ be a finite $p$-group of generator rank $d$, $S$ the set of all finite $p$-groups, and $G$ the $H$-group that is isomorphic to ${\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$ with trivial $H$-action. Then $m(S,d,H,G)=r(H)$ and $m(S,n,H,G)=r(H)+n-d$ for every $n\geq d$. Since the intersection of every normal subgroup and the center of a finite $p$-group is nontrivial, every finite $p$-group acts trivially on all of its minimal normal subgroups, which implies ${\mathcal{CF}}({\bar{S}})=\{({\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}, 1)\}$. Recall that $m(S,n,H,G)$ is defined to be $\lim_{i\to\infty} m(T_i, n, H, G)$, where $T_i$ is an increasing sequence of finite sets of groups such that $\cup T_i=S$. When $i$ is sufficiently large such that $T_i$ contains the $p$-covering group of $H$, the map $\rho:(\hat{F}_d)^{{\bar{T}}_i}\to H$ factors through the $p$-covering group of $H$. Let $1\to R \to F \to H \to 1$ be the fundamental short exact sequence associated to $T_i, d, H$. It’s not hard to check that $R$ is also the maximal quotient of $\ker \rho$ that is an elementary abelian $p$-group with the trivial $F$-action. Therefore, $R$ is the $p$-multiplicator of $H$ and $m(S,d,H,G)=m(T_i, d,H,G)=r(H)$. Assume $n\geq d$. We can find a surjection $\rho_1:\hat{F}_{n+1}\to H$ and generators $x_1, \cdots, x_{n+1}$ of $\hat{F}_{n+1}$ such that $\rho_1(x_{n+1})=1$. Let $\rho_2$ be the restriction of $\rho_1$ on the subgroup generated by $x_1, \cdots, x_n$. Then $\rho_2: \hat{F}_n\to H$ is a surjection. Let $1\to R_1 \to F_1 \overset{\pi_1}{\to} H \to 1$ and $1\to R_2 \to F_2 \overset{\pi_2}{\to} H \to 1$ be the fundamental short exact sequences associated to $T_i, n+1, H$ and $T_i, n, H$ that arise from $\rho_1$ and $\rho_2$ respectively. These constructions allow us to get a surjection $\pi: F_1 \to F_2$ with $\pi_1=\pi_2 \circ \pi$, and a generator set $y_1, \cdots, y_{n+1}$ of $F_1$ such that $\pi(y_{n+1})=1$. Since $y_{n+1}\in\ker \pi_1$ and $F_1$ acts trivially on $R_1$, the subgroup generated by $y_{n+1}$, which is isomorphic to ${\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$, is normal in $F_1$. It implies that $R_1\simeq R_2 \times {\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$ and $m(T_i, n+1, H, G)=m(T_i, n, H, G)+1$. By induction on $n$, we finish the proof of the lemma. Let $H$ be a finite $p$-group of generator rank $d$, and $S$ the set of all finite $p$-groups, and $G$ the $H$-group that is isomorphic to ${\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace}$ with trivial $H$-action. Since ${\mathcal{CF}}({\bar{S}})=({\ensuremath{{\mathbb{Z}}}\xspace}/p{\ensuremath{{\mathbb{Z}}}\xspace},1)$, we have $$\mu_{u}(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u} \prod_{i=1}^{\infty}(1-\frac{\lambda(S,H,G)}{p^{i+u}}).$$ By Equation and Lemma \[L:p-multi\], $\lambda(S,H,G)=p^{r(H)-d}$. So $$\mu_u(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u}\prod_{i=1+u-r(H)+d}^{\infty} (1-p^{-i}),$$ and $\mu_u(V_{S,H})>0$ if and only if $u\geq r(H)-d$. Given $u$, if $X_{n,u}^{{\bar{S}}}$ has generator rank $d$ with $d^2/4\geq d+u$, we have that the $X_{n,u}^{{\bar{S}}}$ is necessarily infinite by the Golod-Shafarevich inequality. We can see from the pro-$p$ abelianization that we get groups $X_{n,u}^{{\bar{S}}}$ with each generator rank $d\geq \min(0,-u)$ with positive probability. All groups in $\mathcal{P}$ have their pro-$p$ quotient finitely generated. When $S$ is the set of all finite nilpotent groups and $H$ is a finite nilpotent group with Sylow $p$-subgroup $H_p$ of generator rank $d_p$, we have $$\mu_u(V_{S,H})=\frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^u}\prod_{p \textrm{ prime}}\prod_{i=1+u-r(H_p)+d_p}^{\infty} (1-p^{-i}).$$ Let $W$ be the set of profinite groups $G$ such that there are only finitely many primes $p$ such that the maximal pro-$p$ quotient of $G$ has generator rank $\geq max(2,-u+1)$. By the Borel-Cantelli lemma, we can see that $\mu_u(W)=1$, and thus $\mu_u$ assigns probability $1$ to the set of groups who pro-nilpotent quotient is finitely generated. \[Ex:inf\] When $u\leq 0$, we have $\mu_u( \{ \textrm{infinite groups} \})=1$ (which can already be seen on the abelianization). When $u>0$, we have $0<\mu_u( \{ \textrm{infinite groups} \})<1$, since $\mu_u( \{ \textrm{trivial group} \})>0$ and there is positive probability of infinite pro-p quotient. This was seen for the $\mu_{u,n}$ in [@Jarden2006]. Which groups appear? {#S:prob0} ==================== In this section, we consider the question of when $\mu_u$ is $0$ on our basic opens $U_{S,H}$. In order for a basic open $U_{S,H}$ to have positive probability for $\mu_{u,n}$, the group $H$ needs to be able to be generated as a pro-${\bar{S}}$ group with $n$ generators and $n+u$ relations. We will see in Proposition \[P:prob0\] that the same criterion holds for $\mu_{u,n}$. We start with a lemma about the number of generators and relations required to present a pro-${\bar{S}}$ group. \[L:present\] Let $S$ be a finite set of finite groups and $u$ an integer. Let $H$ be a finite pro-${\bar{S}}$ group that can be generated by $d$ generators. If $H$ can be presented as a pro-${\bar{S}}$ group by $m$ generators and $m+u$ relations, then $H$ can be presented as a pro-${\bar{S}}$ group by $d$ generators and $d+u$ relations. This is the same as the situation when $S$ is the set of all profinite groups and $H$ is a finite group (see [@Lubotzky2001 Theorem 0.1]), but contrasts to the more general situation of presenting $H$ as a finite group, where the analog is a long-standing open question (see [@Gruenberg1976 Lecture 1: Question 3]). Suppose for the sake of contradiction that we have a counterexample, and consider one with $m$ minimal. We have that $$\begin{aligned} && \mu_{u,m}(U_{S,H})\\ &=& \frac{\|{\operatorname{Sur}}(\hat{F}_m,H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{m+u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{k=0}^{m(S,m,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{m+u}}}) \prod_{\substack{ G\in \mathcal{N}}} (1-\|G\|^{-{m-u}})^{m(S,m,H,G)} \\ &>& 0.\end{aligned}$$ In particular, since $\|G\|$ is a power of $h_H(G)$ this implies that for $G\in \mathcal{A}_H$, we have $$h_H(G)^{m(S,m,H,G)-1}\|G\|^{-m-u}\leq h_H(G)^{-1}.$$ However, since we have a minimal counterexample, we have that $m>d$ and $$\begin{aligned} &&\mu_{u,m-1}(U_{S,H})\\ &=& \frac{\|{\operatorname{Sur}}(\hat{F}_{m-1},H)\|}{\|{\operatorname{Aut}}(H)\|\|H\|^{m-1+u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{k=0}^{m(S,m-1,H,G)-1} (1-\frac{h_H(G)^k}{ \|G\|^{{m-1+u}}}) \prod_{\substack{ G\in \mathcal{N}}} (1-\|G\|^{-m+1-u})^{m(S,m-1,H,G)}\\ &=&0.\end{aligned}$$ By the final statement of Theorem \[T:calc\], we have that one of the factors is $0$. Since $m>d$, we have $\|{\operatorname{Sur}}(\hat{F}_{m-1},H)\|\ne 0$. If $H$ is the trivial group, the lemma is clear. Thus we can assume $d\geq 1$ and $m\geq 2$, and so for $G\in\mathcal{N}$ we have $(1-\|G\|^{-m+1-u})^{m(S,m-1,H,G)}>0$. Thus, for some $G\in \mathcal{A}_H$, we have $$h_H(G)^{m(S,m-1,H,G)-1}\|G\|^{-m+1-u}\geq 1.$$ If $\rho_n : (\hat{F}_n)^{\bar{S}} {\rightarrow}H$ is a surjection, we have $$\begin{aligned} (h_H(G)^{m(S,n,H,G)}-1)\|G\|^{-n} &= (h_H(G)-1) \sum_{\substack{\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\ \textrm{$\ker \pi$ isom. $G$ } \\ \textrm{$\ker \pi$ irred. $E$-group} \\ \textrm{$E$ is level $S$} }} \frac{\|{\operatorname{Sur}}(\rho_n, \pi)\|}{\|{\operatorname{Aut}}_H(E,\pi)\|\|G\|^n} .\end{aligned}$$ Any surjection from $\rho_n$ to $\pi$ can be extended to a surjection from $\rho_{n+1}$ to $\pi$ in $\|G\|$ ways. So, $(h_H(G)^{m(S,n,H,G)}-1)\|G\|^{-n}$ is non-decreasing in $n$. So we have $$\frac{h_H(G)^{m(S,m,H,G)}-1}{\|G\|^{m}} \geq \frac{h_H(G)^{m(S,m-1,H,G)}-1}{\|G\|^{m-1}} ,$$ and then we have $$\begin{aligned} \|G\|^uh_H(G)&\leq h_H(G)^{m(S,m-1,H,G)}\|G\|^{-m+1}\\ &\leq h_H(G)^{m(S,m,H,G)}\|G\|^{-m} +\|G\|^{-m+1}-\|G\|^{-m}\\ &\leq \|G\|^u +\|G\|^{-m+1}-\|G\|^{-m}.\end{aligned}$$ Since $m>d$ and $H$ can be generated by $d$ generators, the number of relations $m+u$ has to be positive. From above, we have $\|G\|^{m+u}(h_H(G)-1)\leq(\|G\|-1)$. Then this is a contradiction, since $\|G\|\geq 2$ and $h_H(G)\geq 2$. Lemma \[L:present\] leads to the following definition. Let $S$ be a finite set of finite groups and $u$ be an integer. We call a finite group $H$ with generator rank $d$ achievable* (with $S$ and $u$ implicit) if it can be generated as a pro-$\bar{S}$ group with $d$ generators and $d+u$ relations. \[P:prob0\] Let $S$ be a finite set of finite groups and $u$ be an integer. Then for a finite group $H$ we have that $\mu_u(U_{S,H})>0$ if $H$ is achievable and $\mu_u(U_{S,H})=0$ otherwise. So given $u$, our measure $\mu_u$ is supported on those groups in $\mathcal{P}$ who pro-$\bar{S}$ completion is achievable (for $u,S$) for every finite set $S$ of finite groups. Note that given $S$, any finite pro-$\bar{S}$ group $H$ is achievable for $u$ sufficiently large. From [@Guralnick2007 Theorem A], we have that every finite simple group can be presented as a profinite group with $2$ generators and $18$ relations. Thus if $u\geq 16$ and $S$ is a finite set of finite groups with $H\in {\bar{S}}$ a simple group, then $H$ is achievable. If $S$ is the set of all groups of order $32$ and $u\leq 0$, we can see that $H={\ensuremath{{\mathbb{Z}}}\xspace}/2{\ensuremath{{\mathbb{Z}}}\xspace}\times {\ensuremath{{\mathbb{Z}}}\xspace}/2{\ensuremath{{\mathbb{Z}}}\xspace}$ is not achievable. To obtain $H$ as a quotient of $F_2$, it is easy to compute we need at least 3 relations (for both generators to be order $2$ and for them to commute with each other). Proposition \[P:prob0\] need not hold for infinite $S$. For example, if $S$ is the set of all finite abelian groups, then any finite abelian group $H$ can be presented as an abelian group with $n$ generators and $n$ relations, but $\mu_0(V_{S,H})=0$. (See Example \[E:abelian\].) This is because the product over $G\in \mathcal{A}_H$ is contains factors $(1-p^{-1})$ for each prime $p$ and thus is $0$ even though no individual factor is $0$. Further, if $S$ is the set of all groups and $H\in\mathcal{P}$, we have $\mu_0(V_{S,H})\leq \mu_0(V_{\textrm{\{abelian groups\}},H^{ab}}) =0$. Some of those groups $H$ can be profinitely presented with $n$ generators and $n$ relations. It is an interesting open question to understand in general for which infinite $S$ and finite $H$ does the product in Equation \[E:lim\] give $\mu_u(V_{S,H})=0$ even when none of the factors in the product is $0$. By Lemma \[L:present\], if $H$ is not achievable, then $\mu_{u,n}(U_{S,H})=0$ for all $n$ and $\mu_u(U_{S,H})=0$. Suppose that $\mu_u(U_{S,H})=0$. Then using Theorem \[T:calc\] and Remark \[R:finprod\], we must have that one of the factors in $$\begin{aligned} \frac{1}{\|{\operatorname{Aut}}(H)\|\|H\|^{u}}\prod_{\substack{G \in \mathcal{A}_H}} \prod_{i=1}^{\infty} (1-\lambda(S,H,G) \frac{h_H(G)^{-i}}{ \|G\|^{{u}}}) \prod_{\substack{ G\in \mathcal{N}}} e^{-\|G\|^{-u}\lambda(S,H,G)}.\nonumber\end{aligned}$$ is $0$, i.e. for some $G\in \mathcal{A}_H$ we have $\lambda(S,H,G)h_H(G)^{-1}\|G\|^{-u}\geq 1$. Recall by Remark \[R:Gpow\] that $\|G\|$ is a power of $h_H(G)$ and thus so is $\lambda(S,H,G)$. In fact, for sufficiently large $n$, we have $\lambda(S,H,G)=h_H(G)^{m(S,n,H,G)}\|G\|^{-n}$. Thus, for sufficiently large $n$, we have $h_H(G)^{m(S,n,H,G)-1}\geq \|G\|^{n+u}$ and $\mu_{u,n}(U_{S,H})=0$. However if we can present $H$ as a pro-$\bar{S}$ group with $d$ generators and $d+u-k$ relations with $k\geq 0$, we can add $m$ generators for any $m$ and $m+k$ relations to trivialize those generators, to present $H$ with $d+m$ generators and $d+m+u$ relations for all $m\geq 0$, which implies $\mu_{u,n}(U_{S,H})>0$ for $n$ sufficiently large. Comparision to non-profinite groups {#S:non-profinite} =================================== Let $Y_{u,n,\ell}$ be $F_n$ modulo $n+u$ random relations uniform from words of length at most $\ell$. In this section, we will compare this model to our $X_{u,n}$. To put the groups on the same footing, we take the profinite completions $\hat{Y}_{u,n,\ell}$ of the $Y_{u,n,\ell}$. Alternatively, we could enlarge our measure space to include non-profinite groups, with the same definition of basic opens. Since our topology would not separate groups with the same profinite completion, we might as well consider only the profinite completions. (Note by [@Ollivier2011] and [@Agol2013], at density $<1/6$, these groups are asymptotically almost surely residually finite and thus inject into their profinite completions.) The following is almost the same as [@Dunfield2006 Lemma 4.4], but we include it here for completeness. \[P:same\] Given integers $n,u$, we have that the distributions $\nu_{u,n,\ell}$ of the $\hat{Y}_{u,n,\ell}$ weakly converge to $X_{u,n}$. As in the proof of Theorem \[T:Main\], it suffices to show that for each finite group $H$ and finite set $S$ of groups that $$\lim_{\ell{\rightarrow}\infty} \nu_{u,n,\ell}(U_{S,H})=\mu_{u,n}(U_{S,H}).$$ Thus we are asked to compare the quotient of the finite group $(\hat{F}_n)^{\bar{S}}$ by the image of random uniform words of length at most $\ell$ versus by uniform random relators. However as $\ell{\rightarrow}\infty$ the image of a random uniform words of length at most $\ell$ converges to the uniform distribution on $(\hat{F}_n)^{\bar{S}}$, by the fundamental theorem on irreducible, aperiodic finite state Markov chains [@Durrett2010 Theorem 6.6.4]. Next we see that taking a number of relations that is going to infinity always gives groups weakly converging to the trivial group in our topology. This includes all positive density Gromov random groups as well as plenty of density $0$ random groups. \[P:alltrivial\] Let $u(\ell)$ be an integer valued function of the positive integers that goes to $\infty$ as $\ell{\rightarrow}\infty$. Then $\nu_{u(\ell),n,\ell}$ weakly converge to the probability measure supported on the trivial group as $\ell{\rightarrow}\infty$. Fix a finite set $S$ of finite groups and a finite group $H$. Fix an integer $v$. For $u(\ell)\geq v$, we have that $${\operatorname{Prob}}(\hat{Y}_{u(\ell),n,\ell} \textrm{ has a surjection to $H$})\leq {\operatorname{Prob}}(\hat{Y}_{v,n,\ell} \textrm{ has a surjection to $H$}).$$ Since the set of groups with a surjection to $H$ is open and closed by Proposition \[P:same\], we have that $$\lim_{\ell{\rightarrow}\infty} {\operatorname{Prob}}(\hat{Y}_{v,n,\ell} \textrm{ has a surjection to $H$}) ={\operatorname{Prob}}(X_{v,n} \textrm{ has a surjection to $H$}).$$ It is easy to see using the approach of our paper that $${\ensuremath{{\mathbb{E}}}}(\|{\operatorname{Sur}}(X_{v,n},H\|)=\frac{\|{\operatorname{Sur}}(F_n,H)\|}{\|H\|^{n+v}}\leq \|H\|^{-v}.$$ Thus $$\limsup_{\ell{\rightarrow}\infty} {\operatorname{Prob}}(\hat{Y}_{u(\ell),n,\ell} \textrm{ has a surjection to $H$}) \leq \|H\|^{-v}$$ for every $v$, and so $\lim_{\ell{\rightarrow}\infty} {\operatorname{Prob}}(\hat{Y}_{u(\ell),n,\ell} \textrm{ has a surjection to $H$})=0$. Thus, for every $U_{S,H}$ with $H$ non-trivial, we have that $$\lim_{\ell{\rightarrow}\infty} \nu_{u(\ell),n,\ell}(U_{S,H})=0.$$ For $u(\ell)\geq v$, we have that $${\operatorname{Prob}}(\hat{Y}_{u(\ell),n,\ell}^{\bar{S}} \textrm{ trivial}) \geq {\operatorname{Prob}}(\hat{Y}_{v,n,\ell}^{\bar{S}} \textrm{ trivial}).$$ By Proposition \[P:same\], we have that $$\lim_{\ell{\rightarrow}\infty}({\operatorname{Prob}}(\hat{Y}_{v,n,\ell}^{\bar{S}} \textrm{ trivial}))={\operatorname{Prob}}(X_{v,n}^{\bar{S}} \textrm{ trivial}).$$ So $$\liminf_{\ell{\rightarrow}\infty} {\operatorname{Prob}}(\hat{Y}_{u(\ell),n,\ell}^{\bar{S}} \textrm{ trivial}) \geq \limsup_{v{\rightarrow}\infty} {\operatorname{Prob}}(X_{v,n}^{\bar{S}} \textrm{ trivial}).$$ From Equation , we have that $\limsup_{v{\rightarrow}\infty} {\operatorname{Prob}}(X_{v,n}^{\bar{S}} \textrm{ trivial})=1$. (We can control the size of the product in Equation , for example, by using the fact that there are at most 2 finite simple groups of any particular order.) Thus, for every $U_{S,1}$, we have that $$\lim_{\ell{\rightarrow}\infty} \nu_{u(\ell),n,\ell}(U_{S,1})=1.$$ \[R:alltrivial\] Proposition \[P:alltrivial\] might seem surprising at first. The groups $Y_{u(\ell),n,\ell}$ are plenty interesting as $\ell{\rightarrow}\infty$. In particularly they are asymptotically almost surely infinite at density $<1/2$ [@Gromov1993], and residually finite at density $<1/6$ [@Ollivier2011; @Agol2013], and so have many finite quotients. The above shows that those quotients are escaping off to infinity, however. Just as a very interesting sequence of numbers might go to $0$, an interesting sequence of random groups can converge to the trivial group. A better analogy might be that a sequence of integers with interesting asymptotic growth that goes to $0$ $p$-adically. 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--- abstract: 'Fluorescent calcium imaging provides a potentially powerful tool for inferring connectivity in neural circuits with up to thousands of neurons. However, a key challenge in using calcium imaging for connectivity detection is that current systems often have a temporal response and frame rate that can be orders of magnitude slower than the underlying neural spiking process. Bayesian inference methods based on expectation-maximization (EM) have been proposed to overcome these limitations, but are often computationally demanding since the E-step in the EM procedure typically involves state estimation for a high-dimensional nonlinear dynamical system. In this work, we propose a computationally fast method for the state estimation based on a hybrid of loopy belief propagation and approximate message passing (AMP). The key insight is that a neural system as viewed through calcium imaging can be factorized into simple scalar dynamical systems for each neuron with linear interconnections between the neurons. Using the structure, the updates in the proposed hybrid AMP methodology can be computed by a set of one-dimensional state estimation procedures and linear transforms with the connectivity matrix. This yields a computationally scalable method for inferring connectivity of large neural circuits. Simulations of the method on realistic neural networks demonstrate good accuracy with computation times that are potentially significantly faster than current approaches based on Markov Chain Monte Carlo methods.' author: - \| Alyson K. Fletcher\ Dept. Electrical Engineering\ University of California, Santa Cruz\ `alyson@ucsc.edu`\ Sundeep Rangan\ Dept. Electrical & Computer Engineering\ New York University Polytechnic Institute\ `srangan@nyu.edu` title: Scalable Inference for Neuronal Connectivity from Calcium Imaging --- Introduction ============ Determining connectivity in populations of neurons is fundamental to understanding neural computation and function. In recent years, calcium imaging has emerged as a promising technique for measuring synaptic activity and mapping neural micro-circuits [@tsien1989fluorescent; @ohki2005functional; @soriano2008development; @vogelstein2009oopsi; @stosiek2003vivo]. Fluorescent calcium-sensitive dyes and genetically-encoded calcium indicators can be loaded into neurons, which can then be imaged for spiking activity either in vivo or in vitro. Current methods enable imaging populations of hundreds to thousands of neurons with very high spatial resolution. Using two-photon microscopy, imaging can also be localized to specific depths and cortical layers [@svoboda2006principles]. Calcium imaging also has the potential to be combined with optogenetic stimulation techniques such as in [@yizhar2011optogenetics]. However, inferring neural connectivity from calcium imaging remains a mathematically and computationally challenging problem. Unlike anatomical methods, calcium imaging does not directly measure connections. Instead, connections must be inferred indirectly from statistical relationships between spike activities of different neurons. In addition, the measurements of the spikes from calcium imaging are indirect and noisy. Most importantly, the imaging introduces significant temporal blurring of the spike times: the typical time constants for the decay of the fluorescent calcium concentration, $\Ca2$, can be on the order of a second – orders of magnitude slower than the spike rates and inter-neuron dynamics. Moreover, the calcium imaging frame rate remains relatively slow – often less than 100 Hz. Hence, determining connectivity typically requires super-resolution of spike times within the frame period. To overcome these challenges, the recent work [@MisVogPan:11] proposed a Bayesian inference method to estimate functional connectivity from calcium imaging in a systematic manner. Unlike “model-free" approaches such as in [@stetter2012model], the method in [@MisVogPan:11] assumed a detailed functional model of the neural dynamics with unknown parameters including a connectivity weight matrix $\Wbf$. The model parameters including the connectivity matrix can then be estimated via a standard EM procedure [@DempLR:77]. While the method is general, one of the challenges in implementing the algorithm is the computational complexity. As we discuss below, the E-step in the EM procedure essentially requires estimating the distributions of hidden states in a nonlinear dynamical system whose state dimension grows linearly with the number of neurons. Since exact computation of these densities grows exponentially in the state dimension, [@MisVogPan:11] uses an approximate method based on blockwise Gibbs sampling where each block of variables consists of the hidden states associated with one neuron. Since the variables within a block are described as a low-dimensional dynamical system, the updates of the densities for the Gibbs sampling can be computed efficiently via a standard particle filter [@doucet2000sequential; @doucet2009tutorial]. However, simulations of the method show that the mixing between blocks can still take considerable time to converge. This paper presents two novel contributions that can potentially significantly improve the computation time of the EM estimation as well as the generality of the model. The first contribution is to employ an approximate message passing (AMP) technique in the computationally difficult EM step. The key insight here is to recognize that a system with multiple neurons can be “factorized” into simple, scalar dynamical systems for each neuron with linear interactions between the neurons. As described below, we assume a standard leaky integrate-and-fire (LIF) model for each neuron [@DayanAbbott:01] and a first-order AR process for the calcium imaging [@vogelstein2009spike]. Under this model, the dynamics of $N$ neurons can be described by $2N$ systems, each with a scalar (i.e. one-dimensional) state. The coupling between the systems will be linear as described by the connectivity matrix $\Wbf$. Using this factorization, approximate state estimation can then be efficiently performed via approximations of loopy belief propagation (BP) [@WainwrightJ:08]. Specifically, we show that the loopy BP updates at each of the factor nodes associated with the integrate-and-fire and calcium imaging can be performed via a scalar standard forward–backward filter. For the updates associated with the linear transform $\Wbf$, we use recently-developed approximate message passing (AMP) methods. AMP was originally proposed in [@DonohoMM:09] for problems in compressed sensing. Similar to expectation propagation [@Minka:01], AMP methods use Gaussian and quadratic approximations of loopy BP but with further simplifications that leverage the linear interactions. AMP was used for neural mapping from multi-neuron excitation and neural receptive field estimation in [@FletcherRVB:11; @KamRanFU:12-nips]. Here, we use a so-called hybrid AMP technique proposed in [@RanganFGS:12-ISIT] that combines AMP updates across the linear coupling terms with standard loopy BP updates on the remainder of the system. When applied to the neural system, we show that the estimation updates become remarkably simple: For a system with $N$ neurons, each iteration involves running $2N$ forward–backward scalar state estimation algorithms, along with multiplications by $\Wbf$ and $\Wbf^T$ at each time step. The practical complexity scales as $O(NT)$ where $T$ is the number of time steps. We demonstrate that the method can be significantly faster than the blockwise Gibbs sampling proposed in [@MisVogPan:11], with similar accuracy. In addition to the potential computational improvement, the AMP-based procedure is somewhat more general. For example, the approach in [@MisVogPan:11] assumes a generalized linear model (GLM) for the spike rate of each neuron. The approach in this work can be theoretically applied to arbitrary scalar dynamics that describe spiking. In particular, the approach can incorporate a physically more realistic LIF model. The second contribution is a novel method for initial estimation of the connectivity matrix. Since we are applying the EM methodology to a fundamentally non-convex problem, the algorithm is sensitive to the initial condition. However, there are now several good approaches for initial estimation the spike times of each neuron from its calcium trace via sparse deconvolution [@vogelstein2010fast; @onativia2013finite; @pnevmatikakis2013sparse]. We show that, under a leaky integrate and fire model, that if the true spike times were known exactly, then the maximum likelihood (ML) estimation of the connectivity matrix can be performed via sparse probit regression – a standard convex programming problem used in classification [@Bishop:06]. We propose to obtain an initial estimate for the connectivity matrix $\Wbf$ by applying the sparse probit regression to the initial estimate of the spike times. System Model {#sec:model} ============ We consider a recurrent network of $N$ spontaneously firing neurons. All dynamics are approximated in discrete time with some time step $\Delta$, with a typical value $\Delta$ = 1 ms. Importantly, this time step is typically smaller than the calcium imaging period, so the model captures the dynamics between observations. Time bins are indexed by $k=0,\ldots,T-1$, where $T$ is the number of time bins so that $T\Delta$ is the total observation time in seconds. Each neuron $i$ generates a sequence of spikes (action potentials) indicated by random variables $s_i^k$ taking values $0$ or $1$ to represent whether there was a spike in time bin $k$ or not. It is assumed that the discretization step $\Delta$ is sufficiently small such that there is at most one action potential from a neuron in any one time bin. The spikes are generated via a standard leaky integrate-and-fire (LIF) model [@DayanAbbott:01] where the (single compartment) membrane voltage $v_i^k$ of each neuron $i$ and its corresponding spike output sequence $s_i^k$ evolve as \[eq:vlif\] \_i\^[1]{} = (1-\_[IF]{})v\_i\^k + q\_i\^k + d\_[v\_i]{}\^k, q\_i\^k = \_[j=1]{}\^N W\_[ij]{}s\_j\^[k-]{} + b\_[IF,i]{}, d\_[v\_i]{}\^k \~[N]{}(0,\_[IF]{}), and \[eq:vireset\] (v\_i\^[1]{},s\_i\^[1]{}) = (\_i\^k, 0) & v\_i\^k < ,\ (0, 1) & \_i\^k , where $\alpha_{IF}$ is a time constant for the integration leakage; $\mu$ is the threshold potential at which the neurons spikes; $b_{IF,i}$ is a constant bias term; $q_i^k$ is the increase in the membrane potential from the pre-synaptic spikes from other neurons and $d_{v_i}^k$ is a noise term including both thermal noise and currents from other neurons that are outside the observation window. The voltage has been scaled so that the reset voltage is zero. The parameter $\delta$ is the integer delay (in units of the time step $\Delta$) between the spike in one neuron and the increase in the membrane voltage in the post-synaptic neuron. An implicit assumption in this model is the post-synaptic current arrives in a single time bin with a fixed delay. To determine functional connectivity, the key parameter to estimate will be the matrix $\Wbf$ of the weighting terms $W_{ij}$ in . Each parameter $W_{ij}$ represents the increase in the membrane voltage in neuron $i$ due to the current triggered from a spike in neuron $j$. The connectivity weight $W_{ij}$ will be zero whenever neuron $j$ has no connection to neuron $i$. Thus, determining $\Wbf$ will determine which neurons are connected to one another and the strengths of those connections. For the calcium imaging, we use a standard model [@MisVogPan:11], where the concentration of fluorescent Calcium has a fast initial rise upon an action potential followed by a slow exponential decay. Specifically, we let $z_i^k=\Ca2_k$ be the concentration of fluorescent Calcium in neuron $i$ in time bin $k$ and assume it evolves as first-order auto-regressive $AR(1)$ model, \[eq:zk\] z\_i\^[1]{} = (1-\_[CA,i]{})z\_i\^k + s\_i\^k, where $\alpha_{CA}$ is the Calcium time constant. The observed net fluorescence level is then given by a noisy version of $z_i^k$, \[eq:yk\] y\_i\^k = a\_[CA,i]{} z\_i\^k + b\_[CA,i]{} + d\_[y\_i]{}\^k, d\_[y\_i]{}\^k \~[N]{}(0,\_y), where $a_{CA,i}$ and $b_{CA,i}$ are constants and $d_{y_i}$ is white Gaussian noise with variance $\tau_y$. Nonlinearities such as saturation described in [@vogelstein2009spike] can also be modeled. As mentioned in the Introduction, a key challenge in calcium imaging is the relatively slow frame rate which has the effect of subsampling of the fluorescence. To model the subsampling, we let $I_F$ denote the set of time indices $k$ on which we observe $F_i^k$. We will assume that fluorescence values are observed once every $T_F$ time steps for some integer period $T_F$ so that $I_F = \left\{ 0, T_F, 2T_F, \ldots, KT_F\right\}$ where $K$ is the number of Calcium image frames. Parameter Estimation via Message Passing ======================================== Problem Formulation ------------------- Let $\theta$ be set of all the unknown parameters, \[eq:thetaDef\] = { , \_[IF]{}, \_[CA]{}, \_[IF]{}, b\_[IF,i]{}, \_[CA]{}, a\_[CA,i]{}, b\_[CA,i]{}, i=1,…,N }, which includes the connectivity matrix, time constants and various variances and bias terms. Estimating the parameter set $\theta$ will provide an estimate of the connectivity matrix $\Wbf$, which is our main goal. To estimate $\theta$, we consider a regularized maximum likelihood (ML) estimate \[eq:thetaML\] = \_ L(\|) + (), L(\|) = -p(\|), where $\ybf$ is the set of observed values; $L(\ybf\|\theta)$ is the negative log likelihood of $\ybf$ given the parameters $\theta$ and $\phi(\theta)$ is some regularization function. For the calcium imaging problem, the observations $\ybf$ are the observed fluorescence values across all the neurons, \[eq:yveci\] = { \_1,…,\_N }, \_i = { y\_i\^k, k I\_F }, where $\ybf_i$ is the set of fluorescence values from neuron $i$, and, as mentioned above, $I_F$ is the set of time indices $k$ on which the fluorescence is sampled. The regularization function $\phi(\theta)$ can be used to impose constraints or priors on the parameters. In this work, we will assume a simple regularizer that only constrains the connectivity matrix $\Wbf$, \[eq:phiell1\] () = \_1, \_1 := \_[ij]{} \|W\_[ij]{}\|, where $\lambda$ is a positive constant. The $\ell_1$ regularizer is a standard convex function used to encourage sparsity [@Donoho:06], which we know in this case must be valid since most neurons are not connected to one another. EM Estimation ------------- Exact computation of $\thetahat$ in is generally intractable, since the observed fluorescence values $\ybf$ depend on the unknown parameters $\theta$ through a large set of hidden variables. Similar to [@MisVogPan:11], we thus use a standard EM procedure [@DempLR:77]. To apply the EM procedure to the calcium imaging problem, let $\xbf$ be the set of hidden variables, \[eq:xdef\] = { , , , }, where $\vbf$ are the membrane voltages of the neurons, $\zbf$ the calcium concentrations, $\sbf$ the spike outputs and $\qbf$ the linearly combined spike inputs. For any of these variables, we will use the subscript $i$ (e.g. $\vbf_i$) to denote the values of the variables of a particular neuron $i$ across all time steps and superscript $k$ (e.g. $\vbf^k$) to denote the values across all neurons at a particular time step $k$. Thus, for the membrane voltage $$\vbf = \left\{v_i^k\right\}, \quad \vbf^k = \left(v_1^k, \ldots, v_{N}^k\right), \quad \vbf_i = \left( v_i^0,\ldots,v_i^{\Tm1}\right).$$ The EM procedure alternately estimates distributions on the hidden variables $\xbf$ given the current parameter estimate for $\theta$ (the E-step); and then updates the estimates for parameter vector $\theta$ given the current distribution on the hidden variables $\xbf$ (the M-step). - E-Step: Given parameter estimates $\thetahat^\ell$, estimate \[eq:estepDist\] P(\|,\^), which is the posterior distribution of the hidden variables $\xbf$ given the observations $\ybf$ and current parameter estimate $\thetahat^\ell$. - M-step Update the parameter estimate via the minimization, \[eq:mstep\] \^[+1]{} = \_+ (), where $L(\xbf,\ybf\|\theta)$ is the joint negative log likelihood, \[eq:Lxytheta\] L(,\|) = - p(,\|). In the expectation is with respect to the distribution found in and $\phi(\theta)$ is the parameter regularization function. The next two sections will describe how we approximately perform each of these steps. E-Step estimation via Approximate Message Passing {#sec:estep} ------------------------------------------------- For the calcium imaging problem, the challenging step of the EM procedure is the E-step, since the hidden variables $\xbf$ to be estimated are the states and outputs of a high-dimensional nonlinear dynamical system. Under the model in Section \[sec:model\], a system with $N$ neurons will require $N$ states for the membrane voltages $v_i^k$ and $N$ states for the bound Ca concentration levels $z_i^k$, resulting in a total state dimension of $2N$. The E-step for this system is essentially a state estimation problem, and exact inference of the states of a general nonlinear dynamical system grows exponentially in the state dimension. Hence, exact computation of the posterior distribution for the system will be intractable even for a moderately sized network. As described in the Introduction, we thus use an approximate messaging passing method that exploits the separable structure of the system. For the remainder of this section, we will assume the parameters $\theta$ in are fixed to the current parameter estimate $\thetahat^\ell$. Then, under the assumptions of Section \[sec:model\], the joint probability distribution function of the variables can be written in a factorized form, \[eq:pxyfact\] P(,) = P(,,,,) = \_[k=0]{}\^[1]{} [\_[ { [\^k = \^k]{} } ]{}]{} \_[i=1]{}\^N \^[IF]{}\_i(\_i,\_i,\_i) \^[CA]{}\_i(\_i,\_i,\_i), where $Z$ is a normalization constant; $\psi^{IF}_i(\qbf_i,\vbf_i,\sbf_i)$ is the potential function relating the summed spike inputs $\qbf_i$ to the membrane voltages $\vbf_i$ and spike outputs $\sbf_i$; $\psi^{CA}_i(\sbf_i,\zbf_i,\ybf_i)$ relates the spike outputs $\sbf_i$ to the bound calcium concentrations $\zbf_i$ and observed fluorescence values $\ybf_i$; and the term ${\mathbbm{1}_{ \{ {\qbf^k = \Wbf\sbf^k} \} }}$ indicates that the distribution is to be restricted to the set satisfying the linear constraints $\qbf^k = \Wbf\sbf^k$ across all time steps $k$. As in standard loopy BP [@WainwrightJ:08], we represent the distribution in a factor graph as shown in Fig. \[fig:factorGraph\]. Now, for the E-step, we need to compute the marginals of the posterior distribution $p(\xbf\|\ybf)$ from the joint distribution . Using the factor graph representation, loopy BP iteratively updates estimates of these marginal posterior distributions using a message passing procedure, where the estimates of the distributions (called beliefs) are passed between the variable and factor nodes in the graph. To reduce the computations in loopy BP further, we employ an approximate message passing (AMP) method for the updates in the factor node corresponding to the linear constraints $\qbf^k = \Wbf\sbf^k$. AMP was originally developed in [@DonohoMM:09] for problems in compressed sensing, and can be derived as Gaussian approximations of loopy BP [@Montanari:12-bookChap; @Rangan:11-ISIT] similar to expectation propagation [@Seeger:08]. In this work, we employ a hybrid form of AMP [@RanganFGS:12-ISIT] that combines AMP with standard message passing. The AMP methods have the benefit of being computationally very fast and, for problems with certain large random transforms, the methods can yield provably Bayes-optimal estimates of the posteriors, even in certain non-convex problem instances. However, similar to standard loopy BP, the AMP and its variants may diverge for general transforms (see [@RanSRFC:13-ISIT; @Krzakala:14-ISITbethe; @RanSchFle:14-ISIT] for some discussion of the convergence). For our problem, we will see in simulations that we obtain fast convergence in a relatively small number of iterations. =\[circle,draw,fill=orange!30\] =\[circle,draw,fill=orange!70\] =\[rectangle,draw,fill=green!30\] (qi) [$\qbf_i$]{}; (psiIF) ------------------------------------- $\psi^{IF}_i(\qbf_i,\vbf_i,\sbf_i)$ Integrate-and-fire dynamics ------------------------------------- ; (vi) [$\vbf_i$]{}; (si) [$\sbf_i$]{}; (W) --------------------- $\qbf^k=\Wbf\sbf^k$ Connectivity between neurons --------------------- ; (psiCA) ------------------------------------- $\psi^{CA}_i(\sbf_i,\zbf_i,\ybf_i)$ Ca imaging dynamics ------------------------------------- ; (yi) [$\ybf_i$]{}; (zi) [$\zbf_i$]{}; at (psiCA) \[below right=0.75cm and 0.0cm\] (nlabel) [Neuron $i$, $i=1,\ldots,N$]{}; at (W) \[below right=0.3cm and 1.75cm\] (klabel) [Time step $k$, $k=0,\ldots,\Tm1$]{}; (qi) – (psiIF); (vi) – (psiIF); (si) – (psiIF); (si) \|- (W); (qi) \|- (W); (si) – (psiCA); (yi) – (psiCA); (zi) – (psiCA); ($(qi.north west)+(-1,2.5)$) rectangle ($(nlabel.south east)+(0.25,0)$); ($(qi.south west)+(-1,-1.2)$) rectangle ($(klabel.south east)$); We provide some details of the hybrid AMP method in Appendix \[sec:estepDetails\], but the basic procedure for the factor node updates and the reasons why these computations are simple can be summarized as follows. At a high level, the factor graph structure in Fig. \[fig:factorGraph\] partitions the $2N$-dimensional nonlinear dynamical system into $N$ scalar systems associated with each membrane voltage $v_i^k$ and an additional $N$ scalar systems associated with each calcium concentration level $z_i^k$. The only coupling between these systems is through the linear relationships $\qbf^k=\Wbf\sbf^k$. As shown in Appendix \[sec:estepDetails\], on each of the scalar systems, the factor node updates required by loopy BP essentially reduces to a state estimation problem for this system. Since the state space of this system is scalar (i.e. one-dimensional), we can discretize the state space well with a small number of points – in the experiments below we use $L=20$ points per dimension. Once discretized, the state estimation can be performed via a standard forward–backward algorithm. If there are $T$ time steps, the algorithm will have a computational cost of $O(TL^2)$ per scalar system. Hence, all the factor node updates across all the $2N$ scalar systems has total complexity $O(NTL^2)$. For the factor nodes associated with the linear constraints $\qbf^k=\Wbf\sbf^k$, we use the AMP approximations [@RanganFGS:12-ISIT]. In this approximation, the messages for the transform outputs $q_i^k$ are approximated as Gaussians which is, at least heuristically, justified since the they are outputs of a linear transform of a large number of variables, $s^k_i$. In the AMP algorithm, the belief updates for the variables $\qbf^k$ and $\sbf^k$ can then be computed simply by linear transformations of $\Wbf$ and $\Wbf^T$. Since $\Wbf$ represents a connectivity matrix, it is generally sparse. If each row of $\Wbf$ has $d$ non-zero values, multiplication by $\Wbf$ and $\Wbf^T$ will be $O(Nd)$. Performing the multiplications across all time steps results in a total complexity of $O(NTd)$. Thus, the total complexity of the proposed E-step estimation method is $O(NTL^2 + NTd)$ per loopy BP iteration. We typically use a small number of loopy BP iterations per EM update (in fact, in the experiments below, we found reasonable performance with one loopy BP update per EM update). In summary, we see that while the overall neural system is high-dimensional, it has a linear + scalar structure. Under the assumption of the bounded connectivity $d$, this structure enables an approximate inference strategy that scales linearly with the number of neurons $N$ and time steps $T$. Moreover, the updates in different scalar systems can be computed separately allowing a readily parallelizable implementation. Approximate M-step Optimization {#sec:mstep} ------------------------------- The M-step is computationally relatively simple. All the parameters in $\theta$ in have a linear relationship between the components of the variables in the vector $\xbf$ in . For example, the parameters $a_{CA,i}$ and $b_{CA,i}$ appear in the fluorescence output equation . Since the noise $d_{y_i}^k$ in this equation is Gaussian, the negative log likelihood is given by $$L(\xbf,\ybf\|\theta) = \frac{1}{2\tau_{y_i}} \sum_{k \in I_F} (y_i^k - a_{CA,i}z_i^k-b_{CA,i})^2 + \frac{T}{2}\log(\tau_{y_i}) + \mbox{other terms},$$ where “other terms" depend on parameters other than $a_{CA,i}$ and $b_{CA,i}$. The expectation $\Exp( L(\xbf,\ybf\|\theta)\|\thetahat^\ell )$ will then depend only on the mean and variance of the variables $y_i^k$ and $z_i^k$, which are provided by the E-step estimation. Thus, the M-step optimization in can be computed via a simple least-squares problem. Using the linear relation , a similar method can be used for $\alpha_{IF,i}$ and $b_{IF,i}$, and the linear relation can be used to estimate the calcium time constant $\alpha_{CA}$. To estimate the connectivity matrix $\Wbf$, let $\rbf^k = \qbf^k-\Wbf\sbf^k$ so that the constraints in is equivalent to the condition that $\rbf^k=0$. Thus, the term containing $\Wbf$ in the expectation of the negative log likelihood $\Exp( L(\xbf,\ybf\|\theta)\|\thetahat^\ell )$ is given by the negative log probability density of $\rbf^k$ evaluated at zero. In general, this density will be a complex function of $\Wbf$ and difficult to minimize. So, we approximate the density as follows: Let $\qbfhat$ and $\sbfhat$ be the expectation of the variables $\qbf$ and $\sbf$ given by the E-step. Hence, the expectation of $\rbf^k$ is $\qbfhat^k-\Wbf\sbfhat^k$. As a simple approximation, we will then assume that the variables $r_i^k$ are Gaussian, independent and having some constant variance $\sigma^2$. Under this simplifying assumption, the M-step optimization of $\Wbf$ with the $\ell_1$ regularizer reduces to \[eq:What\] = \_ \_[k=0]{}\^[1]{} \^k - \^k\^2 + \^2\_1, For a given value of $\sigma^2\lambda$, the optimization is a standard LASSO optimization [@Tibshirani:96] which can be evaluated efficiently via a number of convex programming methods. In this work, in each M-step, we adjust the regularization parameter $\sigma^2\lambda$ to obtain a desired fixed sparsity level in the solution $\Wbf$. Initial Estimation via Sparse Regression {#sec:initW} ---------------------------------------- Since the EM algorithm cannot be guaranteed to converge a global maxima, it is important to pick the initial parameter estimates carefully. The time constants and noise levels for the calcium image can be extracted from the second-order statistics of fluorescence values and simple thresholding can provide a coarse estimate of the spike rate. The key challenge is to obtain a good estimate for the connectivity matrix $\Wbf$. For each neuron $i$, we first make an initial estimate of the spike probabilities $P(s_i^k=1\|\ybf_i)$ from the observed fluorescence values $\ybf_i$, assuming some i.i.d. prior of the form $P(s_i^t)=\lambda\Delta$, where $\lambda$ is the estimated average spike rate per second. This estimation can be solved with the filtering method in [@vogelstein2009spike] and is also equivalent to the method we use for the factor node updates. We can then threshold these probabilities to make a hard initial decision on each spike: $s_i^k=0$ or 1. We then propose to estimate $\Wbf$ from the spikes as follows. Fix a neuron $i$ and let $\wbf_i$ be the vector of weights $W_{ij}$, $j=1,\ldots,N$. Under the assumption that the initial spike sequence $s_i^k$ is exactly correct, it is shown in Appendix \[sec:initWApp\] that a regularized maximum likelihood estimate of $\wbf_i$ and bias term $b_{IF,i}$ is given by \[eq:betaOpt1\] (\_i,\_[IF,i]{}) = \_[\_i,b\_[IF,i]{}]{} \_[k=0]{}\^[1]{} L\_[ik]{}( \_[k]{}\^T \_i +c\_[ik]{}b\_[IF,i]{} - , s\_i\^k) + \_[j=1]{}\^N \|W\_[ij]{}\|, where $L_{ik}$ is a probit loss function and the vector $\ubf_k$ and scalar $c_{ik}$ can be determined from the spike estimates. The optimization is precisely a standard probit regression used in sparse linear classification [@Bishop:06]. This form arises due to the nature of the leaky integrate-and-fire model and . Thus, assuming the initial spike sequences are estimated reasonably accurately, one can obtain good initial estimates for the weights $W_{ij}$ and bias terms $b_{IF,i}$ by solving a standard classification problem. We point out that [@SoltaniGold:14] has recently provided an alternative method for recovery of connectivity matrix from the spikes assuming a LIF model based on maximizing information flow. Parameter Value ------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------- Number of neurons, $N$ 100 Connection sparsity 10% with random connections. All connections are excitatory with the non-zero weights $W_{ij}$ being exponentially distributed. Mean firing rate per neuron 10 Hz Simulation time step, $\Delta$ 1 ms Total simulation time, $T\Delta$ 10 sec (10,000 time steps) Integration time constant, $\alpha_{IF}$ 20 ms Conduction delay, $\delta$ 2 time steps = 2 ms Integration noise, $d_{v_i}^k$ Produced from two unobserved neurons. Ca time constant, $\alpha_{CA}$ 500 ms Fluorescence noise, $\tau_{CA}$ Set to 20 dB SNR Ca frame rate , $1/T_F$ 100 Hz : Parameters for the Ca image simulation. \[tbl:simParam\] ![image](figures/CaNetSim.png){width="60.00000%"} Numerical Example ================= The method was tested using realistic network parameters, as shown in Table \[tbl:simParam\], similar to those found in neurons networks within a cortical column [@sayer1990time]. Similar parameters are used in [@MisVogPan:11]. The network consisted of 100 neurons with each neuron randomly connected to 10% of the other neurons. The non-zero weights $W_{ij}$ were drawn from an exponential distribution. All weights were positive (i.e. the neurons were excitatory – there were no inhibitory neurons in the simulation). However, inhibitory neurons can also be added. A typical random matrix $\Wbf$ generated in this manner would not in general result in a stable system. To stabilize the system, we followed the procedure in [@stetter2012model] where the system is simulated multiple times. After each simulation, the rows of the matrix $\Wbf$ were adjusted up or down to increase or decrease the spike rate until all neurons spiked at a desired target rate. In this case, we assumed a desired average spike rate of 10 Hz. From the parameters in Table \[tbl:simParam\], we can immediately see the challenges in the estimation. Most importantly, the calcium imaging time constant $\alpha_{CA}$ is set for 500 ms. Since the average neurons spike rate is assumed to be 10 Hz, several spikes will typically appear within a single time constant. Moreover, both the integration time constant and inter-neuron conduction time are much smaller than both the image frame rate and Calcium time constants. A typical simulation of the network after the stabilization is shown in Fig. \[fig:CaNetSim\]. Observe that due to the random connectivity, spiking in one neuron can rapidly cause the entire network to fire. This appears as the vertical bright stripes in the lower panel of Fig. \[fig:CaNetSim\]. This synchronization makes the connectivity detection difficult to detect under temporal blurring of Ca imaging since it is hard to determine which neuron is causing which neuron to fire. Thus, the random matrix is a particularly challenging test case. ![Weight estimation accuracy. Left: Normalized mean-squared error as a function of the iteration number. Right: Scatter plot of the true and estimated weights. \[fig:simResults\] ](figures/EMMSE.png "fig:"){width="40.00000%"} ![Weight estimation accuracy. Left: Normalized mean-squared error as a function of the iteration number. Right: Scatter plot of the true and estimated weights. \[fig:simResults\] ](figures/EMscatter.png "fig:"){width="40.00000%"} The results of the estimation are shown in Fig. \[fig:simResults\]. The left panel shows the relative mean squared error defined as \[eq:relMse\] = , where $\What_{ij}$ is the estimate for the weight $W_{ij}$. The minimization over all $\alpha$ is performed since the method can only estimate the weights up to a constant scaling. The relative MSE is plotted as a function of the EM iteration, where we have performed only a single loopy BP iteration for each EM iteration. We see that after only 30 iterations we obtain a relative MSE of 7% – a number at least comparable to earlier results in [@MisVogPan:11], but with significantly less computation. The right panel shows a scatter plot of the estimated weights $\What_{ij}$ against the true weights $W_{ij}$. Conclusions =========== We have presented a scalable method for inferring connectivity in neural systems from calcium imaging. The method is based on factorizing the systems into scalar dynamical systems with linear connections. Once in this form, state estimation – the key computationally challenging component of the EM estimation – is tractable via approximating message passing methods. The key next step in the work is to test the methods on real data and also provide more comprehensive computational comparisons against current techniques such as [@MisVogPan:11]. E-Step Message Passing Implementation Details {#sec:estepDetails} ============================================= As described in Section \[sec:estep\], the E-step inference is performed via an approximate message passing technique [@RanganFGS:12-ISIT]. As in standard sum-product loopy BP [@WainwrightJ:08], the algorithm is based on passing “belief messages" between the variable and factor nodes representing estimates of the posterior marginals of the variables. Referring to the factor graph in Fig. \[fig:factorGraph\], we will use the subscripts $IF$, $CA$ and $W$ to refer respectively to the factor nodes for integrate and fire potential functions $\psi^{IF}_i$, the calcium imaging potential functions $\psi^{CA}_i$ and the linear constraints $\qbf^k=\Wbf\sbf^k$. We use the subscripts $Q$ and $S$ to refer to the variable nodes for $\qbf$ and $\sbf$. We use the notation such as $P_{IF \ra Q}(q_i^k)$ to denote the belief message to the variable node $q_i^k$ from the integrate and factor node $\psi^{IF}_i$. Similarly, $P_{IF \la Q}(q_i^k)$ will denote the reverse message from the variable node to the factor node. The messages to and from the variable nodes $s_i^k$ are binary: $s_i^k=0$ or 1. Hence, they can be parameterized by a single scalar. Similar to expectation propagation [@Minka:01], the messages to and from the variable nodes $q_i^k$ are approximated as Gaussians, so that we only need to maintain the first and second moments. Gaussian approximations are used in the variational Bayes method for calcium imaging inference in [@keshri2013shotgun]. To apply the hybrid AMP algorithm of [@RanganFGS:12-ISIT] to the factor graph in Fig. \[fig:factorGraph\], we use standard loopy BP message updates on the IF and CA factor nodes, and AMP updates on the linear constraints $\qbf^k=\Wbf\sbf^k$. The AMP updates are based on linear-Gaussian approximations. The details of the messages updates are as follows. #### Messages from $\psi^{IF}_i$: This factor node represents the integrate and fire system for the voltages $v_i^k$ and is given by \[eq:psiIF\] \^[IF]{}\_i(\_i,\_i,\_i) = \_[k=0]{}\^[1]{} P(v\_i\^[1]{},s\_i\^[1]{}\|v\_i\^k,q\_i\^k), where the conditional density $P(v_i^{\kp1},s_i^{\kp1}\|v_i^k,q_i^k)$ is given by integrate and fire system and . To describe the output belief propagation messages for this factor node, define the joint distribution,\ &=& \_[k=0]{}\^[1]{} P(v\_i\^[1]{},s\_i\^[1]{}\|v\_i\^k,q\_i\^k) P\_[IF Q]{}(q\_i\^k)P\_[IF S]{}(s\_i\^k), \[eq:pifjoint\] where $P_{IF \la Q}(q_i^k)$ and $P_{IF \la S}(s_i^k)$ are the incoming messages from the variable nodes. To compute the output messages, we must first compute the marginal densities $P(q_i^k)$ and $P(s_i^k)$ of this joint distribution . To compute these marginal densities, define $\xi_i^k = q_i^k + d_{v_i}^k + b_i$. Now recall that the AMP assumption is that each incoming distribution $P_{IF \la Q}(q_i^k)$ is Gaussian. Let $\qhat_i^k$ and $\tau_{q_i}^k$ be the mean and variance of this distribution. Thus, the joint distribution is identical to the posterior distribution of a linear system with a Gaussian input \[eq:vlifi\] \_i\^k = (1-\_[IF]{})\_i\^k + \_i\^k, \_i\^k \~[N]{}(\_i\^k+b\_i,\_[q\_i]{}\^k + \_[IF]{}), with the reset and spike output in and output observations $P(s_i^k\|\phi_i^k)$. This is a nonlinear system with a one-dimensional state $v_i^k$. Hence, one can, in principle, approximately compute the marginal densities $P(q_i^k)$ and $P(s_i^k)$ of with a one-dimensional particle filter [@doucet2009tutorial]. However, we found it computationally faster to simply use a fixed discretization of the set of values $v_i^k$. In the experiments below we used $L=$ 20 values linearly spaced from 0 to the threshold level $\mu$. Using the fixed discretization enables a number of the computations to be computed once for all time steps, and also removes the computations and logic for pruning necessary in particle filtering. After computing the marginals $P(q_i^k)$ and $P(s_i^k)$, we set the output messages as $$P_{IF \ra Q}(q_i^k) \propto P(q_i^k)/P_{IF \la Q}(q_i^k), \quad P_{IF \ra S}(s_i^k) \propto P(s_i^k)/P_{IF \la S}(s_i^k).$$ #### Messages from $\psi^{CA}_i$: In this case, the factor node represents the Ca imaging dynamics and is given by, \[eq:psiCA\] \^[CA]{}\_i(\_i,\_i,\_i) = \_[k=0]{}\^[1]{} P(z\_i\^[1]{}\|z\_i\^k,s\_i\^k) \_[k I\_F]{} P(y\_i\^k\|z\_i\^k), where $P(z_i^{\kp1}\|z_i^k,s_i^k)$ and $P(y_i^k\|z_i^k)$ are given by the relations and describing the fluorescent Ca$^{2+}$ concentration evolution and observed fluorescence. Recall that $I_F$ in is the set of time samples on which the output $y_i^k$ is observed. To compute the output beliefs for the factor node, as before, we define the joint distribution,\ &=& \_[k=0]{}\^[1]{} P(z\_i\^[1]{}\|z\_i\^k,s\_i\^k)P\_[CA S]{}(s\_i\^k) \_[k I\_F]{} P(y\_i\^k\|z\_i\^k), \[eq:pcajoint\] where $P_{CA \la S}(s_i^k)$ are the input messages from the variable nodes $s_i^k$. This distribution $P(\sbf_i,\zbf_i,\ybf_i)$ is identical to a the distribution for a linear system with a scalar state $z_i^k$, Gaussian observations $y_i^k$ and a discrete zero-one input $s_i^k$ with prior $P_{CA \la S}(s_i^k)$. Similar to the integrate-and-fire case, we can approximately compute the posterior marginals $P(s_i^k\|\ybf_i)$ by discretizing the states $z_i^k$ and using a standard forward–backward estimator. From the posterior marginals $P(s_i^k\|\ybf_i)$, we can then compute the belief messages for the factor node back to the variable nodes $s_i^k$: $P_{CA \ra S}(s_i^k) \propto P(s_i^k\|\ybf_i)/P_{CA \la S}(s_i^k)$. #### AMP messages from the linear constraints $\qbf^k=\Wbf\sbf^k$: Standard loopy BP updates for this factor node would be intractable for typical connectivity matrices $\Wbf$. To see this, suppose that in the current estimate for the connection matrix $\Wbf$, each neuron is connected to $d$ other neurons. Hence the rows of $\Wbf$ will have $d$ non-zero entries. Each constraints $q_i^k = (\Wbf\sbf^k)_i$ will thus involve $d$ binary variables, and the complexity of the loopy BP update will then require $O(2^d)$ operations. This computation will be difficult for large $d$. The hybrid AMP algorithm of [@RanganFGS:12-ISIT] uses Gaussian approximations on the messages to reduce the computations to simple linear transforms. First consider the output messages $P_{W \ra Q}(q_i^k)$ to the variable nodes $q_i^k$. These messages are Gaussians. Let $\qhat_i^k$ and $\tau_{q_i^k}$ be their mean and variance and let $\qbfhat^k$ and $\tau_q^k$ be the vector of these quantities. In the hybrid AMP algorithm, these means and variances are given by \[eq:qgamp\] \^k = \^k - \_q\^k\^k, \_q\^k = \|\|\^2\_s\^k, where $\sbfhat^k$ and $\taubf_s^k$ are the vectors of means and variances from the incoming messages $P_{W \la S}(s_i^k)$, and $\|\Wbf\|^2$ is the matrix with components $\|W_{ij}\|^2$. The variables $\pbf^k$ is a real-valued state vector, which is initialized to zero. In , the multiplication $\taubf_q^k\pbf^k$ is to be performed componentwise: $\taubf_q^k\pbf^k)_i = \tau_{q_i^k}p^k_i$. To process the incoming belief messages from the variable nodes $\qbf^k$, let $\overline{\qbf}^k$ and $\overline{\taubf}_q^k$ be the vector of mean and variances of the incoming beliefs $P_{W \la Q}(q^k_i)$. These quantities are to be distinguished from $\qbfhat^k$ and $\tau_q^k$, the mean and variance vectors of the outgoing messages $P_{W \ra Q}(q^k_i)$. We then first compute, \^k = (\^k - \^k)/\_q\^k, \_p\^k = , where the divisions are componentwise. Next, we compute the quantities \^k = \^k + \_s\^k \^T\^k, \_s\^k = 1/(\|\|\^2\_p\^k), where, again, the divisions are componentwise and the multiplication between $\taubf_s^k$ and $\Wbf^T\pbf^k$ is componentwise. The output message to the variable nodes $s^k_i$ is then given by $$P_{W \ra S}(s^k_i) \propto \exp\left( -\frac{1}{2\taubar_{s_i^k}}(s^k_i - \overline{s}^k_i)^2 \right),$$ with possible values $s^k_i = 0$ or 1. #### Variable node updates: The variable node updates are based on the standard sum-product rule [@WainwrightJ:08]. In the factor graph in Fig. \[fig:factorGraph\], each variable nodes $q_i^k$ is only connected to two factor nodes: the factor node for the potential function $\psi^{IF}_i$ and the factor node for the linear constraint $\qbf^k = \Wbf\sbf^k$. Hence, the variable node will simply relay the messages between the nodes: $$P_{IF \la Q}(q_i^k) = P_{W \ra Q}(q^k_i), \quad P_{IF \la W}(q_i^k) = P_{W \ra IF}(q^k_i),$$ Recall that these messages are approximated as Gaussians, so the messages can be represented by mean and variances. Each binary spike variable nodes $s_i^k$ is connected to three factor nodes: the integrate and fire potential function $\psi^{IF}_i$, the calcium imaging potential function $\psi^{CA}_i$ and the linear constraint $\qbf^k = \Wbf\sbf^k$. In the sum-product rule, the output message to any one of these nodes is the product of the incoming messages from the other two. Hence, && P\_[IF S]{}(s\_i\^k) P\_[W S]{}(s\_i\^k)P\_[Q S]{}(s\_i\^k), P\_[CA S]{}(s\_i\^k) P\_[W S]{}(s\_i\^k)P\_[IF S]{}(s\_i\^k),\ && P\_[W S]{}(s\_i\^k) P\_[CA S]{}(s\_i\^k)P\_[IF S]{}(s\_i\^k). The proportionality constant is simple to compute since the variables are binary so that $s_i^k=0$ or 1. Initial Estimation of $\Wbf$ via Sparse Probit Regression {#sec:initWApp} ========================================================= We show that given the spike sequence $s_i^k$, the maximum likelihood estimate of the connectivity weights $\Wbf$ and bias terms $b_{IF,i}$ can be computed approximately via a sparse probit regression of the form . To this end, suppose that we know the true spike sequence $s_i^k$ for all neurons $i$ and times $k$. Let $\{t_i^{\ell}, \ell=1,\ldots,L_i\}$, be the index of time bins $k$ where there is a spike (i.e. $s_i^k=1$ when $k=t_i^{\ell}$ for some $\ell$). Now, consider any time $k$ between two spikes $k \in [t_i^\ell,t_i^{\ell+1})$. Since $s_i^k=1$ at the initial time $k=t_i^\ell$, shows that the voltage must starts at zero: $v_i^k=0$. Integrating from this initial condition, we have that for any $k \in (t_i^\ell,t_i^{\ell+1})$, \[eq:vtildeu\] \_i\^k = \_[j=1]{}\^N W\_[ij]{}u\_j\^k + (k-t\_i\^)b\_[IF,i]{} + \_i\^k, u\_j\^k = \_[m=0]{}\^[k-t\_i\^-1]{} (1-\_[IF]{})\^m s\_i\^[k-m-]{}, where $\xi_i^k$ is the integration of the Gaussian noise $d_{v_i}^k$ up to time $k$. We can rewrite in vector form \[eq:vtildeuvec\] \_i\^k = \_[k]{}\^T \_i + c\_[ik]{}b\_[IF,i]{} + \_i\^k, where $\ubf_{k}$ and $\wbf_i$ are the vectors with the components $W_{ij}$ and $u_j^k$ and $c_{ik} = k-t_i^\ell$. Now, let $\mathcal A^k$ be the set of spikes $s_j^m$ for all $j$ and all time bins $m \leq k$, so that ${\mathcal A}_k$ represents the past spike events. Observe that in the model , the vector $\ubf_k$ can be computed from ${\mathcal A}_k$ and the noise $\xi^k_i$ is independent of $\mathcal A^k$. Also, from , $s_i^{\kp1}=1$ if and only if $\tilde{v}_i^k \geq \mu$. Hence, we have that the conditional probability of the spike event at some time $\kp1$, given the past spikes is \[eq:Psreg\] P(s\_i\^[1]{}=1\|[A]{}\_k) = ( ), where $\sigma^2_{ik}$ is the variance of $\xi_i^k$ in , and $\Phi(z)$ is the cumulative distribution function of a unit Gaussian. Given the conditional probability , we can then estimate the parameters $\betabf$, through the maximization \[eq:betaOpt\] (\_i,\_[IF,i]{}) = \_[\_i,b\_[IF,i]{}]{} \_[k=0]{}\^[1]{} L\_[ik]{}( \_[k]{}\^T \_i +c\_[ik]{}b\_[IF,i]{} - , s\_i\^k) + \_[j=1]{}\^N \|W\_[ij]{}\|, where $L_{ij}(z,s)$ is the probit loss function \[eq:probitLoss\] L\_[ik]{}(z,s) = -( (z/\_[ik]{})) & s\_i\^k=1\ -(1-(z/\_[ik]{})) & s\_i\^k = 0 Given the conditional probabilities , the minimization is precisely the maximum likelihood estimate of the parameters with an additional $\ell_1$ regularization term to encourage sparsity in the weights $\wbf_i$. But this minimization is exactly a sparse probit regression that is standard in linear classification [@Bishop:06]. The only issue is that the optimization function with the probit loss requires knowledge of the threshold $\mu$ and variances $\sigma^2_{ik}$. Since we are only interested in the connectivity weights up to a constant factor, we can arbitrarily set the threshold level $\mu$ to some value, say $\mu=1$. In principle, the noise variances $\sigma^2_{ik}$ can be derived from the integration noise variance $\tau_{IF}$ in . However, the variance $\tau_{IF}$ may itself not be initially known. Instead, we simply select $\sigma^2_{ik}$ to be a constant value that is relatively large to account for initial errors in the $s_i^k$. Acknowledgments {#acknowledgments .unnumbered} =============== This research was supported by NSF grants 1116589 and 1254204. The authors would like to thank Bruno Olshausen, Fritz Sommer, Lav Varshney, Mitya Chlovskii, Peyman Milanfar, Evan Lyall, and Eftychios Pnevmatikakis for their insights and support. This work would not have been possible without the supportive environment and wonderful discussions at the Berkeley Redwood Center for Theoretical Neuroscience – thank you.
--- author: - 'D. Jaffino Stargen,' - 'V. Sreenath[^1],' - 'and L. Sriramkumar' title: 'Quantum-to-classical transition and imprints of wavefunction collapse in bouncing universes' --- Introduction ============ The current cosmological observations seem to be well described by the so-called standard model of cosmology, which consists of the $\Lambda$CDM model, supplemented by the inflationary paradigm [@planck-2015-ccp; @planck-2015-ci]. The primary role played by inflation is to provide a causal mechanism for the generation of the primordial perturbations [@i-reviews], which later lead to the anisotropies in the Cosmic Microwave Background (CMB) and eventually to the inhomogeneities in the Large Scale Structure (LSS) [@cosmology-texts]. The nearly scale invariant power spectrum of primordial perturbations predicted by inflation has been corroborated by the state of the art observations of the CMB anisotropies by the Planck mission [@planck-2015-ci]. Despite the fact that inflation has been successful in helping to overcome some of the problems faced within the hot big bang model, the issue of the big bang singularity still remains to be addressed. Moreover, the remarkable efficiency of the inflationary scenario has led to a situation wherein, despite the constant improvement in the accuracy and precision of the cosmological observations, there seem to exist too many inflationary models that remain consistent with the data [@planck-2015-ci]. This situation has even provoked the question of whether, as a paradigm, inflation can be falsified at all (in this context, see the popular articles [@pa]). Due to these reasons, it seems important, even imperative, to systematically explore alternatives to inflation. One such alternative that has drawn a lot of attention in the literature are the bouncing scenarios [@bs-reviews]. In bouncing models, the universe goes through an initial phase of contraction, until the scale factor reaches a minimum value, and it undergoes expansion thereafter [@bs-reviews]. Driving a bounce often requires one to violate the null energy condition and hence, unlike inflation, they cannot be driven by simple, canonical scalar fields. In fact, the exact content of the universe which is responsible for the bounce remains to be satisfactorily understood. Also, concerns may arise whether quantum gravitational effects can become important at the bounce [@bounces-in-lqg]. To avoid such concerns, one often considers completely classical bounces wherein the energy densities of the matter fields driving the bounce always remain much smaller than the Planckian energy densities. In a fashion similar to slow roll inflation, certain bouncing models referred to as near-matter bounces, can also generate nearly scale invariant power spectra [@nsis-in-b; @rathul-2017], as is demanded by the observations [@planck-2015-ccp; @planck-2015-ci]. However, while proposing an inflationary model seems to be a rather easy task (which is reflected in the multitude of such models), a variety of problems (such as the need for fine tuned initial conditions and the rapid growth of anisotropies, to name just two) plague the bouncing models [@bs-reviews]. It would be fair to say that a satisfactory classical bouncing scenario that is devoid of these various issues is yet to be constructed. The generation of primordial perturbations in the early universe, whether in a bouncing or in an inflationary scenario, is a result of an interplay between quantum and gravitational physics [@dcp; @squeezing; @jerome-2008]. Since it is the quantum perturbations that lead to anisotropies in the CMB and inhomogeneities in the LSS, it provides a unique window to probe fundamental issues pertaining to quantum and gravitational physics. One such issue of interest is the mechanism underlying the transition of the quantum perturbations generated in the early universe to the LSS that can be completely described in terms of correlations involving classical stochastic variables, in other words, the quantum-to-classical transition of the primordial perturbations. While the issue of the quantum-to-classical transition of primordial perturbations has been studied to a good extent in inflation [@dcp; @squeezing; @jerome-2008], we find that there has been hardly any effort in this direction in the context of bouncing scenarios (see, however, Ref. [@leon-2016] which addresses issues similar to what we shall consider here). In this work, we shall investigate this problem for the case of tensor perturbations produced in a class of bouncing scenarios. We shall approach the problem from two different perspectives. Firstly, we shall examine the extent of squeezing of the quantum state associated with the tensor perturbations using the Wigner function [@dcp]. It has been found that, in the context of inflation, the primordial quantum perturbations become strongly squeezed once the modes leave the Hubble radius [@squeezing; @jerome-2008]. In strongly squeezed states, the quantum expectation values can be indistinguishable from classical stochastic averages of the correlation functions, such as those used to characterize the anisotropies in the CMB and the LSS [@dcp]. Specifically, we shall investigate if the Wigner function and the parameter describing the extent of squeezing behave in a similar manner in the bouncing scenarios. Secondly, we shall study the issue from the perspective of a quantum measurement problem. The quantum measurement problem concerns the phenomenon by which a quantum state upon measurement collapses to one of the eigenstates of the observable under measurement. In the cosmological context, this problem translates as to how the quantum state of the primordial perturbations collapse into the eigenstate, say, corresponding to the CMB observed today. This problem is aggravated in the cosmological context due to the fact that there were no observers in the early universe to carry out any measurements [@qm-in-c]. One of the proposals which addresses the quantum measurement problem is the so-called Continuous Spontaneous Localization (CSL) model [@csl]. The advantage of using the CSL model to study the quantum measurement problem in the context of cosmology is that, in this model, the collapse of the wavefunction occurs without the presence of an observer. In the CSL model, the Schrödinger equation is modified by adding non-linear and stochastic terms which suppress the quantum effects in the classical domain, and also reproduce the predictions of quantum mechanics in the quantum regime (for reviews, see Refs. [@csl-reviews]). In the context of inflation, there have been attempts to understand the quantum measurement problem by employing the CSL model [@jerome-2012; @suratna-2013]. Motivated by these efforts in the context of inflation, in this work, we shall investigate the quantum-to-classical transition in bouncing scenarios from the two perspectives described above. The remainder of this paper is organized as follows. In Sec. \[sec:qsp\], working in the Schrödinger picture, we shall quickly review the quantization of the tensor perturbations in an evolving universe and arrive at the wavefunction governing the perturbations. In Sec. \[sec:tps-mb\], we shall describe the evolution of the tensor perturbations in a specific matter bounce scenario and obtain the resulting tensor power spectrum. In Sec. \[sec:stm\], using the Wigner function, we shall examine the squeezing of the quantum state describing the tensor modes as they evolve in a matter bounce. In Sec. \[sec:i-csl-mb\], after a brief summary of the essential aspects of the CSL mechanism, we shall study its imprints on the tensor power spectrum produced in a matter bounce. In Sec. \[sec:i-csl-tps-gb\], we shall discuss the evolution of the tensor perturbations in a more generic bounce and evaluate the corresponding tensor power spectrum, including the effects due to CSL. Finally, in Sec. \[sec:csl-c\], we shall conclude with a brief summary of the main results. Note that we shall work with natural units wherein $\hbar=c=1$, and define the Planck mass to be $M_{_{\rm Pl}}=(8\, \pi\, G)^{-1/2}$. Working in $(3+1)$-spacetime dimensions, we shall adopt the metric signature of $(+,-,-,-)$. Also, overprimes shall denote differentiation with respect to the conformal time coordinate $\eta$. Quantization of the tensor perturbations in the Schrödinger picture {#sec:qsp} =================================================================== We shall consider a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe which is described by the line element $$\d s^2=a^2(\eta)\,\l(\d\eta^2-\delta_{ij}\,\d x^i\,\d x^j\r),$$ where $a(\eta)$ denotes the scale factor, with $\eta$ being the conformal time coordinate. Upon taking into account the tensor perturbations, say, $h_{ij}$, the FLRW metric assumes the form $$\d s^2=a^2(\eta)\,\l[\d\eta^2-\l(\delta_{ij}+h_{ij}\r)\,\d x^i\,\d x^j\r],$$ where $h_{ij}$ satisfies the traceless and transverse conditions ([i.e. ]{} $h^i_i=0$ and $\partial_jh^{ij}=0$). The second order action governing the tensor perturbations $h_{ij}$ is given by (see the following reviews [@i-reviews]) $$\delta_2S=\frac{\Mpl^2}{8}\, \int \d\eta\,\int\d^3 \x\; a^2(\eta)\, \l[h_{ij}'^2-(\partial h_{ij})^2\r]. \label{eq:ag-tp}$$ The homogeneity and isotropy of the background metric permits the following Fourier decomposition of the tensor perturbations: $$h_{ij}(\eta,{\bm x}) =\sum_{s=1}^2\,\int \f{\d^3{\k}}{(2\,\pi)^{3/2}}\, \varepsilon_{ij}^s(\k)\, h_{\k}(\eta)\,{\rm e}^{i\,\k\cdot\x},$$ where $\varepsilon_{ij}^s(\k)$ denotes the polarization tensor, with $s$ representing the helicity. The polarization tensor satisfies the normalization condition: $\varepsilon_{ij}^r(\k)\, \varepsilon_{ij}^{s}(\k)=2\,\delta^{rs}$ [@i-reviews]. In terms of the Fourier modes $h_{\k}$, the second order action (\[eq:ag-tp\]) can be expressed as $$\delta_2S =\frac{\Mpl^2}{2}\,\int \d\eta\, \int \d^3\k\; a^2(\eta)\, \l[h_{\k}'(\eta)\, h_\k'^{}(\eta)-k^2\, h_{\k}(\eta)\,h_{\k}^{}(\eta)\r],$$ where $k =\vert\k\vert$. Note that, since $h_{ij}(\eta,{\bm x})$ is real, the integral over $\k$ runs over only half of the Fourier space, [i.e. ]{} $\mathbb{R}^{3+}$. It proves to be convenient to express the tensor modes in terms of the so-called Mukhanov-Sasaki variable $u_\k$ as $h_{\k}= (\sqrt{2}/\Mpl)\, (u_{\k}/a)$ [@i-reviews]. In terms of the Mukhanov-Sasaki variable, the second order action (\[eq:ag-tp\]) takes the form $$\delta_2S=\int \d \eta\, \int \d^3 {\k}\,\l[u_{{\k}}'\,u_{\k}'^{} -\omega_k^2(\eta)\,u_{\k}\, u_{\k}^{}\r],\label{eq:ag-tp-msv}$$ where $$\omega_k^2(\eta)= k^2-\f{a''}{a}.\label{eq:wk}$$ It should be noted that, upon varying the action with respect to $u_\k$, one obtains the following equation of motion governing $u_\k$: $$u_\k''+\omega_k^2(\eta)\,u_\k=0.\label{eq:uk-eom}$$ The momenta associated with the variables $u_\k$ and $u_\k^$ are given by $$p_\k=u_\k'^,\quad p_\k^=u_\k'.$$ The Hamiltonian associated with the above second order action can be determined to be $${\sf H}=\int \d^3 \k\,\l[p_{\k}\,p_{\k}^{}+\omega_k^2(\eta)\, u_{\k}\,u_{\k}^{}\r].$$ To carry out the quantization procedure, we need to deal with real variables (see, for instance, Refs. [@dcp; @jerome-2008]). Hence, let us write the variables $u_{\k}$ and $p_{\k}$ as $$u_{\k}= \f{1}{\sqrt{2}}\,\l(u_{\k}^{\R}+i\, u_{\k}^{\I}\r),\quad p_{\k}= \f{1}{\sqrt{2}}\,\l(p_{\k}^{\R}+i\, p_{\k}^{\I}\r),$$ where the superscripts $\R$ and $\I$ denote the real and imaginary parts of the corresponding quantities. In terms of these new variables, the Hamiltonian ${\sf H}$ is given by $${\sf H}=\int \d^3 {\k}\;{\sf H}_{\k} =\int \d^3 \k\; \l({\sf H}_{\k}^{\R}+{\sf H}_{\k}^{\I}\r),$$ where $${\sf H}_{{\k}}^{\R,\I} =\f{1}{2}\,(p_{\k}^{\R,\I})^2 +\f{1}{2}\,\omega_k^2(\eta)\,(u_{\k}^{\R,\I})^2.\label{eq:Hk}$$ It is evident from the structure of the Hamiltonian ${\sf H}$ that each variable $u_{\k}^{\R,\I}$ evolves independently as a parametric oscillator with the time-dependent frequency $\omega_k(\eta)$. Therefore, the complete quantum state of the system, say, $\Psi(u_\k,\eta)$, can be written as a product of the wavefunctions of the individual modes, say, $\psi_{\k}(u_{\k},\eta)$, in the following form: $$\Psi(u_\k,\eta)=\prod_{\k}\psi_{\k}(u_{\k},\eta) =\prod_{\k}\psi_{\k}^{\R}(u_{\k}^{\R},\eta)\; \psi_{\k}^{\I}(u_{\k}^{\I},\eta).$$ Quantization of the tensor perturbations can be achieved by promoting the variables $u_{\k}^{\R,\I}$ and $p_{\k}^{\R,\I}$ to quantum operators which satisfy the following non-trivial canonical commutation relations: $$\l[\hat u_{\k}^{\R},\,\hat p_{\k'}^{\R}\r]=i\,\delta^{(3)}(\k-\k'),\quad \l[\hat u_{\k}^{\I},\hat p_{\k'}^{\I}\r]=i\,\delta^{(3)}(\k-\k').$$ The Schrödinger equation governing the evolution of the quantum state $\psi_{\k}^{\R,\I}$ corresponding to the mode $\k$ is given by $$i\,\frac{\partial\psi_\k^{\R,\I} }{\partial \eta} =\hat {\sf H}_{\k}^{\R,\I}\, \psi_{\k}^{\R,\I}.\label{eq:se}$$ Upon using the following representation for $\hat u_{\k}^{\R,\I}$ and $\hat p_{\k}^{\R,\I}$: $$\hat u_{\k}^{\R,\I}\,\Psi=u_{\k}^{\R,\I}\,\Psi,\quad \hat p_{\k}^{\R,\I}\,\Psi=-i\frac{\partial\Psi}{\partial u_{\k}^{\R,\I}},$$ one can write the Hamiltonian operator in Fourier space $\hat {\sf H}_\k^{\R,\I}$ as $$\hat {\sf H}_{\k}^{\R,\I} =-\f{1}{2}\,\frac{\pa^2}{\pa (u_{\k}^{\R,\I})^2} +\f{1}{2}\,\omega_k^2(\eta)\,(\hat u_{\k}^{\R,\I})^2.\label{eq:Hko}$$ It is well known that the wavefunction characterizing a time-dependent oscillator evolving from an initial ground state can be expressed as (see, for instance, Refs. [@dcp]) $$\psi_{\k}^{\R,\I}(u_{\k}^{\R,\I},\eta) =N_{k}(\eta)\,{\rm exp}-\l[\Omega_{k}(\eta)\, (u_{\k}^{\R,\I})^2\r],\label{eq:wf}$$ where $N_k$ is the normalization constant which can be determined (up to a phase) to be $ N_{k}=\l(2\,\Omega_{k}^{\R}/\pi\r)^{1/4}$, with $\Omega_{k}^{\R}$ denoting the real part of $\Omega_{k}$. If we now write $\Omega_{k}=-(i/2)\,f_{k}'/f_{k}$ and substitute the above wave function in the Schrödinger equation (\[eq:se\]), then one finds that the function $f_k$ satisfies the same classical equation of motion \[[i.e. ]{} Eq. (\[eq:uk-eom\])\] as the Mukhanov-Sasaki variable $u_\k$. In other words, if we know the solution to the classical Mukhanov-Sasaki equation, then we can arrive at the complete wavefunction $\psi_{\k}^{\R,\I}$ \[cf. Eq. (\[eq:wf\])\] describing the tensor modes. (Note that, since the equation governing $f_k$ and $u_\k$ are the same, hereafter, we shall often refer to $f_k$ as the tensor mode.) Tensor modes and power spectrum in a matter bounce {#sec:tps-mb} ================================================== We shall be interested in bouncing scenarios where the scale factor $a(\eta)$ is of the form $$a(\eta)=a_0\,\l[1+(\eta/\eta_0)^2\r]^p=a_0\,\l[1+(k_0\,\eta)^2\r]^p, \label{eq:sf}$$ where $a_0$ is the minimum value of the scale factor at the bounce ([i.e. ]{} at $\eta=0$), $p$ is a positive real number, and ${\eta}_0={k_0}^{-1}$ is the time scale associated with the bounce. It is clear from the form of the scale factor that the universe starts in a contracting phase at large negative $\eta$ with the scale factor reaching a minimum at $\eta = 0$, and expands thereafter. Before we discuss the case of evolution of the tensor modes in a bouncing universe characterized by an arbitrary value of $p$, it is instructive to consider the simpler case of $p = 1$. Such a bounce is often referred to as a matter bounce, since, at early times, far away from the bounce, the scale factor behaves in the same manner as in a matter dominated era, [i.e. ]{} as $a(\eta) \propto \eta^2$. The evolution of the tensor modes and the resulting power spectrum in such a matter bounce has been discussed before (see for instance, Refs. [@starobinsky-1979; @wands-1999; @debika-2015]). For the sake of completeness, we shall briefly present the essential derivation here. We need to evolve the modes from early times during the contracting phase, across the bounce until a suitable time after the bounce, when we have to evaluate the power spectrum. In order to arrive at an analytical expression for the tensor modes, it is convenient to divide this period of interest into two domains. Let the time range $-\infty<\eta<-\alpha\,\eta_0$ be the first domain, where the parameter $\alpha$ is a large number, say, $10^5$. This period is far away from the bounce and corresponds to very early times, Since, in this domain, $\eta \ll -\eta_0$, the scale factor behaves as $a(\eta)\simeq a_0\,(k_0\,\eta)^2$. Therefore, the differential equation describing the tensor modes in the first domain reduces to $$f_{k}''+\l(k^2-\f{2}{\eta^2}\r)f_{k}\simeq 0.\label{eq:ms-1}$$ This is exactly the equation of motion satisfied by the tensor modes in de Sitter inflation, whose solutions are well known [@wands-1999]. If we assume that, at very early times during the contracting phase, the oscillator corresponding to each tensor mode is in its ground state, then, we require that $\Omega_k=k/2$ for $\eta\ll-\eta_0$. This, in turn, corresponds to demanding that, for $\eta\ll-\eta_0$, the tensor mode $f_k$ behaves as $$f_k(\eta) \simeq \f{1}{\sqrt{2\,k}}\,{\rm e}^{i\,k\,\eta}, \label{eq:bd-ic}$$ which essentially corresponds to the Bunch-Davies initial condition, had we been working in the Heisenberg picture [@bunch-1978]. Let $\eta_k$ be the time when $k^2=a''/a$, [i.e. ]{} when the modes leave the Hubble radius during the contracting phase. For cosmological modes such that $k/k_0\ll 1$, $\eta_k\simeq -\sqrt{2}/k$. (If, say, $k_0/a_0\simeq \Mpl$, one finds that $k/k_0$ is of the order of $10^{-28}$ or so for scales of cosmological interest.) The Bunch-Davies initial condition can be imposed when $\eta\ll\eta_k$. We shall assume that $\eta_k\ll-\alpha\,\eta_0$, which corresponds to $k\ll k_0/\alpha$. Since, as we mentioned, Eq. (\[eq:ms-1\]) resembles that of the equation in de Sitter inflation, the tensor mode $f_k$ satisfying the Bunch-Davies initial condition can be immediately written down to be [@starobinsky-1979; @wands-1999; @debika-2015] $$f_{k}^{({\rm I})}(\eta)=\f{1}{\sqrt{2\,k}}\, \l(1+\f{i}{k\,\eta}\r)\,{\rm e}^{i\,k\,\eta}.$$ The solution $f_k^{({\rm I})}$ we have obtained above corresponds to the first domain, [i.e. ]{} over $-\infty<\eta<-\alpha\,\eta_0$. Let us now turn to the evolution of the mode during the second domain, which covers the period of bounce. The domain corresponds $-\alpha\,\eta_0<\eta<\beta\,\eta_0$, where we shall assume $\beta$ to be of the order of $10^2$. Over this domain, for scales of our interest ([i.e. ]{} $k\ll k_0/\alpha$), we can ignore the $k^2$ term in Eq. (\[eq:uk-eom\]) which governs $f_k$. In such a case, the equation simplifies to $$f_k''-\f{a''}{a}f_{k}\simeq 0\label{eq:ms-2}$$ or, equivalently, in terms of the original variable $h_k$, to $$h_{k}''+2\,\f{a'}{a}\,h_{k}'\simeq 0.$$ Using the exact form (\[eq:sf\]) of the scale factor, this equation can be immediately integrated to obtain the following solution in the second domain [@debika-2015]: $$f_k^{({\rm II})}(\eta)=a(\eta)\,\l[A_k+B_k\,g(k_0\,\eta)\r],$$ where $A_k$ and $B_k$ are constants, and the function $g(x)$ is given by $$g(x) = \f{x}{1+x^2}+{\rm tan}^{-1}(x).\label{eq:g}$$ The constants $A_k$ and $B_k$ are arrived at by matching the solutions $f_k^{({\rm I})}$ and $f_k^{({\rm II})}$ and their derivatives with respect to $\eta$ at $-\alpha\,\eta_0$. We find that $A_k$ and $B_k$ are given by $$\begin{aligned} A_k &=& \frac{1}{\sqrt{2\,k}}\,\l(\f{1}{a_0\,\alpha^2}\r)\, \l(1-\f{i\,k_0}{\alpha\, k}\r)\,{\rm e}^{-i\,\alpha\, k/k_0} +B_k\, g(\alpha),\\ B_{k} &=&\f{1}{\sqrt{2\,k}}\, \f{\l(1+\alpha^2\r)^2}{2\,a_0\,\alpha ^2}\, \l(\f{i\,k}{k_0}+\f{3}{\alpha}-\f{3\,i\,k_0}{\alpha^2\, k}\r)\, {\rm e}^{-i\,\alpha\, k/k_0}.\end{aligned}$$ Assuming that the universe transits to the conventional radiation dominated epoch at the end of the second domain, we evaluate the tensor power spectrum at $\eta=\beta\,\eta_0$ after the bounce [@debika-2015]. Recall that the tensor power spectrum is defined in terms of the mode function $f_k(\eta)$ as [@i-reviews] $${\cal P}_{_{\rm T}}(k) =\f{8}{\Mpl^2}\,\f{k^3}{2\,\pi^2}\, \f{\vert f_k(\eta)\vert^2}{a^2(\eta)}.\label{eq:tps-d}$$ On using the solution $f_k^{({\rm II})}$ above, the tensor power spectrum at $\eta=\beta/k_0$ can be expressed as $${\cal P}_{_{\rm T}}(k) =\f{8}{\Mpl^2}\,\f{k^3}{2\,\pi^2}\, \vert A_k+B_k\,g(\beta)\vert^2.$$ We have plotted the resulting tensor power spectrum as a function of $k/k_0$ in Fig. \[fig:tps\] for a set of values of the parameters. ![\[fig:tps\]The tensor power spectrum in the matter bounce scenario \[[i.e. ]{} when $p=1$ in Eq. (\[eq:sf\])\] has been plotted as a function of $k/k_0$. Actually, we find that the power spectrum depends only on the combination $k_0/a_0$. We have set $k_0/(a_0\,\Mpl)=10^{-5}$, $\alpha=10^5$ and $\beta=10^2$ in plotting this figure. As expected, the power spectrum is scale invariant for modes such that $k/(k_0/\alpha)\ll 1$, the range over which our analytical approximations are valid.](tps-mb.pdf){width="13.0cm"} Note that our analytical expressions and the resulting power spectrum are valid only for modes such that $k\ll (k_0/\alpha)$. It is clear that the power spectrum is scale invariant over this range of wavenumbers. Such a scale invariant spectrum is indeed expected to arise in a matter bounce as the scenario is ‘dual’ to de Sitter inflation (in this context, see Ref. [@wands-1999]). Squeezing of quantum states associated with tensor modes in the matter bounce {#sec:stm} ============================================================================= Having discussed the evolution of the tensor modes through a matter bounce, let us turn our attention to the behavior of the quantum state $\psi_{\k}$. We shall essentially follow the approach adopted in the context of perturbations generated during inflation [@dcp; @squeezing; @jerome-2012; @suratna-2013]. In classical mechanics, one of the ways of understanding the evolution of a system is to examine its behavior in phase space. However, since canonically conjugate variables cannot be measured simultaneously in quantum mechanics, a method needs to be devised in order to compare the evolution of a quantum system with its classical behavior in phase space. As is well known, one of the ways to understand the evolution of a quantum state is to examine the behavior of the so-called Wigner function, which is a quasi-probability distribution in phase space that can be constructed from a given wave function. Recall that the wave function corresponding to a tensor mode can be expressed as $$\psi_{\k}(u_\k,\eta) =\psi_{\k}^\R(u_\k^\R,\eta)\;\psi_{\k}^\I(u_{\k}^\I,\eta) =N_{k}^2\,{\rm exp}-\l(2\,\Omega_{k}\,u_\k\,u_\k^\r),\label{eq:cwf}$$ where, as mentioned before, $N_{k}=(2\,\Omega_{k}^{\R}/\pi)^{1/4}$, $\Omega_{k} =-(i/2)\,f_{k}'/f_{k}$ and $f_k$ satisfies the differential equation (\[eq:uk-eom\]). The Wigner function associated with the quantum state (\[eq:cwf\]) is defined as [@dcp; @jerome-2012; @suratna-2013] $$\begin{aligned} W(u_{\k}^{\rm R},u_{\k}^{\rm I},p_{\k}^{\rm R},p_{\k}^{\rm I},\eta) &=& \f{1}{(2\,\pi)^2}\, \int_{-\infty}^{\infty}\d x\;\int_{-\infty}^{\infty}\d y\; \psi_{\k}\l(u_{\k}^{\rm R}+\f{x}{2},u_{\k}^{\rm I}+\f{y}{2},\eta\r)\nn\\ & &\qquad\quad \times\,\psi^_{\k}\l(u_{\k}^{\R}-\f{x}{2},u_{\k}^{\rm I}-\f{y}{2},\eta\r)\, {\rm exp} -i\,\l(p_{\k}^{\rm R}\, x+p_{\k}^{\rm I}\, y\r).\end{aligned}$$ The integrals over $x$ and $y$ can be easily evaluated to arrive at the following form for the Wigner function [@jerome-2012] $$\begin{aligned} W(u_{\k}^{\R},u_{\k}^{\I},p_{\k}^{\R},p_{\k}^{\I},\eta) &=&\f{\l\vert\psi_{\k}(u_{\k},\eta)\r\vert^2}{2\,\pi\,\Omega_k^\R}\; {\rm exp}-\l[\f{1}{2\,\Omega_{k}^\R}\, \l(p_{\k}^{\rm R}+2\,\Omega_{\k}^\I\,u_{\k}^{\rm R}\r)^2\r]\nn\\ & &\qquad\qquad\quad\; \times\, {\rm exp}-\l[\f{1}{2\,\Omega_{k}^\R}\, \l(p_{\k}^{\rm I}+2\,\Omega_{\k}^\I u_{\k}^{\rm I}\r)^2\r].\end{aligned}$$ Since we know the mode functions $f_k$, we can evaluate $\Omega_k^\R$ and $\Omega_k^\I$ and thereby determine the above Wigner function as a function of time. Note that, in inflation, to cover a wide range in time, one often works with e-folds, say, $N$, as the time variable. The e-folds are defined through the relation $a(N)=a_{\rm i}\, {\rm exp}\, (N-N_{\rm i})$, where, evidently, $a=a_{\rm i}$ at $N=N_{\rm i}$. However, the exponential function ${\rm e}^{N}$ is a monotonically growing function and hence does not seem appropriate to describe bounces. In the context of bounces, particularly the symmetric ones of our interest, it seems more suitable to introduce a new variable $\cN$ known as e-${\cal N}$-folds, which is defined through the relation $a(\cN)=a_0\,{\rm exp}\l({\cal N}^2/2\r)$ [@sriram-2015]. In the matter bounce, the conformal time coordinate $\eta$ is related to e-${\cal N}$-folds as $$\eta({\cal N})=\pm k_0^{-1}\l(e^{{\cal N}^2/2}-1\r)^{1/2},$$ with $\cN$ being zero at the bounce, while it is negative before the bounce and positive after. Using the above relation $\eta({\cal N})$, we have converted the Wigner function as a function of $\cN$. In Fig. \[fig:wf\], we have illustrated the behavior of the function in terms of contour plots in the $(u_{\k}^{\rm R},p_{\k}^{\rm R})$-plane as a tensor mode (corresponding to a scale of cosmological interest) evolves across the bounce. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -10pt ![\[fig:Wigner\]The evolution of the Wigner function $W(u_{\k}^{\rm R},u_{\k}^{\rm I},p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)$ associated with the quantum state that describes a tensor mode of cosmological interest. Out of the two independent sets of variables $(u_{\k}^{\rm R},p_{\k}^{\rm R})$ and $(u_{\k}^{\rm I},p_{\k}^{\rm I})$, we have chosen the set $(u_{\k}^{\rm R}, ![\[fig:Wigner\]The evolution of the Wigner function $W(u_{\k}^{\rm R},u_{\k}^{\rm I},p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)$ associated with the quantum state that describes a tensor mode of cosmological interest. Out of the two independent sets of variables $(u_{\k}^{\rm R},p_{\k}^{\rm R})$ and $(u_{\k}^{\rm I},p_{\k}^{\rm I})$, we have chosen the set $(u_{\k}^{\rm R}, p_{\k}^{\rm R})$ and have fixed $(u_{\k}^{\rm I},p_{\k}^{\rm I})=(0,0)$ to illustrate the behavior of the quantity $\ln\, [W(u_{\k}^{\rm R},u_{\k}^{\rm I}, p_{\k}^{\rm R})$ and have fixed $(u_{\k}^{\rm I},p_{\k}^{\rm I})=(0,0)$ to illustrate the behavior of the quantity $\ln\, [W(u_{\k}^{\rm R},u_{\k}^{\rm I}, p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)]$. In plotting these figures, we have set $k_0/(a_0\,\Mpl)=10^{-5}$ as in the previous figure, and have chosen the mode corresponding to $k/k_0=10^{-15}$. The plots correspond to the times ${\cal N}=-13$ (top left), ${\cal N} = -12.1$ (top right), ${\cal N}=0$ (bottom left) and ${\cal N} = 5$ (bottom right). The first two instances ([viz. ]{} when ${\cal N}=-13$ and ${\cal N} = -12.1$) correspond to situations when the mode is in the strongly sub-Hubble domain and close to Hubble exit during the contracting phase, respectively. Note that, as time evolves, the Gaussian state that is initially symmetric in $u_{\k}^{\rm R}$ and $p_{\k}^{\rm R}$ (top left) gets increasingly squeezed about about $u_\k=0$ (top right, bottom left) as one approaches the bounce, and remains so (bottom right) as the universe begins to expand. This largely reflects the behavior that occurs in the inflationary scenario.[]{data-label="fig:wf"}](wigner-neq-13.pdf "fig:"){width="7.50cm"} p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)]$. In plotting these figures, we have set $k_0/(a_0\,\Mpl)=10^{-5}$ as in the previous figure, and have chosen the mode corresponding to $k/k_0=10^{-15}$. The plots correspond to the times ${\cal N}=-13$ (top left), ${\cal N} = -12.1$ (top right), ${\cal N}=0$ (bottom left) and ${\cal N} = 5$ (bottom right). The first two instances ([viz. ]{} when ${\cal N}=-13$ and ${\cal N} = -12.1$) correspond to situations when the mode is in the strongly sub-Hubble domain and close to Hubble exit during the contracting phase, respectively. Note that, as time evolves, the Gaussian state that is initially symmetric in $u_{\k}^{\rm R}$ and $p_{\k}^{\rm R}$ (top left) gets increasingly squeezed about about $u_\k=0$ (top right, bottom left) as one approaches the bounce, and remains so (bottom right) as the universe begins to expand. This largely reflects the behavior that occurs in the inflationary scenario.[]{data-label="fig:wf"}](wigner-neq-12p1.pdf "fig:"){width="7.50cm"} -10pt ![\[fig:Wigner\]The evolution of the Wigner function $W(u_{\k}^{\rm R},u_{\k}^{\rm I},p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)$ associated with the quantum state that describes a tensor mode of cosmological interest. Out of the two independent sets of variables $(u_{\k}^{\rm R},p_{\k}^{\rm R})$ and $(u_{\k}^{\rm I},p_{\k}^{\rm I})$, we have chosen the set $(u_{\k}^{\rm R}, ![\[fig:Wigner\]The evolution of the Wigner function $W(u_{\k}^{\rm R},u_{\k}^{\rm I},p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)$ associated with the quantum state that describes a tensor mode of cosmological interest. Out of the two independent sets of variables $(u_{\k}^{\rm R},p_{\k}^{\rm R})$ and $(u_{\k}^{\rm I},p_{\k}^{\rm I})$, we have chosen the set $(u_{\k}^{\rm R}, p_{\k}^{\rm R})$ and have fixed $(u_{\k}^{\rm I},p_{\k}^{\rm I})=(0,0)$ to illustrate the behavior of the quantity $\ln\, [W(u_{\k}^{\rm R},u_{\k}^{\rm I}, p_{\k}^{\rm R})$ and have fixed $(u_{\k}^{\rm I},p_{\k}^{\rm I})=(0,0)$ to illustrate the behavior of the quantity $\ln\, [W(u_{\k}^{\rm R},u_{\k}^{\rm I}, p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)]$. In plotting these figures, we have set $k_0/(a_0\,\Mpl)=10^{-5}$ as in the previous figure, and have chosen the mode corresponding to $k/k_0=10^{-15}$. The plots correspond to the times ${\cal N}=-13$ (top left), ${\cal N} = -12.1$ (top right), ${\cal N}=0$ (bottom left) and ${\cal N} = 5$ (bottom right). The first two instances ([viz. ]{} when ${\cal N}=-13$ and ${\cal N} = -12.1$) correspond to situations when the mode is in the strongly sub-Hubble domain and close to Hubble exit during the contracting phase, respectively. Note that, as time evolves, the Gaussian state that is initially symmetric in $u_{\k}^{\rm R}$ and $p_{\k}^{\rm R}$ (top left) gets increasingly squeezed about about $u_\k=0$ (top right, bottom left) as one approaches the bounce, and remains so (bottom right) as the universe begins to expand. This largely reflects the behavior that occurs in the inflationary scenario.[]{data-label="fig:wf"}](wigner-neq-0.pdf "fig:"){width="7.50cm"} p_{\k}^{\rm R},p_{\k}^{\rm I},\eta)]$. In plotting these figures, we have set $k_0/(a_0\,\Mpl)=10^{-5}$ as in the previous figure, and have chosen the mode corresponding to $k/k_0=10^{-15}$. The plots correspond to the times ${\cal N}=-13$ (top left), ${\cal N} = -12.1$ (top right), ${\cal N}=0$ (bottom left) and ${\cal N} = 5$ (bottom right). The first two instances ([viz. ]{} when ${\cal N}=-13$ and ${\cal N} = -12.1$) correspond to situations when the mode is in the strongly sub-Hubble domain and close to Hubble exit during the contracting phase, respectively. Note that, as time evolves, the Gaussian state that is initially symmetric in $u_{\k}^{\rm R}$ and $p_{\k}^{\rm R}$ (top left) gets increasingly squeezed about about $u_\k=0$ (top right, bottom left) as one approaches the bounce, and remains so (bottom right) as the universe begins to expand. This largely reflects the behavior that occurs in the inflationary scenario.[]{data-label="fig:wf"}](wigner-neq5.pdf "fig:"){width="7.50cm"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- In a time-dependent background, the modes associated with quantum fields are generally expected to get increasingly squeezed as time evolves [@squeezing]. Let us now try to understand the extent to which the tensor modes are squeezed in the matter bounce scenario. If we define $f_k$ as [@jerome-2012] $$f_{k} = \f{1}{\sqrt{2\,k}}\,(\tilde u_{k}+{\tilde v_{k}^}),$$ then the second order differential equation (\[eq:uk-eom\]) governing $f_k$ can be written as two coupled first order differential equations as follows: $$\tilde u_{k}'=i\,k\,\tilde u_{k}+\f{a'}{a}\,\tilde v_{k}^,\quad \tilde v_{k}'=i\,k\,\tilde v_{k}+\f{a'}{a}\,\tilde u_{k}^. \label{eq:ukt-vkt}$$ The Wronskian, say, ${\sf W}$, corresponding to the equation governing $f_k$ is defined as ${\sf W}= f_{k}'\,f_{k}^-f_{k}'^{}\,f_{k}$. It can be readily shown using equation (\[eq:uk-eom\]) that $\d {\sf W}/\d\eta =0$ or, equivalently, ${\sf W}$ is a constant. If we assume that the modes $f_k$ satisfy the Bunch-Davies initial condition (\[eq:bd-ic\]), then one finds that ${\sf W}=i$. In terms of $\tilde{u}_{k}$ and $\tilde{v}_{k}$, the Wronskian can be expressed as ${\sf W}=i\,(\vert\tilde u_{k}\vert^2-\vert\tilde v_{k}\vert^2)$. Since ${\sf W}=i$, we can parametrize the variables $\tilde{u}_{k}$ and $\tilde{v}_{k}$ as [@jerome-2008; @jerome-2012] $$\tilde{u}_k={\rm e}^{i\,\theta_{k}}\,{\rm cosh}\,(r_{k}),\quad \tilde{v}_k={\rm e}^{-i\,\theta_{k}+2\,i\,\phi_{\k}}\,{\rm sinh}\,(r_{k}), \label{eq:rk-d}$$ where $r_k$, $\theta_k$ and $\phi_k$ are known as the squeezing parameter, the rotation and squeezing angles, respectively. On substituting the expressions (\[eq:rk-d\]) in Eqs. (\[eq:ukt-vkt\]), one can arrive at a set of coupled differential equations which determine the behavior of the parameters $r_k$, $\theta_k$ and $\phi_k$ with respect to $\eta$ [@jerome-2012; @jerome-2008]. The coupled differential equations governing these parameters are given by \[eq:s-eom\] $$\begin{aligned} r_{k}'&=&\f{a'}{a}\,{\rm cos}\,(2\phi_{k}),\label{eq:rk-eom}\\ \phi_{k}'&=&k-\f{a'}{a}\,{\rm coth}\,(2r_{k})\,{\rm sin}\,(2\,\phi_{k}),\\ \theta_{k}'&=&k-\f{a'}{a}\,{\rm tanh}\,(r_{k})\,{\rm sin}\,(2\,\phi_{k}).\end{aligned}$$ Our primary quantity of interest is the parameter $r_{k}$ which characterizes the extent of squeezing of the quantum state $\psi_{\k}(u_{\k},\eta)$ as the universe evolves [@squeezing]. By assuming the scale factor of interest, one can attempt to solve the differential equations (\[eq:s-eom\]) to arrive at the behavior of the squeezing parameter. These equations essentially stem from the original equation (\[eq:uk-eom\]) that determines the evolution of the Mukhanov-Sasaki variable $u_k$ or $f_k$. Since, we already know the solution to $f_k$ across the bounce, it would be simpler to express the parameters $r_k$, $\theta_k$ and $\phi_k$ in terms of $f_k$. To begin with, we find that the variables $\tilde u_{k}$ and $\tilde v_{k}$ can be expressed in terms of $f_k$ and its derivative $f_k'$ as follows: $$\tilde u_{k}=\sqrt{\frac{k}{2}}\,\l(1+\f{i}{k}\,\f{a'}{a}\r)\,f_k -\f{i}{\sqrt{2\,k}}\,f_k',\quad \tilde v_{k} =\sqrt{\frac{k}{2}}\,\l(1+\f{i}{k}\,\f{a'}{a}\r)\,f_k^\ast -\f{i}{\sqrt{2\,k}}\,f_k'^\ast$$ and it is straightforward to examine that $\vert\tilde u_{k}\vert^2 -\vert\tilde v_{k}\vert^2=1$, as required. Once we have these two quantities at hand, we can obtain the squeezing parameters $r_k$, $\phi_k$ and $\theta_k$ from the relations $$r_{k}={\rm sinh}^{-1}\l(\vert\tilde v_{k}\vert\r),\quad \phi_k=\f{1}{2}\,{\rm Arg}\,(\tilde u_{k} \tilde v_{k}),\quad \theta_k={\rm Arg}\,(\tilde u_{k}).$$ Using the solutions for $f_k$ we have obtained in the case of the matter bounce in the previous section, we have plotted the behavior of the squeezing parameter $r_k$ as a function of e-$\cN$-folds in Fig. \[fig:s\]. ![The analytical (in red) and the numerical (in blue) solutions for the squeezing parameter $r_k$ have been plotted as a function of e-$\cN$-folds, for a mode corresponding to $k/k_0=10^{-15}$ and values of the parameters of the model mentioned in the earlier figures. Since we begin with the Bunch-Davies initial condition at very early times, the squeezing parameter $r_k$ is close to zero. As the universe contracts, $r_k$ increases till it reaches a maximum at the bounce. Then it decreases to some extent after the bounce, before the universe is assumed to enter the radiation dominated era.[]{data-label="fig:s"}](rk.pdf){width="12.5cm"} We have also independently solved the differential equations (\[eq:s-eom\]) using [Mathematica]{} [@Mathematica] to check the validity of the analytical solution for $r_k$. The numerical solution has also been plotted in the figure. The agreement between the solutions clearly indicate the extent of accuracy of the analytical solutions. It is evident from the figure that the wave function associated with the quantum state $\psi_\k$ is increasingly squeezed as the universe evolves, reaching a maximum at the bounce. In fact, it is this behavior which was reflected in the behavior of the Wigner function (which had peaks about $u_\k=0$) we had considered earlier. While there are similarities in the behavior with what occurs in inflation, there are some crucial differences as well. In inflation, the parameter increases indefinitely with the duration of inflation [@jerome-2012]. For a duration corresponding to about $60$ e-folds of inflation, as is typically required to overcome the horizon problem, $r_k$ is found to grow to about $10^2$ or so. We find that $r_k$ grows to the same order of magnitude in the matter bounce as well. In accordance with the Heisenberg’s uncertainty principle, squeezing of the quantum state about $u_{\k}=0$ gives infinite possibilities of the momentum variable $p_{\k}$. Hence, a squeezed state is not strictly a classical state. But, it has been argued that in a strongly squeezed quantum state, the vacuum expectation values and the stochastic mean are indistinguishable, if the perturbations are assumed to be realizations of a classical stochastic process [@dcp]. In such a sense, one can argue that in the extreme squeezed limit the quantum state ’turns’ classical. However, in contrast to inflation where the growth seems indefinite, in the matter bounce, the parameter $r_k$ begins to decrease as the universe begins to expand. This interesting behavior may point to crucial differences between the quantum-to-classical transition in inflation and bounces and seem to require further study. CSL modified tensor modes and power spectrum in the matter bounce {#sec:i-csl-mb} ================================================================= As we had described in the introduction, one can also view the transition of primordial quantum perturbations into the classical LSS as a quantum measurement problem. In other words, we need to understand as to how the mechanism by which the original state of the primordial perturbations collapsed into a particular eigenstate which corresponds to the realization of the CMB observed today. One of the proposals which addresses this issue is known as the CSL model [@csl]. The crucial advantage of this model is that a specific realization can be attained without the presence of an observer. In the rest of the manuscript we will focus on understanding the effects of CSL on the tensor perturbations in bouncing universes. CSL in brief ------------ The CSL model proposes a unified dynamical description which suppresses the quantum effects, such as the superposition of states in the macroscopic regime, and reproduces the predictions of quantum mechanics in the microscopic regime. In CSL, a unified dynamical description is achieved by appropriately modifying the Schrödinger equation. This modification is carried out by adding nonlinear terms and a stochastic behavior which is encoded through a Wiener process [@csl]. The modified Schrödinger equation encompasses an amplification mechanism which makes the new terms negligible in the quantum regime, hence retrieving the dynamics predicted by quantum mechanics. At the same time, it should make the new terms dominant in the classical regime, so that the classical-like behavior of the system is attained in the classical domain (for reviews, see Refs. [@csl-reviews]). Although, it should be clarified that, in the implementation of CSL for the case of primordial perturbations [@jerome-2012], the above mentioned amplification mechanism does not arise. Upon taking into account such modifications, the modified Schrödinger equation is given by $$\d\psi_\k^{\R,\I} =\l[-i\,\hat{\sf H}_{\k}^{\R,\I}\,\d\eta +\sqrt{\gamma}\,\l(\hat u_{\k}^{\R,\I}-{\bar u}_{\k}^{\R,\I}\r)\,\d {\cal W}_{\eta} -\f{\gamma}{2}\,\l(\hat u_{\k}^{\R,\I}-{\bar u}_{\k}^{\R,\I}\r)^2\,\d \eta\r]\, \psi_{\k}^{\R,\I},\label{eq:mse}$$ where $\hat{\sf H}_{\k}$ is the original Hamiltonian operator (\[eq:Hko\]), $\gamma$ is the CSL parameter, which is a measure of the strength of the collapse and ${\cal W}_\eta$ denotes a real Wiener process, which is responsible for the stochastic behavior. If the CSL modified wavefunction $\psi_{\k}^{\R,\I}$ is assumed to be of the following form [@bassi-2005; @jerome-2012; @suratna-2013] $$\psi_{\k}^{\R,\I}(u_{\k}^{\R,I},\eta) =N_{k}(\eta)\, \ {\rm exp}-\l[\Omega_{k}(\eta)\, \l(u_{\k}^{\R,\I}-{\bar u}_{\k}^{\R,\I}\r)^2 +i\,\chi_{\k}^{\R,\I}(\eta)\,u_{\k}^{\R,\I} +i\,\sigma_{\k}^{\R,\I}(\eta)\r],$$ then the functions $\Omega_{k}(\eta)$, $\chi_{\k}(\eta)$ and $\sigma_{\k}(\eta)$ have to satisfy the following set of differential equations $$\begin{aligned} \Omega_{k}'&=&-2\,i\,\Omega_{k}^2+\f{i}{2}\,\omega_k^2+\f{\gamma}{2},\ \label{eq:m-mse}\\ \f{N_{k}'}{N_{k}}&=&\Omega_{k}^\I,\\ \l({{\bar u}_{\k}^{\R,\I}}\r)' &=&\chi_{\k}^{\R,\I} +\f{\sqrt{\gamma}}{2\,\Omega_{k}^\R}\;{{\cal W}_{\eta}}',\\ \l(\chi_{\k}^{\R,\I}\r)' &=&-\omega_k^2\, \bar{u}_{\k}^{\R,\I} -\sqrt{\gamma}\,\f{\Omega_{k}^\I}{\Omega_{k}^\R}\,{{\cal W}_{\eta}}',\\ \l(\sigma_{\k}^{\R,\I}\r)' &=&\f{\omega_k^2}{2}\,\l(\bar{u}_{\k}^{\R,\I}\r)^2 -\f{1}{2}\,\l(\chi_{\k}^{\R,\I}\r)^2-\Omega_{k}^\R +\sqrt{\gamma}\,\f{\Omega_{k}^\I}{\Omega_{k}^\R}\, \bar{u}_{\k}^{\R,\I}\,{{\cal W}_{\eta}}',\end{aligned}$$ where $\omega_k^2$ is given by Eq. (\[eq:wk\]). In principle, one needs to solve the above set of stochastic differential equations in order to arrive at a complete understanding of the effects of CSL. However, recall that, our primary concern is the imprints of CSL on the tensor power spectrum. Note that, earlier, we had defined $\Omega_k=-(i/2)\,f_k'/f_k$ and the original Schrödinger equation had led to $f_k$ satisfying the Mukhanov-Sasaki equation (\[eq:uk-eom\]). If we now substitute the same expression for $\Omega_k$ in the CSL corresponding modified equation (\[eq:m-mse\]), we find that $f_k$ now satisfies the differential equation [@jerome-2012] $$f_k''+\l(k^2-i\,\gamma-\f{a''}{a}\r)f_k=0,\label{eq:m-uk-eom}$$ [i.e. ]{} the effects of CSL is essentially to replace $k^2$ by $k^2-i\,\gamma$. In the following sub-section, we shall solve this equation in a matter bounce and evaluate the effects of CSL on the tensor power spectrum. CSL modified tensor power spectrum {#eq:e-csl-tps} ---------------------------------- In this sub-section we shall focus on the evaluation of CSL modified tensor power spectrum in the matter bounce scenario. We find that, under certain conditions, even the CSL modified modes can be arrived at using the approximations we had worked with earlier. If we divide the period of our interest into two domains, we find that the CSL modified modes in the first domain ([i.e. ]{} over $-\infty<\eta<-\alpha\, \eta_0$), which satisfy the Bunch-Davies initial conditions, can be expressed as (for a discussion on the initial conditions for the case of CSL modified tensor modes, see Ref. [@jerome-2012]) $$f_{k}^{(\rm I)}(\eta) =\f{1}{\sqrt{2\, z_k\, k}}\l(1+\f{i}{z_k\, k\,\eta}\r)\, {\rm e}^{i\,z_k\, k\,\eta},\label{eq:mfk-1}$$ where $z_k=\l(1-i\,\gamma/k^2\r)^{1/2}$. In the second domain, [i.e. ]{} in the time range $-\alpha\,\eta_0<\eta<\beta\, \eta_0$, the term $a''/a$ in Eq. (\[eq:m-uk-eom\]) behaves as $a''/a \geq 2\,k_0^2/(1+\alpha^2)$. Recall that the modes of cosmological interest are assumed to be very small compared to $k_0/\alpha$. Hence, if the CSL parameter $\gamma$ is also assumed to be very small when compared to $k_0^2$, then Eq. (\[eq:m-uk-eom\]) can be approximated to be $$f_k''-\f{a''}{a}\,f_k\simeq 0,$$ exactly as in the unmodified case. Upon integrating this equation, we obtain that $$f_k^{({\rm II})}(\eta) =a(\eta)\,\l[A_k^{(\gamma)}+B_k^{(\gamma)}\,g(k_0\eta)\r], \label{eq:mfk-2}$$ where $g(x)$ is the same function (\[eq:g\]) we had encountered earlier, while $A_k^{(\gamma)}$ and $B_k^{(\gamma)}$ are given by $$\begin{aligned} A_k^{(\gamma)} &=& \frac{1}{\sqrt{2\,z_k\, k}}\, \f{1}{a_0\,\alpha^2}\,\l(1-\f{i\,k_0}{\alpha\, z_k\, k}\r)\, {\rm e}^{-i\,\alpha\, z_k\, k/k_0} +B_k^{(\gamma)}\, g(\alpha),\\ B_{k}^{(\gamma)} &=&\frac{1}{\sqrt{2\,z_k\, k}}\, \f{\l(1+\alpha^2\r)^2}{2\,a_0\,\alpha^2}\, \l(\f{i\,z_k\, k}{k_0}+\f{3}{\alpha} -\f{3\,i\,k_0}{\alpha^2\, z_k\, k}\r)\, {\rm e}^{-i\,\alpha\, z_k\, k/k_0}.\end{aligned}$$ Note that we have arrived at these expressions for $A_k^{(\gamma)}$ and $B_k^{(\gamma)}$ by matching the solution $f_k^{({\rm II})}$ \[cf. Eq. (\[eq:mfk-1\])\] and its derivative with the corresponding quantities in the first domain ([i.e. ]{} $f_k^{({\rm I})}$ and its derivative) at $\eta=-\alpha\,\eta_0$. We evaluate the tensor power spectrum after the bounce at $\eta=\beta\,\eta_0$ (with $\beta=10^2$), as we had done earlier. It can be expressed as $$\label{tps-fk} {\cal P}_{_{\rm T}}^{(\gamma)}(k) =\f{8}{\Mpl^2}\f{k^3}{2\,\pi^2}\, \vert A_k^{(\gamma)}+B_k^{(\gamma)}\,g(\beta)\vert^2.$$ In Fig. \[fig:tps-csl\] we have plotted logarithm of the ratio of the CSL modified power spectrum to the unmodified power spectrum, [i.e. ]{} $\log\, [{\cal P}_{_{\rm T}}^{(\gamma)}(k)/{\cal P}_{_{\rm T}}(k)]$, as a function of $k/k_0$, for the same set of parameters we have worked with earlier and for a few different choices of $\gamma/k_0^2$. ![The logarithm of the ratio of the CSL modified tensor power spectrum to the standard power spectrum has been plotted as a function of $k/k_0$ for the matter bounce ($p=1$) scenario. We have set $k_0/(a_0\,\Mpl)=10^{-5}$, $\alpha=10^5$ and $\beta=10^2$, as we had done in Fig. \[fig:tps\]. The solid, dashed and dotted lines correspond to $\gamma/k_0^2$ of $10^{-40},~10^{-50}$ and $10^{-60}$, respectively. Note that the introduction of a CSL parameter $\gamma$ leads to a suppression of power in the power spectrum at large scales. In the suppressed part, the power spectrum behaves as $k^3$, which is similar to what occurs in the case of inflation [@jerome-2012].[]{data-label="fig:tps-csl"}](tps-mb-csl-log.pdf){width="12.5cm"} It is evident from the figure that, just as in the case of inflation [@jerome-2012], the effect of CSL on the power spectrum in the matter bounce is to suppress its power at large scales. We find that the power spectrum behaves as $k^3$ in its suppressed part, exactly as observed in inflation [@jerome-2012]. We also note that, larger the value of the dimensionless parameter $\gamma/k_0^2$, smaller is the scale at which the power gets suppressed. Since, the scales of cosmological interest lie in the range $k/k_0 \simeq 10^{-30}$–$10^{-25}$, if we demand a nearly scale invariant power spectrum for the tensor modes, the value of the dimensionless parameter $\gamma/k_0^2$ is constrained to be $\gamma/k_0^2\lesssim 10^{-60}$. Tensor power spectrum in a generic bouncing model {#sec:i-csl-tps-gb} ================================================= Until now, our discussions have focused on the particular case of the bounce referred to as the matter bounce scenario described by the scale factor (\[eq:sf\]), with $p$ set to unity. In this section, we shall turn our attention to a more generic case where $p$ is any positive real number. In order to arrive at the tensor power spectrum in these models, our approach would be the same as in Sec. \[sec:tps-mb\], [viz. ]{} to solve Eq. (\[eq:uk-eom\]) to obtain $f_k$ and then evaluate the power spectrum using the definition (\[eq:tps-d\]). Tensor modes and power spectrum {#subsec:tps-gb} ------------------------------- Following the discussion in Sec. \[sec:tps-mb\], we obtain the tensor modes $f_k$ by dividing the time of interest into two domains and working under suitable approximations. In the first domain, [i.e. ]{} over $-\infty<\eta<-\alpha\, \eta_0$, where $\alpha\gg 1$, the scale factor simplifies to the power law form $a(\eta)\simeq a_0\,(k_0\,\eta)^{2\,p}$. In such a case, the differential equation (\[eq:uk-eom\]) reduces to $$f_k''+\l[k^2-\f{2\,p\,(2\,p-1)}{\eta^2}\r]\,f_k=0,$$ and it is well known that the corresponding solutions can be written in terms of Bessel functions. Upon imposing the Bunch-Davies initial conditions at very early times, we obtain the modes to be $$\begin{aligned} f_k^{({\rm I})}(\eta)=\f{i}{2}\,\sqrt{\f{\pi}{k}}\, \f{{\rm e}^{-i\,p\,\pi}}{{\rm sin}\,(n\,\pi)}\,(-k\,\eta)^{1/2}\, \l[J_{-n}(-k\,\eta)-{\rm e}^{i\,n\,\pi}\,J_{n}(-k\,\eta)\r],\end{aligned}$$ where $n=2\,p-1/2$, while $J_n(z)$ is the Bessel function of first kind [@gradshteyn-2007]. In the second domain, [i.e. ]{} over $-\alpha\,\eta_0<\eta<\beta\,\eta_0$, we can ignore the $k^2$ term in Eq. (\[eq:uk-eom\]) for reasons discussed earlier. For any arbitrary value of the parameter $p$, upon integrating the resulting equation, we find that we can express the modes $f_k$ in the domain as follows: $$f_k^{(\rm II)}(\eta) =a(\eta)\,\l[C_k+D_k\, {\tilde g}(k_0\,\eta)\r],$$ where the function ${\tilde g}(x)$ is given in terms of the hypergeometric function ${}_2F_1[a,b;c;z]$ as $${\tilde g}(x)=x\,{}_2F_1\l[2\,p,\tfrac{1}{2};\tfrac{3}{2};-x^2\r].\label{eq:gt}$$ The constants $C_k$ and $D_k$ are obtained by matching the solutions in the two domains and their derivatives at $\eta=-\alpha\,\eta_0$. They can be determined to be $$\begin{aligned} C_k &=&\f{i}{2\, a_0\,\alpha^n}\, \sqrt{\f{\pi}{k_0}}\, \f{{\rm e}^{-i\,p\,\pi}}{\sin\,(n\,\pi)}\nn\\ & &\times\,\l[J_{-n}(\alpha\, k/k_0)-{\rm e}^{i\,n\,\pi}\,J_{n}(\alpha\, k/k_0)\r] +D_k\, {\tilde g}(\alpha),\\ D_k &=&-\f{i}{2\,a_0\,\alpha^n} \l(\f{k}{k_0}\r)\,\sqrt{\f{\pi}{k_0}}\, \f{{\rm e}^{-i\,p\,\pi}}{\sin\,(n\,\pi)} \l(1+\alpha^2\r)^{2\,p}\nn\\ & &\times\,\l[J_{-(n+1)}(\alpha\, k/k_0) +{\rm e}^{i\,n\,\pi}\,J_{n+1}(\alpha\, k/k_0)\r].\end{aligned}$$ Upon using these expressions in the definition (\[eq:tps-d\]) of the tensor power spectrum and evaluating the spectrum after the bounce at $\eta=\beta/k_0$, we obtain that $${\cal P}_{_{\rm T}}(k)=\f{8}{M_{\rm Pl}^2}\,\f{k^3}{2\pi^2}\, \vert C_k+D_k\,{\tilde g}(\beta)\vert^2.$$ In Fig. \[fig:tps-p\] we have plotted the tensor power spectrum for a set of values $p$ as a function of $k/k_0$ for the same choice of parameters as in Fig. \[fig:tps\]. ![The tensor power spectra in bouncing models corresponding to $p=1$ (in blue), $p=1.001$ (in green) and $p=1.002$ (in red) have been plotted as a function of $k/k_0$. We have set $k_0/(a_0\,\Mpl)=10^{-5}$, $\alpha=10^5$ and $\beta=10^2$ as before. Note that the spectrum exhibits a red tilt for $p>1$.[]{data-label="fig:tps-p"}](tps-arb-p.pdf){width="12.5cm"} As expected, deviations from $p=1$ introduces a tilt to the tensor power spectrum. It is useful to note that the power spectrum is red for $p>1$, as one would expect in inflation. Imprints of CSL {#subsec:-i-csl-tps-gb} --------------- We can now readily compute the effect of CSL mechanism on the tensor power spectrum in a more generic bouncing scenario. In order to calculate the tensor power spectrum, we need solve the differential equation (\[eq:m-uk-eom\]) governing $f_k$ in the presence of CSL mechanism, which effectively replaces $k^2$ in the governing equation by $k^2 - i\, \gamma$. All our previous arguments go through for a general $p$ and hence we shall quickly present the essential results. We find that the CSL modified tensor mode in the first domain is given by $$\begin{aligned} f_k^{({\rm I})}(\eta)=\f{i}{2}\,\sqrt{\f{\pi}{z_k\,k}}\, \f{{\rm e}^{-i\,p\,\pi}}{{\rm sin}(n\,\pi)}\; \f{1}{\sqrt{-z_k\, k\,\eta}}\, \l[J_{-n}(-z_k\, k\,\eta)-{\rm e}^{i\,n\,\pi}\, J_{n}(-z_k\, k\,\eta)\r],\end{aligned}$$ where $n$ and $z_k$ are given by the same expressions as before. Similarly, in the second domain, the CSL modified tensor mode can be found to be $$f_k^{(\rm II)}(\eta) =a(\eta)\,\l[C_k^{(\gamma)}+D_k^{(\gamma)}\, {\tilde g}(k_0\eta)\r],$$ where, as earlier, ${\tilde g}(x)$ is given by Eq. (\[eq:gt\]), while the quantities $C_k^{(\gamma)}$ and $D_k^{(\gamma)}$ are given by $$\begin{aligned} C_k^{(\gamma)} &=&\f{i}{2\,a_0\, \alpha^n}\,\sqrt{\f{\pi}{k_0}}\, \f{{\rm e}^{-i\,p\,\pi}}{\sin\,(n\,\pi)}\nn\\ & &\times\,\l[J_{-n}(\alpha\, z_k\, k/k_0) -{\rm e}^{i\,n\,\pi}J_{n}(\alpha\, z_k\, k/k_0)\r] +D_k^{(\gamma)}\, {\tilde g}(\alpha),\\ D_k^{(\gamma)} &=&-\f{i}{2\,a_0\,\alpha^n}\, \l(\f{z_k\, k}{k_0}\r)\,\sqrt{\f{\pi}{k_0}}\, \f{{\rm e}^{-i\,p\,\pi}}{\sin\,(n\,\pi)}\, \l(1+\alpha^2\r)^{2\,p}\nn\\ & &\times\,\l[J_{-(n+1)}(\alpha\, z_k\, k/k_0) +{\rm e}^{i\,n\,\pi}\,J_{n+1}(\alpha\, z_k\, k/k_0)\r].\end{aligned}$$ The resulting spectrum evaluated after the bounce at $\eta=\beta/k_0$ is given by $${\cal P}_{_{\rm T}}^{(\gamma)}(k) =\f{8}{M_{\rm Pl}^2}\,\f{k^3}{2\,\pi^2}\, \vert C_k^{(\gamma)}+D_k^{(\gamma)}\,{\tilde g}(\beta)\vert^2.$$ As in the case of the matter bounce, in Fig. \[fig:tps-p-gamma\], we have plotted the logarithm of the ratio of the CSL modified tensor power spectrum to the unmodified power spectrum, [i.e. ]{}the quantity $\log\, [{\cal P}_{_{\rm T}}^{(\gamma)}(k) /{\cal P}_{_{\rm T}}(k)]$, for a two different values of $p$ and different values of $\gamma/k_0^2$. ![The behavior of $\log\, [{\cal P}_{_{\rm T}}^{(\gamma)}(k)/{\cal P}_{_{\rm T}}(k)]$ has been plotted as a function of $k/k_0$ for a bounce with scale factor described by the indices $p=1.001$ (in green) and for $p=1.002$ (in red). We have worked with the same values of $k_0/(a_0\,\Mpl)$, $\alpha$ and $\beta$ as in the earlier figures. The solid, dashed and dotted curves correspond to $\gamma/k_0^2$ of $10^{-40}, 10^{-50}$ and $~10^{-60}$, respectively. This figure clearly shows that the CSL mechanism leads to a suppression of power at large scales regardless of the value of $p$.[]{data-label="fig:tps-p-gamma"}](tps-arb-p-csl-log.pdf){width="12.5cm"} It is clear from the figure that the behavior of the power spectrum and the corresponding conclusions are the same as we had arrived at in the case of the matter bounce. Discussion {#sec:csl-c} ========== Generation of perturbations from quantum fluctuations in the early universe and their evolution leading to anisotropies in the CMB and inhomogeneities in the LSS provides a wonderful avenue to understand physics at the interface of quantum mechanics and gravitation. One such fundamental issue that has to be addressed is the mechanism by which the quantum perturbations reduce to be described in terms of classical stochastic variables. In this work, we have investigated the quantum-to-classical transition of primordial quantum perturbations in the context of bouncing universes. Following the footsteps of earlier efforts in this direction [@jerome-2012], we have approached this issue from two perspectives. In the first approach, we have investigated the extent of squeezing of the quantum state associated with a tensor mode as it evolves through a bounce. As in the context of inflation, the extent of squeezing grows as the modes leave the Hubble radius. However, in contrast to inflation where it can grow indefinitely (depending on the duration of inflation), we had found that the squeezing parameter reaches a maximum at the bounce and begins to decrease thereafter. We had found that this behavior is also reflected in the Wigner function. Secondly, we had treated this issue as a quantum measurement problem set in the cosmological context, [i.e. ]{}we had investigated the effects of the collapse of the original quantum state of the perturbations. An approach which have been proposed to achieve such a collapse is the CSL model. Using the model, we had examined if the tensor power spectra are modified due to the collapse in a class of bouncing universes. We had found that CSL mechanism leads to a suppression of the tensor power spectra at large scales, in a manner exactly similar to what occurs in the inflationary context. It would be interesting to extend these analyses to the case of scalar perturbations in bouncing universes [@rathul-2017]. We are currently investigating such issues. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Debika Chowdhury for her comments on the manuscript. VS would also like to acknowledge support from NSF Grant No. PHY-1403943. [99]{} P. A. R. 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--- abstract: 'Emergent pattern formation in self-propelled particle (SPP) systems is extensively studied because it addresses a range of swarming phenomena which occur without leadership. Here we present a dynamic SPP model in which a sensory blind zone is introduced into each particle’s zone of interaction. Using numerical simulations we discovered that the degradation of milling patterns with increasing blind zone ranges undergoes two distinct transitions, including a new, spatially non-homogeneous transition that involves cessation of particles’ motion caused by broken symmetries in their interaction fields. Our results also show the necessity of nearly complete panoramic sensory ability for milling behavior to emerge in dynamic SPP models, suggesting a possible relationship between collective behavior and sensory systems of biological organisms.' author: - 'Jonathan P. Newman' - Hiroki Sayama title: 'The Effect of Sensory Blind Zones on Milling Behavior in a Dynamic Self-Propelled Particle Model' --- Self-organization and pattern formation in self-propelled particle (SPP) systems has been a topic of great interest in theoretical physics, mathematical biology and computational science [@1; @2]. It is well understood that the emergence of cohesive swarming motions requires neither leaders nor globally enforced organizational principles. Various SPP models have been used to explore stability and phase transitions of swarming patterns in response to varying noise levels [@6; @7; @3] and other control parameters [@5; @8; @9; @11], as well as to characterize distinct regimes in the parameter space [@8; @10; @11; @12]. Effort has also been allotted for addressing biological questions concerning swarming behavior [@9; @13; @14; @15; @5; @24; @25] and for designing non-trivial swarming patterns from combinations of different kinetic parameter sets [@17]. SPP models may be classified into two distinct categories: kinematic and dynamic [@12]. Kinematic SPP models typically assume that each particle maintains a speed and orientation in accord with its local neighbors [@2]. In these models, particles cannot halt because they are supplied with a minimal or constant absolute velocity [@2; @3; @13; @14]. Kinematic models have been used for computational modeling of collective behavior of constantly moving groups, such as bird flocks, often implementing empirically constructed complex, spatially discrete interaction zones and behavioral rules that reflect perceptional or locomotive properties of the species being modeled [@14; @13; @15; @24; @25]. On the other hand, dynamic SPP models describe the motion of particles using differential equations based on Newtonian mechanics that involve self-propulsion and pairwise attraction/repulsion forces [@7]. It is known that such models may robustly form coherent milling patterns from initially random conditions even without explicit alignment rules [@6; @7; @10; @11; @12]. Dynamic models have been used for both analytical and numerical research on collective behavior of interacting particles in general, with minimal complexity assumed in particles’ intrinsic behaviors. Here we consider a new dynamic SPP model in which a sensory blind zone is introduced into each particle’s zone of interaction. Although the assumption of sensory blind zones has been widely adopted in kinematic SPP models [@13; @15; @14; @24; @25], it has not been considered within a dynamic framework. We specifically examine the effect of the sensory blind zones on coherent milling behavior in dynamic SPP models. In doing so, we discovered a novel transition that occurred with an increasing range of blind zones, and found the system to be highly sensitive to this type of perturbation. Our model describes the movement, within an open, two-dimensional, continuous space, of $N$ self-propelled particles driven by soft-core interactions whose dynamics are given by $$\begin{aligned} \frac{d x_i}{d t} &=& v_i , \\ m \frac{d v_i}{d t} &=& (\alpha - \beta \|v_i\|^2) v_i - \nabla U_i(x_i) , \label{eq2} \\ U_i(x) &=& \sum_{j \neq i} u(\|x - x_j\|) , \\ u(r) &=& C_r e^{-r/l_r} - C_a e^{-r/l_a} \quad (r \geq 0), \label{pairwisepotential}\end{aligned}$$ where $x_i$ and $v_i$ are the position and the velocity of the $i$-th particle ($i=1 \ldots N$), respectively; $m$ the unit mass of one particle; $\alpha$ and $\beta$ the coefficients of propulsion and friction, respectively; $U_i(x)$ the interaction potential surface for the $i$-th particle; $u(r)$ the pairwise interaction potential function (Fig. \[fig1\] (a)); $C_r$ and $C_a$ the amplitudes of repulsive and attractive pairwise interaction potentials, respectively; and $l_r$ and $l_a$ the characteristic ranges of repulsive and attractive pairwise interaction potentials, respectively. Eq. (\[eq2\]) includes a velocity-dependent locomotory term and an interaction term achieved through a generalized Morse pairwise interaction potential. For $\alpha, \beta > 0$, particles will rapidly approach equilibrium velocity of magnitude $v_{eq} \equiv \sqrt{\alpha / \beta}$ and the system will converge toward a structure for which total dissipation is zero and particles are driven only by conserved forces [@11]. This rule set has been employed, with some mathematical variation, by many previous studies [@10; @11; @12; @6]. In this study, the shape of the pairwise interaction potential falls within the biologically relevant regime defined as $C_r / C_a > 1$ and $l_r / l_a < 1$, as described by [@11; @12]. In the biologically relevant regime, individuals tend to move toward other individuals that are further from, and away from individuals that are closer than, some critical distance from themselves. This rule is generally applicable to the kinetics of many different biological species and natural systems [@1; @15; @19]. ![Model assumptions used in the SPP model employed by this study. (a) Shape of the pairwise interaction potential function $u(r)$ defined by Eq. (\[pairwisepotential\]), where $r$ is the distance between two particles. This study explores the biologically relevant regime of parameter settings, in which particles will accelerate away from neighbors who are closer than, and toward neighbors further than, the equilibrium distance $r_{eq} \equiv \frac{l_a l_r}{l_a-l_r}\log\frac{C_r l_a}{C_a l_r}$ ($\approx 1.39$ with parameter settings used in this paper). (b) A sensory blind zone oriented opposite the direction of forward motion of the particle with angular range $\theta$. Particles are represented by small triangles. In this example, particles 6, 8 and 9 are within particle 1’s blind zone, so their indices are not included in the set $S_{1,t}$ when generating repulsive/attractive forces acting on particle 1.[]{data-label="fig1"}](fig1.eps){width="0.7\columnwidth"} We compare two experimental parameters in the above model: the magnitude of stochastic force (noise) $\gamma$ and the range of sensory blind zones $\theta$, the former analyzed in [@6] and the latter our original extension. Sensory blind zones are incorporated into the design of each particle to mimic the abilities of anisotropic sensory systems observed in nature, such as vision. A sensory blind zone is assumed to exist for each particle with an angular range $\theta$ in a direction opposite to the direction of forward motion (Fig. \[fig1\] (b)). The inclusion of these parameters introduces discrete events into the model, i.e., abrupt changes of velocity by stochastic force and entry and exit of other particles into/out of sensory blind zones. Consequently, we revised the equations of motion using discrete time steps. The difference equations used for numerical simulation are $$\begin{aligned} \frac{x_{i,t+\Delta t} - x_{i,t}}{\Delta t} &=& v_{i, t+\Delta t}, \\ m \frac{v_{i,t+\Delta t} - v_{i,t}}{\Delta t} &=& (\alpha - \beta \|v_{i,t}\|^2) v_{i,t} - \nabla U_{i,t} (x_{i,t}) + \gamma \xi_{i,t}, \\ U_{i,t}(x) &=& \sum_{j \in S_{i,t}} u(\|x-x_{j,t}\|) , \end{aligned}$$ where $x_{i,t}$ and $v_{i,t}$ are the position and the velocity of the $i$-th particle at time $t$, respectively; $\xi_{i,t}$ a randomly oriented vector with length 1 whose orientation is independent for each evaluation; $U_{i,t}(x)$ the interaction potential surface for the $i$-th particle at time $t$; and $S_{i,t}$ the set of indices of all the particles whose positions are outside the blind zone of the $i$-th particle at time $t$ (Fig. \[fig1\] (b)). We conducted numerical simulations of this model to produce a milling pattern similar in structure to those witnessed in schools of teleost fish, insects, and microorganisms [@1; @11; @12; @8; @9; @18; @15]. Specific values of fixed parameters are as follows: $m = 1.0$, $C_r = 1.0$, $C_a = 0.5$, $l_r =0.5$, $l_a = 2.0$, $\alpha = 1.6$, $\beta = 0.5$. Initial conditions of each simulation were such that particles were randomly distributed within a square area of side length $2 l_a$, and each particle was randomly oriented with magnitude of velocity randomly chosen from $[0, v_{eq}]$ as described in [@11; @12; @10]. The model equations were numerically simulated from $t = 0$ to 200 at interval $\Delta t = 0.01$. No spatial boundaries were enforced. For simulations recording the effect of stochastic force, $\gamma$ was varied from 0 to 10 at interval 0.5, while $\theta = 0$. For simulations testing the effect of sensory blind zones, $\theta$ was varied from 0 to $0.2 \pi$ at interval $0.01\pi$, while $\gamma = 0$. Each parameter setting was simulated using several population sizes, $N = 200$, 300, 400, and 500. Ten simulation runs were conducted for each condition. Several metrics were used to characterize the simulation results. These include average absolute velocity $V_{abs}$, ratio of halting particles $H$, normalized angular momentum $M$, and normalized absolute angular momentum $M_{abs}$, defined as follows (same or similar metrics were used in [@11; @12; @15]): $$\begin{aligned} V_{abs} &=& \sum_i \|v_i\| / N \\ H &=& \left\|\left\{ i, {\rm ~s.t.~} \|v_i\| < \mu v_{eq} \right\}\right\| / N \\ M &=& \frac{\|\sum_i r_i \times v_i\|}{\sum_i \|r_i\|\|v_i\|} \\ M_{abs} &=& \frac{\sum_i \|r_i \times v_i\|}{\sum_i \|r_i\|\|v_i\|}\end{aligned}$$ Here $r_i \equiv x_i - x_c$ where $x_c$ is the swarm’s center of mass. To measure $H$ we used 20% of the equilibrium velocity ($\mu=0.2$) as a threshold to determine whether a particle was halting or not. When used comparatively, $M$ and $M_{abs}$ make it possible to distinguish single mill formation from double mill formation in which two mills rotate with opposite sense around similar, but not identical, centers of mass [@11; @12]. All the metrics were averaged over the last 10 time steps of each simulation. Figs. \[fig2\] and \[fig3\] depict the processes of structural decay produced by stochastic force and blind zone perturbations on the milling behavior of 500 particles. A transition from milling state to disordered state was induced by increasing the magnitude of stochastic force across $\gamma \approx 7.0$ (Fig. \[fig2\] (a), Fig. \[fig3\] (a)–(c)). When the range of sensory blind zones $\theta$ was increased, however, structural degradation was very different, involving a new spatially heterogeneous transition (Fig. \[fig2\] (b), Fig. \[fig3\] (d)–(f)). Initiation of collapse occurred at $\theta \approx 0.03 \pi$, where particles near the center of the mill ceased rotation and formed a stationary core that has not previously been described. We call this new state a “carousel” state. This state is different from the rigid-body rotation reported in [@11; @12] and the compact but disordered state of [@6] because it is characterized by a sharp boundary between the milling surface and the central stationary core made of particles with near zero velocity (Fig. \[fig3\] (e)). A secondary transition was observed across $\theta \approx 0.06 \pi$ where the particles moving in the periphery became abruptly disordered and lost coherence in motion while the particles in the central core remained stationary (Fig. \[fig3\] (f)). We call this concluding state a “surface disordered” state. ![A visual comparison of the effects of increasing stochastic force and the effects of increasing range of sensory blind zones on the milling behavior of 500 particles. Each image is a final snapshot of a simulated particle swarm taken at $t=200$. Particles have tails that represent the orientation and magnitude of their velocity. (a) Results with increasing stochastic force $\gamma$ while $\theta=0.0$. Transition from milling to disordered states occurred at $\gamma \approx 7.0$. (b) Results with increasing range of sensory blind zones $\theta$ while $\gamma=0.0$. Transitions from milling to carousel and from carousel to surface disordered states occurred at $\theta \approx 0.03\pi$ and $\theta \approx 0.06\pi$, respectively. See also Fig. \[fig3\].[]{data-label="fig2"}](fig2.eps){height="0.75\textheight"} ![Tangential velocities of particles at a distance $r$ from the center of mass. Data were obtained from numerical simulations of 500 particles at $t=200$. The direction of rotation of the majority was taken as positive.[]{data-label="fig3"}](fig3.eps){width="0.8\columnwidth"} Particles inside the core in the carousel and surface disordered states lose their velocity due to a perceptual and consequent force asymmetry. A sensory blind zone creates a longitudinal imbalance between forces derived from particles ahead and forces from particles behind. Because the pairwise equilibrium distance $r_{eq}$ is much longer than the characteristic distance between neighboring particles in a swarm, the imbalance takes effect in the regime of repulsive interactions and thus results in a net resistance against self-propulsion of particles. A particle near the center of the swarm has more particles to its front and back than does a particle rotating in the periphery, since the density of particles is inversely related to the distance from the center of the swarm under the parameter set used here [@10] (numerically confirmed in our simulation results; data not shown). Thus, the net resistance against forward motion resulting from the blind zone is larger for particles rotating close to the mill’s center. Within a certain distance to the mill’s center, the resistance exceeds the range of self-propulsive force possible in Eq. (\[eq2\]), and consequently particles cease motion. This transition does not occur due to increasing $\gamma$ because the effect of stochastic force is spatially isotropic: it is equally likely to force a particle in any direction and therefore does not lead to cessation of movement. Fig. \[fig4\] summarizes all the simulation results, showing the dependence of the final values of $V_{abs}$, $H$, $M$ and $M_{abs}$ on $\gamma$, $\theta$ and $N$. The onset and the mechanism of structural degradation of milling behavior are different between cases with increasing $\gamma$ and $\theta$. The milling structure is fairly robust to small $\gamma$, and it suddenly collapses at $\gamma \approx 7.0$, nearly independently of $N$. The $H$ plot shows no particles halting in this transition. In contrast, plots of increasing $\theta$ illustrate that structural degradation in increasing $\theta$ is a two-fold process. The first transition from milling to carousel was detected in $V_{abs}$ and $H$ (emergence of halting particles and consequent decrease of average velocity). The second transition from carousel to surface disordered was detected in $M$ and $M_{abs}$ (loss of coherence in angular momentum). It was also observed in our results that the onsets of these transitions depended significantly on $N$. This can be understood in that larger $N$ increased the density at the mill’s center and made particles more reactive to blind zone induced halting. ![image](fig4.eps){width="0.75\columnwidth"} The blind zone ranges used in these simulations were extremely small from a biological viewpoint. The largest value tested, $\theta = 0.2\pi$, was just 10% of the perception range, which was more than sufficient to destroy milling patterns in all cases. Kinematic SPP models, on the other hand, can produce and maintain milling patterns despite considerable sensory blind zones [@15]. To understand this discrepancy in model properties, we note one important difference between kinematic and dynamic frameworks: while particles in kinematic models are always constrained to move with a non-zero velocity, there is a possibility for particles to halt in dynamic models that becomes significant in the presence of rear blind zones like those assumed in our model. This leads us to a hypothesis that milling behavior in an aggregate of organisms may sensitively depend on their ability to maintain constant velocity. Specifically, for organisms that keep moving autonomously at a near constant pace, milling behavior emerges relatively easily even with considerable sensory blind zones. In contrast, for organisms whose motion strongly depends on (either sensory or physical) environmental stimuli, milling behavior requires a nearly complete panoramic range of interaction, especially to perceive the pressure from behind and gain enough forward propulsion to maintain constant velocity. There are several biological observations that directly or indirectly support our hypothesis. Milling behavior is often reported in groups of microorganisms and insects [@10; @11; @9; @1; @3; @5; @8; @18]. These organisms rely chiefly on direct physical contact and chemical sensory input, respectively, when forming mills and therefore use omnidirectional sensory capabilities to form aggregates. They also have the ability to cease motion. Additionally, there is a great deal of evidence supporting the isotropic sensory ability of the lateral line in teleost fish, some of which demonstrate milling behavior [@21]. A recent study that investigated the superficial organization of neuromasts composing the lateral line in goldfish showed that neuromasts’ most sensitive axes were oriented in almost every direction [@22]. Moreover, a recent study on Mormon crickets [@16] reports that the physical, cannibalistic threat of protein and salt deprived individuals from behind plays a critical role in creating a large-scale coherent march. When some crickets are immobilized and therefore unable to respond to a push from behind, the march halts. This study provides clear evidence supporting our conjecture that inputs (physical pressure) from behind a particle can be important in the formation of coherent swarming patterns. In summary, we computationally studied the effects of sensory blind zones on the stability of self-organizing mill formation in a dynamic SPP model. We found that milling behavior collapses through two spatially distinct transitions in response to an increasing range of rear blind zones, characterized by a halting regime emanating from the center of the swarm and then a disorganization of coherent motion in the periphery area. This is quite different from pattern collapse observed with increasing stochastic force described by a spatially uniform transition to a compact but disordered state [@6]. Combined with other results obtained with kinematic SPP models, our results suggest a possible relationship between collective behavior and sensory systems of biological organisms: species that engage in mill formation in nature may necessarily have an omnidirectional sensory system if they do not maintain constant velocity by themselves. This is a hypothesis testable and falsifiable through experimental observation. We thank Boris Chagnaud, Kurt Wiesenfeld and Stefan Boettcher for their insightful comments on this manuscript. [99]{} C. W. 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--- abstract: 'Evolutionary models for populations of constant size are frequently studied using the Moran model, the Wright–Fisher model, or their diffusion limits. When evolution is neutral, a random genealogy given through Kingman’s coalescent is used in order to understand basic properties of such models. Here, we address the use of a genealogical perspective for models with weak frequency-dependent selection, i.e.$Ns =: \alpha$ is small, where $s$ is the fitness advantage of a fit individual and $N$ is the population size. When computing fixation probabilities, this leads either to the approach proposed by [@Rousset2003], who argues how to use Kingman’s coalescent in order to study weak selection, or to extensions of the ancestral selection graph of [@NeuhauserKrone1997] and [@Neuhauser1999]. As an application of this genealogical approach, we re-derive the one-third rule of evolutionary game theory [@Nowak2004]. In addition, we provide the approximate distribution of the genealogical distance of two randomly sampled individuals under linear frequency-dependence.' author: - \| [by P. Pfaffelhuber and B. Vogt]{}\ Albert-Ludwigs-Universität Freiburg title: 'Populations under frequency-dependent selection and genetic drift: a genealogical approach' --- = [^1] = = [^2] = Introduction ============ Evolutionary game theory has long been studied using replicator equations and deterministic systems by using an infinite population limit [@TaylorJonker1978; @Maynard-Smith1982; @Nowak2006]. Recently, the amount of research dealing with repeated games in finite populations increases. In the simplest case, a population of $N$ players (each carrying one out of a possible set of strategies) repeatedly chooses an opponent at random and plays a game with payoff matrix $M$. In addition, players can produce offspring and the fitness of an individual with strategy $A$ is determined by the average payoff the player obtains in the evolutionary game. Since the average payoff depends on the frequencies of the strategies in the population, the fitness is frequency-dependent, and the frequency path is e.g. given by a Moran model or a Wright–Fisher model with frequency-dependent selection. When studying models for populations of individuals carrying different types (or adopting different strategies), several questions are most important (see e.g. [@Ewens2004]). What is the probability of fixation of a type (in absence of mutation), or what is the random distribution of types in mutation-selection balance? If fixation occurs, what is the distribution of the fixation time? When taking a sample from the population, what is its genealogical structure? In addition, extensions to structured populations or to the multi-type case are considered. For the questions on fixation probabilities and times, the theory of birth-death processes and (one-dimensional) diffusions is frequently used [@KarlinTaylor1981]. Here, explicit formulas are available for such quantities, but these can hardly be extended to multi-dimensional cases. Another approach is the use of the genealogical structure by coalescent processes [@Kingman1982a; @Berestycki2009]. The advantage is that it is extensible to populations in structured or multi-type populations. In addition, questions on the genealogical structure can be asked. As an example from evolutionary game theory, [@GokhaleTraulsen2011] study the frequency path of many-player many-strategy games using such a genealogical approach. The current paper was motivated by the one-third rule of evolutionary game theory [@Nowak2004]. It states that in finite populations of players with two strategies $A$ and $B$, strategy $A$ is favored in the sense that the fixation probability of type $A$ in a $B$-population is higher than under neutrality if and only if type $A$ is favored in the infinite population limit at frequency 1/3. Since the one-third rule was found in a Moran model, it has as well been proven to hold in a Wright–Fisher model [@Traulsen2006; @Imhof2006] and in general exchangeable models [@Lessard2007]. An intuitive explanation was given in [@Ohtsuki2007], who argue that an individual interacts on average with $B$-players twice as often as with $A$-players, which implies the one-third rule. [@WuTraulsen2010] discuss violations of the one-third rule for non-linear frequency dependence. The goal of the present paper is to use a genealogical approach in the study of fixation probabilities. Only [@Ladret2007], who use a method developed by [@Rousset2003], already take this route; see also [@Rousset2004], p. 92f. We complement their approach by using the Ancestral Selection Graph (ASG), first introduced in the frequency-independent case by [@NeuhauserKrone1997] and [@KroneNeuhauser1997]. This graph extends the coalescent to the selective case and has been further extended to a simple frequency-dependent case in [@Neuhauser1999]. In addition, we study genealogical distances in models from evolutionary game theory, i.e. frequency-dependent selection. The paper is organized as follows: After stating the models (and their diffusion limits), we re-formulate the one-third rule and our results on genealogical distances. Then, we provide two proofs of the one-third rule. Finally our genealogical result is proved. Models ====== We are studying the solution $\mathcal X = (X_t)_{t\geq 0}$ of the stochastic differential equation $$\begin{aligned} \label{eq:SDE} d X & = ((1-X)\theta_A - X\theta_B)dt + \alpha X(1-X)(\beta - \gamma X)dt + \sqrt{X(1-X)}dW\end{aligned}$$ for $\alpha, \theta_A, \theta_B\geq 0, \beta, \gamma\in\mathbb R$. Here, $\alpha$ is called the scaled selection coefficient, and $\theta_A$ and $\theta_B$ the scaled mutation rates, and $\mathcal X$ gives the frequency path of type (or strategy) $A$ in a population of constant size. For this diffusion, we define in the case $\theta_A = \theta_B=0$ $$T_1 := \inf\{t\geq 0: X_t=1\}$$ and note that $T_1<\infty$ if and only if type-$A$ eventually fixes in the population. In our analysis, the version of without mutation and without fluctuations, $$\begin{aligned} \label{eq:ODE} dx = \alpha x(1-x)(\beta - \gamma x)dt\end{aligned}$$ will also play a role. A finite Moran model {#ss:moran} -------------------- We are turning to the frequency path of strategy (or type) $A$ in a time-continuous Moran model of size $N$, which converges to $\mathcal X$ as $N\to\infty$. We distinguish four cases, corresponding to the signs of $\beta$ and $\gamma$. Setting $x_\ast = \beta/\gamma$ (which is an equilibrium point in ), the four cases are: 1. $\beta>0$ and $\gamma>0$, which leads to a stable fixed point in $x_\ast$ for (if $x_\ast\in(0,1)$), 2. $\beta<0$ and $\gamma<0$, which leads to an unstable fixed point in $x_\ast$ for (if $x_\ast\in(0,1)$), 3. $\beta>0$ and $\gamma<0$, which leads $x=1$ as the only stable fixed point in , 4. $\beta<0$ and $\gamma>0$, which leads $x=0$ as the only stable fixed point in . In case (ii) and (iv), strategy $A$ cannot invade a population consisting only of strategy $B$ individuals in and vice versa. The time-continuous Moran model is a Markov process with state space $\{A,B\}^N$ (indicating the types of $N$ individuals) and the following transition rules: 1. Every pair of individuals $i$ and $j$ resamples at rate 1. Upon a resampling event, the offspring of $i$ (or $j$) replaces $j$ (or $i$) with probability $\tfrac 12$. 2. In cases (i) and (iii), i.e. when $\beta>0$, the offspring of any $A$-individual replaces a randomly chosen individual at rate $\alpha\beta$. 3. In cases (ii) and (iv), i.e. when $\beta<0$, the offspring of any $B$-individual replaces a randomly chosen individual at rate $-\alpha\beta = \alpha\|\beta\|$. 4. In cases (i) and (iv), i.e. when $\gamma>0$, every $B$-individual picks a randomly chosen individual at rate $\alpha\gamma$. If it is an $A$-individual, an offspring of the $B$-individual replaces another, randomly chosen individual. 5. In cases (ii) and (iii), i.e. when $\gamma<0$, every $A$-individual picks a randomly chosen individual at rate $-\alpha\gamma = \alpha\|\gamma\|$. If it is another $A$-individual, an offspring of the first $A$-individual replaces another, randomly chosen individual. 6. Every individual is hit by a mutation event at rate $\bar\theta := \theta_A + \theta_B$. With probability $\theta_A/\bar\theta$, the individual turns into type $A$, and with probability $\theta_B/\bar\theta$, it turns to $B$, independent of the type of the parent. In the graphical construction of the Moran model, the transitions 1.–3. are given by arrows denoted T1-T3, respectively. The T3-arrows are in fact double-arrows since the type of one individual has to be checked (along the line which we will call the checking line; see also [@Neuhauser1999]) while the second arrow leads to reproduction (which will be called the imitating line). See Figures \[fig1\] and \[fig2\] for illustrations of the transitions 1., 2. and 3. By now, it is a classical result that the frequency path of the Moran model converges weakly (with respect to the topology of uniform convergence on compact sets) to solutions of ; see e.g. [@EthierKurtz1993]. (6,10.5)(0,-0.5) (0,.5)[(0,1)[10]{}]{} (1,10.5)[(0,-1)[10]{}]{} (2,10.5)[(0,-1)[10]{}]{} (3,10.5)[(0,-1)[10]{}]{} (4,10.5)[(0,-1)[10]{}]{} (5,10.5)[(0,-1)[10]{}]{} (6,10.5)[(0,-1)[10]{}]{} (4,9.5)[(-1,0)[1]{}]{} (4,9.5) (4,9.7) (0,0) T1 (3,7.7)[(1,0)[1]{} ]{} (3,7.7)[(-1,0)[2]{}]{} (3,7.7) (3,7.9) (0,0) T3 (4,4.5)[(-1,0)[2]{}]{} (4,4.5) (4,4.7) (0,0) T1 (2,3.5)[(1,0)[1]{}]{} (2,3.5) (2,3.7) (0,0) T2 (1,2.3)[(1,0)[1]{}]{} (1,2.3) (1,2.5) (0,0) T1 (3,2.1)[(1,0)[2]{}]{} (3,2.1) (3,2.3) (0,0) T1 (5,1)[(-1,0)[3]{}]{} (5,1) (5,1.2) (0,0) T1 (6,8.5)[(-1,0)[2]{}]{} (6,8.5) (6,8.7) (0,0) T2 (5,6.5)[(-1,0)[2]{}]{} (5,6.5) (5,6.7) (0,0) T1 (5,4.1)[(-1,0)[1]{}]{} (5,4.1)[(1,0)[1]{}]{} (5,4.1) (5,4.3) (0,0) T3 (0, 10.8)[(0,0)[$t$]{}]{} (4.5, 4.2) (0,0) i (5.5, 4.2) (0,0) c (3.5, 7.8) (0,0) i (2.5, 7.8) (0,0) c Connection with evolutionary games ---------------------------------- Consider a game with payoff matrix $A$ $B$ ----- ----- ----- $A$ $a$ $b$ $B$ $c$ $d$ There are $N$ players, each one adopting either strategy (or type) $A$ or $B$. If the frequencies of type $A$ and type $B$ are $X_A$ and $X_B$, respectively, the fitnesses are dependent on the payoff matrix and are given by $$\label{eq:fAB} \begin{aligned} f_A &:= \tfrac 12 + s(aX_A + bX_B) = \tfrac 12 + s(b + (a-b)X_A),\\ f_B &:= \tfrac 12 + s(cX_A + dX_B) = \tfrac 12 + s(d + (c-d)X_A). \end{aligned}$$ Note that $aX_A+bX_B$ is the average payoff of a type-$A$ individual when playing against a random player from the population. Here, the offspring of every individual of type $A$ replaces a randomly chosen individual at rate $f_A$. Accordingly, a type-$B$ offspring replaces a randomly chosen individual at rate $f_B$ (given that $s$ is small enough such that the rates are non-negative). The frequency path of type-$A$, follows in the infinite-population limit, $N\to\infty$, $$\begin{aligned} dx & = sx(1-x)(b+(a-b)x - d - (c-d)x)dt = sx(1-x)(\beta - \gamma x)dt\end{aligned}$$ with $$\begin{aligned} \beta := b-d, \qquad \gamma := b-d+c-a.\end{aligned}$$ Moreover, when $N\to\infty$, $s\to 0$ such that $Ns\to \alpha$, and time is rescaled by a factor of $N$, the frequency path of type-$A$ follows the SDE as given in . Since the Moran model from Section \[ss:moran\] has the same diffusion limit, we use its graphical representation in order to obtain insights into the evolutionary game just described. Results ======= Starting by re-formulating the one-third rule in Section \[S:onethird\], we give our result on the distribution of genealogical distances in the Moran model from Section \[ss:moran\] in Section \[ss:dist\]. The one-third rule {#S:onethird} ------------------ We are now ready to formulate the one-third rule. For $\theta_A=\theta_B=0$, we define $$p_{\text{fix}}(\varepsilon,\alpha) := \mathbf P(T_1 < \infty\| X_0=\varepsilon),$$ which is the fixation probability of type $A$ if it starts in frequency $\varepsilon$ and selection intensity is $\alpha$. Note that $p_{\text{fix}}(\varepsilon,0) = \varepsilon$, i.e. the chance that strategy $A$ fixes equals its starting frequency, if evolution is neutral. \[T:onethird\] Let $\mathcal X = (X_t)_{t\geq 0}$ be the solution of with $\theta_A = \theta_B = 0$. Then, $$\begin{aligned} \lim_{\alpha\to 0} \frac 1 {\alpha\varepsilon}\big(p_{\text{fix}}(\varepsilon,\alpha) - \varepsilon\big) \xrightarrow{\varepsilon\to 0} \beta - \gamma \frac 13 . \end{aligned}$$ In particular, $p_{\text{fix}}(\varepsilon,\alpha) > p_{\text{fix}}(\varepsilon,0)$ for $\alpha, \varepsilon$ small enough, if and only if $$\beta - \gamma \frac 13 > 0,$$ i.e. the fitness of type $A$ is above average at frequency $x=\frac 13$. In Section \[S:twop\], we will provide two different proofs of this result. The first proof goes back to [@Ladret2007], the second one uses the ancestral selection graph which we introduce in Section \[ss:ASG\]. Genealogical distances {#ss:dist} ---------------------- In order to precisely formulate our results on genealogical distances, we are considering the Moran model from Section \[ss:moran\] including mutation, i.e. $\theta_A, \theta_B>0$. Assume that the Moran model (which evolves according to the rules 1.–4.) has run for an infinite amount of time. Then, every pair of individuals has a most recent common ancestor which can be read off from the graphical representation; see Figure \[fig2\]. Here, we denote by $\mathbf P_{N,\alpha}$ the distribution of the Moran model with population size $N$ and selection coefficient $\alpha$ in equilibrium, and by $\mathbf E_{N,\alpha}$ the corresponding expectation. Note that the Moran model is finite and hence, the equilibrium exists and is unique. Moreover, we denote by $R_{12}$ the genealogical distance of two randomly sampled individuals from the equilibrium population. In the case $\alpha=0$, recall that the distance of two randomly sampled individuals is exponentially distributed, and hence, $\mathbf E_{N,\alpha=0}[e^{-\lambda R_{12}}] \xrightarrow{N\to\infty} 1/(1+\lambda)$. The following result generalizes this fact to small $\alpha$, i.e. weak selection. \ \[T:dist\] Let $\mathbf E_{N,\alpha}$ and $R_{12}$ be as above. Then, with $\bar\theta = \theta_A + \theta_B$, $$\begin{aligned} \lim_{N\to\infty} \mathbf E_{N,\alpha} [e^{-\lambda R_{12}}] = \frac{1}{1+\lambda} - \alpha \gamma \frac{\theta_A\theta_B}{\bar \theta} \frac{(2+\bar\theta+\lambda)\lambda}{(6+\bar\theta + \lambda)(3+\bar\theta+\lambda)(1+\bar\theta + \lambda)(1+\bar\theta)(1+\lambda)^2} + \mathcal O(\alpha^2). \end{aligned}$$ In particular, $$\begin{aligned} \lim_{N\to\infty} \mathbf E_{N,\alpha}[R_{12}] & = 1 + \alpha \gamma\frac{\theta_A\theta_B}{\bar\theta} \frac{2+\bar\theta}{(6+\bar\theta)(3+\bar\theta)(1+\bar\theta )^2} + \mathcal O(\alpha^2) \end{aligned}$$ 1. Note that – up to first order in $\alpha$ – the change in the genealogical distance only depends on $\gamma$ but not on $\beta$. This is not surprising since it has been shown that in the case of frequency-independent selection, $\beta=1, \gamma=0$, the change in the genealogical distance is of order $\alpha^2$; see Theorem 4.26 in [@KroneNeuhauser1997]. However, this leads to an interesting effect: assume that $\beta>\gamma>0$. Then, type $A$ is selectively advantageous at any time during the evolution of the population. So, one might guess that type-$A$ individuals have a higher chance to get offspring and genealogical distances are shorter than under neutrality. However, as the result shows, up to first order in $\alpha$, genealogical distances are larger than under neutrality. The reason is that for $\gamma>0$, selection is strongest by frequent interactions between $A$ and $B$ individuals, and such interactions require a high heterozygosity, which in turn requires larger genealogical distance. 2. In our proof, we implicitly take advantage of the recently developed theory of tree-valued stochastic processes from [@GrevenPfaffelhuberWinter2012] and [@DGP2012]. The idea is to describe the evolution of the genealogical distance of two randomly sampled points. If no events hit the two lines, the distance grows at constant speed. If a T1-arrow falls in between the two sampled individuals, their distance is reset to 0. Finally, with high probability, the two sampled lines are hit by T2- or T3-arrows only if the arrows originate from one of the $N-2$ other individuals. Once the evolution of the distance of two randomly sampled lines is given, we only have to find a(n approximate) fixed point in order to show Theorem \[T:dist\]. The details are given in Section \[S:proof2\]. The ASG for linear frequency-dependent selection {#ss:ASG} ================================================ The ancestral selection graph (ASG) was introduced by [@NeuhauserKrone1997] and [@KroneNeuhauser1997] in order to study genealogies in the case of frequency-independent selection. Later, it was extended in [@Neuhauser1999] to a model of minority-advantage, a special form of frequency-dependence in an infinite alleles setting. Here, we introduce the ASG in order to give a proof of Theorem \[T:onethird\]. For its construction, consider again Figure \[fig2\]. If a sample of individuals is drawn at the top of the figure, one can read off all events which finally determine their types. Because there are three different kinds of arrows, the history of the sample as well comes with three different events. First, T1-arrows between lines in the sample lead to coalescence events, because common ancestors are found along such events. Second, T2-arrows mostly origin from lines outside the sample. Since they as well are determinants of the types in the sample, such events lead to branching events into a continuing and an incoming branch. Third, T3-arrows lead to splits of a line into the continuing, the checking and the imitating line. This informal description turns into the following stochastic process: Starting with $n$ lines, the following transitions occur: 1. Every pair of lines coalesces at rate 1. 2. Every single line splits in two (called the continuing and incoming line) at rate $\alpha\|\beta\|$. 3. Every single line splits in three (called the continuing, checking and imitating line) at rate $\alpha\|\gamma\|$. Consider again the graphical representation of the Moran model from Figure \[fig2\]. Here, starting with $n$ lines at the top of the figure, it might be that the last event which hits one of the $n$ lines is a T2- or T3-arrow, which originates from one of the $n$ lines. However, since $N$ is assumed to be large, and the $N\to\infty$ limit gives , this case can be ignored in the limit. Once this graph is run for time $t$ (i.e. from time $t$ down to time 0), and the starting frequency of type $A$ (at time 0) of the forward process is given, we can determine the configuration of the $n$ lines. First, assign randomly each of the lines of the ASG at time 0 with $A$ according to the starting frequency. Then, go through the ASG from time 0 up to time $t$. The types are inherited along every T1-arrow. Moreover, we have to distinguish the inheritance rules in the four cases, (i)–(iv). 1. In cases (i) and (iii), i.e. when $\beta>0$, the type of the incoming branch is inherited if it has type $A$. Otherwise the type of the continuing branch is inherited. 2. In cases (ii) and (iv), i.e. when $\beta<0$, the type of the incoming branch is inherited if it has type $B$. Otherwise the type of the continuing branch is inherited. 3. In cases (i) and (iv), i.e. when $\gamma>0$, the type of the imitating branch is inherited if it has type $B$ and the checking branch has type $A$. Otherwise the type of the continuing branch is inherited. 4. In cases (ii) and (iii), i.e. when $\gamma<0$, the type of the imitating branch is inherited if it has type $A$ and the checking branch has type $A$. Otherwise the type of the continuing branch is inherited. It is important to understand that these rules are reminiscent of the corresponding transitions in the Moran model. As an example, consider Figure \[fig3\](D). (Here, the ASG is given starting with a single line at the top, $n=1$.) In case (i) and (iv), the type of the line at the top is $A$ only if both lines at the bottom have type $A$. If the first line is $B$, the continuing branch of the T3-event is followed which carries type $B$. If the first line is $A$ and the second line is $B$, rule 3A says that the type of the imitating branch is inherited, hence the line at the top has type $B$. In case (ii) and (iii), the line at the top is $A$ if and only if the first line at the bottom is type $A$. In this case, the imitating line can only be used if it has type $A$ as well, leading to type $A$ at the top. If the first line carries $B$, the continuing line is used at the T3-event, leading to type $B$ at the top. In Table \[tab1\], we assume that the frequency of type $A$ at time 0 is $\varepsilon\ll 1$, and we give all possibilities which lead to fixation of type $A$, which require at most one type-$A$ individual in the past, and theire respective probabilities up to order $\varepsilon$. \(A) (B) (C) (D) (E) (F)\ (0.5,3)(0,0) (1,0)[(0,1)[3]{}]{} (1, -0.3)[(0,0)[past]{}]{} (1, 3.3)[(0,0)[present]{}]{} (2,3)(0,0) (1,0)[(0,1)[1]{}]{} (1,2)[(0,1)[1]{}]{} (1,1.2)[(0,1)[.1]{}]{} (1,1.4)[(0,1)[.1]{}]{} (1,1.6)[(0,1)[.1]{}]{} (1,1.8)[(0,1)[.1]{}]{} (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} (2,3)(0,0) (1,3)[(0,-1)[1.5]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[2]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[1]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[2]{}]{} (0,1)[(1,0)[1]{}]{} (0.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[1]{}]{} (1,1)[(1,0)[1]{}]{} (1.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} Two proofs of Theorem \[T:onethird\] {#S:twop} ==================================== [\|l\|\|c\|c\|c\|c\|c\|c\|]{} Genealogy $\mathcal G = g$ & (2,3)(0,0) (1,0)[(0,1)[1]{}]{} (1,2)[(0,1)[1]{}]{} (1,1.2)[(0,1)[.1]{}]{} (1,1.4)[(0,1)[.1]{}]{} (1,1.6)[(0,1)[.1]{}]{} (1,1.8)[(0,1)[.1]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1.5]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[2]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[1]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[2]{}]{} (0,1)[(1,0)[1]{}]{} (0.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[1]{}]{} (1,1)[(1,0)[1]{}]{} (1.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} \ Case (i): in which cases does $A$ fix? & (2,3)(0,0) (1,0)[(0,1)[1]{}]{} (1,2)[(0,1)[1]{}]{} (1,1.2)[(0,1)[.1]{}]{} (1,1.4)[(0,1)[.1]{}]{} (1,1.6)[(0,1)[.1]{}]{} (1,1.8)[(0,1)[.1]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} (0,0)[(0,0)[$\bullet$]{}]{} or (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1.5]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[2]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[1]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[2]{}]{} (0,1)[(1,0)[1]{}]{} (0.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[1]{}]{} (1,1)[(1,0)[1]{}]{} (1.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1.5,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (0,0)[(0,0)[$\bullet$]{}]{} \ Case (i): $\mathbf P[A\text{ fixes}\|\; \mathcal G = g, X_0=\varepsilon]$& $\varepsilon$& $2\varepsilon$& $\varepsilon$& $0$& $\varepsilon$& $\varepsilon$\ Case (ii): in which cases does $A$ fix? & (2,3)(0,0) (1,0)[(0,1)[1]{}]{} (1,2)[(0,1)[1]{}]{} (1,1.2)[(0,1)[.1]{}]{} (1,1.4)[(0,1)[.1]{}]{} (1,1.6)[(0,1)[.1]{}]{} (1,1.8)[(0,1)[.1]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} & (3,3)(0,0) (1,3)[(0,-1)[1.5]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[2]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[1]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[2]{}]{} (0,1)[(1,0)[1]{}]{} (0.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (.5,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[1]{}]{} (1,1)[(1,0)[1]{}]{} (1.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1.5,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (0,0)[(0,0)[$\bullet$]{}]{} or (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} \ Case (ii): $\mathbf P[A\text{ fixes}\| \;\mathcal G = g, X_0=\varepsilon]$& $\varepsilon$& 0& $\varepsilon$& $\varepsilon$& $\varepsilon$& $2\varepsilon$\ Case (iii): in which cases does $A$ fix? & (2,3)(0,0) (1,0)[(0,1)[1]{}]{} (1,2)[(0,1)[1]{}]{} (1,1.2)[(0,1)[.1]{}]{} (1,1.4)[(0,1)[.1]{}]{} (1,1.6)[(0,1)[.1]{}]{} (1,1.8)[(0,1)[.1]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} (0,0)[(0,0)[$\bullet$]{}]{} or (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (3,3)(0,0) (1,3)[(0,-1)[1.5]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[2]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[1]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[2]{}]{} (0,1)[(1,0)[1]{}]{} (0.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (.5,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[1]{}]{} (1,1)[(1,0)[1]{}]{} (1.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1.5,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (0,0)[(0,0)[$\bullet$]{}]{} or (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} \ Case (iii): $\mathbf P[A\text{ fixes}\| \;\mathcal G = g, X_0=\varepsilon]$& $\varepsilon$& $2\varepsilon$& $\varepsilon$& $\varepsilon$& $\varepsilon$& $2\varepsilon$\ Case (iv): in which cases does $A$ fix? & (2,3)(0,0) (1,0)[(0,1)[1]{}]{} (1,2)[(0,1)[1]{}]{} (1,1.2)[(0,1)[.1]{}]{} (1,1.4)[(0,1)[.1]{}]{} (1,1.6)[(0,1)[.1]{}]{} (1,1.8)[(0,1)[.1]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[1]{}]{} (1,2)[(0,-1)[2]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1.5]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[2]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[1]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[2]{}]{} (0,1)[(1,0)[1]{}]{} (0.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} & (2,3)(0,0) (1,3)[(0,-1)[1]{}]{} (0,2)[(1,0)[2]{}]{} (0,2)[(0,-1)[2]{}]{} (1,2)[(0,-1)[1]{}]{} (2,2)[(0,-1)[1]{}]{} (1,1)[(1,0)[1]{}]{} (1.5,1)[(0,-1)[1]{}]{} (0,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (1.5,0)[(0,0)[$\bullet$]{}]{} & (2,3)(0,0) (0,3)[(0,-1)[3]{}]{} (0,2)[(1,0)[2]{}]{} (1,2)[(0,-1)[2]{}]{} (2,2)[(0,-1)[1]{}]{} (2,1)[(-1,0)[1]{}]{} (1,2.2)[(0,0)[c]{}]{} (2,2.2)[(0,0)[i]{}]{} (0,0)[(0,0)[$\bullet$]{}]{} \ Case (iv): $\mathbf P[A\text{ fixes}\| \;\mathcal G = g, X_0=\varepsilon]$& $\varepsilon$& 0& $\varepsilon$& $0$& $\varepsilon$& $\varepsilon$\ A proof based on [@Rousset2003] and [@Ladret2007] ------------------------------------------------- Although the proof of [@Ladret2007], which is based on ideas from [@Rousset2003] is carried out in a time-discrete model, the same argument works for the SDE . Again, we write $\mathbf E_{\varepsilon,\alpha}[.]$ for expectations with $X_0=\varepsilon$ and fitness coefficient $\alpha$. First, is continuous in the parameter $\alpha$, such that $\mathbf E_{\varepsilon,\alpha}[f(X_t)] = \mathbf E_{\varepsilon,0}[f(X_t)](1+\mathcal O(\alpha))$ for $f\in\mathcal C_b^2(\mathbb R)$. Moreover, by the duality of the Wright–Fisher diffusion, i.e. with $\alpha=0$, to Kingman’s coalescent, $$\begin{aligned} \mathbf E_{\varepsilon,0}[X_t(1-X_t)] & = e^{-t} \varepsilon(1-\varepsilon) = e^{-t} \big(\varepsilon + \mathcal O(\varepsilon^2)\big),\\ \mathbf E_{\varepsilon,0}[X_t^2(1-X_t)] & = e^{-3t}\varepsilon^2(1-\varepsilon) + \int_0^t 3e^{-3s} e^{-(t-s)} \frac 13 \varepsilon(1-\varepsilon) ds \\ & = \frac 12 e^{-t}(1-e^{-2t})\big(\varepsilon + \mathcal O(\varepsilon^2)\big)\end{aligned}$$ as $\varepsilon\to 0$. Here, the first equation arises since $2X_t(1-X_t)$ is the probability to obtain two different types when picking from the population. This event requires that the two sampled individuals do not share an ancestor between times 0 and $t$. Moreover, both ancestors have to have different types. For the second equality, when sampling three individuals, two of which are type $A$, there are two possibilities. Either no coalescence of the three sampled lines occurred between times 0 and $t$, or two of three lines coalesced and have a type $A$-ancestor. Combining these results, $$\begin{aligned} \mathbf P_{\varepsilon,\alpha}(A \text{ fixes}) & = \mathbf E_{\varepsilon, \alpha}[X_\infty] = \varepsilon + \int_0^\infty \frac{d}{dt} \mathbf E_{\varepsilon, \alpha}[X_t]dt \\ & = \varepsilon + \alpha \int_0^\infty \mathbf E_{\varepsilon,0}[X_t(1-X_t)(\beta - \gamma X_t)](1+\mathcal O(\alpha)) dt \\ & = \varepsilon + \alpha \int_0^\infty \big(\beta e^{-t}\big(\varepsilon + \mathcal O(\varepsilon^2)\big) - \frac 12 \gamma e^{-t}(1-e^{-2t})\big(\varepsilon + \mathcal O(\varepsilon^2)\big)\big)(1+\mathcal O(\alpha))dt \\ & = \varepsilon\Big( 1 + \alpha \Big(\beta - \gamma \frac 13\Big) \Big) + \mathcal O(\alpha^2, \varepsilon^2),\end{aligned}$$ which finishes the first proof. A proof based on the ASG ------------------------ Let us prove Theorem \[T:onethird\] by using the ASG. Here, type $A$ eventually fixes if and only if the common ancestor of all individuals carries type $A$. Therefore, we have to study the ASG which has run for an infinite amount of time (i.e. from time infinity back to time 0). Since $\alpha$ is assumed to be small, we can assume that the ASG has only a single line at time 0 with high probability. However, with probability of order $\alpha$, the ASG has split close to time 0 but the resulting lines have not fully coalesced yet. Since $\alpha$ is small, we can assume that the last split event happened when the ASG only had a single line. In Figure \[fig3\], we list all possible ASGs near time 0 which we have to consider. Start with case (B), which is a split in two lines. We abbreviate this genealogy by ${ \setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(-.5,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){1}} \put (1,2){\line(0,-1){2}} \end{picture} }$. Noting that such split events occur at rate $\alpha\|\beta\|$, and coalescence of the resulting two lines happens at rate 1, the probability for this case is (by competing exponential clocks) $$\begin{aligned} \label{eq:zwei} \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(-.5,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){1}} \put (1,2){\line(0,-1){2}} \end{picture} }\Big) & = \frac{\alpha\|\beta\|}{1+\alpha\|\beta\|}+ \mathcal O(\alpha^2) = \alpha\|\beta\| + \mathcal O(\alpha^2).\end{aligned}$$ Similarly, consider the case when the last split occurs into three lines, which do not coalesce up to time 0 (case ${ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }$). Since coalescence of any pair happens at rate 3, we obtain $$\begin{aligned} \label{eq:dreinocoal} \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) & = \frac{\alpha\|\gamma\|}{3+\alpha\|\gamma\|} + \mathcal O(\alpha^2)= \frac{\alpha\|\gamma\|}{3} + \mathcal O(\alpha^2).\end{aligned}$$ Finally, the last split can lead to three lines and two of them coalesce up to time 0. Since all three possible genealogies (denoted ${ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){1}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){2}} \put (0,1){\line(1,0){1}} \put (0.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }, { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }$ and ${ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }$) have equal probability, we obtain $$\begin{aligned} \label{eq:dreionecoal} \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){1}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){2}} \put (0,1){\line(1,0){1}} \put (0.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) & = \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) = \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) = \frac 13 \frac{\alpha\|\gamma\|}{1+\alpha\|\gamma\|} + \mathcal O(\alpha^2) = \frac{\alpha\|\gamma\|}{3} + \mathcal O(\alpha^2).\end{aligned}$$ Last, we have for the remaining case ${\setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(0,1) \put (1,0){\line(0,1){1}} \put (1,2){\line(0,1){1}} \put (1,1.2){\line(0,1){.1}} \put (1,1.4){\line(0,1){.1}} \put (1,1.6){\line(0,1){.1}} \put (1,1.8){\line(0,1){.1}} \end{picture} }$ $$\label{eq:eins} \begin{aligned} \mathbf P\Big( {\setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(0,1) \put (1,0){\line(0,1){1}} \put (1,2){\line(0,1){1}} \put (1,1.2){\line(0,1){.1}} \put (1,1.4){\line(0,1){.1}} \put (1,1.6){\line(0,1){.1}} \put (1,1.8){\line(0,1){.1}} \end{picture} }\Big) & = 1 - \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(-.5,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){1}} \put (1,2){\line(0,-1){2}} \end{picture} }\Big) - \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) - \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){1}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){2}} \put (0,1){\line(1,0){1}} \put (0.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) - \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) - \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \mathcal O(\alpha^2)\\ & = 1 - \alpha \Big(\|\beta\| + \frac 43 \|\gamma\|\Big) + \mathcal O(\alpha^2). \end{aligned}$$ Using Table \[tab1\] for the probabilities of fixation conditioned on the genealogy, we can now collect all terms in the four cases. We obtain $$\begin{aligned} \text{Case (i): } & p_\text{fix}(\varepsilon,\alpha) = \varepsilon \cdot \mathbf P\Big( {\setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(0,1) \put (1,0){\line(0,1){1}} \put (1,2){\line(0,1){1}} \put (1,1.2){\line(0,1){.1}} \put (1,1.4){\line(0,1){.1}} \put (1,1.6){\line(0,1){.1}} \put (1,1.8){\line(0,1){.1}} \end{picture} }\Big) + 2\varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(-.5,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){1}} \put (1,2){\line(0,-1){2}} \end{picture} }\Big) + \varepsilon\cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \varepsilon\cdot \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \mathcal O(\alpha^2, \varepsilon^2)\\ \text{Case (ii): } & p_\text{fix}(\varepsilon,\alpha) = \varepsilon \cdot \mathbf P\Big( {\setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(0,1) \put (1,0){\line(0,1){1}} \put (1,2){\line(0,1){1}} \put (1,1.2){\line(0,1){.1}} \put (1,1.4){\line(0,1){.1}} \put (1,1.6){\line(0,1){.1}} \put (1,1.8){\line(0,1){.1}} \end{picture} }\Big) + \varepsilon\cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){1}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){2}} \put (0,1){\line(1,0){1}} \put (0.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \varepsilon \cdot \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + 2\varepsilon\cdot \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \mathcal O(\alpha^2, \varepsilon^2) \\ \text{Case (iii): } & p_\text{fix}(\varepsilon,\alpha) = \varepsilon \cdot \mathbf P\Big( {\setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(0,1) \put (1,0){\line(0,1){1}} \put (1,2){\line(0,1){1}} \put (1,1.2){\line(0,1){.1}} \put (1,1.4){\line(0,1){.1}} \put (1,1.6){\line(0,1){.1}} \put (1,1.8){\line(0,1){.1}} \end{picture} }\Big) + 2\varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(-.5,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){1}} \put (1,2){\line(0,-1){2}} \end{picture} }\Big) + \varepsilon\cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){1}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){2}} \put (0,1){\line(1,0){1}} \put (0.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + 2\varepsilon\cdot \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \mathcal O(\alpha^2, \varepsilon^2) \\ \text{Case (iv): } & p_\text{fix}(\varepsilon,\alpha) = \varepsilon \cdot \mathbf P\Big( {\setlength{\unitlength}{0,2cm} \begin{picture}(2,3)(0,1) \put (1,0){\line(0,1){1}} \put (1,2){\line(0,1){1}} \put (1,1.2){\line(0,1){.1}} \put (1,1.4){\line(0,1){.1}} \put (1,1.6){\line(0,1){.1}} \put (1,1.8){\line(0,1){.1}} \end{picture} }\Big) + \varepsilon\cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1.5}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){2}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \varepsilon \cdot \mathbf P\Big( { \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (1,3){\line(0,-1){1}} \put (0,2){\line(1,0){2}} \put (0,2){\line(0,-1){2}} \put (1,2){\line(0,-1){1}} \put (2,2){\line(0,-1){1}} \put (1,1){\line(1,0){1}} \put (1.5,1){\line(0,-1){1}} \put (0,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \varepsilon\cdot \mathbf P\Big({ \setlength{\unitlength}{0,2cm} \begin{picture}(4,4)(-1,1) \put (0,3){\line(0,-1){3}} \put (0,2){\line(1,0){2}} \put (1,2){\line(0,-1){2}} \put (2,2){\line(0,-1){1}} \put (2,1){\line(-1,0){1}} \put (1,2.2){\makebox(0,0){\tiny c}} \put (2,2.2){\makebox(0,0){\tiny i}} \end{picture} }\Big) + \mathcal O(\alpha^2, \varepsilon^2)\end{aligned}$$ Plugging in the probabilities from , , and , we obtain in all cases $$\begin{aligned} p_\text{fix}(\varepsilon,\alpha) = \varepsilon\Big( 1 + \alpha \Big( \beta - \gamma \frac 13 \Big)\Big) + \mathcal O(\alpha^2, \varepsilon^2) \end{aligned}$$ which finishes the second proof of Theorem \[T:onethird\]. Proof of Theorem \[T:dist\] {#S:proof2} =========================== The proof is based on the graphical construction of the Moran model; see Figure \[fig2\]. We have to take into account all three different kinds of arrows. Let $R_{12} = R_{12}(t)$ be the random distance of two individuals taken at random from the Moran model at time $t$. In addition, $Z_3 = Z_3(t)$ and $Z_4 = Z_4(t)$ are the types of two additionally sampled individuals. We claim that in case (i) $$\label{eq:distbasic} \begin{aligned} \frac{d}{dt} \mathbf E_{N,\alpha}[e^{-\lambda R_{12}}] & = -\lambda \mathbf E_{N,\alpha}[e^{-\lambda R_{12}}] + 1 - \mathbf E_{N,\alpha}[e^{-\lambda R_{12}}] \\ & \qquad + \alpha \|\beta\| \cdot \mathbf E_{N,\alpha}[1_{Z_3=A} (e^{-\lambda R_{23}} - e^{-\lambda R_{12}})] \\ & \qquad \qquad \qquad + \alpha\|\gamma\| \cdot \mathbf E_{N,\alpha} [1_{Z_3=B}1_{Z_4=A} (e^{-\lambda R_{23}} - e^{-\lambda R_{12}})] + \mathcal O(1/N). \end{aligned}$$ Here, the first term describes the increase of the genealogical distance of individuals 1 and 2 if no events occur. Second, T1-arrows between 1 and 2 lead to $R_{12}=0$ (hence $e^{-\lambda R_{12}}=1$) at rate 1. Third, the origin of T2-arrows is with high probability ($1-\mathcal O(1/N)$) neither individual 1 nor 2, but a third individual, with type $Z_3$. However, this arrow only takes effect if $Z_3=A$. Last, T3-arrows again originate from a third individual with type $Z_3$. It picks a fourth individual with type $Z_4$ and if $Z_3=B$ and $Z_4=A$, the arrow takes effect. All terms require that the two/three/four chosen individuals are distinct; hence the error term $\mathcal O(1/N)$. Our goal is to approximate the right hand side of . We note that the ASG implies that for all expectations on the right hand side we have that $\mathbf E_{N,\alpha}[.] = \mathbf E_{N,0}[.] + \mathcal O(\alpha)$ for small $\alpha$ and large $N$. Hence, we can use $\mathbf E_{N,0}[.]$ as an approximation which is valid up to order $\alpha$. Recall that the mutation rate from $A$ to $B$ is $\theta_B/2$ and from $B$ to $A$ is $\theta_A/2$, as well as $\bar\theta := \theta_A + \theta_B$. We note that for $N\to\infty$, we can use Kingman’s coalescent, since it is known to give genealogies in the large population limit of the neutral Moran model. We write $\mathbf E[.]$ for the expectation when using Kingman’s coalescent and finally argue that $\mathbf E_{N,0}[.] = \mathbf E[.] + \mathcal O(1/N)$. We obtain $$\begin{aligned} \mathbf E[1_{Z_1=B}1_{Z_2=A}] & = \frac{\theta_B/2}{1+\bar\theta} \cdot \frac{\theta_A}{\bar\theta} + \frac{\theta_A/2}{1+\bar\theta} \cdot \frac{\theta_B}{\bar\theta} = \frac{\theta_A\theta_B}{\bar\theta}\frac{1}{1+\bar\theta},\\ \mathbf E[1_{Z_1=A} e^{-\lambda R_{12}}] & = \mathbf E[1_{Z_3=A}e^{-\lambda R_{12}}] = \frac{\theta_A}{\bar\theta} \frac{1}{1+\lambda},\\ \mathbf E[1_{Z_1=B} e^{-\lambda R_{12}}] & = \mathbf E[1_{Z_3=B}e^{-\lambda R_{12}}] = \frac{\theta_B}{\bar\theta} \frac{1}{1+\lambda}.\end{aligned}$$ Here, the first equality holds since $Z_1=B$ and $Z_2=A$ if and only if the first mutation event occurs before the two sampled lines coalesce. This mutation event must either be $B\to A$ in individual 1 or $B\to A$ in individual 2. Using similar arguments, we obtain $$\begin{aligned} \mathbf E[1_{Z_1=A} 1_{Z_2=B} e^{-\lambda R_{12}}] & = \int_0^\infty (1+\bar\theta)e^{-(1+\bar\theta)t} \cdot\Big(\frac{1 + \theta_A/2 + \theta_B/2}{1+\bar\theta}\cdot 0 \\ & \qquad\qquad+ \frac{\theta_A/2}{1+\bar\theta} \mathbf E[1_{Z_2=B}e^{-\lambda(t+R_{12})}] + \frac{\theta_B/2}{1+\bar\theta}\mathbf E[1_{Z_1=A}e^{-\lambda(t+R_{12})}]\Big)dt \\ & = \frac{\theta_A\theta_B}{\bar\theta}\frac{1}{(1+\bar\theta + \lambda)(1+\lambda)},\\ \mathbf E[1_{Z_2=B} 1_{Z_3=A} e^{-\lambda R_{12}}] & = \int_0^\infty (3+\bar\theta)e^{-(3+\bar\theta)t} \\ & \qquad \cdot\Big( \frac{1}{3+\bar\theta} \mathbf E[1_{Z_1=A} 1_{Z_2=B} e^{-\lambda (t+R_{12})}] + \frac{1}{3+\bar\theta} \mathbf E[1_{Z_2=A} 1_{Z_3=B} e^{-\lambda t}] \\ & \qquad \qquad + \frac{\theta_A/2}{3+\bar\theta}\mathbf E[1_{Z_1=B}e^{-\lambda (t+R_{12})}] + \frac{\theta_B/2}{3+\bar\theta}\mathbf E[1_{Z_3=A}e^{-\lambda (t+R_{12})}]\Big) dt \\ & = \frac{\theta_A\theta_B}{\bar\theta}\frac{1}{3+\bar\theta+\lambda}\Big( \frac{1}{(1+\bar\theta+\lambda)(1+\lambda)} + \frac{1}{1+\bar\theta} + \frac{1}{1+\lambda}\Big) \\ & = \frac{\theta_A\theta_B}{\bar\theta}\Big( \frac{2+\bar\theta+\lambda}{3+\bar\theta+\lambda}\frac{1}{(1+\bar\theta+\lambda)(1+\lambda)} + \frac{1}{(3+\bar\theta+\lambda)(\bar\theta+1)}\Big),\\ \mathbf E[1_{Z_2=A} 1_{Z_3=B} e^{-\lambda R_{12}}] & = \mathbf E[1_{Z_2=B} 1_{Z_3=A} e^{-\lambda R_{12}}],\end{aligned}$$ $$\begin{aligned} \mathbf E[1_{Z_3=B} 1_{Z_4=A} e^{-\lambda R_{12}}] & = \int_0^\infty (6+\bar\theta)e^{-(6+\bar\theta)t} \\ & \qquad \cdot\Big( \frac{1}{6+\bar\theta} \mathbf E[1_{Z_3=B} 1_{Z_4=A}e^{-\lambda t}] + \frac{4}{6+\bar\theta} \mathbf E[1_{Z_2=A} 1_{Z_3=B} e^{-\lambda (t+R_{12})}] \\ & \qquad \qquad + \frac{\theta_A/2}{6+\bar\theta} \mathbf E[1_{Z_3=B} e^{-\lambda (t+R_{12})}] + \frac{\theta_B/2}{6+\bar\theta} \mathbf E[1_{Z_3=A} e^{-\lambda (t+R_{12})}]\Big) dt \\ & = \frac{\theta_A\theta_B}{\bar\theta} \frac{1}{6+\bar\theta + \lambda} \Big(\frac{1}{1+\bar\theta} + \frac{2+\bar\theta+\lambda}{3+\bar\theta+\lambda}\frac{4}{(1+\bar\theta+\lambda)(1+\lambda)} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \frac{4}{(3+\bar\theta + \lambda)(1+\bar\theta)}+ \frac{1}{1+\lambda} \Big) \\ & = \frac{\theta_A\theta_B}{\bar\theta} \Big(\frac{7+\bar\theta+\lambda}{6 + \bar\theta + \lambda}\frac{1}{(3+\bar\theta+\lambda)(1+\bar\theta)} \\ & \qquad \qquad \qquad + \frac{2+\bar\theta+\lambda}{3+\bar\theta+\lambda} \frac{5 + \bar\theta + \lambda}{6 + \bar\theta + \lambda}\frac{1}{(1+\bar\theta+\lambda)(1+\lambda)} \\ & \qquad \qquad \qquad \qquad \qquad \qquad + \frac{1}{(6+\bar\theta+\lambda)(3+\bar\theta+\lambda)(1+\lambda)}\Big)\end{aligned}$$ Plugging the last two terms in we find that in equilibrium $$\begin{aligned} \mathbf E_{N,\alpha} [e^{-\lambda R_{12}}] & = \frac{1}{1+\lambda} + \frac{\alpha\|\gamma\|}{1+\lambda} \mathbf E_{N,0}[1_{Z_2=A}1_{Z_3=B}e^{-\lambda R_{12}}-1_{Z_3=A}1_{Z_4=B}e^{-\lambda R_{12}}] + \mathcal O(\alpha^2) \\ & = \frac{1}{1+\lambda} + \frac{\alpha\|\gamma\|}{1+\lambda}\frac{\theta_A\theta_B}{\bar \theta} \frac{1}{6+\bar\theta + \lambda} \Big( \frac{2 + \bar\theta + \lambda}{3+\bar\theta+\lambda} \frac{1}{(1+\bar\theta+\lambda)(1+\lambda)} \\ & \qquad \qquad \qquad \qquad \qquad \qquad - \frac{1}{(3+\bar\theta+\lambda)(1+\bar\theta)} - \frac{1}{(3+\bar\theta+\lambda)(1 + \lambda)}\Big) + \mathcal O(\alpha^2) \\ & = \frac{1}{1+\lambda} + \frac{\alpha\|\gamma\|}{1+\lambda}\frac{\theta_A\theta_B}{\bar \theta} \frac{1}{6+\bar\theta + \lambda} \frac{2+\bar\theta+\lambda}{3+\bar\theta+\lambda} \Big( \frac{1}{(1+\bar\theta+\lambda)(1+\lambda)} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad - \frac{1}{(1+\bar\theta)(1+\lambda)}\Big) + \mathcal O(\alpha^2) \\ & = \frac{1}{1+\lambda} - \alpha\|\gamma\|\frac{\theta_A\theta_B}{\bar \theta} \frac{(2+\bar\theta+\lambda)\lambda}{(6+\bar\theta + \lambda)(3+\bar\theta+\lambda)(1+\bar\theta + \lambda)(1+\bar\theta)(1+\lambda)^2} + \mathcal O(\alpha^2).\end{aligned}$$ In particular, using a derivative according to $\lambda$ at $\lambda=0$, $$\begin{aligned} \mathbf E_{N,\alpha}[R_{12}] & = - \frac{\partial}{\partial\lambda} \mathbf E_{N,\alpha} [e^{-\lambda R_{12}}]\Big\|_{\lambda=0} = 1 + \alpha\|\gamma\|\frac{\theta_A\theta_B}{\bar\theta} \frac{2+\bar\theta}{(6+\bar\theta)(3+\bar\theta)(1+\bar\theta )^2} + \mathcal O(\alpha^2)\end{aligned}$$ In case (ii), changes to $$\begin{aligned} \frac{d}{dt} \mathbf E_{N,\alpha}[e^{-\lambda R_{12}}] & = -\lambda \mathbf E_{N,\alpha}[e^{-\lambda R_{12}}] + 1 - \mathbf E_{N,\alpha}[e^{-\lambda R_{12}}] + \alpha \|\beta\| \mathbf E_{N,\alpha}[1_{Z_3=B} (e^{-\lambda R_{12}} - e^{-\lambda R_{23}})] \\ & \qquad \qquad \qquad \qquad \qquad + \alpha\|\gamma\| \mathbf E_{N,\alpha} [1_{Z_3=A}1_{Z_4=A} (e^{-\lambda R_{23}} - e^{-\lambda R_{12}})] + \mathcal O(1/N).\end{aligned}$$ As above, the term behind $\alpha\|\beta\|$ vanishes. Moreover, $$\begin{aligned} \mathbf E_{N,\alpha} [1_{Z_2=A}1_{Z_3=A} e^{-\lambda R_{12}}] & = \mathbf E_{N,\alpha} [1_{Z_2=A} e^{-\lambda R_{12}}] - \mathbf E_{N,\alpha} [1_{Z_2=A}1_{Z_3=B} e^{-\lambda R_{12}}],\\ \mathbf E_{N,\alpha} [1_{Z_3=A}1_{Z_4=A} e^{-\lambda R_{12}}] & = \mathbf E_{N,\alpha} [1_{Z_4=A} e^{-\lambda R_{12}}] - \mathbf E_{N,\alpha} [1_{Z_3=B} 1_{Z_4=A}e^{-\lambda R_{12}}].\end{aligned}$$ Therefore, in equilibrium, $$\begin{aligned} \mathbf E_{N,\alpha} [e^{-\lambda R_{12}}] & = \frac{1}{1+\lambda} + \frac{\alpha\|\gamma\|}{1+\lambda} \mathbf E_{N,0}[1_{Z_2=A}1_{Z_3=A}e^{-\lambda R_{12}}-1_{Z_3=A}1_{Z_4=A}e^{-\lambda R_{12}}] + \mathcal O(\alpha^2, 1/N) \\ & = \frac{1}{1+\lambda} - \frac{\alpha \|\gamma\|}{1+\lambda} \mathbf E_{N,0}[1_{Z_2=A}1_{Z_3=B}e^{-\lambda R_{12}}-1_{Z_3=A}1_{Z_4=B}e^{-\lambda R_{12}}] + \mathcal O(\alpha^2, 1/N).\end{aligned}$$ Using the result from case (i), we are done with case (ii). 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--- abstract: 'We prove the local existence and uniqueness of solutions to a system of quasi-linear wave equations involving a jump discontinuity in the lower order terms. A continuation principle is also established.' address: - 'Albert Einstein Institute, Am Mühlenberg 1, D-14476 Potsdam, Germany' - \| School of Mathematical Sciences\ Monash University, VIC 3800\ Australia author: - Lars Andersson - 'Todd A. Oliynyk' bibliography: - 'trans.bib' title: 'A transmission problem for quasi-linear wave equations' --- introduction {#intro} ============ Most of the visible matter in our Universe is composed of gravitating relativistic elastic matter; for example, asteroids, comets, planets and stars, including neutron stars, are all thought of as being accurately described as elastic bodies [@AnderssonComer:2007; @ChamelHaensel:2008]. Due to this, it is of clear theoretical and even practical interest to have a good analytic understanding of gravitating relativistic elastic bodies with the first step being to establish local existence and uniqueness results. In the non-relativistic setting of Newtonian gravity, local existence and uniqueness theorems are available. In the approximation of a compact (non-fluid) elastic body moving in an external gravitational field, where the gravitational self-interaction and interaction with the object generating the external field are ignored, local existence and uniqueness has been established in [@ShibataNakamura:1989]. Local existence and uniqueness results for the general case, which includes gravitational self and mutual interactions between adjacent (non-fluid) elastic bodies, are given in [@Andersson_et_al:2011]. See also [@LindbladNordgren:2009] for related results on self-gravitating, incompressible fluid bodies. In contrast, much less is known in the relativistic setting where local existence and uniqueness theorems are lacking except in certain restricted situations [@ChoquetFriedrich:2006; @KindEhlers:1993; @Rendall:1992]. Relativistic compact elastic bodies are governed by the Einstein field equations coupled through the stress-energy tensor to the field equations of relativistic elasticity. The difficulty in establishing local existence and uniqueness results can be attributed to two sources: the free boundary arising from the evolving matter-vacuum interface, and the irregularity in the stress-energy tensor across the matter-vacuum interface. For elastic bodies, there are essentially two distinct types of irregularities. The first type corresponds to gaseous fluid bodies where the proper energy density monotonically decreases in a neighborhood of the vacuum boundary and vanishes identically there. In this situation, the fluid evolution equations become degenerate and are no longer hyperbolic at the boundary leading to severe analytic difficulties. The second type of irregularity that occurs for elastic bodies is where the proper energy density has a finite (positive) limit at the vacuum boundary. Examples of this type are liquid fluids and solid elastic bodies. This case leads to a jump discontinuity in the stress energy tensor across the vacuum boundary. Our main motivation for this article is to develop local existence and uniqueness results that are applicable to the gravitational part of the initial value problem (IVP) for gravitating relativistic elastic bodies that are not fluids[^1] and have the second type of discontinuity. For such elastic bodies, it is well known from [@BeigSchmidt:2003; @BeigWernig:2007], see also Section \[disc\], when harmonic coordinates are employed and the material representation is employed, that the gravitational component of the field equations consists of a system of non-linear wave equations with a jump discontinuity at the matter-vacuum boundary while the elastic component consists of a non-linear system of wave equations with Neumann boundary conditions. This leads us to consider IVPs of the form[^2] [ $$\begin{aligned} {\partial_{\mu}}\bigl(A^{\mu \nu}(U) {\partial_{\nu}} U\bigr) &= F(U,{\partial_{}}U) + \chi_{\Omega} H(U, {\partial_{}} U) \quad \text{in $[0,T]\times {\mathbb{R}{}}^n$}, \label{waveA.1} \\ (U,{\partial_{t}}U)\|_{t=0} & = ({\tilde{U}{}}_0,{\tilde{U}{}}_1) \quad \text{in ${\mathbb{R}{}}^n$} \label{waveA.2}, \end{aligned}$$ ]{} where 1. $\Omega$ is a bounded open set in ${\mathbb{R}{}}^n$ with smooth boundary, 2. $U({\mathbf{x}{}}) = (U^1({\mathbf{x}{}}),\ldots,U^{N}({\mathbf{x}{}}))$ is vector valued, 3. $A(U)=(A^{\mu\nu}(U))$, $F=(F^I(U,{\partial_{}}U))$ and $H=(H^I(U,{\partial_{}}U))$ (I=1,…,N) are smooth maps with $F(0,0)=0$, and 4. for some $\gamma,\kappa > 0$, $A^{\mu\nu}(U)$ satisfies [$$\frac{1}{\gamma} \|\xi\|^2 \leq A^{ij}(U)\xi_i \xi_j \leq \gamma \|\xi\|^2 {{\quad\text{and}\quad}}A^{00}(U) \leq -\kappa \label{waveAa}$$]{} for all $(U,\xi) \in {\mathbb{R}{}}^N\times {\mathbb{R}{}}^n$. In Section \[disc\], we describe how the results of this article can be used in conjunction with the local existence theory from [@Koch:1993] to establish the local existence and uniqueness of solutions that represent gravitating relativistic compact elastic bodies. The complete local existence and uniqueness proof will be provided in a separate article [@Andersson_et_al:2013]. Aside from this application, we believe that the results of this paper are of independent interest and may be useful for other initial value problems involving systems of wave equations with lower order coefficients that have a jump discontinuity across a fixed boundary. Due to the discontinuity in the wave equation arising from the term $\chi_{\Omega} H(U, {\partial_{}} U)$, the initial value problem (IVP) - is a transmission problem, that is, a problem where we can view that total solution as comprised of an interior solution and an exterior solution that are appropriately “matched” across the dividing interface ${\partial_{}}\Omega$. Due to the jump discontinuity across ${\partial_{}}\Omega$, standard $L^2$ Sobolev spaces $H^s({\mathbb{R}{}}^n)$ do not provide a suitable setting for establishing the local existence and uniqueness of solutions, and we use instead the intersection spaces ${\mathcal{H}{}}^{k,s}({\mathbb{R}{}}^n) = H^s(\Omega)\cap H^s({\mathbb{R}{}}^n\setminus \Omega)\cap H^k({\mathbb{R}{}}^n)$. Similar to the situation that arises for initial boundary value problems, we also find it necessary to choose initial data [$$({\tilde{U}{}}_0,{\tilde{U}{}}_1) \in {\mathcal{H}{}}^{2,s+1}({\mathbb{R}{}}^n)\times {\mathcal{H}{}}^{2,s}({\mathbb{R}{}}^n) \quad s\in {\mathbb{Z}{}}_{>n/2} \label{compat1}$$]{} that satisfy compatibility conditions given by [$${\tilde{U}{}}_\ell := {\partial_{t}}^\ell U \|_{t=0} \in {\mathcal{H}{}}^{m_{s+1-\ell},s+1-\ell}({\mathbb{R}{}}^n) \quad \ell=2,\ldots,s. \label{compat2}$$]{} Here the time derivatives ${\partial_{t}}^\ell U \|_{t=0}$ $\ell \geq 2$ are generated from the initial data by formally differentiating with respect to $t$ at $t=0$. To see how this works, we note that ${\partial_{t}}^2 U\|_{t=0}$ can be computed by substituting the initial data in and then solving for ${\partial_{t}}^2 U\|_{t=0}$. Differentiating formally with respect to $t$ while substituting in the lower time derivatives $\ell=0,1,2$ at $t=0$ then uniquely determines the $\ell=3$ time derivative at $t=0$ in terms of the initial data. Continuing on by formally differentiating the evolution equations with respect to $t$, it is not difficult to see that the higher time derivatives $\ell=2,\ldots,s$ at $t=0$ are uniquely determined in terms of the initial data. We are now ready to state the main local existence and uniqueness result. \[mainthmA\] Suppose $n\geq 3$, $s\in {\mathbb{Z}{}}_{>n/2}$ and $({\tilde{U}{}}_0,{\tilde{U}{}}_1) \in {\mathcal{H}{}}^{s+1}({\mathbb{R}{}}^n)\times {\mathcal{H}{}}^{s}({\mathbb{R}{}}^n)$ satisfy the compatibility conditions . Then there exist a $T>0$ and a map $U\in CX_{T}^{s+1}({\mathbb{R}{}}^n)$ such that $U$ is the unique solution in $CX_{T}^2({\mathbb{R}{}}^n) \cap \bigcap_{\ell=0}^1 C^\ell\bigl([0,T),W^{1-\ell,\infty}({\mathbb{R}{}}^n)\bigr)$ to the initial value problem -. Moreover, if ${\\|u\\|}_{W^{1,\infty}((0,T)\times {\mathbb{T}{}}^n)} < \infty$, then there exists a $T^>T$ such that the $u$ can be continued uniquely to a solution on $[0,T^)\times {\mathbb{T}{}}^n$. The proof of this theorem can be found in Section \[exist\] and relies on a strategy similar to the one employed by Koch in [@Koch:1993] to establish the existence and uniqueness of solutions to fully non-linear wave equations on bounded domains with Neumann or Dirichlet boundary conditions. Koch’s method involves differentiating the evolution equation $s$ times with respect to $t$ for $s$ sufficiently large. He then views the equations involving the lower order time derivatives ${\partial_{t}}^\ell u$ $(\ell=0,\ldots,s-1)$ as a system of coupled elliptic equations for the purpose of obtaining estimates and estimates the top time derivative ${\partial_{t}}^s u$ using hyperbolic energy estimates. This allows him to avoid directly differentiating in directions normal to the boundary. For us, this strategy allows us to avoid differentiating the term $\chi_\Omega H(U,{\partial_{}} U)$ across ${\partial_{}}\Omega$ where it is discontinuous. Although the arguments used in this article are structurally similar to those employed in Koch, there are some differences. One difference is that the elliptic equations that arise in this article are not of a standard type due to the presence of the discontinuous term $\chi_\Omega H(U,{\partial_{}} U)$. As a consequence, we cannot, as did Koch, appeal to standard elliptic estimates, and instead we employ potential theory to derive the desired estimates. Another distinction is that we are not able to obtain estimates for all of the derivatives by differentiating tangentially to the space-time boundary $[0,T]\times{\partial_{}}\Omega$ and then using the evolution equations to recover the missing estimate for the derivative normal to the boundary as was done by Koch in [@Koch:1993]. One immediate consequence of this is that we cannot employ Koch’s strategy to derive a continuation principle and instead must argue differently. \[waveArem\] $\;$ - The assumptions on the IVP - can easily be relaxed so that - $A$, $F$ and $H$ depend explicitly on ${\mathbf{x}{}}\in {\mathbb{R}{}}^{n+1}$, i.e. $A=A({\mathbf{x}{}},U)$, $F=F({\mathbf{x}{}},U,{\partial_{}}U)$, and $H=H({\mathbf{x}{}},U,{\partial_{}}U)$, and are defined for $({\mathbf{x}{}},U,{\partial_{}} U) \in {\mathbb{R}{}}^{n+1}\times \mathcal{U}\times \mathcal{V}$ with $\mathcal{U}$ and $\mathcal{V}$ open in ${\mathbb{R}{}}^N$ and ${\mathbb{R}{}}^{(n+1)\times N}$, respectively, - $A$ and $\{F,H\}$ are $s+1$ and $s$ times continuously differentiable in all variables, respectively, where $s\in {\mathbb{Z}{}}_{>n/2}$, and - the inequality holds for $({\mathbf{x}{}},U,\xi) \in {\mathbb{R}{}}^{n+1}\times\mathcal{U}\times {\mathbb{R}{}}^n$. - The following generalizations of Theorem \[mainthmA\] also hold. - The continuous dependence of the solutions from Theorem \[mainthmA\] on the initial data satisfying the compatibility conditions can be established using similar arguments as in [@Koch:1993]. - Theorem \[mainthmA\] is also valid for quasi-linear wave equations [ $$A^{\mu \nu}(U,{\partial_{}}U) {\partial_{\mu}}{\partial_{\nu}} U = F(U,{\partial_{}}U) + \chi_{\Omega} H(U, {\partial_{}} U) \quad \text{in $[0,T]\times {\mathbb{R}{}}^n$},$$ ]{} provided that we take $s>n/2+1$ and change the continuation principle to that of bounding ${\\|U\\|}_{W^{2,\infty}((0,T)\times {\mathbb{R}{}}^n)}$. Preliminaries {#prelim} ============= Notation -------- In this article, we use $(x^{\mu})_{\mu=0}^n$ to denote Cartesian coordinates on ${\mathbb{R}{}}^{n+1}$, and we use $x^0$ and $t$, interchangeability, to denote the time coordinate, and $(x^i)_{i=1}^n$ to denote the spatial coordinates. We also use $x=(x^1,\ldots,x^n)$ and ${\mathbf{x}{}}= (x^0,\ldots,x^n)$ to denote spatial and spacetime points, respectively. Partial derivatives are denoted by [ $${\partial_{\mu}} = \frac{\partial \;}{\partial x^\mu},$$ ]{} and we use $Du(x) = ({\partial_{1}}u(x),\ldots,{\partial_{n}}u(x))$ and ${\partial_{}}u({\mathbf{x}{}}) = ({\partial_{0}}u({\mathbf{x}{}}),Du({\mathbf{x}{}}))$ to denote the spatial and spacetime gradients, respectively. For time derivatives, we often employ the notation [ $$u_r := {\partial_{t}}^r u,$$ ]{} and use [ $${\mathbf{u}{}}_r = (u_1, u_2, \ldots, u_r )^{\text{Tr}}$$ ]{} to denote the collection of partial derivatives of $u$ with respect to $t$. Sets ---- The following subsets of ${\mathbb{R}{}}^n$ will be of interest: [ $$\begin{aligned} Q^-_\delta &= \{\: (x^1,\ldots,x^n) \: \|\: -\delta < x^1,x^2,\ldots,x^{n-1} < \delta, \quad -\delta < x^n < 0 \: \}, \\ Q^+_\delta &= \{\: (x^1,\ldots,x^n) \: \|\: -\delta < x^1,x^2,\ldots,x^{n-1} < \delta, \quad 0 < x^n < \delta \: \} \\ \intertext{and} Q_\delta &= \{\: (x^1,\ldots,x^n) \: \|\: -\delta \leq x^1,x^2,\ldots,x^{n} \leq \delta \:\}. \end{aligned}$$ ]{} We will also need to identify the opposite sides of the box $Q_\delta$ so that[^3] [ $$Q_\delta/\sim \approx{\mathbb{T}{}}^n .$$ ]{} We note that under this identification, the Carestian coordinates $x=(x^i)$ on ${\mathbb{R}{}}^n$ define periodic coordinates on ${\mathbb{T}{}}^n$. The following open and connected subset of ${\mathbb{T}{}}^n$ with smooth boundary will also be of interest [ $$\Omega_\delta = Q^+_{\delta}/\sim.$$ ]{} Finally, given an open set $\Omega$ of ${\mathbb{G}{}}^n$, where [ $$\text{${\mathbb{G}{}}^n$ = ${\mathbb{T}{}}^n$ or ${\mathbb{R}{}}^n$,}$$ ]{} we let $\chi_\Omega$ denote the characteristic function, and we use $\Omega^c$ to denote the interior of its complement, that is [ $$\Omega^c := {\mathbb{G}{}}^n \setminus \overline{\Omega}.$$ ]{} Function spaces {#funct} --------------- ### Spatial function spaces {#sfunct} Given an open set $\Omega \subset {\mathbb{G}{}}^n$, we define the Banach spaces [ $${\mathcal{H}{}}^{k,s}({\mathbb{G}{}}^n) = H^k({\mathbb{G}{}}^n)\cap H^s(\Omega)\cap H^s(\Omega^c) \quad (s\geq k; k,s\in{\mathbb{Z}{}}_{\geq 0})$$ ]{} with norm [ $${\\|u\\|}_{{\mathcal{H}{}}^{k,s}({\mathbb{G}{}}^n)}^2 = {\\|u\\|}_{H^s(\Omega)}^2 + {\\|u\\|}_{ H^k({\mathbb{G}{}}^n)}^2 + {\\| u\\|}^2_{H^s(\Omega^c)},$$ ]{} and [ $${\mathcal{H}{}}^{k,s}(Q_\delta) = H^k(Q_\delta)\cap H^s(Q^+_\delta)\cap H^s(Q^-_\delta) \quad (s\geq k; k,s\in{\mathbb{Z}{}}_{\geq 0})$$ ]{} with norm [ $${\\|u\\|}_{{\mathcal{H}{}}^{k,s}(Q_\delta)}^2 = {\\|u\\|}_{H^s(Q^+_\delta)}^2 + {\\|u\\|}_{ H^k(Q_\delta)}^2 + {\\| u\\|}^2_{H^s(Q^-_\delta)}.$$ ]{} We also define the following auxiliary spaces [ $$\begin{gathered} X^{k,r} = \prod_{\ell=0}^{r} {\mathcal{H}{}}^{2,k-\ell}({\mathbb{T}{}}^n) \quad (k-r\geq 2), \qquad {\mathcal{X}{}}^{k,r} = \prod_{\ell=0}^{r} {\mathcal{H}{}}^{0,k-\ell}({\mathbb{T}{}}^n) \quad (k-r\geq 0) \intertext{and} Y^{k,r} = \prod_{\ell=0}^{r} H^{k-\ell}(\Omega) \quad (k-r\geq 0,\; \Omega \subset {\mathbb{T}{}}^n) \end{gathered}$$ ]{} with norms [ $$\begin{gathered} {\\|{\mathbf{u}{}}_r\\|}_{X^{k,r}}^2 = \sum_{\ell=0}^r {\\|u_{\ell}\\|}^2_{{\mathcal{H}{}}^{2,k-\ell}({\mathbb{T}{}}^n)},\qquad {\\|{\mathbf{u}{}}_r\\|}_{{\mathcal{X}{}}^{k,r}}^2 = \sum_{\ell=0}^r {\\|u_{\ell}\\|}^2_{{\mathcal{H}{}}^{0,k-\ell}({\mathbb{T}{}}^n)}, \intertext{and} {\\|{\mathbf{u}{}}_r\\|}_{Y^{k,r}}^2 = \sum_{\ell=0}^r {\\|u_{\ell}\\|}^2_{H^{k-\ell}(\Omega)}, \end{gathered}$$ ]{} respectively, where, as above, we employ the vector notation [ $${\mathbf{u}{}}_r = (u_1,\ldots,u_r)^{\text{Tr}}.$$ ]{} ### Spacetime function spaces {#stfunct} Given an open subset $\Omega \subset {\mathbb{G}{}}^n$ and a $T>0$, we define the spaces [$$X^s_T({\mathbb{G}{}}^n) = \bigcap_{\ell=0}^s W^{\ell,\infty}\bigl([0,T),{\mathcal{H}{}}^{m_{s-\ell},s-\ell}({\mathbb{G}{}}^n)\bigr), \label{XTdef}$$]{} where [ $$m_\ell = \begin{cases} 2 & \text{if $\ell \geq 2$} \\ \ell & \text{ otherwise } \end{cases},$$ ]{} [$${\mathcal{X}{}}^s_T({\mathbb{G}{}}^n) = \bigcap_{\ell=0}^s W^{\ell,\infty}\bigl([0,T),{\mathcal{H}{}}^{0,s-\ell}({\mathbb{G}{}}^n)\bigr) \label{XcTdef}$$]{} and [$$Y^s_T(\Omega) = \bigcap_{\ell=0}^s W^{\ell,\infty}\bigl([0,T),H^{s-\ell}(\Omega)\bigr). \label{YTdef}$$]{} We also define the following energy norms: [ $$\begin{aligned} {\\|u\\|}_{E^s({\mathbb{G}{}}^n)}^2 &= \sum_{\ell=0}^s {\\|{\partial_{t}}^\ell u\\|}^2_{{\mathcal{H}{}}^{m_{s-\ell},s-\ell}({\mathbb{G}{}}^n)},\\ {\\|u\\|}_{{\mathcal{E}{}}^s({\mathbb{G}{}}^n)}^2 &= \sum_{\ell=0}^s {\\|{\partial_{t}}^\ell u\\|}^2_{{\mathcal{H}{}}^{0,s-\ell}({\mathbb{G}{}}^n)}, \\ {\\|u\\|}^2_{E^s(\Omega)} &= \sum_{\ell=0}^s {\\|{\partial_{t}}^\ell u\\|}^2_{H^{s-\ell}(\Omega)}, \\ {\\|u\\|}_{E^{s,r}({\mathbb{G}{}}^n)}^2 &= \sum_{\ell=0}^r {\\|{\partial_{t}}^\ell u\\|}^2_{{\mathcal{H}{}}^{m_{s-\ell},s-\ell}({\mathbb{G}{}}^n)} \quad (s\geq r) \intertext{and} {\\|u\\|}_{E({\mathbb{G}{}}^n)}^2 &= {\\|u\\|}_{E^1({\mathbb{G}{}}^n)}^2 = {\\|u\\|}_{H^1({\mathbb{G}{}}^n)}^2 + {\\|{\partial_{t}}u\\|}^2_{L^2({\mathbb{G}{}}^n)}. \end{aligned}$$ ]{} In terms of these energy norms, we can write the norms of the spaces , and as [ $$\begin{aligned} {\\|u\\|}_{X^s_T({\mathbb{G}{}}^n)} & = \sup_{0\leq t < T} {\\|u(t)\\|}_{E^s({\mathbb{G}{}}^n)}, \\ {\\|u\\|}_{{\mathcal{X}{}}^s_T({\mathbb{G}{}}^n)} & = \sup_{0\leq t < T} {\\|u(t)\\|}_{{\mathcal{E}{}}^s({\mathbb{G}{}}^n)} \intertext{and} {\\|u\\|}_{Y^s_T(\Omega)} & = \sup_{0\leq t < T} {\\|u(t)\\|}_{E^s(\Omega)}, \end{aligned}$$ ]{} respectively. Finally, we define the following subspace of : [ $$\begin{aligned} CX^s_T({\mathbb{G}{}}^n) &= \bigcap_{\ell=0}^s C^{\ell}\bigl([0,T),{\mathcal{H}{}}^{m_{s-\ell},s-\ell}({\mathbb{G}{}}^n)\bigr). \end{aligned}$$ ]{} Estimates and constants {#cost} ----------------------- We employ that standard notation [ $$a \lesssim b$$ ]{} for inequalities of the form [ $$a \leq C b$$ ]{} in situations where the precise value or dependence on other quantities of the constant $C$ is not required. On the other hand, when the dependence of the constant on other inequalities needs to be specified, for example if the constant depends on the norms ${\\|u\\|}_{L^\infty({\mathbb{T}{}}^n)}$ and ${\\|v\\|}_{L^\infty(\Omega)}$, we use the notation [ $$C = C({\\|u\\|}_{L^\infty({\mathbb{T}{}}^n)},{\\|v\\|}_{L^\infty(\Omega)}).$$ ]{} Constants of this type will always be non-negative, non-decreasing, continuous functions of their arguments. A simple extension operator {#domain} --------------------------- Given an open set $\Omega$ in ${\mathbb{G}{}}^n$, we define the trivial extension operator by [ $$\chi_\Omega u(x) = \begin{cases} u(x) & \text{if $x\in \Omega$} \\ 0 & \text{otherwise} \end{cases}.$$ ]{} Clearly, this defines a bounded linear operator from $L^p(\Omega)$ to $L^p({\mathbb{G}{}}^n)$. Smoothing operator {#mollifier} ------------------ \[mollprop\] Suppose $\Omega$ is an open set in ${\mathbb{T}{}}^n$ with a smooth boundary, $1\leq p < \infty$ and $s \in {\mathbb{Z}{}}_{\geq 0}$. Then there exists a family of continuous linear maps [ $$J_\lambda \: :\: W^{s,p}({\mathbb{T}{}}^n) \longrightarrow W^{s,p}({\mathbb{T}{}}^n) \quad \lambda > 0$$ ]{} satisfying [ $$\begin{gathered} {\\|J_\lambda \chi_\Omega u\\|}_{W^{k,p}({\mathbb{T}{}}^n)} < \infty \quad k\geq s,\\ {\\|J_\lambda \chi_\Omega u\\|}_{W^{s,p}(\Omega)} \lesssim {\\|u\\|}_{W^{s,p}(\Omega)} \quad 0<\lambda \leq 1 \\ \intertext{and} \lim_{\lambda\searrow 0} {\\|J_\lambda \chi_\Omega u - u\\|}_{W^{s,p}(\Omega)} = 0 \end{gathered}$$ ]{} for all $u\in W^{s,p}(\Omega)$. On ${\mathbb{R}{}}^n$, the proof follows directly from [@AdamsFournier:2003 Theorem 2.29] and the discussion in the section Approximation by Smooth Functions on $\Omega$ starting on p. 65 of the same reference. On ${\mathbb{T}{}}^n$, the proof follows from using a smooth partition of unity to decompose functions into a finite sum of functions to which the results on ${\mathbb{R}{}}^n$ apply. \[mollcor\] Suppose $1\leq p < \infty$, $m\in {\mathbb{Z}{}}_{\geq 0}$, $s \in {\mathbb{Z}{}}_{\geq m}$, and let $J_\lambda$ be as defined in Proposition \[mollprop\]. Then $J_\lambda$ is a well-defined, continuous linear operator on ${\mathcal{H}{}}^{m,s}({\mathbb{T}{}}^n)$ satisfying [ $$\begin{gathered} {\\|J_\lambda u\\|}_{{\mathcal{H}{}}^{\ell,k}({\mathbb{T}{}}^n)} < \infty \quad k\geq s,\; \ell \geq m, \; k\geq \ell, \\ {\\|J_\lambda u\\|}_{{\mathcal{H}{}}^{m,s}({\mathbb{T}{}}^n)} \lesssim {\\|u\\|}_{{\mathcal{H}{}}^{m,s}(\Omega)} \quad 0<\lambda \leq 1 \intertext{and} \lim_{\lambda\searrow 0} {\\|J_\lambda u - u\\|}_{{\mathcal{H}{}}^{m,s}({\mathbb{T}{}}^n)} = 0 \end{gathered}$$ ]{} for all $u \in {\mathcal{H}{}}^{m,s}({\mathbb{T}{}}^n)$. Linear wave equations {#linear} ===================== Initial value problem {#linit} --------------------- Our proof of the existence and uniqueness of solutions to the IVP - relies on first analyzing the following linear IVP: [ $$\begin{aligned} {\partial_{\mu}}(A^{\mu\nu}{\partial_{\nu}} U) &= F + \chi_\Omega H \quad \text{in $[0,T)\times {\mathbb{R}{}}^n$,} \label{linIVP.1} \\ (U,{\partial_{t}} U)\|_{t=0} &= ({\tilde{U}{}}_0,{\tilde{U}{}}_1) \quad \text{ in ${\mathbb{R}{}}^n$,} \label{linIVP.2} \end{aligned}$$ ]{} where $\Omega$ is a bounded open set in ${\mathbb{R}{}}^n$ with smooth boundary, the initial data [$$({\tilde{U}{}}_0,{\tilde{U}{}}_1) \in {\mathcal{H}{}}^{2,s+1}({\mathbb{R}{}}^n)\times {\mathcal{H}{}}^{2,s}({\mathbb{R}{}}^n) \quad s\in {\mathbb{Z}{}}_{>n/2} \label{lincompat1a}$$]{} satisfies the compatibility conditions[^4] [$${\tilde{U}{}}_\ell := {\partial_{t}}^\ell U \|_{t=0} \in {\mathcal{H}{}}^{m_{s+1-\ell},s+1-\ell}({\mathbb{R}{}}^n) \quad \ell=2,\ldots,s, \label{lincompat1}$$]{} and there exist constants $\gamma,\kappa >0$ for which the matrix $A^{\mu\nu}$ satisfies [ $$\begin{gathered} \frac{1}{\gamma} \|\xi\|^2 \leq A^{ij}\xi_i \xi_j \leq \gamma \|\xi\|^2 \quad \text{for all $\xi \in {\mathbb{R}{}}^n$} \label{linAa.1} \intertext{and} A^{00} \leq -\kappa. \label{linAa.2} \end{gathered}$$ ]{} Away from the boundary of $\Omega$, the existence, uniqueness and regularity of solutions to - can be obtained by appealing to standard results on hyperbolic equations. So this leaves us with analyzing the problem in a neighborhood of the boundary ${\partial_{}}\Omega$ where standard results do not apply due to the jump discontinuity in the term $\chi_\Omega H$. Appealing to the property of finite speed of propagation, we can, using a smooth change of (spatial) coordinates, locally straighten out the boundary of $\Omega$ and localize the problem to a spacetime region of the form $[0,T)\times Q_\delta$ where $\delta$ can be chosen as small as we like. To be specific, we fix a point $x_0 \in \partial{}\Omega$, and choose an open neighborhood ${\mathcal{N}{}}_{x_0,\delta}$ of $x_0$ that is diffeomorphic to $Q_\delta$ such that the diffeomorphism [ $$\Phi_{x_0,\delta} \: : \: {\mathcal{N}{}}_{x_0,\delta} \longrightarrow Q_\delta$$ ]{} satisfies [ $$\begin{gathered} \Phi_{x_0,\delta}(x_0) = 0 \intertext{and} \Phi_{x_0,\delta}\bigl(\partial{}\Omega\cap {\mathcal{N}{}}_{x_0,\delta}\bigr) = \{\: (x^1,\ldots,x^{n-1},0) \: \|\: -\delta < x^1,x^2,\ldots,x^{n-1} < \delta \: \}. \end{gathered}$$ ]{} We also demand that all the derivatives of $\Phi_{x_0,\delta}^{-1}$ are uniformly bounded pointwise on $Q_\delta$ for $\delta \in (0,1]$. To see that this is possible, we fix a diffeomorphism $\Phi_{x_0,1}$ from ${\mathcal{N}{}}_{x_0,1}$ to $Q_1$. We then define diffeomorphisms [ $$\Psi_{x_0,\delta}:= \Phi_{x_0,1}^{-1}\|_{Q_\delta} \: : \: Q_\delta \longrightarrow {\mathcal{N}{}}_{x_0,\delta}:= \Phi_{x_0,\delta}^{-1}(Q_\delta)$$ ]{} for $\delta\in (0,1]$. Clearly, this family of diffeomorphisms satisfies the required properties. We extend $\Psi_{x_0,\delta}$ to a spacetime map by defining [$$\psi_{x_0,\delta} \: :\: [0,T) \times Q_\delta \longrightarrow [0,T)\times {\mathcal{N}{}}_{x_0,\delta} \: :\: (x^0,x) \longmapsto (x^0,\Psi_{x_0,\delta}(x)), \label{PsidefD}$$]{} and we let [ $$J^\mu_\nu = {\partial_{\nu}}\psi_{x_0,\delta}^\mu$$ ]{} denote the Jacobian matrix of the diffeomorphism and [ $${\check{J}{}}= J^{-1}$$ ]{} its inverse. Next, we define [ $$\begin{aligned} {\bar{U}{}}({\mathbf{x}{}}) &= U(\psi_{x_0,\delta}({\mathbf{x}{}})), \label{barvars.1} \\ {\hat{U}{}}_j(x) &= {\tilde{U}{}}_j(\Psi_{x_0,\delta}(x)) \quad j=0,1, \label{barvars.2} \\ {\bar{A}{}}^{\mu\nu}({\mathbf{x}{}}) &= \det{J({\mathbf{x}{}})}{\check{J}{}}^\mu_\alpha({\mathbf{x}{}}) A^{\alpha\beta}(\psi_{x_0,\delta}({\mathbf{x}{}})) {\check{J}{}}^\nu_\beta({\mathbf{x}{}}), \label{barvars.3} \\ {\bar{F}{}}({\mathbf{x}{}}) &= \det{J({\mathbf{x}{}})}F(\psi_{x_0,\delta}({\mathbf{x}{}})), \label{barvars.4} \intertext{and} {\bar{H}{}}(t,x) &= \det{J({\mathbf{x}{}})}H(\psi_{x_0,\delta}({\mathbf{x}{}})). \label{barvars.5} \end{aligned}$$ ]{} Letting, [ $$g=\eta_{\mu\nu}dx^\mu dx^\nu \qquad (\eta_{\mu\nu}) = \operatorname{diag}(-1,1,1,1)$$ ]{} denote the Minkowski metric, we recall the following pullback formula for a vector field $X=X^\mu{\partial_{\mu}}$: [$$\frac{1}{\sqrt{\|{\bar{g}{}}\|}}{\partial_{\mu}}\bigl(\sqrt{\|{\bar{g}{}}\|}{\bar{X}{}}^\mu\bigr) = \left(\frac{1}{\sqrt{\|g\|}}{\partial_{\mu}}\bigl(\sqrt{\|g\|}X^\mu\bigr) \right)\circ \psi_{x_0,\delta} \label{pback1}$$]{} where [ $$\begin{aligned} {\bar{X}{}}^\mu & := (\psi_{x_0,\delta}^X)^\mu = {\check{J}{}}^\mu_\nu {\bar{X}{}}^\nu \circ \psi_{x_0,\delta},\\ \|g\| &:= -\det(\eta_{\mu\nu}) = 1 \intertext{and} \|{\bar{g}{}}\| & := \|\psi_{x_0,\delta}^g\| = \det(J)^2. \end{aligned}$$ ]{} Setting [ $$X^\mu = A^{\mu\nu}{\partial_{\nu}}U$$ ]{} in , a short calculation using the chain rule and the definitions - shows that ${\bar{U}{}}$ satisfies the IVP [ $$\begin{aligned} {\partial_{\mu}}({\bar{A}{}}^{\mu\nu}{\partial_{\nu}} {\bar{U}{}}) &= {\bar{F}{}}+ \chi_{Q_1^+} {\bar{H}{}}\quad \text{in $[0,T)\times Q_\delta$ ,} \label{linP.1} \\ ({\bar{U}{}},{\partial_{t}} {\bar{U}{}})\|_{t=0} &= ({\hat{U}{}}_0,{\hat{U}{}}_1) \quad \text{ in $Q_\delta$,} \label{linP.2} \end{aligned}$$ ]{} and the compatibility conditions [$${\hat{U}{}}_\ell := {\partial_{t}}^\ell {\bar{U}{}}\|_{t=0} \in {\mathcal{H}{}}^{m_{s+1-\ell},s+1-\ell}(Q_\delta) \quad \ell=0,\ldots,s. \label{linPcompat1}$$]{} Projection to the $n$-Torus {#linrs} --------------------------- Before proceeding with the analysis of the IVP -, we first introduce two technical refinements. The first is to exploit the freedom to choose $\delta$ small by rescaling the fields and working on a fixed domain $Q_1$ instead. With this in mind, we define [ $$\begin{aligned} u({\mathbf{x}{}}) &= \frac{{\bar{U}{}}(\delta {\mathbf{x}{}})-{\bar{U}{}}({\mathbf{0}})}{\delta}, \label{epscale.1} \\ m^{\mu\nu} & = {\bar{A}{}}^{\mu\nu}({\mathbf{0}}), \label{epscale.2} \\ b^{\mu\nu}({\mathbf{x}{}}) & = \frac{{\bar{A}{}}^{\mu\nu}(\delta {\mathbf{x}{}})-m^{\mu\nu}}{\delta^\sigma}, \label{epscale.3} \\ f({\mathbf{x}{}}) &= \delta{\bar{F}{}}(\delta {\mathbf{x}{}}), \label{epscale.4} \intertext{and} h({\mathbf{x}{}}) &= \delta {\bar{H}{}}(\delta {\mathbf{x}{}}). \label{epscale.5} \end{aligned}$$ ]{} We note that by making a linear change of coordinates we can, due to the conditions - satisfied by $A^{\mu\nu}$, always arrange that [ $$(m^{\mu\nu}) = \operatorname{diag}(-1,1,\ldots,1).$$ ]{} A short calculation then shows that $u$ satisfies [ $$\begin{aligned} {\partial_{\mu}}((m^{\mu\nu}+{\epsilon}b^{\mu\nu}){\partial_{\nu}} u ) &= f+ \chi_{Q_1^+} h \quad \text{in $[0,T/\delta)\times Q_1$,} \label{linQ.1} \\ (u,{\partial_{t}}u)\|_{t=0} &= ({\tilde{u}{}}_0,{\tilde{u}{}}_1) := \left(\frac{{\hat{U}{}}_0(\delta x)-{\hat{U}{}}_0(0)}{\delta},{\hat{U}{}}_1(\delta x)\right) \quad \text{ in $Q_1$,} \label{linQ.2} \end{aligned}$$ ]{} where [ $${\epsilon}= \delta^\sigma.$$ ]{} The second technical refinement is to avoid the analytic difficulties that arise from boundary of $Q_1$ and, at the same time, put the equations in a suitable form using potential theory estimates. This is accomplished by using appropriate cutoff functions and appealing to the finite speed of propagation in order to “project” the evolution equations to a suitable form defined on [ $${\mathbb{T}{}}^n \cong Q_1/\sim.$$ ]{} We now describe the projected IVP. First, we fix points $x_+ \in { {\Omega_1}}$ and $x_- \in { {\Omega_1}}^c$ and choose a $\rho >0$ so that $B_{3\rho}(x_+) \in \Omega_1$ and $B_{3\rho}(x_-) \in { {\Omega_1}}^c$. Then we let $\psi$ denote a smooth non-negative function such that $\psi\|_{B_{\rho}(x_{\pm})} =1$ and $\psi\|_{{\mathbb{T}{}}^n\setminus (B_{2\rho}(x_+)\cup B_{2\rho}(x_-))}=0$. The projected IVP is then defined by [ $$\begin{aligned} {\partial_{\mu}}((m^{\mu\nu}+{\epsilon}\phi_1 b^{\mu\nu}){\partial_{\nu}} u ) -\psi u &= \phi_1 f + \phi_1 \chi_{ {\Omega_1}}h + \mu \quad \text{in $[0,T/\delta)\times {\mathbb{T}{}}^n$,} \label{linPN.1} \\ (u,{\partial_{t}}u)\|_{t=0} &= (\phi_1{\tilde{u}{}}_0,\phi_1{\tilde{u}{}}_1) \quad \text{ in ${\mathbb{T}{}}^n$,} \label{linPN.2} \end{aligned}$$ ]{} where - [ $$\phi_\eta(x^1,\ldots,x^n) := \phi\left(\frac{4x^1}{\eta}\right)\phi\left(\frac{4x^2}{\eta}\right)\cdots \phi\left(\frac{4x^n}{\eta}\right)$$ ]{} with $\phi \in C^\infty({\mathbb{R}{}})$ a cutoff function satisfying $\phi(\tau)=1$ for $\|\tau\|\leq 1$, $\phi(\tau)=0$ for $\|\tau\|\geq 2$ and $\phi(\tau)\geq 0$ for all $\tau \in {\mathbb{R}{}}$, and - [ $$\mu = \sum_{\ell=0}^{s-1} \frac{t^\ell}{\ell !} \mu_\ell$$ ]{} where the $\mu_\ell$ are determined in Proposition \[muprop\] below. \[muprop\] Suppose $n\geq 3$ and $s\in {\mathbb{Z}{}}_{>n/2}$, $0 < \delta \leq 1$, $\sigma = \min\{1,s-n/2\}$ and let $u_\ell = {\partial_{t}}^\ell u\|_{t=0}$ and ${\tilde{u}{}}_\ell = {\partial_{t}}^\ell u\|_{t=0}$, where the $\ell^{\text{th}}$ time derivative of $u$ is generated from formally differentiating and , respectively. Then there exist a $\delta_0 \in (0,1]$, $\eta_0 \in (0,1/4]$ and a sequence $\mu_\ell \in {\mathcal{H}{}}^{0,s-1-\ell}({\mathbb{T}{}}^n)$ $\ell=0,1,\ldots s-1$ such that [ $$u_\ell = \phi_1{\tilde{u}{}}_\ell \quad \ell=0,1, \quad \mu_\ell\|_{Q_{\eta_0}} = 0 \quad \ell=0,1,\ldots,s-1$$ ]{} and [ $$\begin{aligned} {\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} &\lesssim {\\|{\bar{U}{}}(0)\\|}_{E^{s+1}(Q_1)},\\ {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} &\lesssim \bigl(1+ {\\|{\bar{A}{}}(0)\\|}_{{\mathcal{E}{}}^s(Q_1)}+ {\\|D{\bar{A}{}}(0)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)}\bigr) {\\|{\bar{U}{}}(0)\\|}_{E^{s+1}(Q_1)} \\ &\text{\hspace{4.0cm}}+ \delta\bigl({\\|{\bar{F}{}}(0)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)}+ {\\|{\bar{H}{}}(0)\\|}_{E^{s-1}(Q_1^+)}\bigr) \end{aligned}$$ ]{} for all $\delta \in (0,\delta_0]$. In the following, we will use the notation [ $$(\cdot)_\ell = {\partial_{t}}^\ell (\cdot)\|_{t=0}$$ ]{} to denote the $\ell^{\text{th}}$ time derivative of a quantity evaluated at $t=0$, where the time derivatives of $u$ are generated from formally differentiating . Similarly, we use the [ $$\tilde{(\cdot)}_\ell = {\partial_{t}}^\ell (\cdot)\|_{t=0}$$ ]{} to denote the $\ell^{\text{th}}$ time derivative of a quantity evaluated at $t=0$ that depends on $u$ where the time derivatives of $u$ are generated from formally differentiating . We begin by defining [ $$\begin{gathered} {\bar{g}{}}^{\mu\nu} = m^{\mu\nu}+{\epsilon}\phi_1 b^{\mu\nu},\quad {\bar{f}{}}= \phi_1 f - {\partial_{i}}\phi_1 b^{i\nu}{\partial_{\nu}}u -\phi_1 {\partial_{\mu}}b^{\mu\nu} {\partial_{\nu}}u \intertext{and} {\bar{h}{}}= \phi_1 h, \end{gathered}$$ ]{} which allow us to write as [$${\bar{g}{}}^{\mu\nu}{\partial_{\mu}}{\partial_{\nu}}u -\psi u = {\bar{f}{}}+ \chi_{{ {\Omega_1}}}{\bar{h}{}}+ \mu. \label{muprop11}$$]{} Since $n>n/2$, we have by Sobolev’s inequality, see Theorem \[Sobolev\], that [ $${\\|\phi_1 b^{00}(0)\\|}_{L^\infty({\mathbb{T}{}}^n)} = \max \bigl\{{\\|\phi_1 b^{00}(0)\\|}_{L^\infty({ {\Omega_1}})},{\\|\phi_1 b^{00}(0)\\|}_{L^\infty(\Omega_1^c)}\bigr\} \lesssim {\\|b^{00}(0)\\|}_{H^{0,s}({\mathbb{T}{}}^n)}.$$ ]{} By Proposition \[scalepropA\], it follows from this inequality that [ $${\\|\phi_1 b^{00}(0)\\|}_{L^\infty({\mathbb{T}{}}^n)} \leq C {\\|{\bar{A}{}}(0)\\|}_{{\mathcal{E}{}}^{s}(Q_1)}$$ ]{} for some constant $C>0$ independent of $\delta \in (0,1]$. Therefore, since ${\epsilon}=\delta^\sigma$ and $\sigma >0$, we can choose $\delta_0 \in (0,1]$ small enough so that [ $${\epsilon}{\\|\phi_1 b^{00}(0)\\|}_{L^\infty({\mathbb{T}{}}^n)} \leq \frac{1}{2}$$ ]{} for $\delta \in (0,\delta_0]$, and this, in turn, guarantees that [ $$\frac{1}{2} \leq {\bar{g}{}}_{00} \leq \frac{3}{2}.$$ ]{} Using this, we can solve for the $2^{\text{nd}}$ order time derivative to get [$${\partial_{t}}^2u = \frac{1}{{\bar{g}{}}^{00}}\Bigr(-2{\bar{g}{}}^{i0} {\partial_{i}} {\partial_{t}}u - {\bar{g}{}}^{ij}{\partial_{i}}{\partial_{j}} u + \psi u + {\bar{f}{}}+ \chi_{\Omega_1}{\bar{h}{}}+ \mu \Bigr). \label{muprop12}$$]{} Setting [$$\mu_{0} = 2{\bar{g}{}}_0^{i0}{\partial_{i}}(\phi_1{\tilde{u}{}}_1) + {\bar{g}{}}_0^{ij}{\partial_{i}}{\partial_{j}}(\phi_1 {\tilde{u}{}}_0) -\psi \phi_1 {\tilde{u}{}}_0 -\phi_{1}\bigl({\bar{f}{}}_0 +\chi_{\Omega_1}{\bar{h}{}}_0 \bigr) + {\bar{g}{}}_0^{00}\phi_1 {\tilde{u}{}}_2, \label{muprop13}$$]{} we see from that [$$u_2 = \phi_1 {\tilde{u}{}}_2. \label{muprop14}$$]{} Moreover, it follows directly from the assumption $s>n/2$ and the multiplication inequality from Proposition \[elemE\] that [ $$\begin{aligned} {\\|\mu_{0}\\|}_{{\mathcal{H}{}}^{0,s-1}({\mathbb{T}{}}^n)} \lesssim \bigl(1+ {\epsilon}{\\|b(0)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)}+{\epsilon}{\\|Db(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}&\bigr){\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} \\ &+ {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}+ {\\|h(0)\\|}_{E^{s-1}(\Omega_1)}. \end{aligned}$$ ]{} Calculating the $2^{\text{nd}}$ order time derivative of $u$ using , we find that [$${\partial_{t}}^2 u= \frac{1}{g^{00}}\Bigr(-2g^{i0} {\partial_{i}}{\partial_{t}}u - g^{ij}{\partial_{i}}{\partial_{j}} u + k+ \chi_{\Omega_1}h \Bigr), \label{muprop16}$$]{} where [ $$g^{\mu\nu} = m^{\mu\nu}+{\epsilon}b^{\mu\nu} {{\quad\text{and}\quad}}k = f(u,{\partial_{}}u,v) -{\partial_{\mu}} b^{\mu\nu}{\partial_{\nu}} u.$$ ]{} Since [$$\phi_1\|_{Q_\eta}=1 \quad \eta\in [0,1/4], \label{muprop18}$$]{} and [$$\psi\|_{\{\|x^n\|\leq \eta_0\}} =0 \label{muprop19}$$]{} for $\eta_0$ small enough, we see, with the help of , and , that $\mu_0$, see , satisfies [ $$\mu_{0}\|_{Q_{\eta_0}} = 0$$ ]{} for some $\eta_0 \in (0,1/4]$. With the base case satisfied, we proceed by induction and assume that [ $$\begin{gathered} u_\ell = \phi_1{\tilde{u}{}}_\ell \quad \ell=0,1,\ldots,r+1, \label{muprop21.1} \\ \mu_\ell\|_{Q_{\eta_0}} = 0 \quad \ell=0,1,\ldots,r-1 \notag \intertext{and} {\\|\mu_\ell\\|}_{{\mathcal{H}{}}^{0,s-1-\ell}} \lesssim \bigl(1+ {\epsilon}{\\|b(0)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)}+{\epsilon}{\\|Db(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}\bigr){\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} \notag \\ \hspace{6.0cm} + {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}+ {\\|h(0)\\|}_{E^{s-1}(\Omega_1)}, \notag \end{gathered}$$ ]{} where $r<s$. Differentiating $r$-times with respect to $t$ and evaluating at $t=0$, we find that [ $$\begin{aligned} u_{2+r} = \frac{1}{{\bar{g}{}}^{00}_0}\Biggl( -\sum_{\ell=0}^{r-1}{r \choose \ell} {\bar{g}{}}^{00}_{r-\ell} u_{2+\ell} - \sum_{\ell=0}^r {r\choose \ell} \Bigl({\bar{g}{}}^{0i}_{r-\ell}& {\partial_{i}}u_{\ell+1} + {\bar{g}{}}^{ij}_{r-\ell} {\partial_{i}}{\partial_{j}}u_{\ell}\Bigr) \notag \\ &+\psi u_r + {\bar{f}{}}_r +\chi_{ {\Omega_1}}{\bar{h}{}}_r + \mu_{r} \Biggr). \label{muprop22.1} \end{aligned}$$ ]{} Setting [ $$\begin{aligned} \mu_{r} = \sum_{\ell=0}^{r-1}{r \choose \ell} {\bar{g}{}}^{00}_{r-\ell} u_{2+\ell} + \sum_{\ell=0}^r {r\choose \ell}& \Bigl({\bar{g}{}}^{0i}_{r-\ell} {\partial_{i}}u_{\ell+1} + {\bar{g}{}}^{ij}_{r-\ell} {\partial_{i}}{\partial_{j}}u_{\ell}\Bigr) \notag \\ &-\psi u_r - {\bar{f}{}}_r -\chi_{ {\Omega_1}}{\bar{h}{}}_r + {\bar{g}{}}^{00}_0\phi_1{\tilde{u}{}}_{2+r}, \label{muprop23.1} \end{aligned}$$ ]{} it follows immediately from that [ $$u_{2+r} = \phi_1{\tilde{u}{}}_{2+r}.$$ ]{} Similarly, differentiating $r$-times with respect to $t$ and evaluating at $t=0$, we obtain [ $$\begin{aligned} {\tilde{u}{}}_{2+r} = \frac{1}{{\tilde{g}{}}^{00}_0}\Biggl( -\sum_{\ell=0}^{r-1}{r \choose \ell} {\tilde{g}{}}^{00}_{r-\ell} {\tilde{u}{}}_{2+\ell} - \sum_{\ell=0}^r {r\choose \ell} \Bigl({\tilde{g}{}}^{0i}_{r-\ell} {\partial_{i}}{\tilde{u}{}}_{\ell+1} + {\tilde{g}{}}^{ij}_{r-\ell} {\partial_{i}}{\partial_{j}}{\tilde{u}{}}_{\ell}\Bigr) + {\tilde{k}{}}_r +\chi_{ {\Omega_1}}{\tilde{h}{}}_r \Biggr). \label{muprop24.1} \end{aligned}$$ ]{} Clearly, the induction hypothesis together with implies that [ $${\tilde{g}{}}_\ell\|_{Q_{\eta_0}} = {\tilde{g}{}}_\ell\|_{Q_{\eta_0}}, \quad {\bar{f}{}}_\ell\|_{Q_{\eta_0}} = {\tilde{k}{}}_\ell\|_{Q_{\eta_0}} {{\quad\text{and}\quad}}{\bar{h}{}}_\ell\|_{Q_{\eta_0}} = {\tilde{h}{}}_\ell\|_{Q_{\eta_0}}$$ ]{} for $\ell=0,\ldots,r+1$. Consequently, it follows directly from , and that [ $$\mu_r\|_{Q_{\eta_0}} = 0.$$ ]{} Furthermore, applying the multiplication inequality from Proposition \[elemE\] to , see the proof of Lemma \[linlemA\] for similar estimates, it is not difficult to verify that [ $$\begin{aligned} {\\|\mu_r\\|}_{{\mathcal{H}{}}^{0,s-1-r}({\mathbb{T}{}}^n)} \lesssim \bigl(1+ {\epsilon}{\\|b(0)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)}+&{\epsilon}{\\|Db(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}\bigr){\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)}\notag \\ &+ {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}+ {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} \label{muprop26.1}. \end{aligned}$$ ]{} This completes the induction step. Finally, observing the scaling definitions -, it is clear from Propositions \[scalepropA\] and \[scalepropB\] that the estimates [ $$\begin{aligned} {\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} &\lesssim {\\|{\bar{U}{}}(0)\\|}_{E^{s+1}(Q_1)} \intertext{and} {\\|\mu_\ell\\|}_{{\mathcal{H}{}}^{0,s-1-\ell}({\mathbb{T}{}}^n)} &\lesssim \bigl(1+ {\\|{\bar{A}{}}(0)\\|}_{{\mathcal{E}{}}^s(Q_1)}+ {\\|D{\bar{A}{}}(0)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)}\bigr){\\|{\bar{U}{}}(0)\\|}_{E^{s+1}(Q_1)} \\ &\text{\hspace{4.0cm}}+ \delta\bigl({\\|{\bar{F}{}}(0)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)}+ {\\|{\bar{H}{}}(0)\\|}_{E^{s-1}(Q_1^+)}\bigr) \end{aligned}$$ ]{} for $\ell=0,1,\ldots,s$ are a direct consequence of . Existence and uniqueness for the linear system - {#existA} ------------------------------------------------ In light of the form of the projected system -, we now turn our attention to the following class of linear IVPs: [ $$\begin{aligned} {\partial_{\mu}}((m^{\mu\nu}+{\epsilon}b^{\mu\nu}){\partial_{\nu}} u ) &= f + \chi_{Q_1^+} h + \mu \quad \text{in $[0,T)\times {\mathbb{T}{}}^n$,} \label{linM.1} \\ (u,{\partial_{t}}u)\|_{t=0} &= ({\tilde{u}{}}_0,{\tilde{u}{}}_1) \quad \text{ in ${\mathbb{T}{}}^n$,} \label{linM.2} \end{aligned}$$ ]{} where the initial data is chosen so that the compatibility conditions [$${\tilde{u}{}}_\ell := {\partial_{t}}^\ell u \|_{t=0} \in {\mathcal{H}{}}^{m_{s+1-\ell},s+1-\ell}({\mathbb{R}{}}^n) \quad \ell=2,\ldots,s \label{linMcompat}$$]{} are satisfied. \[linthm\] Suppose $n\geq 3$, $\delta \in (0,1]$, $\sigma = \min\{1,s-n/2\}$, ${\epsilon}=\delta^\sigma$, $s\in {\mathbb{Z}{}}_{>n/2}$, $T>0$, $b = (b^{\mu\nu}), {\partial_{t}}b\in {\mathcal{X}{}}_T^{s}({\mathbb{T}{}}^n)$, $Db \in {\mathcal{X}{}}_T^{s-1}({\mathbb{T}{}}^n)$, $f \in {\mathcal{X}{}}_T^{s}({\mathbb{T}{}}^n)$, $h \in Y^{s}_T(\Omega_1)$, $\mu \in {\mathcal{X}{}}_T^{s}({\mathbb{T}{}}^n)$ with ${\partial_{t}}^s \mu = 0$, $({\tilde{u}{}}_0,{\tilde{u}{}}_1) \in {\mathcal{H}{}}^{2,s+1}({\mathbb{T}{}}^n)\times {\mathcal{H}{}}^{2,s}({\mathbb{T}{}}^n)$ satisfy the compatibility conditions , and let [ $$R= \sup_{0\leq t < T}( {\\|b(t)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}})} + {\\|Db(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}})}) {{\quad\text{and}\quad}}\beta(t) = 1+{\\|b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}+{\\|{\partial_{t}}b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}.$$ ]{} Then there exists a constant $c_L=c_L(n,s) >0$ such that the IVP - has a unique solution $u \in CX_T^{s+1}({\mathbb{T}{}}^n)$ whenever $\delta$ is chosen so that ${\epsilon}$ satisfies $0\leq {\epsilon}\leq \min\left\{\frac{1}{3 c_L R},\frac{1}{3}\right\}$. Moreover, $u$ satisfies the following estimate [ $$\begin{aligned} {\\|u(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)}&\leq C(c_L) e^{C(c_L)\int_{0}^T \beta(\tau)\,d\tau} \biggl[ \beta(0){\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} \notag \\ +& {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + \int_{0}^T \beta(\tau)\Bigl({\\|f(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{E^s(\Omega_1)} + {\\|\mu(\tau)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \Bigr)\, d\tau \biggr] \end{aligned}$$ ]{} for $0\leq t < T$. Instead of solving the IVP - directly. We first regularize the problem using a mollifier to smooth the coefficients and the initial data with the resulting regularized IVP being [ $$\begin{aligned} {\partial_{\mu}}((m^{\mu\nu}+{\epsilon}J_\lambda(b^{\mu\nu})){\partial_{\nu}} u^\lambda) -\psi u^\lambda &= J_\lambda f + J_\lambda (\chi_{ {\Omega_1}}h) + J_\lambda \mu \quad \text{in $[0,T]\times {\mathbb{T}{}}^n$,} \label{linR.1} \\ (u^\lambda,{\partial_{t}}u^\lambda_\lambda)\|_{t=0} &= (J_\lambda {\tilde{u}{}}_0, J_\lambda {\tilde{u}{}}_1) \quad \text{ in ${\mathbb{T}{}}^n$,} \label{linR.2} \end{aligned}$$ ]{} where $\lambda \in (0,1]$. Applying a standard existence theorem for linear hyperbolic equations, for example see [@TaylorIII:1996 Ch. 16, Proposition 1.7], we obtain a 1-parameter family of (unique) solutions [ $$u^\lambda \in \bigcap_{\ell=0}^{s+100} C^\ell([0,T),H^{s+100-\ell}({\mathbb{T}{}}^n))\quad 0 < \lambda \leq 1.$$ ]{} Our goal now is to derive $\lambda$-independent estimates for $u^\lambda$ and then use these estimates to obtain a solution to the IVP - by letting $\lambda \searrow 0$ and extracting a (weakly) convergent subsequence that converges to a solution. The proof of the $\lambda$-independent estimates involves using elliptic estimates to estimate the first $s-1$ time derivatives of $u^\lambda$ followed by hyperbolic estimates to estimate the remaining $s$ and $s+1$ time derivatives. *Elliptic estimates:* Differentiating $k$ times with respect to $t$, we observe that $u_k={\partial_{t}}^k u$ satisfies the elliptic system [$$\Delta u^\lambda_k - \psi u^\lambda_k = u^\lambda_{k+2} + {\epsilon}\bigl(q^0_{k} + q^1_{k} + q^2_{k}\bigl)+ J_\lambda f_k+ J_\lambda \bigl(\chi_{\Omega_1} h_k \bigr) + J_\lambda \mu_k, \label{linB}$$]{} where [ $$\begin{aligned} q^0_{k} &= -\sum_{\ell=0}^{k} \binom{k}{\ell}\bigl( J_\lambda(b^{ij}_{k-\ell}) {\partial_{i}}{\partial_{j}}u^\lambda_\ell + J_\lambda ( {\partial_{i}} b^{ij}_{k-\ell}) {\partial_{j}} u^\lambda_\ell + J_\lambda (b^{0j})_{k-\ell+1}{\partial_{j}} u^\lambda_\ell \bigr) \notag \\ & -\sum_{\ell=1}^{k}\binom{k}{\ell-1} \bigl( 2 J_\lambda( b^{0j}_{k-\ell+1}){\partial_{j}} u^\lambda_\ell + J_\lambda({\partial_{j}} b^{0j}_{k-\ell+1})u^\lambda_\ell + b^{00}_{k-\ell+2} u^\lambda_\ell \bigr) - \sum_{\ell=2}^{k}\binom{k}{\ell-2} J_\lambda(b^{00}_{k-\ell+2}) u^\lambda_\ell, \notag \\q^1_{k} & = - k J_\lambda(b^{00}_1) u^\lambda_{k+1} - 2 J_\lambda(b^{0j}_0) {\partial_{j}} u^\lambda_{k+1} - J_\lambda({\partial_{j}} b^{0j}_0) u^\lambda_{k+1} - J_\lambda(b_1^{00}) u^\lambda_{k+1}, \notag \\q^2_{k} &= -J_\lambda(b^{00}_0) u^\lambda_{k+2} \notag \end{aligned}$$ ]{} and [ $$\Delta = \delta^{ij}{\partial_{i}}{\partial_{j}}$$ ]{} is the Euclidean Laplacian. \[linlemA\] The following estimates hold: [ $$\begin{aligned} &{\\|q^0_{k}\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \lesssim \bigl({\\|b\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} + {\\|Db\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \bigr) {\\|u^\lambda\\|}_{E^{s+1,k}({\mathbb{T}{}}^n)},\\ &{\\|q^1_{k}\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \lesssim \bigl({\\|b\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} + {\\|Db\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \bigr) {\\|u^\lambda_{k+1}\\|}_{{\mathcal{H}{}}^{1,s+1-(k+1)}({\mathbb{T}{}}^n)} \intertext{and} &{\\|q^2_{k}\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \lesssim {\\|b\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} {\\|u^\lambda_{k+2}\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \end{aligned}$$ ]{} for $0\leq k \leq s-1$. To begin, we observe that [ $${\\| J_\lambda(b^{ij}_{k-\ell}) {\partial_{i}}{\partial_{j}}u^\lambda_\ell\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \lesssim {\\|b_{k-\ell} \\|}_{{\mathcal{H}{}}^{0,s-(k-\ell)}({\mathbb{T}{}}^n)}{\\|D^2u^\lambda_\ell\\|}_{{\mathcal{H}{}}^{0,s-1-\ell}({\mathbb{T}{}}^n)} \quad 0\leq \ell \leq k \leq s-1$$ ]{} follows directly from Proposition \[elemE\] since $s>n/2$. Similar arguments show that the estimates [ $$\begin{aligned} &{\\| J_\lambda({\partial_{i}}b^{ij}_{k-\ell}){\partial_{j}}u^\lambda_\ell\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \lesssim {\\|Db_{k-\ell} \\|}_{{\mathcal{H}{}}^{0,s-1-(k-\ell)}({\mathbb{T}{}}^n)}{\\|D u^\lambda_\ell\\|}_{{\mathcal{H}{}}^{0,s-\ell}({\mathbb{T}{}}^n)} \\ &{\\| J_\lambda(b^{0j}_{k-\ell+1}){\partial_{j}} u^\lambda_\ell\\|}_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \lesssim {\\|b_{k-\ell+1} \\|}_{{\mathcal{H}{}}^{0,s-(k-\ell+1)}({\mathbb{T}{}}^n)}{\\|D u^\lambda_\ell\\|}_{{\mathcal{H}{}}^{0,s-\ell}({\mathbb{T}{}}^n)} \\ \end{aligned}$$ ]{} hold for $0\leq \ell \leq k \leq s-1$. These inequalities give us the following estimate for the first sum in $q^0_k$: [ $$\begin{aligned} \biggl\\| \sum_{\ell=0}^{k} \binom{k}{\ell}\bigl( J_\lambda(b^{ij}_{k-\ell}) {\partial_{i}}{\partial_{j}}u^\lambda_\ell + J_\lambda({\partial_{i}} b^{ij}_{k-\ell}) {\partial_{j}} u^\lambda_\ell + & J_\lambda(b^{0j}_{k-\ell+1}){\partial_{j}} u^\lambda_\ell \bigr) \biggr\\|_{{\mathcal{H}{}}^{0,s+1-(k+2)}({\mathbb{T}{}}^n)} \\ &\lesssim \bigl({\\|b\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} + {\\|Db\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \bigr) {\\|u^\lambda\\|}_{E^{s+1,k}({\mathbb{T}{}}^n)} \end{aligned}$$ ]{} for $0\leq k\leq s-1$. The required estimates for the rest of $q^0_k$ and $q^1_{k}$, $q^2_{k}$ follow from similar arguments. We collect the equations $(0\leq k \leq s-1)$ into a single system [$$L_{\epsilon}({\mathbf{u}{}}^\lambda_{s-1}) = {\mathbf{K}{}}_{s-1}, \label{linC}$$]{} where [ $${\mathbf{K}{}}_{s-1} = {\mathbf{f}{}}_{s-1}+ J_\lambda \bigl(\chi_{\Omega_1} {\mathbf{h}{}}_{s-1}\bigr) + {\boldsymbol{\mu}{}}_{s-1} + {\epsilon}\bigl(0,\ldots,0,q^2_{s-2},q^1_{s-1}+q^2_{s-1}\bigr)^\text{Tr}$$ ]{} and [ $$L_{\epsilon}= L_0 - {\epsilon}L_1$$ ]{} with [ $$L_0= \begin{pmatrix} \Delta-\psi & 0 & -1 & 0 & \cdots & 0 \\ 0 & \Delta - \psi & 0 & -1 & & 0 \\ 0 & 0 & \Delta-\psi & 0 & \ddots & \vdots \\ 0 & 0 & 0 & \Delta-\psi & \ddots & -1 \\ & \vdots & & & \ddots & 0 \\ 0 & 0 & 0 & 0 & \cdots & \Delta-\psi \\ \end{pmatrix}$$ ]{} and [ $$L_1({\mathbf{u}{}}_{s-1}) = {\mathbf{q}{}}^0_{s-1} + (q^1_0 + q^2_0,\ldots,q^1_{s-3}+q^2_{s-3},q^1_{s-1},0)^\text{Tr}.$$ ]{} Next, we observe that Lemma \[linlemA\] shows that the operator norm of [ $$L_1\: : \: X^{s+1,s-1} \longrightarrow {\mathcal{X}{}}^{s-1,s-1}$$ ]{} is bounded by [ $${\\|L_1\\|}_{\operatorname{op}} \leq C_{L_1}( {\\|b\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}})} + {\\|Db\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}})})$$ ]{} for some constant $C_{L_1} > 0$. Also, it is clear from Proposition \[potprop\] and the tridiagonal structure of $L_0$ that $L_0 \, :\, X^{s+1,s-1} \rightarrow {\mathcal{X}{}}^{s-1,s-1}$ is an isomorphism. From these facts, we get, via the Born series [ $$({\mathord{{\mathrm 1}\kern-0.27em{\mathrm I}}\kern0.35em}- {\epsilon}L_0^{-1}L_1)^{-1} = \sum_{k=0}^\infty {\epsilon}^k (L_0^{-1}L_1)^k \quad {\epsilon}{\\|L_0^{-1}L_1\\|} < 1,$$ ]{} that $L_{{\epsilon}} \, :\, X^{s+1,s-1} \rightarrow {\mathcal{X}{}}^{s-1,s-1}$ is invertible and the estimate [$${\\|L_{\epsilon}^{-1}\\|}_{\operatorname{op}} \leq \frac{{\\|L_0^{-1}\\|}_{\operatorname{op}}}{1-{\epsilon}( {\\|b\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)} + {\\|Db\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}){\\|L_0^{-1}\\|}_{\operatorname{op}}C_{L_1}}. \label{linlemB6}$$]{} holds whenever $\delta \in (0,1]$ is chosen small enough so that ${\epsilon}=\delta^\sigma$ satisfies [ $${\epsilon}( {\\|b\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}})} + {\\|Db\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}){\\|L_0^{-1}\\|}_{\operatorname{op}}C_{L_1} < 1.$$ ]{} It then follows directly from equation , the estimate and Lemma \[linlemA\] that there exists a constant $c_L=c_L({\\|L_0^{-1}\\|}_{\operatorname{op}},C_{L_1})>0$ such that [ $$\begin{aligned} {\\|u^\lambda(t)\\|}_{E^{s+1,s-1}({\mathbb{T}{}}^n)} \leq \frac{c_L}{1-{\epsilon}c_L R}\Bigl(&{\epsilon}R\bigl({\\|u^\lambda(t)\\|}_{E^{s+1,s-1}({\mathbb{T}{}}^n)} + {\\|u^\lambda_s(t)\\|}_{E({\mathbb{T}{}}^n)} \bigr) \notag \\ &+ {\\|f(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}+ {\\|h(t)\\|}_{E^{s-1}(\Omega_1)} + {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \Bigr) \label{uvest1.1} \end{aligned}$$ ]{} for $0\leq t < T$, where [ $$R = \sup_{0\leq t < T}( {\\|b(t)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}})} + {\\|Db(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}})})$$ ]{} and $\delta$ is chosen small enough so that ${\epsilon}=\delta^\sigma$ satisfies [$${\epsilon}c_L R < 1. \label{Rbound}$$]{} Choosing $\delta$ so that [ $${\epsilon}= \min\left\{\frac{1}{3c_L R},\frac{1}{3}\right\},$$ ]{} we can write as [$${\\|u^\lambda(t)\\|}_{E^{s+1,s-1}({\mathbb{T}{}}^n)} \leq {\\|u^\lambda_s(t)\\|}_{E({\mathbb{T}{}}^n)}+ 2c_L \bigl( {\\|f(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}+ {\\|h(t)\\|}_{E^{s-1}(\Omega_1)} + {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \bigr) \label{uvest2}$$]{} for $0\leq t < T$. Writing $f(t)$ and $h(t)$ as [ $$f(t) = f(0) + \int_{0}^t {\partial_{t}}f(\tau)\, d\tau {{\quad\text{and}\quad}}h(t) = h(0) + \int_{0}^t {\partial_{t}}h(\tau)\, d\tau,$$ ]{} respectively, we see that [ $${\\|f(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + {\\|h(t)\\|}_{E^{s-1}(\Omega_1)} \leq {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}+ {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} + \int_{0}^t\bigl( {\\|f(\tau)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{{\mathcal{E}{}}^{s}(\Omega_1)}\bigr) \, d\tau.$$ ]{} Combining this inequality with , we arrive at the estimate [ $$\begin{aligned} {\\|u^\lambda(t)\\|}_{E^{s+1,s-1}({\mathbb{T}{}}^n)} \leq {\\|u^\lambda_s(t)\\|}_{E({\mathbb{T}{}}^n)}&+ 2c_L\Bigl({\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} \notag \\ &+ \int_{0}^t\bigl( {\\|f(\tau)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{{\mathcal{E}{}}^{s}(\Omega_1)}\bigr) \, d\tau + {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \Bigr), \label{uvest3.1} \end{aligned}$$ ]{} which holds for $0\leq t < T$. *Hyperbolic estimates:* By assumption ${\partial_{t}}^s \mu = 0$. Therefore, differentiating $s$-times with respect to $t$ yields [$${\partial_{\mu}}\bigl( \bigl(m^{\mu\nu} + {\epsilon}J_\lambda (b^{\mu\nu})\bigr){\partial_{\nu}} u^\lambda_s \bigr) - \psi u^\lambda_s = {\epsilon}{\partial_{\mu}}\bigl( p^\mu - s J_\lambda(b^{\mu 0}_1) u_s^\lambda \bigr)+ J_\lambda f_s + J_\lambda (\chi_{ {\Omega_1}}h_s), \label{linH}$$]{} where [ $$p^\mu = -\sum_{\ell=0}^{s-1} \binom{s}{\ell}J_\lambda(b^{\mu j}_{s-\ell}) {\partial_{j}} u_\ell^\lambda - \sum_{\ell=0}^{s-2} \binom{s}{\ell} J_\lambda(b^{\mu 0}_{s-\ell})u_{\ell+1}^\lambda.$$ ]{} Recalling that $s>n/2$, it follows directly from Sobolev’s inequality, see Theorem \[Sobolev\], and Corollary \[mollcor\] that the inequalities [ $$\begin{aligned} {\\|J_\lambda b^{\mu \nu}(t)\\|}_{L^\infty({\mathbb{T}{}}^n)} &\lesssim {\\|J_\lambda b^{\mu \nu}(t)\\|}_{{\mathcal{H}{}}^{0,s}({\mathbb{T}{}}^n)} \lesssim {\\|b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} , \label{best.1}\\ {\\|{\partial_{t}}J_\lambda b^{\mu \nu}(t)\\|}_{L^\infty({\mathbb{T}{}}^n)} &\lesssim {\\|J_\lambda b^{\mu \nu}_1(t)\\|}_{{\mathcal{H}{}}^{0,s}({\mathbb{T}{}}^n)} \lesssim {\\|b(t)\\|}_{{\mathcal{E}{}}^{s+1}({\mathbb{T}{}}^n)}, \label{best.2} \\ {\\|{\partial_{t}}J_\lambda b^{\mu 0}_1(t)\\|}_{L^n({\mathbb{T}{}}^n)} &\lesssim {\\|J_\lambda b^{\mu 0}_2(t)\\|}_{{\mathcal{H}{}}^{0,s-1}({\mathbb{T}{}}^n)} \lesssim {\\|{\partial_{t}}b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} \label{best.3} \intertext{and} {\\|J_\lambda b^{\mu 0}_1(t)\\|}_{L^\infty({\mathbb{T}{}}^n)} &\lesssim {\\|J_\lambda b^{\mu 0}_1(t)\\|}_{{\mathcal{H}{}}^{0,s}({\mathbb{T}{}}^n)} \lesssim {\\|{\partial_{t}}b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)} \end{aligned}$$ ]{} are satisfied for $0\leq t < T$. Also, a slight adaptation of the arguments used to prove Lemma \[linlemA\] show that [ $$\begin{aligned} {\\|p^\mu(t)\\|}_{L^2({\mathbb{T}{}}^n)} &\lesssim {\\|b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}{\\|u^\lambda(t)\\|}_{E^{s+1,s-1}({\mathbb{T}{}}^n)} \label{pest.1} \intertext{and} {\\|{\partial_{t}} p^\mu (t)\\|}_{L^2({\mathbb{T}{}}^n)} &\lesssim \bigl({\\|b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}+{\\|{\partial_{t}}b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}\bigr)\bigl({\\|u^\lambda(t)\\|}_{E^{s+1,s-1}({\mathbb{T}{}}^n)} + {\\|u^\lambda_s(t)\\|}_{H^1({\mathbb{T}{}}^n)}\bigr) \label{pest.2}{\\|A(t)\\|}_{{\mathcal{E}{}}^{s+1}({\mathbb{R}{}}^n)} \end{aligned}$$ ]{} hold for $0\leq t < T$. We now are in a position to apply the energy estimates from Theorem \[weakthm\] to $u_s^\lambda$ since it solves the wave equation . Doing so, we see that these energy estimates in conjunction with the inequalities , - and Corollary \[mollcor\] show that $u^\lambda_s$ satisfies [ $$\begin{aligned} {\\|u^\lambda_s(t)\\|}_{E({\mathbb{T}{}}^n)} \leq C(c_L)\Bigl[\beta(0)&{\\|u^\lambda(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + \int_{0}^t \beta(\tau)\Bigl({\\|u^\lambda(\tau)\\|}_{E^{s+1,s-1}} \\ &+ {\\|u^\lambda(\tau)\\|}_{E({\mathbb{T}{}}^n)}\Bigr) + {\\|f_s(\tau)\\|}_{L^2({\mathbb{T}{}}^n)} + {\\|h_s(\tau)\\|}_{L^2(\Omega_1)} \, d\tau \Bigr] \end{aligned}$$ ]{} for $0\leq t < T$, where [ $$\beta(t) = 1+{\\|b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}+{\\|{\partial_{t}}b(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{T}{}}^n)}.$$ ]{} Combining this estimate with gives [ $$\begin{aligned} {\\|u^\lambda_s(t)\\|}_{E({\mathbb{T}{}}^n)} &\leq C(c_L)\biggl[\beta(0){\\|u^\lambda(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} \notag \\ + \int_{0}^t \beta(\tau)& \Bigl({\\|u^\lambda(\tau)\\|}_{E({\mathbb{T}{}}^n)} + {\\|f(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{E^s(\Omega_1)} + {\\|\mu(\tau)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \Bigr) \, d\tau \biggr] \label{usest2.1} \end{aligned}$$ ]{} for $0\leq t < T$. Together, the estimates and show that $u^\lambda$ satisfies the uniform bound [ $$\begin{aligned} {\\|u^\lambda(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} &\leq C(c_L)\Bigl[ \beta(0){\\|u^\lambda(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} \notag \\ + {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}& + \int_{0}^t \beta(\tau)\Bigl({\\|u^\lambda(\tau)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + {\\|f(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{E^s(\Omega_1)} + {\\|\mu(\tau)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}\Bigr) \, d\tau \Bigr] \end{aligned}$$ ]{} which, in turn, implies via Gronwall’s inequality that [ $$\begin{aligned} {\\|u^\lambda(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)}&\leq C(c_L) e^{C(c_L)\int_{0}^T \beta(\tau)\,d\tau} \biggl[ \beta(0){\\|u^\lambda(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} \notag \\ +& {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + \int_{0}^T \beta(\tau)\Bigl({\\|f(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{E^s(\Omega_1)} + {\\|\mu(\tau)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \Bigr)\, d\tau \biggr] \label{uvest5.1} \end{aligned}$$ ]{} for $0\leq t < T$. This inequality implies that $u^\lambda$ is a bounded 1-parameter family of solutions in $X^{s+1}_T({\mathbb{T}{}}^n)$ to the IVP -, and consequently, we know, by standard arguments, that there exists a weakly convergent subsequence, again denoted $u^\lambda$, that converges as $\lambda \searrow 0$ to the unique weak solution $u\in X^{s+1}_T({\mathbb{T}{}}^n)$ of -. Moreover, by the uniqueness of weak limits, we see, by sending $\lambda \searrow 0$ in the estimate , that $u$ satisfies [ $$\begin{aligned} {\\|u(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)}&\leq C(c_L) e^{C(c_L)\int_{0}^T \beta(\tau)\,d\tau} \biggl[ \beta(0){\\|u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + {\\|f(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + {\\|h(0)\\|}_{E^{s-1}(\Omega_1)} \notag \\ +& {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + \int_{0}^T \beta(\tau)\Bigl({\\|f(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)} + {\\|h(\tau)\\|}_{E^s(\Omega_1)} + {\\|\mu(\tau)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \Bigr)\, d\tau \biggr] \label{uvest6.1} \end{aligned}$$ ]{} for $0\leq t < T$. Finally, a straightforward calculation shows that the difference $u-u_\lambda$ satisfies a linear equation of the type -, and hence, also an estimate of the type from which it follows that [ $${\\|u-u_\lambda\\|}_{X^{s+1}_T({\mathbb{T}{}}^n)} \lesssim {\\|u(0)-J_\lambda u(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + c(\lambda)$$ ]{} for some constant $c(\lambda)$ satisfying $\lim_{\lambda \searrow 0} c(\lambda) = 0$. Since $u_\lambda \in CX^{s+1}_T({\mathbb{T}{}}^n)$, this estimate implies that $u_\lambda(t)$ converges uniformly on $[0,T]$ to $u(t)$, thereby showing that $u\in CX^{s+1}_T({\mathbb{T}{}}^n)$. With the existence and uniqueness for the system - established, it is now straightforward, using the finite speed of propagation, to conclude an analogous uniqueness and existence result for the original system -. \[linGthm\] Suppose $n\geq 3$, $s\in {\mathbb{Z}{}}_{>n/2}$, $T>0$, $A = (A^{\mu\nu}),{\partial_{t}}A \in {\mathcal{X}{}}_T^{s}({\mathbb{R}{}}^n)$, $DA \in {\mathcal{X}{}}_T^{s-1}({\mathbb{R}{}}^n)$, $F \in {\mathcal{X}{}}_T^{s}({\mathbb{R}{}}^n)$, $H \in Y^{s}_T(\Omega)$, $({\tilde{U}{}}_0,{\tilde{U}{}}_1) \in {\mathcal{H}{}}^{2,s+1}({\mathbb{R}{}}^n)\times {\mathcal{H}{}}^{2,s}({\mathbb{R}{}}^n)$ satisfy the compatibility conditions , and let [ $$\begin{gathered} \alpha = \sup_{0\leq t < T}\bigl({\\|A(t)\\|}_{{\mathcal{E}{}}^s({\mathbb{R}{}}^n)}+{\\|D A (t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{R}{}}^n)}\bigr), \quad \beta(t) = 1+ {\\|A(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{R}{}}^n)}+ {\\|{\partial_{t}}A(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{R}{}}^n)} \intertext{and} \gamma = \int_{0}^{T}\beta(\tau) \,d\tau. \end{gathered}$$ ]{} Then the IVP - has a unique solution $U \in CX_T^{s+1}({\mathbb{R}{}}^n)$ and satisfies the estimate [ $$\begin{aligned} {\\|U(t)\\|}_{ E^{s+1}({\mathbb{R}{}}^n)}& \leq c(\alpha,\beta(0),\gamma)\biggl[ {\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)} + {\\|F(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{R}{}}^n)} \notag \\ & + {\\|H(0)\\|}_{E^{s-1}(\Omega)} + \int_{0}^{T}\beta(\tau)\bigl( {\\|F(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{R}{}}^n)} + {\\|H(\tau)\\|}_{E^s(\Omega)}\bigr)\, d\tau \biggr] \end{aligned}$$ ]{} for $0\leq t < T$. We begin by letting $u \in CX^{s+1}_T({\mathbb{T}{}}^n)$ be the unique solution to the IVP - from Theorem \[linthm\] with $\delta \in (0,1]$ chosen small enough so that ${\epsilon}=\delta^\sigma$ satisfies [$${\epsilon}= \min\left\{\frac{1}{3 c_L R},\frac{1}{3}\right\} \label{linGthm3}$$]{} with [$$R = \sup_{0\leq t < T}\bigl({\\|b(t)\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)}+{\\|Db(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)}\bigr). \label{linGthm5}$$]{} We know from Proposition \[muprop\] that there exists an $\eta_0 \in (0,1/4]$ such that the initial data $(u\|_{t=0},{\partial_{t}}u\|_{t=0})$, the source term $\mu(t)$ and the coefficient $\psi$ satisfy [ $$\begin{aligned} \bigl(u\|_{\{0\}\times Q_{\eta_0}},{\partial_{t}}u\|_{\{0\}\times Q_{\eta_0}}\bigr) &= \bigl({\tilde{u}{}}_0\|_{Q_{\eta_0}},{\tilde{u}{}}_1\|_{Q_{\eta_0}}\bigr),\\ \psi\|_{Q_{\eta_0}} &= 0 \intertext{and} \mu\|_{[0,T]\times Q_{\eta_0}} & = 0, \end{aligned}$$ ]{} respectively. Appealing to the finite propagation speed for solutions of wave equations, we see, shrinking $T$ if necessary, that $u$ solves the IVP - on the spacetime region $[0,T)\times Q_{\eta_0/2}$. Moreover, it follows directly from Theorem \[linthm\] and Propositions \[muprop\], \[scalepropA\] and \[scalepropB\] that $u$ satisfies the estimate [ $$\begin{aligned} {\\|u(t)\\|}_{E^{s+1}(Q_{\eta_0/2})}\leq & c({\bar{\alpha}{}},{\bar{\beta}{}}(0),{\bar{\gamma}{}}) \Biggl[ {\\|U(0)\\|}_{E^{s+1}(Q_1)} + \delta {\\|{\bar{F}{}}(0)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)} \notag \\ & + \delta {\\|{\bar{H}{}}(0)\\|}_{E^{s-1}(Q_1^+)} + \int_{0}^{\delta T}{\bar{\beta}{}}(\tau)\bigl( {\\|{\bar{F}{}}(\tau)\\|}_{{\mathcal{E}{}}^s(Q_1)} + {\\|{\bar{H}{}}(\tau)\\|}_{E^s(Q^+_1)}\bigr)\, d\tau \Biggr] \label{linGthm7.1} \end{aligned}$$ ]{} for $0\leq t < T$, where [ $$\begin{aligned} {\bar{\alpha}{}}&= \sup_{0\leq t < \delta T}\bigl({\\|{\bar{A}{}}(t)\\|}_{{\mathcal{E}{}}^s(Q_1)}+{\\|D{\bar{A}{}}(t)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)}\bigr), \label{abgdef.1}\\ {\bar{\beta}{}}(t) &= 1+ {\\|{\bar{A}{}}(t)\\|}_{{\mathcal{E}{}}^{s}(Q_1)}+{\\|{\partial_{t}}{\bar{A}{}}(t)\\|}_{{\mathcal{E}{}}^{s}(Q_1)}, \notag \\ {\bar{\gamma}{}}&= \frac{1}{\delta}\int_{0}^{\delta T}{\bar{\beta}{}}(\tau) \,d\tau \notag \end{aligned}$$ ]{} and ${\bar{U}{}}$,${\bar{A}{}}$,${\bar{F}{}}$ and ${\bar{H}{}}$ are as defined previously by -. Next, a simple change of variable argument shows that [ $${\\|{\bar{U}{}}(t)\\|}_{ E^{s+1}(Q_{\delta \eta_0/2} )} \leq c(1/\delta){\\|u(t/\delta)\\|}_{E^{s+1}(Q_{\eta_0/2})}$$ ]{} for $0 \leq t < \delta T$, while the inequality [ $$\frac{1}{\delta} \lesssim 1+{\bar{\alpha}{}}^{1/\sigma}$$ ]{} follows from , and Proposition \[scalepropA\]. Using these estimates together with , we see that [ $$\begin{aligned} {\\|{\bar{U}{}}(t)\\|}_{ E^{s+1}(Q_{\delta\eta_0/2} )} \leq & c({\bar{\alpha}{}},{\bar{\beta}{}}(0),{\bar{\gamma}{}}) \biggl[ {\\|{\bar{U}{}}(0)\\|}_{E^{s+1}(Q_1)} + {\\|{\bar{F}{}}(0)\\|}_{{\mathcal{E}{}}^{s-1}(Q_1)} \notag \\ & + {\\|{\bar{H}{}}(0)\\|}_{E^{s-1}(Q_1^+)} + \int_{0}^{T^}{\bar{\beta}{}}(\tau)\bigl( {\\|{\bar{F}{}}(\tau)\\|}_{{\mathcal{E}{}}^s(Q_1)} + {\\|{\bar{H}{}}(\tau)\\|}_{E^s(Q^+_1)}\bigr)\, d\tau \biggr] \label{linGthm10.1} \end{aligned}$$ ]{} for $0\leq t < T^$, where $T^=\delta T$. Since $u$ solves the IVP - on the spacetime region $[0,T)\times Q_{\eta_0/2}$, ${\bar{U}{}}$ must solve the IVP - on $[0,T^)\times Q_{\delta\eta/2}$. Recalling the definitions -, it is clear that $U(t,x)={\bar{U}{}}(t,\Phi_{x_0,\delta\eta_0/2}(x))$ satisfies the IVP - on $[0,T^)\times {\mathcal{N}{}}_{x_0,\delta\eta_0/2}$. We also see, with the help of , that $U$ satisfies the estimate [ $$\begin{aligned} {\\|U(t)\\|}_{ E^{s+1}({\mathcal{N}{}}_{x_0,\eta_0/2})}& \leq c(\alpha,\beta(0),\gamma) \biggl[ {\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)} + {\\|F(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{R}{}}^n)} \notag \\ & + {\\|H(0)\\|}_{E^{s-1}(\Omega)} + \int_{0}^{T^}\beta(\tau)\bigl( {\\|F(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{R}{}}^n)} + {\\|H(\tau)\\|}_{E^s(\Omega)}\bigr)\, d\tau \biggr] \label{linGthm11.1} \end{aligned}$$ ]{} for $0\leq t < T^$, where [ $$\begin{gathered} \alpha = \sup_{0\leq t \leq T^}\bigl({\\|A(t)\\|}_{{\mathcal{E}{}}^s({\mathbb{R}{}}^n)}+{\\|D A (t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{R}{}}^n)}\bigr), \quad \beta(t) = 1+ {\\|A(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{R}{}}^n)}+{\\|{\partial_{t}}A(t)\\|}_{{\mathcal{E}{}}^{s}({\mathbb{R}{}}^n)} \intertext{and} \gamma = \int_{0}^{T^}\beta(\tau) \,d\tau. \end{gathered}$$ ]{} Since $x_0\in \Omega$ was chosen arbitrarily and $\Omega$ is bounded, we can, using the finite propagation speed and the uniqueness of solutions, piece together a finite number of solutions $\{U_{j}\}_{j=1}^M$ to - defined on regions $\{[0,T^)$$\times$ ${\mathcal{N}{}}_{x_j,\delta\eta_0/2}\}_{j=1}^M$ such that $\partial\Omega$ $\subset$ $\cup_{j=1}^M {\mathcal{N}{}}_{x_j,\delta\eta_0/2}$ to obtain a solution $\hat{U}$ to - defined on $[0,T^)\times {\mathcal{N}{}}$, where ${\mathcal{N}{}}$ is an open neighborhood of ${\partial_{}}\Omega$. Away from the boundary ${\partial_{}}\Omega$, the existence and uniqueness of solutions to - satisfying the usual energy estimates is guaranteed by standard results. Piecing together this solution with $\hat{U}$, we obtain a solution to - on a time interval $[0,T^)$ with $T^* > 0$ independent of the initial data. Moreover, it is clear from and the familiar energy estimates for wave equations, that $U$ satisfies the estimate [ $$\begin{aligned} {\\|U(t)\\|}_{ E^{s+1}({\mathbb{R}{}}^n)}& \leq c(\alpha,\beta(0),\gamma)\biggl[ {\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)} + {\\|F(0)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{R}{}}^n)} \\ & + {\\|H(0)\\|}_{E^{s-1}(\Omega)} + \int_{0}^{T^}\beta(\tau)\bigl( {\\|F(\tau)\\|}_{{\mathcal{E}{}}^s({\mathbb{R}{}}^n)} + {\\|H(\tau)\\|}_{E^s(\Omega)}\bigr)\, d\tau \biggr] \end{aligned}$$ ]{} for $0\leq t < T^$. Iterating this estimate a finite number of times shows that we can take $T^=T$. Finally, we note that uniqueness follows directly from Theorem \[weakthm\]. Proof of Theorem \[mainthmA\] {#exist} ============================== We are now ready to prove Theorem \[mainthmA\]. Existence and uniqueness {#eandu} ------------------------ We establish existence and uniqueness of solutions using the well-known strategy, also used by Koch [@Koch:1993], of setting up an appropriate iterative approximation scheme and showing convergence by establishing boundedness in a high norm followed by contraction in a low norm. ### Boundedness in the high norm {#bound} To establish the boundedness in the high norm, we begin by defining the set [ $${\mathcal{B}{}}_R = \{ \, Z \in CX^{s+1}_T({\mathbb{R}{}}^3) \, \|\, {\partial_{t}}^\ell Z(0)={\tilde{U}{}}_\ell {{\quad\text{and}\quad}}{\\|Z\\|}_{X^{s+1}_T({\mathbb{R}{}}^3)} \leq R \, \},$$ ]{} where the ${\tilde{U}{}}_\ell$ are as defined by the initial data that satisfy the compatibility conditions . We consider the map [ $$J_T : {\mathcal{B}{}}_R \longrightarrow CX^{s+1}_{T}({\mathbb{R}{}}^3)$$ ]{} defined by [ $$J_T(U) = Z,$$ ]{} where $Z$ is the unique solution to the IVP: [ $$\begin{aligned} {\partial_{\mu}}\bigl(A^{\mu \nu}(U) {\partial_{\nu}} Z\bigr) &= F(U,{\partial_{}}U) + \chi_{\Omega} H(U, {\partial_{}} U) \quad \text{in $[0,T)\times {\mathbb{R}{}}^n$}, \notag \\ (Z,{\partial_{t}}Z)\|_{t=0} & = ({\tilde{U}{}}_0,{\tilde{U}{}}_1) \quad \text{in ${\mathbb{R}{}}^n$} \notag . \end{aligned}$$ ]{} Next, we define [ $$\mu = {\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)}.$$ ]{} Then the bounds [ $$\begin{aligned} {\\|A(U)\\|}_{{\mathcal{X}{}}^{s+1}_T({\mathbb{R}{}}^n)} &\leq C(R), \label{FHbnds.0}\\ {\\|F(U,{\partial_{}}U)\\|}_{{\mathcal{X}{}}^s_T({\mathbb{R}{}}^n)} &\leq C(R), \label{FHbnds.1} \\ {\\|F(U(0),{\partial_{}}U(0))\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{R}{}}^n)} &\leq C(\mu), \label{FHbnds.2} \\ {\\|H(U,{\partial_{}}U)\\|}_{X^s_T(\Omega)} &\leq C(R), \label{FHbnds.3} \intertext{and} {\\|H(U(0),{\partial_{}}U(0))\\|}_{E^{s-1}(\Omega)} &\leq C(\mu) \label{FHbnds.4} \end{aligned}$$ ]{} follow directly from Proposition \[fpropB\]. Writing $A(U)$ as [ $$\begin{aligned} A(U(t)) = A(U(0)) + \int_{0}^t DA(U(\tau))\cdot {\partial_{t}}U(\tau) \, d\tau, \end{aligned}$$ ]{} we see with the help of Proposition \[fpropB\] that [$${\\|A(U)\\|}_{{\mathcal{X}{}}^{s}_T({\mathbb{R}{}}^n)} + {\\|D[A(U)]\\|}_{{\mathcal{X}{}}^{s-1}_T({\mathbb{R}{}}^n)} \leq C(\mu) + T C(R). \label{AbndB}$$]{} Theorem \[linGthm\] in conjunction with the bounds - then implies that $Z$ satisfies the estimate [ $${\\|Z\\|}_{CX^{s+1}_T({\mathbb{R}{}}^n)} \leq c(\mu, TC(R)).$$ ]{} From this estimate, it is clear that we can arrange that [ $${\\|Z\\|}_{CX^{s+1}_T({\mathbb{R}{}}^n)} < R$$ ]{} by choosing $R$ large enough and $T$ sufficiently small. This shows that $J_T$ satisfies [$$J_T\bigl({\mathcal{B}{}}_R\bigr) \subset {\mathcal{B}{}}_R, \label{JmapD}$$]{} thereby establishing the boundedness in the high norm. ### Contraction in the low norm {#contract} Choosing $U_0,U_1 \in {\mathcal{B}{}}_R$, we set [ $$Z_0 = J_T(U_0) {{\quad\text{and}\quad}}Z_1 = J_T(U_1).$$ ]{} Then $Z_0-Z_1$ satisfies [ $$\begin{aligned} {\partial_{\mu}}\bigl(A^{\mu\nu}(U_0){\partial_{\nu}}(Z_0-Z_1)\bigr) &= F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1) + \chi_\Omega\bigl( H(U_0,{\partial_{}}U_0)-H(U_1,{\partial_{}}U_1)\bigr) \notag \\ & \text{\hspace{2.0cm}} - {\partial_{\mu}}\bigl( \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]{\partial_{\nu}}Z_1\bigr) \quad \text{in $[0,T)\times {\mathbb{R}{}}^n$} , \label{lnormA.1} \\ {\partial_{t}}^\ell (Z_0-Z_1)\|_{t=0} & = 0 \quad \text{in ${\mathbb{R}{}}^n$ for $\ell=0,1,\ldots,s$.} \label{lnormA.2} \end{aligned}$$ ]{} Writing $F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1)$ as [ $$F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1) = \int_{0}^1 DF\bigl(U_1 + \tau (U_0-U_1),{\partial_{}} U_1 + \tau ( {\partial_{}}U_0-{\partial_{}}U_1)\bigr)\, d\tau \cdot (U_0-U_1,{\partial_{}} U_0-{\partial_{}} U_1),$$ ]{} we see that [ $$\begin{aligned} &{\\|F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1) \\|}_{L^2({\mathbb{R}{}}^n)} \\ &\text{\hspace{1.0cm}} \leq \left\\|\int_{0}^1 DF\bigl(U_1 + \tau (U_0-U_1),{\partial_{}} U_1 + \tau ( {\partial_{}}U_0-{\partial_{}}U_1)\bigr)\, d\tau \right\\|_{L^\infty({\mathbb{R}{}}^n)}{\\|U_0-U_1\\|}_{H^{1}({\mathbb{R}{}}^n)} \\ &\text{\hspace{1.0cm}} \leq \left\\|\int_{0}^1 DF\bigl(U_1 + \tau (U_0-U_1),{\partial_{}} U_1 + \tau ( {\partial_{}}U_0-{\partial_{}}U_1)\bigr)\, d\tau \right\\|_{{\mathcal{H}{}}^{0,s}({\mathbb{R}{}}^n)}{\\|U_0-U_1\\|}_{H^{1}({\mathbb{R}{}}^n)}. \end{aligned}$$ ]{} where in deriving the last line we have used Sobolev’s inequality, see Theorem \[Sobolev\]. Applying Proposition \[fpropB\] to the above expression, we obtain the estimate [$${\\|F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1) \\|}_{L^2({\mathbb{R}{}}^n)} \leq C(R){\\|U_0-U_1\\|}_{E({\mathbb{R}{}}^n)}. \label{contractD}$$]{} Similar calculations together with the bound also show that [ $$\begin{aligned} {\\|\chi_\Omega\bigl(H(U_0,{\partial_{}}U_0)-H(U_1,{\partial_{}}U_1) \bigr) \\|}_{L^2({\mathbb{R}{}}^n)} &\leq C(R){\\|U_0-U_1\\|}_{E({\mathbb{R}{}}^n)}, \label{contractE.1} \\ {\\|{\partial_{\mu}}\bigl( \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]{\partial_{\nu}}Z_1\bigr)\\|}_{L^2({\mathbb{R}{}})} &\leq C(R){\\|U_0-U_1\\|}_{E({\mathbb{R}{}}^n)} \label{contractE.2} \end{aligned}$$ ]{} and [$${\\|A^{\mu\nu}(U_0)-A^{\mu\nu}(0)\\|}_{L^\infty({\mathbb{R}{}}^n)} + {\\|{\partial_{t}}[A^{\mu\nu}(U_0)]\\|}_{L^\infty({\mathbb{R}{}}^n)} \leq C(R). \label{contractF}$$]{} Since $Z_0-Z_1$ satisfies the IVP -, we are in a position to apply the energy estimates for weak solutions of wave equation from Theorem \[weakthm\] to conclude, with the help of the bounds -, that $Z_0-Z_1$ satisfies the estimate [ $${\\|Z_0(t)-Z_1(t)\\|}_{E({\mathbb{R}{}}^n)} \leq C(R)\int_{0}^T {\\|Z_0(\tau)-Z_1(\tau)\\|}_{E({\mathbb{R}{}}^n)} + {\\|U_0(\tau)-U_1(\tau)\\|}_{E({\mathbb{R}{}}^n)}\, d\tau$$ ]{} for $0\leq t < T$. Appealing to Gronwall’s inequality, we see that [ $${\\|Z_0-Z_1\\|}_{X_T^1({\mathbb{R}{}}^n)} \leq C(R)e^{C(R)T} T \sup_{0\leq t < T}{\\|U_0-U_1\\|}_{X_T^1({\mathbb{R}{}}^n)}.$$ ]{} Choosing $T>0$ small enough, we get that [ $${\\|J_T(U_0)-J_T(U_1)\\|}_{X_T^1({\mathbb{R}{}}^n)} \leq {\ensuremath{\textstyle\frac{1}{2}}}{\\|U_0-U_1\\|}_{X_T^1({\mathbb{R}{}}^n)},$$ ]{} and so, $J_T$ defines a contraction map on the subset [ $${\mathcal{B}{}}_R \subset CX^{1}_T({\mathbb{R}{}}^n).$$ ]{} In particular, for any $U_0\in {\mathcal{B}{}}_R$, the sequence [ $$U_n = \overset{\text{n times}}{\overbrace{J_T \circ \cdots \circ J_T}}(U_0) \quad n=1,2\ldots$$ ]{} converges to a unique fixed point $U \in CX^{1}_T({\mathbb{R}{}}^n)$ of $J_T$, that is $J_T(U) = U$ or in other words, a weak solution of the IVP -. Since the sequence $U_n$ is bounded in $CX^{s+1}_{T}({\mathbb{R}{}}^n)$ by virtue of the mapping property of $J_T$, we have, after passing to a subsequence, that $U_n$ converges weakly in $X^{s+1}_T({\mathbb{R}{}}^n)$ to a limit that must coincide with $U$ by the uniqueness property of weak limits. Consequently, $U$ satisfies the additional regularity $U \in X^{s+1}_T({\mathbb{R}{}}^n)$, which can be upgraded to $U \in CX^{s+1}_T({\mathbb{R}{}}^n)$ with the help of Theorem \[linGthm\]. ### Uniqueness {#unique} Before we establish uniqueness, we first note that the inclusion [ $$CX_{T}^{s+1}({\mathbb{R}{}}^n) \subset CX_{T}^2({\mathbb{R}{}}^n) \cap \bigcap_{\ell=0}^1 C^\ell\bigl([0,T),W^{1-\ell,\infty}({\mathbb{R}{}}^n)\bigr)$$ ]{} follows directly from Sobolev’s inequality since $s>n/2$ by assumption. To prove uniqueness, we suppose that [ $$\begin{gathered} U_0 \in CX_{T}^2({\mathbb{R}{}}^n) \cap \bigcap_{\ell=0}^1 C^\ell\bigl([0,T),W^{1-\ell,\infty}({\mathbb{R}{}}^n)\bigr) \intertext{and} U_1 \in CX^{s+1}_T({\mathbb{R}{}}^n) \end{gathered}$$ ]{} are two solutions of the IVP -. Then the difference $U_0-U_1$ satisfies [ $$\begin{aligned} &{\partial_{\mu}}\bigl(A^{\mu\nu}(U_0){\partial_{\nu}}(U_0-U_1)\bigr) = F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1) + \chi_\Omega\bigl( H(U_0,{\partial_{}}U_0)-H(U_1,{\partial_{}}U_1)\bigr)\notag\\ & \text{\hspace{6.0cm}} - {\partial_{\mu}}\bigl( \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]{\partial_{\nu}}U_1\bigr) \quad \text{in $[0,T)\times {\mathbb{R}{}}^n$,} \label{uniqueB.1} \\ &\bigl( (U_0-U_1), {\partial_{t}}(U_0-U_1)\bigr)\|_{t=0} = (0,0) \quad \text{in ${\mathbb{R}{}}^n$.} \label{uniqueB.2} \end{aligned}$$ ]{} Using similar arguments as in the previous section, it is not difficult to derive the bounds [ $$\begin{aligned} {\\|F(U_0,{\partial_{}}U_0)-F(U_1,{\partial_{}}U_1) \\|}_{L^2({\mathbb{R}{}}^n)} &\leq C(\rho_0,\rho_1){\\|U_0-U_1\\|}_{E({\mathbb{R}{}}^n)}, \label{uniqueC.1}\\ {\\|\chi_\Omega\bigl(H(U_0,{\partial_{}}U_0)-H(U_1,{\partial_{}}U_1) \bigr) \\|}_{L^2({\mathbb{R}{}}^n)}&\leq C(\rho_0,\rho_1){\\|U_0-U_1\\|}_{E({\mathbb{R}{}}^n)} \label{uniqueC.2} \end{aligned}$$ ]{} and [$${\\|A^{\mu\nu}(U_0)-A^{\mu\nu}(0)\\|}_{L^\infty({\mathbb{R}{}}^n)} + {\\|{\partial_{t}}[A^{\mu\nu}(U_0)]\\|}_{L^\infty({\mathbb{R}{}}^n)} \leq C(\rho_0), \label{uniqueD}$$]{} where [ $$\rho_0 = \sup_{0\leq t < T}\Bigl[ {\\|U_0(t)\\|}_{W^{1,\infty}({\mathbb{R}{}}^n)} + {\\|{\partial_{t}}U_0(t)\\|}_{L^\infty({\mathbb{R}{}}^n)} \Bigr] {{\quad\text{and}\quad}}\rho_1 = {\\|U_1\\|}_{X^{s+1}_T({\mathbb{R}{}}^n)}.$$ ]{} We also observe that [ $$\begin{aligned} {\\|{\partial_{\mu}}\bigl( \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]{\partial_{\nu}}U_1\bigr)\\|}_{L^2({\mathbb{R}{}}^n)} \leq &{\\|{\partial_{\mu}} \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]\\|}_{L^2({\mathbb{R}{}}^n)} {\\|{\partial_{}}U_1\\|}_{L^\infty({\mathbb{R}{}}^n)}\notag \\ &+ {\\|A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1)\\|}_{L^\infty({\mathbb{R}{}}^n)}{\\|{\partial_{}}^2U_1\\|}_{L^\infty({\mathbb{R}{}}^n)} \notag\\ \leq {\\|{\partial_{\mu}} \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]\\|}_{L^2({\mathbb{R}{}}^n)}&\rho_1 + {\\|A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1)\\|}_{L^{2n/(n-2)}({\mathbb{R}{}}^n)}{\\|{\partial_{}}^2U_1\\|}_{L^n({\mathbb{R}{}}^n)} \notag \\ \leq {\\|{\partial_{\mu}} \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]\\|}_{L^2({\mathbb{R}{}}^n)}&\rho_1 + {\\|A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1)\\|}_{H^1({\mathbb{R}{}}^n)}{\\|{\partial_{}}^2U_1\\|}_{L^n({\mathbb{R}{}}^n)}\label{uniqueF.1}, \end{aligned}$$ ]{} where in deriving the last inequality we used Sobolev’s inequality. Again, using similar arguments as in the previous section, it is not difficult to verify that [$${\\|{\partial_{\mu}} \bigl[A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1) \bigr]\\|}_{L^2({\mathbb{R}{}}^n)} + {\\|A^{\mu\nu}(U_0)-A^{\mu\nu}(U_1)\\|}_{H^1({\mathbb{R}{}}^n)} \leq C(\rho_0,\rho_1){\\|U_0-U_1\\|}_{E({\mathbb{R}{}}^n)}. \label{uniqueG}$$]{} Finally, we estimate [ $$\begin{aligned} {\\|{\partial_{}}^2U_1\\|}_{L^n({\mathbb{R}{}}^n)} &\leq \max\{{\\|{\partial_{}}^2U_1\\|}_{L^n(\Omega)},{\\|{\partial_{}}^2U_1\\|}_{L^n(\Omega^c)}\} \notag \\ & \lesssim \max\{{\\|{\partial_{}}^2U_1\\|}_{H^{s-1}(\Omega)},{\\|{\partial_{}}^2U_1\\|}_{H^{s-1}(\Omega^c)}\} \notag \\ & \lesssim {\\|{\partial_{}}^2 U_1\\|}_{H^{0,s-1}({\mathbb{R}{}}^n)} \notag \\ & \lesssim \sum_{\ell=0}^2 {\\|{\partial_{t}}^\ell U_1\\|}_{H^{2,s+1-\ell}({\mathbb{R}{}}^n)} \notag \\ & \lesssim \rho_1, \label{uniqueH.1} \end{aligned}$$ ]{} where we have again used Sobolev’s inequality and the assumption $s>n/2$. Since $U_0-U_1$ satisfies the IVP -, we can apply the energy estimates for weak solutions of linear wave equation from Theorem \[weakthm\] to conclude, with the help of the bounds -, that $U_0-U_1$ satisfies the estimate [ $${\\|U_0(t)-U_1(t)\\|}_{E({\mathbb{R}{}}^n)} \leq C(\rho_0,\rho_1)\int_{0}^T {\\|U_0(\tau)-U_1(\tau)\\|}_{E({\mathbb{R}{}}^n)}\, d\tau$$ ]{} for $0\leq t < T$, which in turn, implies that [ $${\\|U_0(t)-U_1(t)\\|}_{E({\mathbb{R}{}}^n)} = 0 \quad 0\leq t < T,$$ ]{} by Gronwall’s inequality. We conclude that $U_0=U_1$ and uniqueness holds. The continuation principle {#cont} -------------------------- In order to establish the continuation principle, we assume, by way of contradiction, that $U \in CX^{s+1}_T({\mathbb{R}{}}^n)$ $(s\in {\mathbb{Z}{}}_{> n/2})$ is a solution of the wave equation satisfying [$$\limsup_{t\nearrow T} {\\|U(t)\\|}_{E^{s+1}({\mathbb{R}{}}^n)} = \infty \label{contassumpA}$$]{} and [$${\\|U\\|}_{W^{1,\infty}((0,T)\times{\mathbb{R}{}}^n)} \leq K < \infty. \label{contassumpB}$$]{} Using the property of finite speed of propagation, it is enough to show that $U$ cannot locally satisfy both and , where we can, by suitably shifting the origin of the time coordinate, assume that $T$ is small as we like. We note that away from the boundary ${\partial_{}}\Omega$ where there are no singular terms in the wave equation , we can appeal to the standard continuation principle, see for example [@Majda:1984 Theorem 2.2], to conclude that, locally, the solution cannot satisfy both and . In light of this observation, we need only worry about the behavior of $U$ in a neighborhood of the boundary ${\partial_{}}\Omega$ for arbitrarily small times. Furthermore, since [ $${\\|u\\|}_{W^{1,\infty}((0,T),Q_1)} \lesssim {\\|{\bar{U}{}}\\|}_{W^{1,\infty}((0,\delta T)\times Q_\delta)} \lesssim K$$ ]{} for all $ \delta \in (0,1]$ where $u$ and ${\bar{U}{}}$ are as defined previously by and , respectively, it is enough to consider the solution $U$ on an arbitrary small spacetime neighborhoods $(x_0,0)$ with $x_0 \in {\partial_{}}\Omega$ that satisfy the bound . We can, therefore, use the scaling and projection technique from Sections \[linit\] and \[linrs\] to reduce the continuation question to that of proving a continuation principle for the following scaled and projected system where we may choose $\delta$ as small as we like: [ $$\begin{aligned} {\partial_{\nu}}\bigl((m^{\mu\nu} + \delta b^{\mu\nu}({\mathbf{x}{}},u)){\partial_{\mu}} u\bigr) - \psi u &= \delta(f({\mathbf{x}{}},u{\partial_{}}u)+ \chi_{{ {\Omega_1}}}h({\mathbf{x}{}},u{\partial_{}}u)) + \mu \quad \text{in $[0,T)\times {\mathbb{T}{}}^n$,}\label{contev.1} \\ (u,{\partial_{t}}u)\|_{t=0} &= (\hat{u}_0,\hat{u}_1) \quad \text{in ${\mathbb{T}{}}^n$,} \label{contev.2} \end{aligned}$$ ]{} where - [ $$\begin{aligned} \hat{u}_0(x) &= \phi_1(x)\frac{U(\psi_{x_0,\delta}(0,\delta x))-U(\psi_{x_0,\delta}({\mathbf{0}}))}{\delta},\\ \hat{u}_1(x) &= \phi_1(x) {\partial_{t}}U(0,\Psi_{x_0,\delta}(\delta x)),\\ (m^{\mu\nu}) &= \bigl(\det(J({\mathbf{0}})){\check{J}{}}^\mu_\alpha({\mathbf{0}}){\check{J}{}}^\nu_\beta({\mathbf{0}})A^{\alpha\beta}(U(\psi_{x_0,\delta}({\mathbf{0}}))) \bigr) = \text{diag}(-1,1,\ldots,1), \\ b^{\mu\nu}({\mathbf{x}{}},u) &= \phi_1(x)\frac{\det(J(\delta{\mathbf{x}{}})){\check{J}{}}^\mu_\alpha(\delta{\mathbf{x}{}}){\check{J}{}}^\nu_\beta(\delta{\mathbf{x}{}})A^{\alpha\beta}\bigl(U(\psi_{x_0,\delta}({\mathbf{0}}))+\delta u)-m^{\mu\nu}}{\delta}, \\ f({\mathbf{x}{}},u,{\partial_{}}u) &= \phi_1(x) \det(J(\delta{\mathbf{x}{}}))F\bigl(U(\psi_{x_0,\delta}({\mathbf{0}}))+\delta u,{\check{J}{}}(\delta{\mathbf{x}{}}){\partial_{}}u\bigr), \intertext{and} H({\mathbf{x}{}},u,{\partial_{}}u) &= \phi_1(x) \det(J(\delta{\mathbf{x}{}}))H\bigl(U(\psi_{x_0,\delta}({\mathbf{0}}))+\delta u,{\check{J}{}}(\delta{\mathbf{x}{}}){\partial_{}}u\bigr), \end{aligned}$$ ]{} - $0<\delta \leq 1$, - the initial data $(\hat{u}_0,\hat{u}_1)$ satisfies [ $$\begin{aligned} {\\|\hat{u}_0\\|}_{{\mathcal{H}{}}^{s+1,2}({\mathbb{T}{}}^n)} \lesssim {\\|U(0)\\|}_{{\mathcal{H}{}}^{s+1,2}({\mathbb{R}{}}^n)} \intertext{and} {\\|\hat{u}_0\\|}_{{\mathcal{H}{}}^{s,2}({\mathbb{T}{}}^n)} \lesssim {\\|{\partial_{t}}U(0)\\|}_{{\mathcal{H}{}}^{s,2}({\mathbb{R}{}}^n)}, \end{aligned}$$ ]{} - and [ $$\mu = \sum_{\ell=0}^{s-1} \frac{t^\ell}{\ell !} \mu_\ell$$ ]{} with the $\mu_\ell$ determined as in Proposition \[muprop\] so that [ $$\begin{aligned} {\\|u(0)\\|}_{E^{s+1}} &\lesssim {\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)} \intertext{and} {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} & \leq (1+t^{s-1})C\bigl({\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)}\bigr). \end{aligned}$$ ]{} In light of the above discussion, the following proposition completes the proof of Theorem \[mainthmA\]. \[contprop\] Suppose $u_\delta \in CX^{s+1}_{T_{\delta}({\mathbb{R}{}}^n)}$ is a family of solutions depending on $\delta \in (0,1]$ to the IVP - satisfying the conditions (i)-(iv) above. If [ $${\\|u_\delta\\|}_{W^{1,\infty}((0,T_\delta)\times{\mathbb{T}{}}^n)} \leq K < \infty,$$ ]{} then there exists a $\delta_0>0$ and a time $T^_\delta > T_\delta$ for each $\delta \in (0,\delta_0]$ such that the solution $u_\delta$ can be continued to a solution of - on $[0,T^_\delta)\times {\mathbb{T}{}}^n$. Before proceeding with the proof, we will, in order to simplify calculations, suppress the explicit ${\mathbf{x}{}}$-dependence of the functions $b^{\mu\nu}$, $f$ and $h$. Since $u_\delta$ satisfies [$${\partial_{\nu}}\bigl((m^{\mu\nu} + \delta b^{\mu\nu}(u_\delta)){\partial_{\mu}} u_\delta\bigr) - \psi u_\delta = \delta(f(u_\delta{\partial_{}}u_\delta)+ \chi_{{ {\Omega_1}}}h(u_\delta{\partial_{}}u_\delta)) \quad \text{in $[0,T_\delta)\times {\mathbb{T}{}}^n$,} \label{contprop2}$$]{} we see after differentiating $k$-times, where $0\leq k \leq s-1$, with respect to $t$ that ${\partial_{t}}^k u_\delta$ satisfies [ $$\begin{aligned} (\Delta -\psi){\partial_{t}}^k u_\delta = {\partial_{t}}^{k+2} u_\delta +\delta\bigl[-{\partial_{t}}^{k+1}&\bigl(b^{00}(u_\delta){\partial_{t}}u_\delta + b^{0i}(u_\delta){\partial_{i}}u_\delta \bigr) -{\partial_{i}} {\partial_{t}}^k\bigl( b^{i0}(u_\delta){\partial_{t}}u_\delta \notag\\ &+ b^{ij}(u_\delta){\partial_{j}}u_\delta \bigr) + {\partial_{t}}^kf(u_\delta,{\partial_{}}u_\delta) + \chi_{ {\Omega_1}}{\partial_{t}}^k h(u,{\partial_{}}u) \bigr] + {\partial_{t}}^k \mu. \label{contprop3.1} \end{aligned}$$ ]{} Since [ $${\\|{\partial_{t}}^k u_\delta\\|}_{{\mathcal{H}{}}^{2,s+1-k}({\mathbb{T}{}}^n)} \lesssim {\\|(\Delta-\psi){\partial_{t}}^k u_\delta\\|}_{{\mathcal{H}{}}^{0,s-k-1}({\mathbb{T}{}}^n)} \quad 0\leq k \leq s-1$$ ]{} by Proposition \[potprop\], it follows from Proposition \[STpropC\] and that ${\partial_{t}}^k u_\delta$ satisfies [ $$\begin{aligned} {\\|{\partial_{t}}^k u(t)\\|}_{{\mathcal{H}{}}^{2,s+1-k}({\mathbb{T}{}}^n)} \leq c\bigl(&{\\|{\partial_{t}}^{k+2}u\\|}_{{\mathcal{H}{}}^{0,s-k-1}({\mathbb{T}{}}^n)}+ {\\|{\partial_{t}}^k \mu(t)\\|}_{{\mathcal{H}{}}^{0,s-k-1}({\mathbb{T}{}}^n)} \bigr) \notag \\ & + \delta C(K) \bigl(1+ {\\|u(t)\\|}_{{\mathcal{H}{}}^{2,s+1}({\mathbb{T}{}}^n)} + {\\|{\partial_{t}}u(t)\\|}_{{\mathcal{H}{}}^{2,s}({\mathbb{T}{}}^n)} \bigr)\quad 0\leq t < T_\delta \label{contprop4.1} \end{aligned}$$ ]{} where [$${\\|u_\delta\\|}_{W^{1,\infty}((0,T_\delta)\times{\mathbb{T}{}}^n)} \leq K \qquad 0 < \delta \leq 1. \label{contprop5}$$]{} We collect the estimates , for $0\leq k \leq s-1$, into the single matrix inequality [$$\|M_\delta X_\delta(t)\| \lesssim {\\|{\partial_{t}}^s u_\delta(t)\\|}_{E({\mathbb{T}{}}^n)} + {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + \delta C(K) \quad 0\leq t < T_\delta, \label{contprop6}$$]{} where [ $$M_\delta = \begin{pmatrix}1-\delta C(K) & -\delta C(K) & 0 & c & 0 & 0 & & \cdots & &0 \\ -\delta C(K) & 1-\delta C(K) & 0 & 0 & c & 0 & & & &0 \\ -\delta C(K) & -\delta C(K) & 1 & 0 & 0 & c& & & & \\ -\delta C(K) & -\delta C(K) & 0 & 1 & 0 & 0& & \ddots & &0 \\ & \vdots & & & \ddots & & & & &c \\ & & & & & & & & &0 \\ & & & & & & & & & \\ -\delta C(K) & -\delta C(K) & 0 & 0 & 0 & 0 & & \cdots & &1 \end{pmatrix},$$ ]{} and [ $$X_\delta(t) = \bigl({\\|u_\delta(t)\\|}_{{\mathcal{H}{}}^{2,s+1}({\mathbb{T}{}}^n)}, {\\|{\partial_{t}}u_\delta(t)\\|}_{{\mathcal{H}{}}^{2,s}({\mathbb{T}{}}^n)}, \ldots , {\\|{\partial_{t}}^{s-1}u_\delta(t)\\|}_{{\mathcal{H}{}}^{2,2}({\mathbb{T}{}}^n)} \bigr)^T.$$ ]{} Since $M_0$ is tri-diagonal, it follows that $M_0$ is invertible, and hence, that there exists a $\delta_0 \in (0,1]$ such that $M_\delta$ is invertible with a uniformly bounded inverse for all $\delta \in (0,\delta_0]$. This fact together with shows that $X_\delta(t)$ satisfies [$$\|X_\delta(t)\| \lesssim {\\|{\partial_{t}}^s u_\delta(t)\\|}_{E({\mathbb{T}{}}^n)} + {\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} + \delta C(K) \label{contprop9}$$]{} for all $(t,\delta) \in [0,T_\delta)\times (1,\delta_0]$. Next, differentiating $s$-times with respect to $t$, we see that ${\partial_{t}}^s u_\delta$ is a weak solution of [ $${\partial_{\nu}}\bigl((m^{\mu\nu} + \delta b^{\mu\nu}(u_\delta)){\partial_{\mu}} {\partial_{t}}^s u_\delta\bigr) - \psi {\partial_{t}}^s u_\delta = -\delta {\partial_{\mu}}\bigl([{\partial_{t}}^s,b^{\mu\nu}(u_\delta){\partial_{\nu}}]u_\delta\bigr) + \delta\bigl( {\partial_{t}}^s f(u_\delta{\partial_{}}u_\delta)+ \chi_{{ {\Omega_1}}} {\partial_{t}}^s h(u_\delta{\partial_{}}u_\delta)\bigr)$$ ]{} in $[0,T_\delta)\times {\mathbb{T}{}}^n$. Applying the estimates from Propositions \[STpropA\] and \[STpropB\], and Theorem \[weakthm\], we obtain, with the help of the bound , the following energy estimate for ${\partial_{t}}^s u_\delta$: [$${\\|{\partial_{t}}^s u_\delta(t)\\|}_{E({\mathbb{T}{}}^n)} \leq C(K)\left({\\|{\partial_{t}}^s u_\delta(0)\\|}_{E({\mathbb{T}{}}^n)} + \int_{0}^t {\\|u_\delta(\tau)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} + 1 + {\\|\mu(\tau)\\|}_{{\mathcal{E}{}}^{s-1}({\mathbb{T}{}}^n)} \, d\tau\right) \label{contprop11}$$]{} for $t\in (0,T_\delta)$. By assumption, $u_\delta$ and $\mu$ satisfy the bounds [$${\\|u_\delta(0)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} \lesssim {\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)} {{\quad\text{and}\quad}}{\\|\mu(t)\\|}_{{\mathcal{E}{}}^{s-1}} \leq (1+t^{s-1})C\bigl({\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)}\bigr) \label{contprop10}$$]{} for $\delta \in (0,1]$. Combining these bounds with the estimates and , we get that [ $${\\|u_\delta(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} \leq C\bigl(K,T_\delta,{\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)}\bigr)\left(1 + \int_{0}^t {\\|u_\delta(\tau)\\|}_{E^{s+1}({\mathbb{T}{}}^n)}\, d\tau\right)$$ ]{} for $t\in (0,T_\delta)$, and hence, by Gronwall’s inequality, that [ $${\\|u_\delta(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} \leq C\bigl(K,T_\delta,{\\|U(0)\\|}_{E^{s+1}({\mathbb{R}{}}^n)}\bigr) \qquad 0\leq t < T_\delta.$$ ]{} In particular, this implies that [ $$\limsup_{t\nearrow T_\delta}{\\|u_\delta(t)\\|}_{E^{s+1}({\mathbb{T}{}}^n)} < \infty$$ ]{} for each $\delta \in (0,\delta_0]$. From this point, we can follow standard arguments, for example, see the proof of Theorem 2.2, p. 46 of [@Majda:1984] , to conclude that for each $\delta \in (0,\delta_0]$, there exists a $T^_\delta > 0$ such that the solution $u_\delta$ extends to a solution on $[0,T^_\delta)\times {\mathbb{T}{}}^n$. Discussion and outlook {#disc} ====================== As discussed in the introduction, the main application that we have in mind for the results presented in this article is to establish the local existence and uniqueness of solutions to the Einstein equations coupled to elastic matter that describe the motion of self-gravitating compact elastic bodies. While the complete details of the local existence and uniqueness proof will be presented in a separate article [@Andersson_et_al:2013], we give here the main ideas of the proof in order to illustrate the role that the results of this article play in the proof. Following [@BeigSchmidt:2003], a single compact relativistic elastic body[^5], locally in time, is characterized by a map [ $$f \: : \: W \longrightarrow \Omega$$ ]{} from a space-time cylinder $W \cong [0,T]\times \Omega$ to a 3-dimensional compact manifold $\Omega$ with boundary, known as the material manifold. The body world tube $W$ is taken to be contained in an ambient Lorentzian spacetime $(M,g)$, where $M\cong [0,T]\times \Sigma$ for some 3-manifold $\Sigma$. For simplicity of presentation, we assume that both $\Omega$ and $M$ can each be covered by a single coordinate chart given by $(X^I)$ $(I=1,2,3)$ and $(x^\lambda)$ $(\lambda=0,1,2,3)$, respectively. In these local coordinates, we can express $f$ and $g$ as [ $$X^I=f^I(x^\lambda) {{\quad\text{and}\quad}}g=g_{\mu\nu}(x^\lambda)dx^\mu dx^\nu.$$ ]{} The field equations satisfied by $\{f^I,g_{\mu\nu}\}$ are then given by [ $$\begin{aligned} G^{\mu\nu} &= 2\kappa T^{\mu\nu} \text{\hspace{0.3cm} in $M$,} \label{EinElasA.1} \\ \nabla_{\mu} T^{\mu\nu} &= 0 \text{\hspace{1.1cm} in $W$,} \label{EinElasA.2} \end{aligned}$$ ]{} where $G^{\mu\nu}$ is the Einstein tensor and [ $$T_{\mu\nu} = 2\frac{{\partial_{}}\rho}{{\partial_{}} g^{\mu\nu}} - \rho g_{\mu\nu}$$ ]{} is the stress-energy of the elastic body with [ $$\rho = \rho(f,H) \qquad (H^{IJ} := g^{\mu\nu}{\partial_{\mu}}f^I{\partial_{\nu}}f^J )$$ ]{} defining the proper energy density of the elastic body. By definition, $\rho$ is non-zero inside $W$ and vanishes outside. Letting $\Gamma$ denote the space-like boundary of $W$, the elastic field must also satisfy the boundary conditions [$$n^\mu T_{\mu\nu} = 0 \text{\hspace{0.3cm} in $\Gamma$,} \label{EinElasB}$$]{} where here $n^\mu$ denotes the outward pointing unit normal to $\Gamma$. Initial conditions for - are given by [ $$\begin{aligned} (g_{\mu\nu},{\mathcal{L}{}}_t g_{\mu\nu}) &= (g^0_{\mu\nu},g^1_{\mu\nu}) \text{\hspace{0.3cm} in $\Sigma$,} \label{EinElasC.1} \\ (f^I,{\mathcal{L}{}}_t f^I) &= (f_0^I,f_1^I) \text{\hspace{0.6cm} in $\Sigma\cap W$,} \label{EinElasC.2} \end{aligned}$$ ]{} where $\Sigma$ forms the “bottom” of the spacetime slab $M\cong [0,T]\times \Sigma$, $\Sigma\cap W$ forms the bottom of the spacetime cylinder $W\cong [0,T]\times \Omega$, $t=t^\mu{\partial_{\mu}}$ is a future pointing time-like vector field tangent to $\Gamma$, and the initial data satisfies the constraint equations* [$$t_{\mu} G^{\mu\nu} = 2\kappa t_{\mu} T^{\mu\nu} \text{\hspace{0.3cm} in $\Sigma$.} \label{EinElasD}$$]{} The method we use to solve the initial value boundary problem (IVBP), given by -, begins with introducing harmonic coordinates as this allows us to replace the full Einstein equations with the reduced equations given by [$$R_{\mu\nu} - \nabla_{(\mu}\xi_{\nu)} = 2\kappa\bigl( T_{\mu\nu}-{\ensuremath{\textstyle\frac{1}{3}}}T g_{\mu\nu}\bigr) \qquad (\xi^\gamma := g^{\mu\nu}\Gamma_{\mu\nu}^\gamma). \label{EinElasE}$$]{} For this method to work, we must choose the initial data so that the constraint [$$\xi^\mu = 0 \quad \text{in $\Sigma$} \label{xidata}$$]{} is also satisfied in addition to . The next step is to introduce the material representation via the map [ $$x^i = \phi^i(X^0,X^I) \qquad (i=1,2,3),$$ ]{} which is uniquely determined by the requirement [ $$f^I(X^0,\phi(X^0,X^I)) = X^I \quad \forall \; (X^0,X^I)\in [0,T]\times \Omega.$$ ]{} In the material representation, the elastic field is completely characterized by the map $\phi$ while the gravitational field is determined by the components of the metric expressed in the material representation as follows [ $$\gamma_{\mu\nu}(X^0,X^I) = g_{\mu\nu}(X^0,\phi(X^0,X^I)).$$ ]{} A straightforward calculation then shows that the field equations -, the boundary conditions and the initial conditions -, when expressed in terms of the variables $\{\gamma_{\mu\nu},\phi^i\}$, take the form [ $$\begin{aligned} \frac{{\partial_{}}\;\;}{{\partial_{}}X^\Delta}\left(a^{\Delta\Lambda}\bigl(\gamma,{\partial_{}}{\tilde{\phi}{}}\bigr)\frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^\Lambda}\right)& = q_{\mu\nu}\bigl(\gamma,{\partial_{}}\gamma,{\partial_{}}{\tilde{\phi}{}}\bigr) + \chi_{\Omega}p_{\mu\nu}\bigl(\gamma,\phi,{\partial_{}}\phi\bigr) &&\text{in $[0,T]\times \tilde{\Sigma} $,} \label{EinElasF.1} \\ \frac{{\partial_{}}\;\;}{{\partial_{}}X^\Delta}\left(F^{\Delta}_i \bigl(\gamma,\phi,{\partial_{}}\phi\bigr)\right) &= w_i(\gamma,\phi,{\partial_{}}\phi) &&\text{in $[0,T]\times \Omega $,} \label{EinElasF.2} \\ \nu_J F^{J}_i \bigl(\gamma,\phi,{\partial_{}}\phi\bigr) &= 0 &&\text{in $[0,T]\times {\partial_{}}\Omega $,} \label{EinElasF.3} \\ \left(\gamma_{\mu\nu}, \frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^0} \right) &= (\gamma^0_{\mu\nu},\gamma^1_{\mu\nu}) &&\text{in $\{0\}\times \tilde{\Sigma}$,} \label{EinElasF.4} \\ \left(\phi^i, \frac{{\partial_{}}\phi^i}{{\partial_{}}X^0}\right) &= (\phi_0^i,\phi_1^i) &&\text{in $\{0\} \times \Omega$,} \label{EinElasF.5} \end{aligned}$$ ]{} where $\Omega \subset \tilde{\Sigma}$ with $\tilde{\Sigma}$ defined by ${\tilde{\phi}{}}(0,\tilde{\Sigma}) = \Sigma$, ${\tilde{\phi}{}}=E(\phi)$ with $E$ a suitable extension operator from $\Omega$ to $\tilde{\Sigma}$, $\nu_J$ is the outward pointing unit normal to ${\partial_{}}\Omega$ and [ $${\partial_{}}(\cdot) = \frac{{\partial_{}}(\cdot)}{{\partial_{}}X^\Delta} \qquad (\Delta=0,1,2,3)$$ ]{} is the spacetime gradient. From the point of view of local existence, we lose nothing by assuming that $\tilde{\Sigma} \cong {\mathbb{R}{}}^3$ and the $(X^I)$ are Cartesian coordinates on $\tilde{\Sigma}$. \[discremA\] The dependence of the coefficients $a^{\Delta\Lambda}$ in on ${\partial_{}}\phi$ is problematic from a regularity perspective for the hyperbolic estimate of the top time derivative $\left(\frac{{\partial_{}}\;\;}{{\partial_{}}X^0}\right)^s \!\gamma_{\mu\nu}$ from the proof of Theorem \[linthm\]. This is because if we were to estimate the top $X^0$-derivative using the wave equation as in the proof of Theorem \[linthm\], we would require an estimate on the $(s+1)^{\text{th}}$ $X^0$-derivative of ${\partial_{}}\phi$, and this is one too many derivatives to be compatible with Koch’s [@Koch:1993] estimates for equation . To avoid this loss of derivatives scenario, we instead use a first order formulation of the gravitational field equations based on the variables $\{\gamma,\lambda\}$, where [ $$\lambda_{\sigma\mu\nu} := ({\partial_{\sigma}}g_{\mu\nu})(X^0,\phi(X^0,X^I)),$$ ]{} to estimate the top $X^0$-derivative. We note that the lower $X^0$-derivatives are still estimated using elliptic estimates based on the wave formulation as in the proof of Theorem \[linthm\]. A straightforward calculation shows that the reduced equations can be expressed in terms of the $\{\gamma_{\mu\nu},\lambda_{\sigma\mu\nu}\}$ variables as a symmetric hyperbolic system of the form [ $$b^{\alpha\beta\kappa}(\gamma,{\partial_{}}\phi)\frac{{\partial_{}}\lambda_{\beta\mu\nu}}{{\partial_{}}X^\kappa} = f(\gamma,\lambda,{\partial_{}}\phi) + \chi_{\Omega}h(\gamma,\phi,{\partial_{}}\phi),$$ ]{} with the point being that, unlike , after differentiating this equation $s$-times with respect to $X^0$, we obtain an $L^2$ estimate for $\left(\frac{{\partial_{}}\;\;}{{\partial_{}}X^0}\right)^s\! \lambda_{\mu\nu}$ with the highest $X^0$-derivative of ${\partial_{}}\phi$ appearing in the estimate being the $s^{\text{th}}$ one. Importantly, this $L^2$ estimate is, with the help the estimates on $\phi$ coming from , enough to obtain an appropriate $L^2$ estimate for $\left(\frac{{\partial_{}}\;\;}{{\partial_{}}X^0}\right)^s\! {\partial_{}}\gamma_{\mu\nu}$ thereby avoiding any loss of derivatives. To proceed, we assume that the initial data - satisfy the constraint equations , and also the compatibility conditions [ $$\begin{aligned} \gamma^\ell_{\mu\nu} := \left(\frac{{\partial_{}}\;\;}{{\partial_{}}X^0}\right)^\ell\Bigl\|_{X^0=0} \! \gamma_{\mu\nu} &\in {\mathcal{H}{}}^{m_{s+1-\ell},s+1-\ell}(\tilde{\Sigma}) \quad \ell=0,1,\ldots,s+1, \label{EinElasG.1} \\ \phi_\ell^i := \left(\frac{{\partial_{}}\;\;}{{\partial_{}}X^0}\right)^\ell\Bigl\|_{X^0=0} \! \phi^i & \in H^{s+1-\ell}(\Omega) \quad \ell=0,1,\ldots,s+1 \label{EinElasG.2} \intertext{and} \left(\frac{{\partial_{}}\;\;}{{\partial_{}}X^0}\right)^\ell \!\bigl(\nu_J F^J_i(\gamma,{\partial_{}}\phi)\bigr)\Bigr\|_{X^0=0} &\in H^{s-\ell}(\Omega)\cap H^1_0(\Omega), \label{EinElasG.3} \end{aligned}$$ ]{} where $s \in {\mathbb{Z}{}}_{> 5/2}$, [ $$m_j = \begin{cases} 2 & \text{if $j\geq 2$} \\ j & \text{otherwise} \end{cases}$$ ]{} and [ $${\mathcal{H}{}}^{k,r}(\tilde{\Sigma}) = H^{r}(\Omega)\cap H^{k}(\tilde{\Sigma})\cap H^{r}(\tilde{\Sigma}\setminus \Omega)$$ ]{} We know from the results of [@Andersson_et_al:2013] that the set of initial data satisfying the constraint equations and the compatibility conditions is non-empty. However, a complete classification of the space of initial data satisfying these conditions appears to be very difficult, and in fact, the classification of the space of initial data satisfying just the constraint equations is far from complete. Rather than solving the elastic boundary value problem - directly, we follow [@Koch:1993] and differentiate it once with respect to $X^0$ to obtain the system [ $$\begin{aligned} \frac{{\partial_{}}\;\;}{{\partial_{}}X^\Delta}\left(L^{\Delta\Lambda}_{ij} \bigl(\gamma,{\partial_{}}\phi\bigr)\frac{{\partial_{}}\psi^j}{{\partial_{}}X^\Lambda} + Z^{\Delta\mu\nu}_i(\gamma,{\partial_{}}\phi)\frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^0}\right) &= Y_i\left(\gamma,\frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^0},\phi,{\partial_{}}\phi,{\partial_{}}\psi\right), && \text{in $[0,T]\times \Omega $,} \label{EinElasH.1} \\ {\partial_{0}}\phi^i &= \psi^i && \text{in $[0,T]\times \Omega $,} \label{EinElasH.2} \\ \nu_J \left(L^{J\Lambda}_{ij} \bigl(\gamma,{\partial_{}}\phi\bigr)\frac{{\partial_{}}\psi^j}{{\partial_{}}X^\Lambda} + Z^{J\mu\nu}_i(\gamma,{\partial_{}}\phi)\frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^0}\right) &= 0 && \text{ in $[0,T]\times {\partial_{}}\Omega $,} \label{EinElasH.3} \end{aligned}$$ ]{} where [$$L^{\Delta\Lambda}_{ij}(\gamma,{\partial_{}}\phi) = \frac{{\partial_{}}F^\Delta_i}{{\partial_{}}\frac{{\partial_{}}\phi^j}{{\partial_{}}X^\Lambda}}(\gamma,{\partial_{}}\phi) \label{Ldef}$$]{} is the elasticity tensor, as expressed in the material frame. In particular, we restrict ourself to elastic materials for which the elasticity tensor satisfies Koch’s coercivity condition[^6] [@Koch:1993 Assumption 3, p. 12]. We note that Koch’s other assumptions, Assumptions 1,2 and 4 in [@Koch:1993 pp. 12-13], are satisfied automatically by the elasticity tensor of reasonable relativistic materials. In order to solve the IVBP defined by , - and -, we employ an iteration scheme, analogous to the one used in Section \[eandu\], defined by the map [$$J_T(\mu_{\mu\nu},\alpha^i,\beta^i) = (\gamma_{\mu\nu},\phi^i,\psi^i), \label{EEJmapA}$$]{} which maps the triple [ $$(\mu_{\mu\nu},\alpha^i,\beta^i) \in {\mathcal{B}{}}_R,$$ ]{} where [ $$\begin{aligned} {\mathcal{B}{}}_R:= &\Bigl\{(\gamma,\phi,\psi)\in CX^{s+1}_T({\mathbb{R}{}}^3)\times CY^{s+1}_T(\Omega) \times CY^{s}_T(\Omega) \: \Bigl\| \: {\\|(\gamma,\phi,\psi)\\|}\leq R, \\ &({\partial_{X^0}}^\ell\gamma,{\partial_{X^0}}^\ell \phi)\|_{X^0=0}=(\gamma^\ell,\phi_\ell) \; \; \ell=0,1,\ldots,s+1 \quad \& \quad {\partial_{X^0}}^\ell \psi\|_{X^0=0} = \phi_{\ell+1} \; \; \ell=0,1,\ldots s \Bigr\} \end{aligned}$$ ]{} to a solution [ $$(\gamma_{\mu\nu},\phi^i,\psi^i) \in CX^{s+1}_T({\mathbb{R}{}}^3)\times CY^{s+1}_T(\Omega) \times CY^{s}_T(\Omega)$$ ]{} of the IVBP [ $$\begin{aligned} \frac{{\partial_{}}\;\;}{{\partial_{}}X^\Delta}\left(a^{\Delta\Lambda}\bigl(\lambda,{\tilde{\beta}{}},D{\tilde{\alpha}{}}\bigr)\frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^\Lambda}\right) - q_{\mu\nu}\bigl(\lambda,{\partial_{}}\lambda,{\tilde{\beta}{}},D{\tilde{\alpha}{}}\bigr) \hspace{1.5cm} & \notag \\ - \chi_{\Omega}p_{\mu\nu}\bigl(\lambda,\beta,D\alpha\bigr) &= 0 &&\text{in $[0,T]\times \tilde{\Sigma} $,} \label{EinElasK.1} \\ \frac{{\partial_{}}\;\;}{{\partial_{}}X^\Delta}\left(L^{\Delta\Lambda}_{ij} \bigl(\lambda,\beta,D\alpha\bigr)\frac{{\partial_{}}\psi^j}{{\partial_{}}X^\Lambda} + Z^{\Delta\mu\nu}_i(\lambda,\beta,D\alpha)\frac{{\partial_{}}\lambda_{\mu\nu}}{{\partial_{}}X^0}\right) \hspace{1.5cm} & \notag \\ -Y_i\left(\gamma,\frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^0},\phi,{\partial_{}}\phi,{\partial_{}}\psi\right) &= 0 &&\text{in $[0,T]\times \Omega $,} \label{EinElasK.2}\\ {\partial_{0}}\phi^i &= \beta^i &&\text{in $[0,T]\times \Omega $,} \label{EinElasK.3} \\ \nu_J \left(L^{J\Lambda}_{ij} \bigl(\lambda,\beta,D\alpha\bigr)\frac{{\partial_{}}\psi^j}{{\partial_{}}X^\Lambda} + Z^{J\mu\nu}_i(\lambda,\beta,D\alpha)\frac{{\partial_{}}\lambda_{\mu\nu}}{{\partial_{}}X^0}\right) &= 0 &&\text{in $[0,T]\times {\partial_{}}\Omega $,} \label{EinElasK.4} \\ \left(\gamma_{\mu\nu}, \frac{{\partial_{}}\gamma_{\mu\nu}}{{\partial_{}}X^0} \right) &= (\gamma^0_{\mu\nu},\gamma^1_{\mu\nu}) &&\text{in $\{0\}\times \tilde{\Sigma}$,} \label{EinElasK.5} \\ \phi^i &= \phi_0^i &&\text{in $\{0\} \times \Omega$,} \label{EinElasF.6}\\ \left(\psi^i, \frac{{\partial_{}}\psi^i}{{\partial_{}}X^0}\right) &= (\phi_1^i,\phi_2^i) &&\text{in $\{0\} \times \Omega$,} \label{EinElasF.7} \end{aligned}$$ ]{} where we are using [ $$D(\cdot) = \frac{{\partial_{}}(\cdot)}{{\partial_{}}X^I}$$ ]{} to denote the spatial gradient. The mapping property [ $$J_T\: :\: {\mathcal{B}{}}_R \longrightarrow CX^{s+1}_T({\mathbb{R}{}}^3)\times CY^{s+1}_T(\Omega) \times CY^{s-1}_T(\Omega)$$ ]{} is a consequence of the linear estimates contained in Theorem \[linGthm\] of this article and Theorem 2.4 of [@Koch:1993]. Furthermore, it follows from these estimates and the calculus inequalities of this article and those of [@Koch:1993] that $J_T$ satisfies [ $$J_T({\mathcal{B}{}}_R) \subset {\mathcal{B}{}}_R$$ ]{} for $T>0$ small enough. This establishes boundedness in a high norm. Mimicking the arguments used in Section \[contract\] of this article and those in Section 3 of [@Koch:1993], it can be shown that $J_T$, shrinking $T>0$ if necessary, defines a contraction in a suitable low norm, and this, in turn, yields the existence of a unique solution [ $$(\gamma_{\mu\nu},\phi^i,\psi^i) \in CX^{s+1}_T({\mathbb{R}{}}^3)\times CY^{s+1}_T(\Omega) \times CY^{s}_T(\Omega)$$ ]{} of the IVBP , - and -. It is then a simple consequence of the above definitions that the pair [ $$(\gamma_{\mu\nu},\phi^i) \in CX^{s+1}_T({\mathbb{R}{}}^3)\times CY^{s+1}_T(\Omega)$$ ]{} is the unique solution to the IVBP -. Inverting the transformation used to define the material representation, it is not difficult to verify that this solution yields a (unique) solution $(g_{\mu\nu},f^I)$ to the reduced IVBP -. The final step is to show that the vector field $\xi^\mu$ vanishes so that the solution $(g_{\mu\nu},f^I)$ also satisfies the full Einstein equations . This is accomplished by realizing that the boundary condition together with the elasticity field equations imply that the stress energy tensor $T^{\mu\nu}$ satisfies [ $$\nabla_\mu T^{\mu\nu} = 0 \quad \text{in $M$,}$$ ]{} in the distributional sense. This is enough to conclude from the reduced equations , with the help of the contracted Bianchi identity, that $\xi^\mu$ weakly solves a linear wave equation of the form [ $$\nabla_\mu \nabla^\mu \xi^\nu + C^\nu_\mu \xi^\mu = 0 \quad \text{in $M$.}$$ ]{} Moreover, it is a consequence of the constraint equations and that [ $$(\xi^\mu,{\mathcal{L}{}}_t \xi^\mu) = 0 \quad \text{in $\Sigma$}.$$ ]{} By uniqueness of weak solutions to linear wave equations, it follows that [ $$\xi^\mu = 0 \quad \text{in $M$,}$$ ]{} completing our local existence and uniqueness argument. *Acknowledgments* This work was partially supported by the ARC grants DP1094582 and FT1210045. Part of this work was completed during a visit of the author T.A.O. to the Albert Einstein Institute. We are grateful to the Institute for its support and hospitality during these visits. We also thank B. Schmidt for many illuminating and productive discussions. Finally, we thank the referee for their comments and criticisms, which have served to improve the content and exposition of this article. Calculus inequalities {#calculus} ===================== In this appendix we state, for the convenience of the reader, some well known calculus inequalities for the standard Sobolev spaces $W^{s,p}(\Omega)$, and we derive a number of related inequalities for the $H^{k,s}({\mathbb{G}{}}^n)$ spaces. In the following, $\Omega$ will always denote a bounded, open subset of ${\mathbb{G}{}}^n$ with a smooth boundary. Spatial inequalities {#Sineq} -------------------- The proof of the following inequalities are well known and may be found, for example, in the books [@AdamsFournier:2003], [@Friedman:1976] and [@TaylorIII:1996]. Alternatively, one can also consult Appendix A of Koch’s thesis [@Koch:1990] for detailed proofs. [\[Hölder’s inequality\]]{} \[Holder\] If $0< p,q,r \leq \infty$ satisfy $1/p+1/q = 1/r$, then [ $${\\|uv\\|}_{L^r(\Omega)} \leq {\\|u\\|}_{L^p(\Omega)}{\\|v\\|}_{L^q(\Omega)}$$ ]{} for all $u\in L^p(\Omega)$ and $v\in L^q(\Omega)$. [\[Sobolev’s inequality\]]{} \[Sobolev\] Suppose $s\in {\mathbb{Z}{}}_{\geq 1}$ and $1\leq p < \infty$. - If $sp<n$, then [ $${\\|u\\|}_{L^q(\Omega)} \lesssim {\\|u\\|}_{W^{s,p}(\Omega)} \qquad p\leq q \leq np/(n-s p)$$ ]{} for all $u\in W^{s,p}(\Omega)$. - (Morrey’s inequality) If $sp > n$, then [ $${\\|u\\|}_{C^{0,\mu}(\overline{\Omega})} \lesssim {\\|u\\|}_{W^{s,p}(\Omega)} \qquad 0 < \mu \leq \min\{1,s-n/p\}$$ ]{} for all $u\in W^{s,p}(\Omega)$. [\[Interpolation\]]{} \[interp\] Suppose ${\epsilon}_0 >0$, $1\leq p \leq \infty$, $k,s\in {\mathbb{Z}{}}_{\geq 0}$ and $k\leq s$. Then there exists a constant $K>0$ such that [ $$\|u\|_{k,p} \leq K\bigl({\epsilon}\|u\|_{s,p} + {\epsilon}^{-k/(s-k)}{\\|u\\|}_{L^p(\Omega)}\bigr)$$ ]{} for $0<{\epsilon}\leq {\epsilon}_0$, where $\|\cdot\|_{k,p}$ is the seminorm defined by [ $$\|u\|_{k,p} = \left(\sum_{\|\alpha\|=k}{\\|D^\alpha u\\|}_{L^p(\Omega)}^p\right)^{1/p}.$$ ]{} [\[Multiplication inequality\]]{} \[calcpropB\] Suppose $1\leq p <\infty$, $s_1,s_2,\ldots s_{\ell+1}\in {\mathbb{Z}{}}$, $s_1,s_2,\ldots,s_\ell \geq s_{\ell+1}\geq 0$, and $\sum_{j=1}^\ell s_j -n/p > s_{\ell+1}$. Then [ $${\\|u_1 u_2 \cdots u_\ell\\|}_{W^{p,s_{\ell+1}}(\Omega)} \lesssim {\\|u_1\\|}_{W^{p,s_1}(\Omega)} {\\|u_2\\|}_{W^{p,s_2}(\Omega)} \cdots {\\|u_\ell\\|}_{W^{p,s_\ell}(\Omega)}$$ ]{} for all $u_i \in W^{p,s_i}(\Omega)$ $i=1,2,\ldots,\ell$. [\[Gagliardo-Nirenberg’s inequality\]]{} \[GNMa\] If $1\leq p,q,r\leq \infty$, $s\in {\mathbb{Z}{}}_{\geq 1}$ and $\|\alpha\|\leq s$, then [ $${\\|D^\alpha u\\|}_{L^r(\Omega)} \lesssim {\\|u\\|}^{1-\|\alpha\|/s}_{L^q(\Omega)} {\\|u\\|}_{W^{s,p}(\Omega)}^{\|\alpha\|/s}$$ ]{} for all $u\in L^q(\Omega) \cap W^{s,p}(\Omega)$, where [ $$\frac{s-\|\alpha\|}{sq} + \frac{\|\alpha\|}{sp} = \frac{1}{r}.$$ ]{} In particular [ $${\\|D^\alpha u\\|}_{L^{\frac{sp}{\|\alpha\|}}(\Omega)} \lesssim {\\|u\\|}_{L^\infty(\Omega)}^{1-\frac{\|\alpha\|}{s}} {\\|u\\|}^{\frac{\|\alpha\|}{s}}_{W^{s,p}(\Omega)}.$$ ]{} [\[Moser’s inequality\]]{} \[GNMb\] Suppose $s\in {\mathbb{Z}{}}_{\geq 1}$, $1\leq p \leq \infty$, $\|\alpha\|\leq s$, $f\in C^s({\mathbb{R}{}})$, $f(0)=0$, $u\in C^0(\Omega)\cap L^\infty(\Omega)\cap W^{s,p}(\Omega)$, and $u(x) \in V$ for all $x\in \Omega$ where $V$ is open and bounded in ${\mathbb{R}{}}$. Then [ $${\\|D^\alpha f(u)\\|}_{L^p(\Omega)} \leq C({\\|f\\|}_{C^s(\overline{V})})(1+{\\|u\\|}^{s-1}_{L^\infty(\Omega)}){\\|u\\|}_{W^{s,p}(\Omega)}.$$ ]{} Spacetime inequalities {#STineq} ---------------------- We now prove spacetime versions of the multiplication and Moser inequalities adapted to the ${\mathcal{H}{}}^{0,s}({\mathbb{G}{}}^n)$, ${\mathcal{X}{}}_T^{s}({\mathbb{G}{}}^n)$ and $X_T^{s}(\Omega)$ spaces. \[elemE\] Suppose $s_1,s_2,\ldots s_{\ell+1}\in {\mathbb{Z}{}}$, $s_1,s_2,\ldots,s_\ell \geq s_{\ell+1}\geq 0$, and $\sum_{j=1}^\ell s_j -n/2 > s_{\ell+1}$. Then [ $${\\|u_1 u_2 \cdots u_\ell\\|}_{{\mathcal{H}{}}^{0,s_{\ell+1}}({\mathbb{T}{}}^n)} \lesssim {\\|u_1\\|}_{{\mathcal{H}{}}^{0,s_1}({\mathbb{T}{}}^n)} {\\|u_2\\|}_{{\mathcal{H}{}}^{0,s_2}({\mathbb{T}{}}^n)} \cdots {\\|u_\ell\\|}_{{\mathcal{H}{}}^{0,s_\ell}({\mathbb{T}{}}^n)}$$ ]{} for all $u_i \in {\mathcal{H}{}}^{0,s_i}({\mathbb{T}{}}^n)$ $i=1,2,\ldots,\ell$. By Theorem \[calcpropB\], we have that [ $$\begin{aligned} {\\|u_1 u_2\cdots u_\ell\\|}_{H^{s_{\ell+1}}(\Omega)} \lesssim {\\|u_1\\|}_{H^{s_1}(\Omega)}{\\|u_2\\|}_{H^{s_2}(\Omega)} \cdots {\\|u_\ell\\|}_{H^{s_\ell}(\Omega)} \intertext{and} {\\|u_1 u_2\cdots u_\ell\\|}_{H^{s_3}(\Omega^c)} \lesssim {\\|u_1\\|}_{H^{s_1}(\Omega^c)}{\\|u_2\\|}_{H^{s_2}(\Omega^c)} \cdots {\\|u_\ell\\|}_{H^{s_\ell}(\Omega^c)}. \end{aligned}$$ ]{} Moreover, it is obvious that [ $${\\|u_1 u_2 \cdots u_\ell\\|}^2_{L^2({\mathbb{T}{}}^n)} = {\\|u_1 u_2\cdots u_\ell\\|}^2_{L^2(\Omega)} + {\\|u_1 u_2\cdots u_\ell\\|}^2_{L^2(\Omega^c)} \leq {\\|u_1 u_2 \cdots u_\ell\\|}^2_{H^{s_{\ell+1}}(\Omega)} + {\\|u_1 u_2 \cdots u_\ell\\|}^2_{H^{s_{\ell+1}}(\Omega^c)}.$$ ]{} The desired inequality [ $${\\|u_1 u_2 \cdots u_\ell\\|}_{{\mathcal{H}{}}^{0,s_{\ell+1}}({\mathbb{T}{}}^n)} \lesssim {\\|u_1\\|}_{{\mathcal{H}{}}^{0,s_1}({\mathbb{T}{}}^n)} {\\|u_2\\|}_{{\mathcal{H}{}}^{0,s_2}({\mathbb{T}{}}^n)} \cdots {\\|u_\ell\\|}_{{\mathcal{H}{}}^{0,s_\ell}({\mathbb{T}{}}^n)}$$ ]{} now follows directly from the above inequalities. The next four propositions are closely related to Lemma 3.2 and Theorem A.6 from [@Koch:1993]. Since proofs of Lemma 3.2 and Theorem A.6 are not provided in [@Koch:1993], we, for the convenience of the reader, provide some of the details here. \[fpropB\] Suppose $s\in {\mathbb{Z}{}}_{> n/2}$, $f\in C^s({\mathbb{R}{}})$, $f(0)=0$, $u\in {\mathcal{X}{}}_T^s({\mathbb{G}{}}^n)$ and $v\in Y_T^s(\Omega)$ with $\Omega \subset {\mathbb{G}{}}^n$. Then [ $${\\|{\partial_{t}}^\ell f(u)\\|}_{{\mathcal{H}{}}^{0,s-\ell}({\mathbb{G}{}}^n)} \leq C({\\|u\\|}_{{\mathcal{E}{}}^s({\mathbb{G}{}}^n)}) {{\quad\text{and}\quad}}{\\|{\partial_{t}}^\ell f(v)\\|}_{H^{s-\ell}(\Omega)} \leq C({\\|u\\|}_{E^s(\Omega)}).$$ ]{} for $0\leq \ell \leq s$. We begin by differentiating $f(u)$ $\ell$-times $(0\leq \ell \leq s)$ with respect to $t$ to get [$${\partial_{t}}^\ell f(u) = \sum_{k_1+\cdots+k_m=\ell} f_{k_1,\ldots,k_m}(u) {\partial_{t}}^{k_1} u \cdots {\partial_{t}}^{k_m} u, \label{fpropB3}$$]{} where $f_{k_1,\ldots,k_m} \in C^{s-\ell}({\mathbb{R}{}})$. Noting that $s+ \sum_{j=1}^m (s-k_j) - n/2 = ms - \ell + s-n/2> s - \ell$, we see that we can apply Proposition \[elemE\] to to get [ $${\\|{\partial_{t}}^\ell f(u)\\|}_{{\mathcal{H}{}}^{0,s-\ell}({\mathbb{T}{}}^n)} = {\\|f_{k_1,\ldots,k_m}(u)\\|}_{{\mathcal{H}{}}^{0,s}({\mathbb{T}{}}^n)} {\\|{\partial_{t}}u^{k_1}\\|}_{{\mathcal{H}{}}^{0,s-k_1}({\mathbb{T}{}}^n)} \cdots {\\|{\partial_{t}}^{k_m} u \\|}_{{\mathcal{H}{}}^{0,s-k_m}({\mathbb{T}{}}^n)}.$$ ]{} Combining this estimates together with Theorems \[Sobolev\] and \[GNMb\], we arrive at the desired estimate [ $${\\|{\partial_{t}}^\ell f(u)\\|}_{{\mathcal{H}{}}^{0,s-\ell}({\mathbb{T}{}}^n)} \leq C_\ell({\\|u\\|}_{{\mathcal{E}{}}^s({\mathbb{T}{}}^n)}).$$ ]{} The other estimate [ $${\\|{\partial_{t}}^\ell f(v)\\|}_{H^{s-\ell}(\Omega)} \leq C({\\|v\\|}_{E^s(\Omega)})$$ ]{} is proved in a similar manner. \[STpropA\] Suppose $s\in {\mathbb{Z}{}}_{\geq 1}$, $1\leq p \leq \infty$, $f\in C^s({\mathbb{R}{}}\times {\mathbb{R}{}}^{n+1},{\mathbb{R}{}})$, $u\in W^{p,s+1}(\Omega)$, ${\partial_{t}}u \in W^{p,s}(\Omega)$ and the higher time derivatives ${\partial_{t}}^\ell u$ $(\ell \geq 2)$ are obtained by formally differentiating [ $${\partial_{t}}^2 u = a^{ij}(u,{\partial_{}}u){\partial_{i}}{\partial_{j}}u + b^i(u,{\partial_{}}u){\partial_{i}}{\partial_{t}}u + g(u,{\partial_{}}u) +h,$$ ]{} where $h\in W^{p,s+1}(\Omega)$, ${\partial_{t}}^\ell h =0$ and $a^{ij},b^j,g \in C^s({\mathbb{R}{}}\times {\mathbb{R}{}}^{n+1},{\mathbb{R}{}})$. Then $u$ satisfies the estimate [ $${\\|{\partial_{t}}^j f(u,{\partial_{}} u)\\|}_{W^{p,s-j}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{p,s+1}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{p,s}(\Omega)})$$ ]{} for $0\leq j \leq s$. First, we observe that estimate [$${\\|{\partial_{t}}^\ell f(u,Du,{\partial_{t}}u)\\|}_{W^{p,s-\ell}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{p,s+1}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}) \label{STpropA4}$$]{} holds for $\ell=0$, $1\leq p \leq \infty$ and $s\in {\mathbb{Z}{}}_{\geq 1}$ thanks to Theorem \[GNMb\]. We now proceed by induction and assume that holds for $1\leq p \leq \infty$, $s\in {\mathbb{Z}{}}_{\geq 1}$, $\ell=0,\ldots,\min\{s-1,j-1\}$, and maps $f\in C^s$, where the constant $C$ in also depends implicitly on the $C^s$ norm of $f$. In particular, this implies that [$${\\|{\partial_{t}}^{j-1}(D_1 f(u,Du,{\partial_{t}}u) {\partial_{t}}u)\\|}_{W^{s-j+1,p}(\Omega)} \leq C (1+{\\|u\\|}_{W^{s+1,p}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}), \label{STpropA5}$$]{} where [$$C = C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr). \label{STpropA6}$$]{} To proceed, we write the $j^\text{th}$ time derivative of $f(u,Du,{\partial_{t}}u)$ as [$${\partial_{t}}^j\bigl[f(u,Du,{\partial_{t}}u)\bigr] = {\partial_{t}}^{j-1}\bigl(D_1 f(u,Du,{\partial_{t}}u) {\partial_{t}} u\bigr) + {\partial_{t}}^{j-1}\bigl(D_2 f(u,Du,{\partial_{t}}u) \cdot {\partial_{t}}Du \bigr) + {\partial_{t}}^{j-1}\bigl(D_3 f(u,Du,{\partial_{t}}u) {\partial_{t}}^2 u\bigr). \label{STpropA6a}$$]{} Since we can already estimate the first term of the right hand side of using , we turn to estimating the second term, which we write as follows [$${\partial_{t}}^{j-1}(D_{2}f(u,Du,{\partial_{t}}u)\cdot{\partial_{t}}Du) = \sum^{j-1}_{k=0} a_k \bigr({\partial_{t}}^k D_{2}f\bigr) \cdot \bigl({\partial_{t}}^{j-1-k} D{\partial_{t}} u\bigr). \label{STpropA7}$$]{} Next, we observe that [$$D^\alpha\bigl[{\partial_{t}}^k D_{2}f \cdot {\partial_{t}}^{j-1-k} D{\partial_{t}} u\bigr] = \sum_{\beta+\gamma = \alpha} a_{\alpha\beta} \bigl(D^\beta {\partial_{t}}^k D_{2}f\bigr)\cdot\bigl( D^\gamma {\partial_{t}}^{j-1-k} D {\partial_{t}}u\bigr) \qquad \|\alpha\|= s-j, \label{STpropA8}$$]{} for appropriate constants $a_k$ and $a_{\alpha\beta}$. Letting $C$ denote a constant of the form , we now estimate as follows: [ $$\begin{aligned} &{\\|D^\alpha\bigl[{\partial_{t}}^k D_{2}f \cdot {\partial_{t}}^{j-1-k} D{\partial_{t}} u\bigr]\\|}_{L^p(\Omega)} \lesssim \sum_{\|\beta\|+\|\gamma\|=s-j} {\\|\bigl(D^\beta {\partial_{t}}^k D_{2}f\bigr)\cdot\bigl( D^\gamma {\partial_{t}}^{j-1-k} D {\partial_{t}}u\bigr)\\|}_{L^p(\Omega)} \notag \\ &\text{\hspace{0.1cm}}\lesssim \sum_{\|\beta\|+\|\gamma\|=s-j}{\\|D^\beta {\partial_{t}}^k D_{2}f\\|}_{L^{\frac{ps}{\|\beta\|+k}}(\Omega)} {\\|D^\gamma {\partial_{t}}^{j-1-k} D {\partial_{t}}u\\|}_{L^{\frac{ps}{\|\gamma\|+j-k}}(\Omega)} \text{\hspace{0.2cm} by H\"{o}lder's inequality} \notag \\ &\text{\hspace{0.1cm}}\lesssim \sum_{\|\beta\|+\|\gamma\|=s-j} {\\|{\partial_{t}}^k D_{2}f\\|}_{W^{\|\beta\|+k-k,\frac{ps}{\|\beta\|+k}}(\Omega)} {\\|{\partial_{t}}^{j-1-k}{\partial_{t}}u\\|}_{W^{\|\gamma\|+j-k- (j-k-1),\frac{ps}{\|\gamma\|+j-k}}(\Omega)} \notag\\ &\text{\hspace{0.1cm}}\leq C \sum_{\|\beta\|+\|\gamma\|=s-j}\biggl(1+{\\|u\\|}_{W^{\|\beta\|+k+1,\frac{ps}{\|\beta\|+k}}(\Omega)}+ {\\|{\partial_{t}}u\\|}_{W^{\|\beta\|+k,\frac{ps}{\|\beta\|+k}}(\Omega)}\biggr) \notag \\ &\text{\hspace{0.2cm}} \times \biggl(1+{\\|u\\|}_{W^{\|\gamma\|+j-k+1,\frac{ps}{\|\gamma\|+j-k}}(\Omega)}+ {\\|{\partial_{t}}u\\|}_{W^{\|\gamma\|+j-k,\frac{ps}{\|\gamma\|+j-k}}(\Omega)}\biggr) \text{\hspace{0.3cm} by induction hypothesis } \notag \\ &\text{\hspace{0.1cm}}\leq C \sum_{\|\beta\|+\|\gamma\|=s-j}\biggl(1+{\\|u\\|}_{W^{s+1,p}(\Omega)}^{\frac{\|\beta\|+k}{s}}+ {\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}^{\frac{\|\beta\|+k}{s}}\biggr) \notag \\ &\text{\hspace{5.4cm}} \times \biggl(1+{\\|u\\|}_{W^{s,p}(\Omega)}^{\frac{\|\gamma\|+j-k}{s}}+ {\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}^{\frac{\|\gamma\|+j-k}{s}}\biggr) \text{\hspace{0.3cm} by Theorem \ref{GNMa}} \notag \\ &\text{\hspace{0.1cm}}\leq C (1+{\\|u\\|}_{W^{s+1,p}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}). \notag \end{aligned}$$ ]{} This estimate together with the formula , shows that we can, with the help of Theorem \[interp\], estimate by [$${\\|{\partial_{t}}^{j-1}(D_{2}f(u,Du,{\partial_{t}}u)\cdot {\partial_{t}}Du)\\|}_{W^{s-j,p}(\Omega)} \leq C (1+{\\|u\\|}_{W^{s+1,p}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}), \label{STpropA10}$$]{} where the constant $C$ is of the form . With the second term in estimated, we use relation [$${\partial_{t}}^2 u = a^{ij}(u,{\partial_{}}u){\partial_{i}}{\partial_{j}}u + b^i(u,{\partial_{}}u){\partial_{i}}{\partial_{t}}u + g(u,{\partial_{}}u) +h, \label{STpropA11}$$]{} to write the third term in as [ $${\partial_{t}}^{j-1}(D_3 f(u,Du,{\partial_{t}}u){\partial_{t}}^2 u) = {\partial_{t}}^{j-1}\bigl[D_3 f(u,Du,{\partial_{t}}u) \bigl(a^{ij}(u,{\partial_{}}u){\partial_{i}}{\partial_{j}}u + b^i(u,{\partial_{}}u){\partial_{i}}{\partial_{t}}u + g(u,{\partial_{}}u) +h\bigr)\bigr].$$ ]{} Similar arguments employed above to derive show also that [$${\\|{\partial_{t}}^{j-1}(D_{3}f(u,Du,{\partial_{t}}u){\partial_{t}}^2u)\\|}_{W^{s-j,p}(\Omega)} \leq C (1+{\\|u\\|}_{W^{s+1,p}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}) \label{STpropA13}$$]{} for a constant $C$ of the form . Together, the estimates , and show that [ $${\\|{\partial_{t}}^j f(u,Du,{\partial_{t}}u)\\|}_{W^{p,s-j}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{p,s+1}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}).$$ ]{} This completes the induction argument and the proof of the proposition. Similar arguments can be used to prove the following variant of Proposition \[STpropA\]. \[STpropC\] Suppose $s\in {\mathbb{Z}{}}_{\geq 1}$, $1\leq p \leq \infty$, $f\in C^{s+1}({\mathbb{R}{}},{\mathbb{R}{}})$, $u\in W^{p,s+1}(\Omega)$, ${\partial_{t}}u \in W^{p,s}(\Omega)$ and the higher time derivatives ${\partial_{t}}^\ell u$ $\ell \geq 2$ are obtained by formally differentiating [ $${\partial_{t}}^2 u = a^{ij}(u,{\partial_{}}u){\partial_{i}}{\partial_{j}}u + b^i(u,{\partial_{}}u){\partial_{i}}{\partial_{t}}u + g(u,{\partial_{}}u) +h,$$ ]{} where $h\in W^{p,s+1}(\Omega)$, ${\partial_{t}}^\ell h =0$ and $a^{ij},b^j,g \in C^s({\mathbb{R}{}}\times {\mathbb{R}{}}^{n+1},{\mathbb{R}{}})$. Then $u$ satisfies the estimate [ $${\\|{\partial_{t}}^j f(u)\\|}_{W^{p,s+1-j}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{p,s+1}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{p,s}(\Omega)})$$ ]{} for $0\leq j \leq s+1$. \[STpropB\] Suppose $s\in {\mathbb{Z}{}}_{\geq 1}$, $1\leq p \leq \infty$, $f^{\mu\nu}\in C^{s+1}({\mathbb{R}{}},{\mathbb{R}{}})$, $u\in W^{p,s+1}(\Omega)$, ${\partial_{t}}u \in W^{p,s}(\Omega)$ and the higher time derivatives ${\partial_{t}}^\ell u$ $\ell \geq 2$ are obtained by formally differentiating [ $${\partial_{t}}^2 u = a^{ij}(u,{\partial_{}}u){\partial_{i}}{\partial_{j}}u + b^i(u,{\partial_{}}u){\partial_{i}}{\partial_{t}}u + g(u,{\partial_{}}u) +h,$$ ]{} where $h\in W^{p,s+1}(\Omega)$, ${\partial_{t}}^\ell h =0$ and $a^{ij},b^j,g \in C^s({\mathbb{R}{}}\times {\mathbb{R}{}}^{n+1},{\mathbb{R}{}})$. Then [ $${\\|{\partial_{\mu}}[{\partial_{t}}^k,f^{\mu\nu}(u){\partial_{\nu}}]u\\|}_{W^{p,s-k}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{p,s+1}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{p,s}(\Omega)})$$ ]{} for $0\leq k \leq s$. Differentiating the formula [ $$[{\partial_{t}}^k,f^{\nu\mu}(u){\partial_{\mu}}]u = \sum_{\ell=0}^{k-1}\binom{k}{\ell}\bigl[{\partial_{t}}^{k-\ell}f^{\nu i}{\partial_{i}}{\partial_{t}}^\ell u + {\partial_{t}}^{k-\ell}f^{\nu 0} {\partial_{t}}^{\ell+1}u\bigr],$$ ]{} we see that [ $$\begin{aligned} {\partial_{j}}\bigl([{\partial_{t}}^k,f^{j\mu}(u){\partial_{\mu}}]u\bigr) = \sum_{\ell=0}^{k-1}\binom{k}{\ell}\bigl[&{\partial_{t}}^{k-\ell}(Df^{j i}{\partial_{j}}u){\partial_{i}}{\partial_{t}}^\ell u + {\partial_{t}}^{k-\ell}f^{ji}{\partial_{j}}{\partial_{i}}{\partial_{t}}^\ell u \notag \\ & + {\partial_{t}}^{k-\ell}(Df^{j 0}{\partial_{j}}u) {\partial_{t}}^{\ell+1}u + {\partial_{t}}^{k-\ell}f^{j 0} {\partial_{t}}^{\ell+1}{\partial_{j}}u \bigr] \label{StpropB6.1} \end{aligned}$$ ]{} and [ $$\begin{aligned} {\partial_{0}}\bigl([{\partial_{t}}^k,f^{0\mu}(u){\partial_{\mu}}]u\bigr) = \sum_{\ell=0}^{k-1}\binom{k}{\ell}\bigl[&{\partial_{t}}^{k-\ell}(Df^{0 i}{\partial_{t}}u){\partial_{i}}{\partial_{t}}^\ell u + {\partial_{t}}^{k-\ell}f^{0i}{\partial_{t}}{\partial_{i}}{\partial_{t}}^\ell u \notag \\ & + {\partial_{t}}^{k-\ell}(Df^{0 0}{\partial_{t}}u) {\partial_{t}}^{\ell+1}u + {\partial_{t}}^{k-\ell}f^{0 0} {\partial_{t}}^{\ell+2} u \bigr]. \label{StpropB7.1} \end{aligned}$$ ]{} To estimate and , we start by differentiating the term [$${\partial_{t}}^{k-\ell}f^{ji}{\partial_{j}}{\partial_{i}}{\partial_{t}}^\ell u \label{STpropB8}$$]{} $s-k$ times to get [$$D^\alpha\bigl({\partial_{t}}^{k-\ell}f^{ji}{\partial_{j}}{\partial_{i}}{\partial_{t}}^\ell u\bigr) = \sum_{\beta+\gamma=\alpha} a_{\alpha\beta} D^\beta\bigl({\partial_{t}}^{k-\ell}f^{ji}\bigr)D^\gamma \bigl({\partial_{j}}{\partial_{i}}{\partial_{t}}^\ell u\bigr) \quad \|\alpha\|=s-1-k \label{STpropB8a}$$]{} for appropriate constants $a_{\alpha\beta}$. Letting $C$ denote a constant of the form, [ $$C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr),$$ ]{} we estimate as follows: [ $$\begin{aligned} &{\\|D^\alpha\bigl[{\partial_{t}}^{k-\ell}f^{ij} {\partial_{i}}{\partial_{j}}{\partial_{t}}^{\ell} u\bigr]\\|}_{L^p(\Omega)} \lesssim \sum_{\|\beta\|+\|\gamma\|=s-k} {\\|\bigl(D^\beta{\partial_{t}}^{k-\ell} f^{ij}\bigr)\bigl( D^\gamma {\partial_{i}}{\partial_{j}}{\partial_{t}}^{\ell}u\bigr)\\|}_{L^p(\Omega)} \notag \\ &\text{\hspace{0.1cm}}\lesssim \sum_{\|\beta\|+\|\gamma\|=s-k}{\\|D^\beta {\partial_{t}}^{k-\ell}f\\|}_{L^{\frac{ps}{\|\beta\|+k-\ell-1}}(\Omega)} {\\|D^{\gamma}D^2 {\partial_{t}}^{\ell} u\\|}_{L^{\frac{ps}{\|\gamma\|+1+\ell}}(\Omega)} \text{\hspace{0.2cm} by H\"{o}lder's inequality} \notag \\ &\text{\hspace{0.1cm}}\lesssim \sum_{\|\beta\|+\|\gamma\|=s-k} {\\|{\partial_{t}}^{k-\ell}f\\|}_{W^{\|\beta\|,\frac{ps}{\|\beta\|+k-\ell-1}}(\Omega)} {\\|{\partial_{t}}^{\ell}u\\|}_{W^{\|\gamma\|+2,\frac{ps}{\|\gamma\|+\ell+1}}(\Omega)} \notag\\ &\text{\hspace{0.1cm}}\leq C \sum_{\|\beta\|+\|\gamma\|=s-k}\biggl(1+{\\|u\\|}_{W^{\|\beta\|+k-\ell,\frac{ps}{\|\beta\|+k-\ell-1}}(\Omega)}+ {\\|{\partial_{t}}u\\|}_{W^{\|\beta\|+k-\ell-1,\frac{ps}{\|\beta\|+k-\ell-1}}(\Omega)}\biggr) \notag \\ &\text{\hspace{0.2cm}} \times \biggl(1+{\\|u\\|}_{W^{\|\gamma\|+2+\ell,\frac{ps}{\|\gamma\|+1+\ell}}(\Omega)}+ {\\|{\partial_{t}}u\\|}_{W^{\|\gamma\|+1+\ell,\frac{ps}{\|\gamma\|+1+\ell}}(\Omega)}\biggr) \text{\hspace{0.3cm} by Proposition \ref{STpropA} } \notag \\ &\text{\hspace{0.1cm}}\leq C \sum_{\|\beta\|+\|\gamma\|=s-k}\biggl(1+{\\|u\\|}_{W^{s+1,p}(\Omega)}^{\frac{\|\beta\|+k-\ell-1}{s}}+ {\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}^{\frac{\|\beta\|+k-\ell-1}{s}}\biggr) \notag \\ &\text{\hspace{5.4cm}} \times \biggl(1+{\\|u\\|}_{W^{s,p}(\Omega)}^{\frac{\|\gamma\|+1+\ell}{s}}+ {\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}^{\frac{\|\gamma\|+1+\ell}{s}}\biggr) \text{\hspace{0.3cm} by Theorem \ref{GNMa}} \notag \\ &\text{\hspace{0.1cm}}\leq C (1+{\\|u\\|}_{W^{s+1,p}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}). \notag \end{aligned}$$ ]{} This estimate together with the formula shows that we can, with the help of Theorem \[interp\], estimate by [ $${\\|{\partial_{t}}^{k-\ell}f^{ji}{\partial_{j}}{\partial_{i}}{\partial_{t}}^\ell u\\|}_{W^{s-k}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{s+1,p}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{s,p}(\Omega)}),$$ ]{} for $0\leq \ell \leq k-1$. Using the same arguments, it is not difficult to verify similar estimates hold for the remaining terms in and , which allows us to conclude that [ $${\\|{\partial_{\mu}}[{\partial_{t}}^k,f^{\mu\nu}(u){\partial_{\nu}}]u\\|}_{W^{p,s-k}(\Omega)} \leq C\bigl({\\|u\\|}_{W^{1,\infty}(\Omega)},{\\|{\partial_{t}}u\\|}_{L^\infty(\Omega)}\bigr) (1+{\\|u\\|}_{W^{p,s+1}(\Omega)}+{\\|{\partial_{t}}u\\|}_{W^{p,s}(\Omega)})$$ ]{} for $0\leq k \leq s$. Potential Theory {#potential} ================ In this appendix, we recall some results from potential theory that we require to prove energy estimates. We begin by recalling the following well known result.[^7] \[Lisoprop\] Suppose $p\in (1,\infty)$, $s\geq0$ and $\psi \in C^\infty({\mathbb{T}{}}^n)$ satisfies $\psi \geq 0$ on ${\mathbb{T}{}}^n$ and $\psi(x_0) > 0$ for some $x_0 \in {\mathbb{T}{}}^n$. Then the map [ $$\Delta - \psi : W^{s+1,p}({\mathbb{T}{}}^n) \longrightarrow W^{s-1,p}({\mathbb{T}{}}^n) \quad (s\geq 0)$$ ]{} is an isomorphism. Letting [ $${\mathcal{L}{}}= (\Delta-\psi)^{-1} : W^{s-1,p}({\mathbb{T}{}}^n) \longrightarrow W^{s,p}({\mathbb{T}{}}^n)$$ ]{} denote the inverse of $\Delta-\psi$, we can represent ${\mathcal{L}{}}$ as [ $${\mathcal{L}{}}v(x) = \int_{{\mathbb{T}{}}^n} E(x,y)v(y)\, d^n x$$ ]{} where $E$ is the integral kernel of ${\mathcal{L}{}}$. Fixing an open set $\Omega \subset {\mathbb{T}{}}^n$ with $C^\infty$ boundary, we then define the single and double layer potentials by [$$\mathcal{S} v (x) = \int_{\partial \Omega} E(x,y)v(y)\, d\sigma (y) \quad x \notin \partial \Omega \label{Spotdef}$$]{} and [$$\mathcal{D} v (x) = \int_{\partial \Omega} \frac{\partial E}{\partial \nu_y}(x,y)v(y)\, d\sigma (y) \quad x \notin \partial \Omega, \label{Dpotdef}$$]{} respectively. Here, $d\sigma$ is the natural area element on $\partial \Omega$ and $\nu$ is the outward unit conormal to $\Omega$. \[potprop\] Suppose $p\in (1,\infty)$, $k\in {\mathbb{Z}{}}_{\geq 1}$, and $\psi(x_0) > 0$ for some[^8] $x_0 \in \Omega^c$. Then the linear map[^9] [ $${\mathcal{R}{}}_\Omega \circ{\mathcal{L}{}}\circ \chi_\Omega \: : \: W^{k,p}(\Omega) \longrightarrow W^{k+2,p}(\Omega)$$ ]{} is continuous (i.e. bounded). The proof follows from a straightforward adaptation of Proposition 3.6 in [@Andersson_et_al:2011]. Here, one simply needs to use the analogous mapping properties for the single and double layer potential, as defined above in \[Spotdef\] and \[Dpotdef\], as a replacement for the potential theory used in [@Andersson_et_al:2011] that was based on the (flat) Laplacian on ${\mathbb{R}{}}^3$. With this replacement, the proof from [@Andersson_et_al:2011] goes through directly without any further changes needed. Weak solutions of wave equations {#weak} ================================ We recall some basic facts about weak solutions to linear wave equations. We begin with the definition of a weak solution. Suppose $a^{\mu\nu} \in W^{1,\infty}([0,T],L^\infty({\mathbb{G}{}}^n))$, $a^{\mu\nu}=a^{\nu\mu}$, $p^\mu\in H^1([0,T],L^2({\mathbb{G}{}}^n))$, $q^{\nu}\in W^{1,\infty}\bigl([0,T],L^n({\mathbb{G}{}}^n)\bigr)\cap L^\infty([0,T]\times{\mathbb{G}{}}^n)$, and there exists a $\kappa > 0$ such that [ $$\kappa \|\xi\|^2 \leq a^{ij}\xi_i\xi_j \quad \text{for all $\xi = (\xi_i) \in {\mathbb{R}{}}^n$} {{\quad\text{and}\quad}}a^{00} \leq -\kappa.$$ ]{} Then we say that $u \in H^1([0,T]\times {\mathbb{G}{}}^n)$ is a weak solution of [ $$\begin{aligned} {\partial_{\mu}}(a^{\mu\nu} {\partial_{\nu}} u) &= f + {\partial_{\mu}}( p^\mu + q^{\mu} u), \label{weakB.1}\\ (u\|_{t=0},{\partial_{t}}u\|_{t=0})& = (u_0,u_1) \in H^1({\mathbb{G}{}}^n)\times L^2({\mathbb{G}{}}^n), \label{weakB.2} \end{aligned}$$ ]{} if[^10] [ $$(u(t),{\partial_{t}}u(t)) \rightharpoonup (u_0,u_1) \quad \text{in $H^1({\mathbb{G}{}}^n)\times L^2({\mathbb{G}{}}^n)$}$$ ]{} and [ $${ \langle a^{\mu\nu}{\partial_{\mu}}u \| \phi \rangle}_{L^2([0,T]\times {\mathbb{G}{}}^n)} = -{ \langle f \| \phi \rangle}_{L^2([0,T]\times {\mathbb{G}{}}^n)} + { \langle p^\mu + q^{\mu} u \| {\partial_{\mu}}\phi \rangle}_{L^2([0,T]\times {\mathbb{G}{}}^n)}$$ ]{} for all $\phi \in H^1([0,T],L^2({\mathbb{G}{}}^n))\cap L^2([0,T],H^1({\mathbb{G}{}}^n))$. With the above notion of a weak solution, the proof of the next theorem is just a special case of Theorem 2.2 from [@Koch:1993]. \[weakthm\] Suppose $u$ is a weak solution of -. Then $u \in C([0,T],H^1({\mathbb{G}{}}^n))\cap C^1([0,T],L^2({\mathbb{G}{}}^n))$ and $u$ satisfies the estimate [ $${\\|u(t_2)\\|}_{E({\mathbb{G}{}}^n)} \leq c\left({\\|u(t_1)\\|}_{E({\mathbb{G}{}}^n)} + d_1 + \int_{t_1}^{t_2} d_2(\tau){\\|u(\tau)\\|}_{E({\mathbb{G}{}}^n)} +d_3(\tau) \, d\tau \right)$$ ]{} for all $0\leq t_1 \leq t_2 \leq T$, where [ $$\begin{aligned} d_1 &= {\\|p(t_1)\\|}_{L^2({\mathbb{G}{}}^n)} +(1+{\\|q(t_1)\\|}_{L^\infty({\mathbb{G}{}}^n)}){\\|u(t_1)\\|}_{L^2({\mathbb{G}{}}^n)},\\ d_2(t) &= 1+ {\\|{\partial_{t}}a(t)\\|}_{L^\infty({\mathbb{G}{}}^n)} + {\\|q(t)\\|}_{L^\infty({\mathbb{G}{}}^n)} + {\\|{\partial_{t}}q(t)\\|}_{L^n({\mathbb{G}{}}^n)} \\ d_3(t) &= {\\|f(t)\\|}_{L^2({\mathbb{G}{}}^n)} + {\\|{\partial_{t}}p(t)\\|}_{L^n({\mathbb{G}{}}^n)} \end{aligned}$$ ]{} and $c=c(\kappa,{\\|a\\|}_{L^\infty([0,T]\times {\mathbb{G}{}}^n)})$. Field rescalings {#scale} ================ In this appendix, we establish the behavior of the norms ${\mathcal{H}{}}^{k,s}(Q_1)$ and $H^s(Q^+_1)$ under rescaling. These results are used repeatedly in Section \[linear\] when we exploit the freedom to localize the estimates for the linear IVP -. \[scalepropA\] Suppose $0<\delta\leq 1$, $s,\ell\in {\mathbb{Z}{}}_{\geq 0}$, $n\geq 3$, $0\leq \sigma \leq 1$, $0\leq \sigma < s-n/2$, $s-\ell \geq 0$, $f\in {\mathcal{H}{}}^{2,s}(Q_1)$, $g\in {\mathcal{H}{}}^{0,s-\ell}(Q_1)$, $h\in {\mathcal{H}{}}^{m_{s-\ell},s-\ell}(Q_1)$ and let [ $$f_\delta(x) = \frac{f(\delta x)-f(0)}{\delta^\sigma}, \quad g_\delta(x) = g(\delta x) {{\quad\text{and}\quad}}h_\delta(x) = h(\delta x).$$ ]{} Then [ $$\begin{aligned} {\\|f_\delta\\|}_{{\mathcal{H}{}}^{2,s}(Q_1)} &\lesssim {\\|f\\|}_{{\mathcal{H}{}}^{2,s}(Q_1)}, \\ {\\|g_\delta\\|}_{{\mathcal{H}{}}^{0,s-\ell}(Q_1)} &\lesssim \frac{1}{\delta^\ell}{\\|g\\|}_{{\mathcal{H}{}}^{0,s-\ell}(Q_1)} \intertext{and} {\\|h_\delta\\|}_{{\mathcal{H}{}}^{m_{s-\ell},s-\ell}(Q_1)} &\lesssim \frac{1}{\delta^\ell}{\\|h\\|}_{{\mathcal{H}{}}^{m_{s-\ell},s-\ell}(Q_1)}. \end{aligned}$$ ]{} First, a short calculation shows that [ $${\\|D^s f_\delta\\|}_{L^2(Q^\pm_1)}^2 = \delta^{2(s-\sigma)-n}{\\|D^s f\\|}_{L^2(Q^\pm_\delta)}^2 \leq \delta^{2(s-\sigma)-n} {\\|D^s f\\|}_{L^2(Q^\pm_1)}^2,$$ ]{} and in particular, that [$${\\|D^s f_\delta\\|}_{L^2(Q^\pm_1)} \leq {\\|D^s f\\|}_{L^2(Q^\pm_1)} \label{fep2}$$]{} since $s-\sigma -n/2 \geq 0$ by assumption. Next, we observe that [ $$\begin{aligned} {\\|D^2 f_\delta\\|}_{L^2(Q_1)}^2 &= \delta^{2(2-\sigma)-n} {\\|D^2 f\\|}_{L^2(Q_\delta)}^2 \\ &= \delta^{2(2-\sigma)-n}\left( \int_{Q^-_\delta}\|D^2 f\| \, d^n x + \int_{Q^+_\delta}\|D^2 f\|^2 \, d^n x\right) \end{aligned}$$ ]{} which implies via Hölder’s inequality, Theorem \[Holder\], that [ $$\begin{aligned} {\\|D^2 f_\delta\\|}_{L^2(Q_1)}^2 & \leq \delta^{2-n} \bigl( {\\|1\\|}_{L^p(Q^-_\delta)}{\\|D^2 f\\|}_{L^{2q}(Q^-_\delta)}^2 + {\\|1\\|}_{L^p(Q^+_\delta)}{\\|D^2 f\\|}_{L^{2q}(Q^+_\delta)}^2\bigr)\notag \\ & \leq 2^{(n-1)/p}\delta^{2(2-\sigma)-n+n/p} \bigl({\\|D^2 f\\|}_{L^{2q}(Q^-_1)}^2 + {\\|D^2 f\\|}_{L^{2q}(Q^+_1)}^2\bigr) \notag \end{aligned}$$ ]{} for $1/q + 1/p=1$. Choosing $p=n/n-2$ and hence $q=n/2$, the above inequality becomes [ $${\\|D^2 f_\delta\\|}_{L^2(Q_1)}^2 \leq 2^{(n-1)(n-2)/n}\delta^{2(1-\sigma)}\bigl({\\|D^2 f\\|}_{L^{n}(Q^-_1)}^2 + {\\|D^2 f\\|}_{L^{n}(Q^+_1)}^2\bigr)$$ ]{} However, by Sobolev’s inequality, Theorem \[Sobolev\], we have that [ $${\\|D^2 u\\|}_{L^n(Q^\pm_1)} \lesssim {\\|D^2 f\\|}_{H^{n/2-1}(Q^\pm_1)},$$ ]{} and this allows us to conclude that [$${\\|D^2 f_\delta\\|}_{L^2(Q_1)} \lesssim \bigl({\\|f\\|}_{H^{s}(Q^-_1)} + {\\|f\\|}_{H^{s}(Q^+_1)}\bigr), \label{fep7}$$]{} since $s> n/2$ and $\sigma \leq 1$. Next, we observe that [ $$\|f_{\delta}(x)\| \leq \|x\|^\sigma \frac{\|f(\delta x)-f(0)\|}{\|\delta x - 0\|^\sigma} \leq \max\bigl\{{\\|f\\|}_{C^{0,\sigma}(Q_{\delta}^{+})},{\\|f\\|}_{C^{0,\sigma}(Q_{\delta}^{-})} \bigr\} \leq \max\bigl\{{\\|f\\|}_{C^{0,\sigma}(Q_{1}^{+})},{\\|f\\|}_{C^{0,\sigma}(Q_{1}^{-})} \bigr\}$$ ]{} for all $x\in Q_1$. This together with Sobolev’s inequality gives [ $${\\|f_{\delta}\\|}_{L^\infty(Q_1)} \lesssim {\\|f\\|}_{{\mathcal{H}{}}^{0,s}(Q_1)},$$ ]{} which, in turn, implies, with the help of Hölder’s inequality, that [$${\\|f_{\delta}\\|}_{L^2(Q_1)} \lesssim {\\|f\\|}_{{\mathcal{H}{}}^{0,s}(Q_1)}. \label{fep10}$$]{} The inequalities , and together with interpolation, see Theorem \[interp\], then show that [ $${\\|f_\delta\\|}_{{\mathcal{H}{}}^{2,s}(Q_1)} \lesssim {\\|f\\|}_{{\mathcal{H}{}}^{2,s}(Q_1)},$$ ]{} while, a short calculation shows that [$${\\|D^{s-\ell}g_\delta\\|}^2_{L^2(Q^{\pm}_1)} = \delta^{2(s-\ell)-n}{\\|D^{s-\ell}g\\|}^2_{L^2(Q^\pm_\delta)} \leq \frac{1}{\delta^{2\ell}} {\\|D^{s-\ell}g\\|}^2_{L^2(Q^\pm_1)}, \label{fep12}$$]{} since $s>n/2$ implies that $2 s-n>0$. We now consider the two cases[^11] Case 1: $s-\ell > n/2$ Suppose now that $s-\ell > n/2$. Then [ $${\\|g_\delta\\|}_{L^\infty(Q^\pm_1)} \leq {\\|g\\|}_{L^\infty(Q^\pm_1)} \lesssim {\\|g\\|}_{H^{s-\ell}(Q^\pm_1)}$$ ]{} by Sobolev’s inequality, and so [$${\\|g_\delta\\|}_{L^2(Q^\pm_1)} \leq {\\|g\\|}_{L^\infty(Q^\pm_1)}{\\|1\\|}_{L^2(Q^\pm_1)} \lesssim {\\|g\\|}_{H^{s-\ell}(Q^\pm_1)} \label{fep14}$$]{} follows by Hölder’s inequality. Case 2: $s-\ell < n/2$ Suppose now that $s-\ell < n/2$. Then [$${\\|g_\delta\\|}_{L^2(Q^\pm_1)} = \delta^{-n/2}{\\|g\\|}_{L^2(Q^\pm_\delta)} \leq \delta^{-n/2}{\\|1\\|}_{L^{n/(s-\ell)}(Q^\pm_\delta)} {\\|g\\|}_{L^q}(Q^\pm_\delta) \label{fep15}$$]{} for [ $$\frac{1}{q} = \frac{1}{2} - \frac{(s-\ell)}{n}$$ ]{} by Hölder’s inequality. But ${\\|1\\|}_{L^{n/(s-\ell)}} = 2^{(n-1)(s-\ell)/n}\delta^{s-\ell}$, and so, we see from that [ $${\\|g_\delta\\|}_{L^2(Q^\pm_1)} \lesssim \frac{1}{\delta^\ell} {\\|g\\|}_{L^q(Q^\pm_1)}$$ ]{} since $s-n/2 > 0$. However, [ $${\\|g\\|}_{L^q(Q^\pm_1)} \lesssim {\\|g\\|}_{H^{s-\ell}(Q^\pm_1)}$$ ]{} by Sobolev’s inequality, and therefore, we have that [$${\\|g_\delta\\|}_{L^2(Q^\pm_1)} \lesssim \frac{1}{\delta^\ell} {\\|g\\|}_{H^{s-\ell}(Q^\pm_1)}. \label{fep20}$$]{} In either case, the inequalities and when combined with and interpolation show that [ $${\\|g_\delta\\|}_{H^{s-\ell}(Q^\pm_1)} \lesssim \frac{1}{\delta^\ell}{\\|g\\|}_{H^{s-\ell}(Q^\pm_1)},$$ ]{} and so we see that [$${\\|g_\delta\\|}_{H^{0,s-\ell}(Q_1)} \lesssim \frac{1}{\delta^\ell}{\\|g\\|}_{H^{0,s-\ell}(Q_1)}. \label{fep21}$$]{} Continuing on, a simple calculation yields [$${\\|D^{m_{s-\ell}} h_\delta\\|}_{L^2(Q_1)} \lesssim \delta^{m_{s-\ell}-n/2} {\\|D^{m_{s-\ell}} h\\|}_{L^2(Q_1)}. \leq \delta^{ s-n/2}\frac{1}{\delta^{s-m_{s-\ell}}}{\\|D^{m_{s-\ell}} h\\|}_{L^2(Q_1)}. \label{fep22}$$]{} Four cases $s-\ell = 0$, $s-\ell =1$, $s-\ell =2$ and $s-\ell \geq 3$ now follow. Case 1: $s-\ell = 0$ If $s=\ell$, then $m_{s-\ell}=0$ and the estimate [$${\\|h_\delta\\|}_{L^2(Q_1)} \lesssim \frac{1}{\delta^s} {\\|h\\|}_{{\mathcal{H}{}}^{0,0}(Q_1)} \label{fep23}$$]{} is a direct consequence of since $s-n/2 > 0$. Case 2: $s-\ell = 1$ If $\ell = s-1$, then we see from and $m_1=1$ that [$${\\|Dh_\delta\\|}_{L^2(Q_1)} \lesssim \delta^{s-n/2}\frac{1}{\delta^{s-1}}{\\|Dh\\|}_{L^2(Q_1)} \lesssim \frac{1}{\delta^{s-1}} {\\|Dh\\|}_{{\mathcal{H}{}}^{1,1}(Q_1)} \label{fep24}$$]{} since $s-n/2 > 0$. Case 3: $s-\ell = 2$ If $\ell = s-2$, then we see from and $m_2=2$ that [$${\\|D^2 h_\delta\\|}_{L^2(Q_1)} \lesssim \delta^{s -n/2}\frac{1}{\delta^{s-2}}{\\|D^2h\\|}_{L^2(Q_1)} \lesssim \frac{1}{\delta^{s-2}} {\\|D^2h\\|}_{{\mathcal{H}{}}^{2,2}(Q_1)} \label{fep25}$$]{} since $s-n/2 > 0$. Case 4: $s-\ell \geq 3$ If $s-\ell \geq 3$, then $m_{s-\ell}=2$ and two cases $s-2-\ell < n/2$ and $s-2-\ell > n/2$ follow. Case 4a: $s-2-\ell < n/2$ [ $$\begin{aligned} {\\|D^2 h_\delta\\|}_{L^2(Q_1)}^2 &= \delta^{4-n} {\\|D^2 h\\|}_{L^2(Q_\delta)}^2 \\ &= \delta^{4-n}\left( \int_{Q^-_\delta}\|D^2 h\| \, d^n x + \int_{Q^+_\delta}\|D^2 h\|^2 \, d^n x\right). \end{aligned}$$ ]{} Using Hölder’s inequality, we can write this as [ $$\begin{aligned} {\\|D^2 h_\delta\\|}_{L^2(Q_1)}^2 & \leq \delta^{4-n} \bigl( {\\|1\\|}_{L^p(Q^-_\delta)}{\\|D^2 h\\|}_{L^{2q}(Q^-_\delta)}^2 + {\\|1\\|}_{L^p(Q^+_\delta)}{\\|D^2 h\\|}_{L^{2q}(Q^+_\delta)}^2\bigr) \notag \\ & \leq 2^{(n-1)/p}\delta^{4-n+n/p} \bigl({\\|D^2 h\\|}_{L^{2q}(Q^-_1)}^2 + {\\|D^2 h\\|}_{L^{2q}(Q^+_1)}^2\bigr) \label{fep27.1} \end{aligned}$$ ]{} for $1/q + 1/p=1$. But, we notice that [$${\\|D^2 h\\|}_{L^{\frac{2n}{n-2(s-2-\ell)}}(Q^\pm_1)} \lesssim {\\|D^2 h\\|}_{H^{s-2-\ell}(Q^\pm_1)} \lesssim {\\|h\\|}_{H^{s-\ell}(Q^\pm_1)} \label{fep28}$$]{} where in obtaining the first inequality we used Sobolev’s inequality, while [$${\\|D^2 h_\delta\\|}_{L^2(Q_1)}^2 \lesssim \frac{1}{\delta^{2\ell}} \biggl({\\|D^2 h\\|}_{L^{\frac{2n}{n-2(s-2-\ell)}}(Q^-_1)}^2 + {\\|D^2 h\\|}_{L^{\frac{2n}{n-2(s-2-\ell)}}(Q^+_1)}^2\biggr) \label{fep29}$$]{} follows from setting [ $$q =\frac{n}{n-2(s-2-\ell)} {{\quad\text{and}\quad}}p = \frac{n}{2(s-2-\ell)}$$ ]{} in the inequality and recalling that $s>n/2$. Combining the two inequalities and , we arrive at [$${\\|D^2 h_\delta\\|}_{L^2(Q_1)} \lesssim \frac{1}{\delta^{\ell}}{\\|h\\|}_{H^{0,s-\ell}(Q_1)} \lesssim \frac{1}{\delta^{\ell}}{\\|h\\|}_{H^{2,s-\ell}(Q_1)}. \label{fep31}$$]{} Case 4b: $s-2-\ell>n/2$ Suppose now that $s-2-\ell>n/2$. Then [ $${\\|D^2 h_\delta\\|}_{L^\infty(Q^\pm_1)} \lesssim \delta^2 {\\|D^2 h\\|}_{L^\infty(Q^\pm_1)} \lesssim {\\|D^2 h\\|}_{H^{s-2-\ell}(Q^\pm_1)} \lesssim {\\|h\\|}_{H^{s-\ell}(Q^\pm_1)}$$ ]{} by Sobolev’s inequality. Using Hölder’s inequality, it is not difficult to see that the above inequality implies that [$${\\|D^2 h_\delta\\|}_{L^2(Q_1)} \lesssim {\\|h\\|}_{H^{0,s-\ell}(Q_1)} \lesssim \frac{1}{\delta^{\ell}}{\\|h\\|}_{H^{2,s-\ell}(Q_1)}. \label{fep33}$$]{} In either case, and show that [$${\\|D^2 h_\delta\\|}_{L^2(Q_1)} \lesssim \frac{1}{\delta^{\ell}}{\\|h\\|}_{H^{2,s-\ell}(Q_1)} \label{fep34}$$]{} holds. From the inequalities , , , , and interpolation, we conclude that [ $${\\|h_\delta\\|}_{{\mathcal{H}{}}^{m_{s-\ell},s-\ell}(Q_1)} \lesssim \frac{1}{\delta^\ell}{\\|h\\|}_{{\mathcal{H}{}}^{m_{s-\ell},s-\ell}(Q_1)}.$$ ]{} We will also need the following version of Proposition \[scalepropA\] for the $H^s(Q^+_1)$ spaces. Since it can be established using similar arguments, we omit the details. \[scalepropB\] Suppose $0<\delta\leq 1$, $s,\ell\in {\mathbb{Z}{}}_{\geq 0}$, $n\geq 3$, $0\leq \sigma \leq 1$, $0\leq \sigma < s-n/2$, $s-\ell \geq 0$ $f\in H^s(Q^+_1)$, $g\in H^{s-\ell}(Q^+_1)$ and let [ $$f_\delta(x) = \frac{f(\delta x)-f(0)}{\delta^\sigma} {{\quad\text{and}\quad}}g_\delta(x) = g(\delta x).$$ ]{} Then [ $$\begin{aligned} {\\|f_\delta\\|}_{H^{s}(Q^+_1)} &\lesssim {\\|f\\|}_{H^{s}(Q^+_1)} \\ \intertext{and} {\\|g_\delta\\|}_{H^{s-\ell}(Q^+_1)} &\lesssim \frac{1}{\delta^\ell}{\\|g\\|}_{H^{s-\ell}(Q^+_1)}. \end{aligned}$$ ]{} [^1]: We recall that relativistic fluids are a special case of relativistic elastic matter. [^2]: The notation used in this article for coordinates, indices, partial derivatives, function spaces, and the like can be found in Section \[prelim\]. [^3]: Here, $\sim$ denotes the equivalence relation on $Q_\delta$ determined by the identification of the opposite sides of the boundary. [^4]: As described in the introduction, the time derivatives ${\partial_{t}}^\ell U \|_{t=0}$ $\ell \geq 2$ are generated from the initial data by formally differentiating with respect to $t$ and evaluating at $t=0$. [^5]: The extension to non-colliding multiple interacting bodies is straightforward. [^6]: We note that this condition rules out perfect fluids. [^7]: Here, $\Delta = \delta^{ij}{\partial_{i}}{\partial_{j}}$ is the flat Laplacian [^8]: Recall that $\Omega^c = {\mathbb{T}{}}^n\setminus \overline{\Omega}$. [^9]: Here ${\mathcal{R}{}}_\Omega$ denotes the restriction operator, i.e. for a function $f$ defined on ${\mathbb{T}{}}^n$, ${\mathcal{R}{}}_\Omega f (x) := f(x)$ for all $x\in \Omega$. [^10]: Here, following standard notation, “$\rightharpoonup$” denotes weak convergence. [^11]: We can avoid the case $s-\ell = n/2$ for $n$ even by replacing $s$ by a non-integral ${\tilde{s}{}}$ which is slightly less than $s$ while using the version of Sobolev’s inequality that is valid for the fractional Sobolev spaces.
--- abstract: 'We study the relation of Yangians and spinons in the SU($n$) Haldane–Shastry model. The representation theory of the Yangian is shown to be intimately related to the fractional statistics of the spinons. We construct the spinon Hilbert space from tensor products of the fundamental representations of the Yangian.' author: - Dirk Schuricht date: 'November 9, 2007' title: \| Many-spinon states and representations of Yangians in the\ SU(n) Haldane–Shastry model --- Introduction ============ Quantum groups [@Drinfeld85; @Jimbo85] first arose from the quantum inverse scattering method [@Faddeev82; @KorepinBogoliubovIzergin93], which had been developed to construct and solve integrable quantum systems. In particular, quantum groups provide a way to construct and study the solutions, called R-matrices, of the quantum Yang–Baxter equation. Mathematically, quantum groups are deformations of the universal enveloping algebra of the classical Lie algebras. In general, they depend on a parameter $h$ and the underlying Lie algebra is recovered in the limit $h\rightarrow 0$. Yangians are special quantum groups which were first introduced by Drinfel’d [@Drinfeld85] in 1985. Their representation theory is intimately related [@ChariPressley90; @ChariPressley98] to the rational R-matrices. Later it was discovered that Yangians also appear as additional symmetries of quantum field theories [@Bernard91; @Haldane-92], and furthermore, that Yangians are part of the symmetry algebra of special integrable spin systems. In particular, the one-dimensional nearest-neighbour Heisenberg model possesses a Yangian symmetry in the limit of a chain of infinite length [@Bernard93], whereas the Haldane–Shastry model possesses a Yangian symmetry even for a chain of finite length [@Haldane-92; @Hikami95npb]. In addition, a Yangian symmetry exists for the one-dimensional Hubbard model on an infinite chain [@UglovKorepin94] as well as for a finite chain with suitable hopping amplitudes [@GoehmannInozemtsev96; @EsslerFrahmGoehmannKluemperKorepin05]. From a physical point of view, the Haldane–Shastry model (HSM) [@Haldane88] owes its special importance to two reasons. The first and more technical one is that the model is exactly solvable even for a chain of finite length. It is possible to derive explicit wave functions for the ground state and the elementary spinon excitations [@Haldane91prl1]. The second and more fundamental reason is that the HSM possesses non-interacting or free spinon excitations [@Haldane94], a conclusion which is in particular supported by the trivial spinon-spinon scattering matrix calculated by Essler [@Essler95] using the asymptotic Bethe Ansatz. In 2001 this picture was challenged by Bernevig et al. [@Bernevig-01prb], who studied the explicit two-spinon wave functions and claimed to have identified effects of an interaction between the spinons. A critical re-examination [@GS05prb] of their conclusions, however, showed that these alleged interaction effects are in fact due to the fractional statistics of the spinons [@Haldane91prl2], which results in non-trivial quantisation rules for the individual spinon momenta [@Greiter]. This debate showed that free particles may appear interacting at first sight if an inappropriate representation is chosen. In this article we investigate the relation between the Yangian symmetry and the physical properties of the spinons. We show that individual spinons in the HSM transform under the fundamental representation of the Yangian. We then study the implications of the Yangian symmetry on many-spinon states. The main result of this analysis is the derivation of a general rule governing the fractional statistics of the spinons. This rule states that in the spinon Hilbert space only the irreducible subrepresentations of the tensor products of certain fundamental representations of the Yangian exists. This enables us to derive, starting from a set of individual spinon momenta, the allowed values of the total spin of the corresponding many-spinon states, which are subject to highly non-trivial restrictions due to the fractional statistics of the spinons. All results are generalised to the elementary excitations of the SU(3) HSM. Before we discuss the main topic of this article, we will briefly review the HSM and its most important physical features, and give a concise introduction to Yangians and their representation theory. Haldane–Shastry model {#sec:hsm} ===================== In 1988 Haldane and Shastry discovered independently [@Haldane88] that a trial wave function proposed by Gutzwiller [@Gutzwiller63] in 1963 provides the exact ground state to a Heisenberg type spin Hamiltonian whose interaction strength falls off as the inverse square of the distance between two spins on the chain. Later the model was generalised to an SU($n$) spins by Kawakami [@Kawakami92prb2]. The HSM is most conveniently formulated by embedding the one-dimensional chain with periodic boundary conditions into the complex plane by mapping it onto the unit circle with the (SU($n$)) spins located at complex positions $\eta_\alpha=\exp\!\left(i\frac{2\pi}{N}\alpha\right)$, where $N$ denotes the number of sites and $\alpha=1,\ldots,N$. The Hamiltonian is given by [@Haldane88] $$\label{eq:ham} H_{\mathrm{HS}} =\frac{2\pi^2}{N^2} \sum^N_{\alpha\neq\beta} \frac{\boldsymbol{S}_{\alpha}\!\cdot\!\boldsymbol{S}_{\beta}} {\vert \eta_{\alpha}-\eta_{\beta}\vert^2}.$$ The SU(3) HSM [@Kawakami92prb2] is given by replacing $\boldsymbol{S}_\alpha$ by the eight-dimensional SU(3) spin vector $\boldsymbol{J}_{\alpha}=\frac{1}{2}\sum_{\sigma\tau} c_{\alpha\sigma}^{\dagger} \boldsymbol{\lambda}_{\sigma\tau} c_{\alpha\tau}^{\phantom{\dagger}}$, where $\boldsymbol{\lambda}$ a vector consisting of the eight Gell-Mann matrices [@Cornwell84vol2], and $\sigma$ and $\tau$ are SU(3) spin or colour indices which take the values blue (b), red (r), or green (g) (see Fig. \[fig:weightdiagrams\].a). The spins on the lattice sites transform under the fundamental representation $\boldsymbol{n}$ of SU($n$), $S=1/2$ for SU(2). (32,11)(1,0) (1,10)[a)]{} (12,10)[3]{} (2,5)[(1,0)[10]{}]{} (7,0)[(0,1)[10]{}]{} (12,4.3)[$J^3$]{} (7.2,9.6)[$J^8$]{} (7,1.5) (6,1.5)[g]{} (9.93,6.75) (10.43,6.75)[b]{} (4.07,6.75) (4.57,6.75)[r]{} (6.8,6.75) ------------------------------------------------------------------------ (7.3,6.5)[$\textstyle\frac{1}{2\sqrt{3}}$]{} (7.3,1.25)[$\textstyle\frac{-1}{\sqrt{3}}$]{} (4.07,4.8) ------------------------------------------------------------------------ (3.06,3.8)[$-\textstyle\frac{1}{2}$]{} (9.93,4.8) ------------------------------------------------------------------------ (9.7,3.8)[$\textstyle\frac{1}{2}$]{} (19,10)[b)]{} (30,10)[$\boldsymbol{\bar{3}}$]{} (20,5)[(1,0)[10]{}]{} (25,0)[(0,1)[10]{}]{} (30,4.3)[$J^3$]{} (25.2,9.6)[$J^8$]{} (25,8.5) (23.7,8.5)[m]{} (27.93,3.25) (28.43,2.75)[c]{} (22.07,3.25) (22.57,2.75)[y]{} (24.8,3.25) ------------------------------------------------------------------------ (25.3,3)[$\textstyle\frac{-1}{2\sqrt{3}}$]{} (25.3,8.25)[$\textstyle\frac{1}{\sqrt{3}}$]{} (22.07,4.8) ------------------------------------------------------------------------ (21.06,5.7)[$-\textstyle\frac{1}{2}$]{} (27.93,4.8) ------------------------------------------------------------------------ (27.7,5.7)[$\textstyle\frac{1}{2}$]{} The ground state ($N=2M$, $M$ integer) of the SU(2) model is given by $$\label{eq:groundstate} {\left\|\Psi_0\right\rangle}=P_{\mathrm{G}}{\left\|\Psi_{\mathrm{SD}}^N\right\rangle},\quad {\left\|\Psi_{\mathrm{SD}}^N\right\rangle}\equiv \prod_{q\in\mathcal{I}} c_{q{\uparrow}}^\dagger c_{q{\downarrow}}^\dagger{\left\|0\right\rangle},$$ where the Gutzwiller projector $P_{\mathrm{G}}$ eliminates configurations with more than one particle on any site and the interval $\mathcal{I}$ contains $M$ adjacent momenta. For SU($n$), each momentum in $\mathcal{I}$ has to be occupied by $n$ particles with different spins [@Kawakami92prb3]. The model is invariant under global SU(2) or SU(3) rotations generated by $$\boldsymbol{S}=\sum_{\alpha=1}^N\boldsymbol{S}_{\alpha} \quad\mathrm{(for\ SU(2))} \quad\mathrm{or}\quad \boldsymbol{J}=\sum_{\alpha=1}^N\boldsymbol{J}_\alpha \quad\mathrm{(for\ SU(3))}, \label{eq:su3-Jsymmetry}$$ respectively. The system possesses an additional symmetry [@Haldane-92; @Hikami95npb], which is given by $$\boldsymbol{\Lambda}=\frac{\mathrm{i}}{2}\sum_{\alpha\neq\beta}^N\, \frac{\eta_\alpha + \eta_\beta}{\eta_\alpha - \eta_\beta}\, (\boldsymbol{S}_\alpha\times\boldsymbol{S}_\beta)\quad\mathrm{or}\quad \Lambda^a=\frac{1}{2}\sum^N_{\alpha\neq\beta} \frac{\eta_\alpha +\eta_\beta}{\eta_\alpha -\eta_\beta}\; f^{abc}J^b_\alpha J^c_\beta, \label{eq:su3-Upsilon}$$ (we use the Einstein summation convention) where $a,b,c=1,\ldots,8$ and $f^{abc}$ denote the structure constants of SU(3). The total spin (\[eq:su3-Jsymmetry\]) and rapidity operator (\[eq:su3-Upsilon\]) generate the Yangian, which we will discuss in detail below. The elementary excitations of the SU($n$) HSM are constructed by annihilation of a particle from the Slater determinant state before Gutzwiller projection [@Haldane91prl1; @SG05epl], $${\left\|\Psi_{p\bar\sigma}\right\rangle}=P_{\mathrm{G}}\;\! c_{-p\sigma}^{\phantom{\dagger}}\!{\left\|\Psi_{\mathrm{SD}}^{N+1}\right\rangle}, \quad N=nM-1. \label{eq:loccoloron}$$ Here $p$ denotes the momentum, $\sigma$ either the spin (for $n=2$) or one of the colons blue, red or green (for $n=3$). In order to ensure that every site is occupied by a spin after the projection, we annihilate a particle from the Slater determinant state with $N+1$ particles. Note that for SU(2) the annihilation of an up-spin electron creates a down-spin spinon and vice versa. The spinons possess spin $1/2$ like the electrons on the lattice sites. For SU(3) the situation is, however, fundamentally different. Here, the annihilation of a, say, blue particle creates an anti-blue SU(3) spinon or coloron. (We will use the terms SU(3) spinon and coloron simultaneously.) This means that colorons transform under the conjugate representation $\boldsymbol{\bar{3}}$ (see Fig. \[fig:weightdiagrams\].b), if the particles on the sites transform under $\boldsymbol{3}$ [@BouwknegtSchoutens96; @SG05epl]. A non-orthogonal but complete basis for spin-polarised two-spinon eigenstates with total momentum $p=-k_1-k_2$ is given by (we assume $k_1>k_2$) $${\left\|\Psi_{p_1\bar{\sigma},p_2\bar{\sigma}}\right\rangle}= P_{\mathrm{G}}\;\!c_{k_1\sigma}c_{k_2\sigma} \!{\left\|\Psi_{\mathrm{SD}}^{N+2}\right\rangle}, \quad N=nM-2. \label{eq:twospinons}$$ These states are not energy eigenstates, but as $H_{\mathrm{HS}}$ scatters $k_1$ and $k_2$ in only one direction (increasing $k_1-k_2$), there is a one-to-one correspondence between these basis states and the exact eigenstates of the HSM. The total energy of the eigenstates takes the form of free particles if and only if the single-spinon momenta are shifted with respect to $k_{1,2}$ [@GS05prb; @SG05epl]: $$\label{eq:twospinonmomenta} p_{1,2}=-k_{1,2}\pm\frac{1}{2n}\frac{2\pi}{N},\quad p_1<p_2.$$ The shift is due to the fractional statistics of the spinons [@Haldane91prl2; @Greiter]. In general, all two-spinon states with the same single-spinon momenta are obtained by acting with the total spin and rapidity operators on the polarised states (\[eq:twospinons\]). In particular, for SU(2) the action of $\Lambda^zS^-$ yields the two-spinon singlet states. However, this two-spinon singlet state $\Lambda^zS^-{\left\|\Psi_{p_1{\uparrow},p_2{\uparrow}}\right\rangle}$ exists only for $p_2-p_1>\frac{1}{2}\frac{2\pi}{N}$, as (\[eq:twospinons\]) is annihilated [@Haldane-92] by $\Lambda^zS^-$ for $p_2-p_1=\frac{1}{2}\frac{2\pi}{N}$. For general many-spinon states or spinons in the SU($n$) chain these restrictions on the possible values of the total spin become highly non-trivial. Tableau formalism {#sec:yt} ================= Recently, Greiter and I introduced a formalism to obtain all existing many-spinon states starting from a given set of single-spinon momenta [@GS07]. The formalism works as follows. To begin with, the Hilbert space of a system of $N$ identical SU($n$) spins can be decomposed into representations of the total spin (\[eq:su3-Jsymmetry\]), which commutes with (\[eq:ham\]) and hence can be used to classify the energy eigenstates. These representations are compatible with the representations of the symmetric group S$_N$ of $N$ elements, which may be expressed in terms of Young tableaux [@InuiTanabeOnodera96]. In order to obtain these Young tableaux, we draw for each of the $N$ spins a box numbered consecutively from left to right. The representations of SU($n$) are constructed by putting the boxes together such that the numbers assigned to them increase in each row from left to right and in each column from top to bottom. Each tableau obtained in this way represents an irreducible representation of SU($n$), it further indicates symmetrisation over all boxes in the same row, and antisymmetrisation over all boxes in the same column. This implies that we cannot have more than $n$ boxes on top of each other for SU($n$) spins. (31,5)(1,0) (0.2,3)[(2,1)[rep]{}]{} (7,3)[(3,1)[$S_\mathrm{tot}$]{}]{} (17.8,3)[(7,1)[$L$]{}]{} (25.2,3)[(5,1)[$a_1,\ldots,a_L$]{}]{} (0,0)(0,1)[3]{}[(1,0)[3]{}]{} (0,0)(1,0)[4]{}[(0,1)[2]{}]{} (0,1)[(1,1)[1]{}]{} (0,0)[(1,1)[2]{}]{} (1,1)[(1,1)[3]{}]{} (1,0)[(1,1)[4]{}]{} (2,1)[(1,1)[5]{}]{} (2,0)[(1,1)[6]{}]{} (8,1)[(1,1)[0]{}]{} (10.5,0.9)[(1,1)[$\rightarrow$]{}]{} (13,0)(0,1)[3]{}[(1,0)[3]{}]{} (13,0)(1,0)[4]{}[(0,1)[2]{}]{} (13,1)[(1,1)[1]{}]{} (13,0)[(1,1)[2]{}]{} (14,1)[(1,1)[3]{}]{} (14,0)[(1,1)[4]{}]{} (15,1)[(1,1)[5]{}]{} (15,0)[(1,1)[6]{}]{} (21,1)[(1,1)[0]{}]{} (24,1.5)[(1,0)[7]{}]{} (25,1.35)(1,0)[6]{}[(0,1)[0.3]{}]{} (31,3)(1,0) (0,0)(0,1)[3]{}[(1,0)[3]{}]{} (0,0)(1,0)[4]{}[(0,1)[2]{}]{} (0,1)[(1,1)[1]{}]{} (0,0)[(1,1)[2]{}]{} (1,1)[(1,1)[3]{}]{} (2,1)[(1,1)[4]{}]{} (1,0)[(1,1)[5]{}]{} (2,0)[(1,1)[6]{}]{} (8,1)[(1,1)[0]{}]{} (10.5,0.9)[(1,1)[$\rightarrow$]{}]{} (13,2)[(1,0)[3]{}]{} (13,1)[(1,0)[4]{}]{} (13,0)[(1,0)[1]{}]{} (15,0)[(1,0)[2]{}]{} (13,1)(1,0)[4]{}[(0,1)[1]{}]{} (13,0)(1,0)[5]{}[(0,1)[1]{}]{} (13,1)[(1,1)[1]{}]{} (13,0)[(1,1)[2]{}]{} (14,1)[(1,1)[3]{}]{} (15,1)[(1,1)[4]{}]{} (15,0)[(1,1)[5]{}]{} (16,0)[(1,1)[6]{}]{} (14.5,0.5)(2,1)[2]{} (21,1)[(1,1)[2]{}]{} (24,1.5)[(1,0)[7]{}]{} (25,1.35)(1,0)[6]{}[(0,1)[0.3]{}]{} (27,1.5)(3,0)[2]{} (26.5,0)[(1,1)[3]{}]{} (29.5,0)[(1,1)[6]{}]{} (31,3)(1,0) (0,1)(0,1)[2]{}[(1,0)[4]{}]{} (0,0)[(1,0)[2]{}]{} (0,1)(1,0)[5]{}[(0,1)[1]{}]{} (0,0)(1,0)[3]{}[(0,1)[1]{}]{} (0,1)[(1,1)[1]{}]{} (0,0)[(1,1)[2]{}]{} (1,1)[(1,1)[3]{}]{} (2,1)[(1,1)[4]{}]{} (1,0)[(1,1)[5]{}]{} (3,1)[(1,1)[6]{}]{} (8,1)[(1,1)[1]{}]{} (10.5,0.9)[(1,1)[$\rightarrow$]{}]{} (13,1)(0,1)[2]{}[(1,0)[4]{}]{} (13,0)(2,0)[2]{}[(1,0)[1]{}]{} (13,1)(1,0)[5]{}[(0,1)[1]{}]{} (13,0)(1,0)[4]{}[(0,1)[1]{}]{} (13,1)[(1,1)[1]{}]{} (13,0)[(1,1)[2]{}]{} (14,1)[(1,1)[3]{}]{} (15,1)[(1,1)[4]{}]{} (15,0)[(1,1)[5]{}]{} (16,1)[(1,1)[6]{}]{} (14.5,0.5)(2,0)[2]{} (21,1)[(1,1)[2]{}]{} (24,1.5)[(1,0)[7]{}]{} (25,1.35)(1,0)[6]{}[(0,1)[0.3]{}]{} (27,1.5)(3,0)[2]{} (26.5,0)[(1,1)[3]{}]{} (29.5,0)[(1,1)[6]{}]{} (31,3)(1,0) (0,1)(0,1)[2]{}[(1,0)[4]{}]{} (0,0)[(1,0)[2]{}]{} (0,1)(1,0)[5]{}[(0,1)[1]{}]{} (0,0)(1,0)[3]{}[(0,1)[1]{}]{} (0,1)[(1,1)[1]{}]{} (1,1)[(1,1)[2]{}]{} (2,1)[(1,1)[3]{}]{} (0,0)[(1,1)[4]{}]{} (1,0)[(1,1)[5]{}]{} (3,1)[(1,1)[6]{}]{} (8,1)[(1,1)[1]{}]{} (10.5,0.9)[(1,1)[$\rightarrow$]{}]{} (13,2)[(1,0)[3]{}]{} (17,2)[(1,0)[1]{}]{} (13,1)[(1,0)[5]{}]{} (15,0)[(1,0)[2]{}]{} (13,1)(1,0)[6]{}[(0,1)[1]{}]{} (15,0)(1,0)[3]{}[(0,1)[1]{}]{} (13,1)[(1,1)[1]{}]{} (14,1)[(1,1)[2]{}]{} (15,1)[(1,1)[3]{}]{} (15,0)[(1,1)[4]{}]{} (16,0)[(1,1)[5]{}]{} (17,1)[(1,1)[6]{}]{} (13.5,0.5)(1,0)[2]{} (16.5,1.5)(1,-1)[2]{} (21,1)[(1,1)[4]{}]{} (24,1.5)[(1,0)[7]{}]{} (25,1.35)(1,0)[6]{}[(0,1)[0.3]{}]{} (25,1.5)(1,0)[2]{} (29,1.5)(1,0)[2]{} (24.5,0)[(1,1)[1]{}]{} (25.5,0)[(1,1)[2]{}]{} (28.5,0)[(1,1)[5]{}]{} (29.5,0)[(1,1)[6]{}]{} (31,3)(1,0) (0,1)(0,1)[2]{}[(1,0)[5]{}]{} (0,0)[(1,0)[1]{}]{} (0,1)(1,0)[6]{}[(0,1)[1]{}]{} (0,0)(1,0)[2]{}[(0,1)[1]{}]{} (0,1)[(1,1)[1]{}]{} (1,1)[(1,1)[2]{}]{} (2,1)[(1,1)[3]{}]{} (3,1)[(1,1)[4]{}]{} (0,0)[(1,1)[5]{}]{} (4,1)[(1,1)[6]{}]{} (8,1)[(1,1)[2]{}]{} (10.5,0.9)[(1,1)[$\rightarrow$]{}]{} (13,1)(0,1)[2]{}[(1,0)[5]{}]{} (16,0)[(1,0)[1]{}]{} (13,1)(1,0)[6]{}[(0,1)[1]{}]{} (16,0)(1,0)[2]{}[(0,1)[1]{}]{} (13,1)[(1,1)[1]{}]{} (14,1)[(1,1)[2]{}]{} (15,1)[(1,1)[3]{}]{} (16,1)[(1,1)[4]{}]{} (16,0)[(1,1)[5]{}]{} (17,1)[(1,1)[6]{}]{} (13.5,0.5)(1,0)[3]{} (17.5,0.5) (21,1)[(1,1)[4]{}]{} (24,1.5)[(1,0)[7]{}]{} (25,1.35)(1,0)[6]{}[(0,1)[0.3]{}]{} (25,1.5)(1,0)[3]{} (30,1.5) (24.5,0)[(1,1)[1]{}]{} (25.5,0)[(1,1)[2]{}]{} (26.5,0)[(1,1)[3]{}]{} (29.5,0)[(1,1)[6]{}]{} Now, there is a one-to-one correspondence between these Young tableaux and the non-interacting many-spinon states, the eigenstates of the HSM. The principle is illustrated for a few representations of an SU(2) chain with six sites in Fig. \[fig:foursitesu2\], and for an SU(3) chain in Fig. \[fig:sixsitesu3\]. In each Young tableau we shift boxes to the right such that each box is below or in the column to the right of the box with the preceding number. Each missing box in the resulting, extended tableaux represents a spinon, to which we assign a spinon momentum number (SMN) $a_i$ as follows: For an SU(2) chain, it is simply given by the number in the box in the same column. For a general SU($n$) chain, the SMN’s for the spinons in each column are given by a sequence of numbers (half-integers for $n$ odd, integers for $n$ even) with integer spacings such that the arithmetic mean equals the arithmetic mean of the numbers in the boxes of the column, as illustrated in the examples presented in Fig. \[fig:sixsitesu3\]. The extended tableaux provide us with the total SU($n$) spin of each multiplet (given by the original Young tableau), as well as the number $L$ of spinons present and the individual single-spinon momenta $p_1,\ldots,p_L$ given in terms of the SMN’s as $$\label{eq:singlespinonmom} p_i=\frac{2\pi}{N}\:\frac{a_i-\frac{1}{2}}{n},$$ which implies $0\le p_i\le\frac{2\pi}{n}$ for $N\to\infty$. The total momentum and Haldane–Shastry energies of the many-spinon states are $$p=p_0+\sum_{i=1}^L p_i,\quad E=E_0+\sum_{i=1}^L \epsilon(p_i), \label{eq:Lspinonenergy}$$ where $p_0$ and $E_0$ denote the ground-state momentum and energy given by $$\label{eq:pzeroezero} p_0=-\frac{(n\!-\!1)\pi}{n}\:N,\ \ E_0=-\frac{\pi^2}{12}\!\left(\frac{n\!-\!2}{n}N+\frac{2n\!-\!1}{N}\right)\!,$$ and the single-spinon dispersion is $$\label{eq:spinonenergyepsilon} \epsilon(p)=\frac{n}{4}\,p\left(\frac{2\pi}{n}-p\right) +\frac{n^2-1}{12n}\,\frac{\pi^2}{N^2}.$$ This formalism provides the complete spectrum of the HSM [@GS07]. It is easy to see that the momentum spacings for spin-polarised spinons predicted by this formalism reproduce (\[eq:twospinonmomenta\]) for general $n$, and that spinons transform under the representation $\boldsymbol{\bar{n}}$ of SU($n$). (31,6)(1,0) (0.2,4)[(2,1)[rep]{}]{} (7,4)[(3,1)[dim]{}]{} (17.8,4)[(7,1)[$L$]{}]{} (25.2,4)[(5,1)[$a_1,\ldots,a_L$]{}]{} (0,0)(0,1)[4]{}[(1,0)[2]{}]{} (0,0)(1,0)[3]{}[(0,1)[3]{}]{} (0,2)[(1,1)[1]{}]{} (0,1)[(1,1)[2]{}]{} (1,2)[(1,1)[3]{}]{} (1,1)[(1,1)[4]{}]{} (0,0)[(1,1)[5]{}]{} (1,0)[(1,1)[6]{}]{} (8,2)[(1,1)[1]{}]{} (10.5,1.9)[(1,1)[$\rightarrow$]{}]{} (13,2)(0,1)[2]{}[(1,0)[2]{}]{} (13,1)[(1,0)[3]{}]{} (14,0)[(1,0)[2]{}]{} (13,2)(1,0)[3]{}[(0,1)[1]{}]{} (13,1)(1,0)[3]{}[(0,1)[1]{}]{} (14,0)(1,0)[3]{}[(0,1)[1]{}]{} (13,2)[(1,1)[1]{}]{} (13,1)[(1,1)[2]{}]{} (14,2)[(1,1)[3]{}]{} (14,1)[(1,1)[4]{}]{} (14,0)[(1,1)[5]{}]{} (15,0)[(1,1)[6]{}]{} (13.5,0.5) (15.5,1.5)(0,1)[2]{} (21,2)[(1,1)[3]{}]{} (24,2.3)[(1,0)[7]{}]{} (25,2.15)(1,0)[6]{}[(0,1)[0.3]{}]{} (25.5,2.3) (29.5,2.3)(1,0)[2]{} (25,0.6)[(1,1)[$\frac{3}{2}$]{}]{} (29,0.6)[(1,1)[$\frac{11}{2}$]{}]{} (30,0.6)[(1,1)[$\frac{13}{2}$]{}]{} (31,4)(1,0) (0,2)(0,1)[2]{}[(1,0)[3]{}]{} (0,1)[(1,0)[2]{}]{} (0,0)[(1,0)[1]{}]{} (0,0)(1,0)[2]{}[(0,1)[3]{}]{} (2,1)[(0,1)[2]{}]{} (3,2)[(0,1)[1]{}]{} (0,2)[(1,1)[1]{}]{} (0,1)[(1,1)[2]{}]{} (1,2)[(1,1)[3]{}]{} (1,1)[(1,1)[4]{}]{} (0,0)[(1,1)[5]{}]{} (2,2)[(1,1)[6]{}]{} (8,2)[(1,1)[8]{}]{} (10.5,1.9)[(1,1)[$\rightarrow$]{}]{} (13,2)(0,1)[2]{}[(1,0)[3]{}]{} (13,1)[(1,0)[2]{}]{} (14,0)[(1,0)[1]{}]{} (13,2)(1,0)[4]{}[(0,1)[1]{}]{} (13,1)(1,0)[3]{}[(0,1)[1]{}]{} (14,0)(1,0)[2]{}[(0,1)[1]{}]{} (13,2)[(1,1)[1]{}]{} (13,1)[(1,1)[2]{}]{} (14,2)[(1,1)[3]{}]{} (14,1)[(1,1)[4]{}]{} (14,0)[(1,1)[5]{}]{} (15,2)[(1,1)[6]{}]{} (13.5,0.5)(2,0)[2]{} (15.5,1.5) (21,2)[(1,1)[3]{}]{} (24,2.3)[(1,0)[7]{}]{} (25,2.15)(1,0)[6]{}[(0,1)[0.3]{}]{} (25.5,2.3) (29.5,2.3)(1,0)[2]{} (25,0.6)[(1,1)[$\frac{3}{2}$]{}]{} (29,0.6)[(1,1)[$\frac{11}{2}$]{}]{} (30,0.6)[(1,1)[$\frac{13}{2}$]{}]{} (31,4)(1,0) (0,2)(0,1)[2]{}[(1,0)[3]{}]{} (0,1)[(1,0)[2]{}]{} (0,0)[(1,0)[1]{}]{} (0,0)(1,0)[2]{}[(0,1)[3]{}]{} (2,1)[(0,1)[2]{}]{} (3,2)[(0,1)[1]{}]{} (0,2)[(1,1)[1]{}]{} (0,1)[(1,1)[2]{}]{} (1,2)[(1,1)[3]{}]{} (0,0)[(1,1)[4]{}]{} (1,1)[(1,1)[5]{}]{} (2,2)[(1,1)[6]{}]{} (8,2)[(1,1)[8]{}]{} (10.5,1.9)[(1,1)[$\rightarrow$]{}]{} (13,3)[(1,0)[2]{}]{} (16,3)[(1,0)[1]{}]{} (13,2)[(1,0)[4]{}]{} (13,1)[(1,0)[3]{}]{} (14,0)[(1,0)[1]{}]{} (13,2)(1,0)[5]{}[(0,1)[1]{}]{} (13,1)(1,0)[4]{}[(0,1)[1]{}]{} (14,0)(1,0)[2]{}[(0,1)[1]{}]{} (13,2)[(1,1)[1]{}]{} (13,1)[(1,1)[2]{}]{} (14,2)[(1,1)[3]{}]{} (14,0)[(1,1)[4]{}]{} (15,1)[(1,1)[5]{}]{} (16,2)[(1,1)[6]{}]{} (13.5,0.5)(1,1)[2]{} (15.5,0.5)(0,2)[2]{} (16.5,0.5)(0,1)[2]{} (21,2)[(1,1)[6]{}]{} (24,2.3)[(1,0)[7]{}]{} (25,2.15)(1,0)[6]{}[(0,1)[0.3]{}]{} (25.5,2.3) (27.5,2.3)(1,0)[4]{} (29.5,2.8) (25,0.6)[(1,1)[$\frac{3}{2}$]{}]{} (27,0.6)[(1,1)[$\frac{7}{2}$]{}]{} (28,0.6)[(1,1)[$\frac{9}{2}$]{}]{} (29,0.6)[(1,1)[$\frac{11}{2}$]{}]{} (30,0.6)[(1,1)[$\frac{13}{2}$]{}]{} (31,4)(1,0) (0,2)(0,1)[2]{}[(1,0)[3]{}]{} (0,1)[(1,0)[2]{}]{} (0,0)[(1,0)[1]{}]{} (0,0)(1,0)[2]{}[(0,1)[3]{}]{} (2,1)[(0,1)[2]{}]{} (3,2)[(0,1)[1]{}]{} (0,2)[(1,1)[1]{}]{} (1,2)[(1,1)[2]{}]{} (0,1)[(1,1)[3]{}]{} (1,1)[(1,1)[4]{}]{} (2,2)[(1,1)[5]{}]{} (0,0)[(1,1)[6]{}]{} (8,2)[(1,1)[8]{}]{} (10.5,1.9)[(1,1)[$\rightarrow$]{}]{} (13,3)[(1,0)[2]{}]{} (16,3)[(1,0)[1]{}]{} (13,2)[(1,0)[4]{}]{} (14,1)[(1,0)[3]{}]{} (16,0)[(1,0)[1]{}]{} (13,2)(1,0)[5]{}[(0,1)[1]{}]{} (14,1)(1,0)[3]{}[(0,1)[1]{}]{} (16,0)(1,0)[2]{}[(0,1)[1]{}]{} (13,2)[(1,1)[1]{}]{} (14,2)[(1,1)[2]{}]{} (14,1)[(1,1)[3]{}]{} (15,1)[(1,1)[4]{}]{} (16,2)[(1,1)[5]{}]{} (16,0)[(1,1)[6]{}]{} (13.5,0.5)(0,1)[2]{} (14.5,0.5)(2,1)[2]{} (15.5,0.5)(0,2)[2]{} (21,2)[(1,1)[6]{}]{} (24,2.3)[(1,0)[7]{}]{} (25,2.15)(1,0)[6]{}[(0,1)[0.3]{}]{} (24.5,2.3)(1,0)[6]{} (24,0.6)[(1,1)[$\frac{1}{2}$]{}]{} (25,0.6)[(1,1)[$\frac{3}{2}$]{}]{} (26,0.6)[(1,1)[$\frac{5}{2}$]{}]{} (27,0.6)[(1,1)[$\frac{7}{2}$]{}]{} (28,0.6)[(1,1)[$\frac{9}{2}$]{}]{} (29,0.6)[(1,1)[$\frac{11}{2}$]{}]{} (31,4)(1,0) (0,2)(0,1)[2]{}[(1,0)[4]{}]{} (0,0)(0,1)[2]{}[(1,0)[1]{}]{} (0,0)(1,0)[2]{}[(0,1)[3]{}]{} (2,2)(1,0)[3]{}[(0,1)[1]{}]{} (0,2)[(1,1)[1]{}]{} (1,2)[(1,1)[2]{}]{} (0,1)[(1,1)[3]{}]{} (2,2)[(1,1)[4]{}]{} (3,2)[(1,1)[5]{}]{} (0,0)[(1,1)[6]{}]{} (8,2)[(1,1)[10]{}]{} (10.5,1.9)[(1,1)[$\rightarrow$]{}]{} (13,2)(0,1)[2]{}[(1,0)[4]{}]{} (14,1)[(1,0)[1]{}]{} (16,0)(0,1)[2]{}[(1,0)[1]{}]{} (13,2)(1,0)[5]{}[(0,1)[1]{}]{} (14,1)(1,0)[2]{}[(0,1)[1]{}]{} (16,0)(1,0)[2]{}[(0,1)[1]{}]{} (13,2)[(1,1)[1]{}]{} (14,2)[(1,1)[2]{}]{} (14,1)[(1,1)[3]{}]{} (15,2)[(1,1)[4]{}]{} (16,2)[(1,1)[5]{}]{} (16,0)[(1,1)[6]{}]{} (15.5,1.5)(1,0)[2]{} (13.5,1.5) (13.5,0.5)(1,0)[3]{} (21,2)[(1,1)[6]{}]{} (24,2.3)[(1,0)[7]{}]{} (25,2.15)(1,0)[6]{}[(0,1)[0.3]{}]{} (24.5,2.3)(1,0)[6]{} (24,0.6)[(1,1)[$\frac{1}{2}$]{}]{} (25,0.6)[(1,1)[$\frac{3}{2}$]{}]{} (26,0.6)[(1,1)[$\frac{5}{2}$]{}]{} (27,0.6)[(1,1)[$\frac{7}{2}$]{}]{} (28,0.6)[(1,1)[$\frac{9}{2}$]{}]{} (29,0.6)[(1,1)[$\frac{11}{2}$]{}]{} Yangians {#sec:yangians} ======== In this section we discuss the Yangian of sl$_n$. Let $\{I^a\}$ be an orthonormal basis with respect to a scalar product $\langle .\,,.\rangle$ of sl$_n$. We will use $$\langle X,Y\rangle =\mathrm{tr}(XY),\quad X,Y\in\mathrm{sl}_n. \label{eq:yangianslnscp}$$ For example, an orthonormal basis of sl$_2$ is then given by $I^a=\sqrt{2}S^a=\sigma^a/\sqrt{2}$ with the Pauli matrices $\sigma^a, a=1,2,3$. The operators $I^a$ fulfil the algebra $${\left[I^a,I^b\right]}=c^{abc}I^c,\quad a,b,c=1,\ldots,n^2-1, \label{eq:yangiansunalgebra}$$ where $c^{abc}$ are the structure constants, $c^{abc}=\mathrm{i}\sqrt{2}\varepsilon^{abc}$ for sl$_2$. The Yangian Y(sl$_n$) [@Drinfeld85] of sl$_n$ is the infinite-dimensional associative algebra over $\mathbb{C}$ generated by the elements $I^a$, $\mathcal{I}^a$ with defining relations $$\begin{aligned} &&\hspace{-20mm}{\left[I^a,I^b\right]}=c^{abc}I^c, \qquad{\left[I^a,\mathcal{I}^b\right]}=c^{abc}\,\mathcal{I}^c, \label{eq:yangiandefeq1}\\ &&\hspace{-20mm}\Bigl[\mathcal{I}^a,{\left[\mathcal{I}^b,I^c\right]}\Bigr]- \Bigl[I^a,{\left[\mathcal{I}^b,\mathcal{I}^c\right]}\Bigr]= \mathfrak{c}^{abcdef}\left\{I^d,I^e,I^f\right\}\!, \label{eq:yangiandefeq2}\\ &&\hspace{-20mm} \Bigl[{\left[\mathcal{I}^a,\mathcal{I}^b\right]},{\left[I^i,\mathcal{I}^j\right]}\Bigr]+ \Bigl[{\left[\mathcal{I}^i,\mathcal{I}^j\right]},{\left[I^a,\mathcal{I}^b\right]}\Bigr]= \left(\mathfrak{c}^{abcdef}c^{ijc}+\mathfrak{c}^{ijcdef}c^{abc}\right) \left\{I^d,I^e,\mathcal{I}^f\right\}\!, \label{eq:yangiandefeq3}\end{aligned}$$ where repeated indices are summed over and $$\hspace{-10mm} \mathfrak{c}^{abcdef}=\frac{1}{24}\,c^{adi}c^{bej}c^{cfk}c^{ijk},\quad \left\{X^a,X^b,X^c\right\}=\sum_{\pi\in S_3} X^{\pi(a)}X^{\pi(b)}X^{\pi(c)}.$$ Y(sl$_n$) has a Hopf algebra structure with comultiplication $\Delta: \mathrm{Y(sl}_n)\rightarrow \mathrm{Y(sl}_n)\otimes \mathrm{Y(sl}_n)$ given by $$\hspace{-10mm} \Delta(I^a)= I^a\otimes 1 + 1\otimes I^a,\quad \Delta(\mathcal{I}^a)=\mathcal{I}^a\otimes 1+1\otimes\mathcal{I}^a -\frac{1}{2}\,c^{abc}I^b\otimes I^c.\label{eq:comultiplication}$$ The counit $\epsilon: \mathrm{Y(sl}_n)\rightarrow\mathbb{C}$ and the antipode $\mathcal{S}: \mathrm{Y(sl}_n)\rightarrow \mathrm{Y(sl}_n)$ will not be used in this article, there definitions can be found in the literature [@Drinfeld85; @ChariPressley98]. The defining relations (\[eq:yangiandefeq1\])–(\[eq:yangiandefeq3\]) depend on the choice of the scalar product (\[eq:yangianslnscp\]) but, up to isomorphism, the Hopf algebra Y(sl$_n$) does not [@ChariPressley98]. We have chosen (\[eq:yangianslnscp\]) in order to match the notations of [@ChariPressley90; @ChariPressley96] for the representation theory of Y(sl$_n$). The algebra generated by the total spin (\[eq:su3-Jsymmetry\]) and rapidity operator (\[eq:su3-Upsilon\]) is recovered with the identifications $$S^a=\frac{1}{\sqrt{2}}\,I^a \mbox{ or } J^a=\frac{1}{\sqrt{2}}\,I^a,\quad \Lambda^a=\frac{1}{\sqrt{2}}\,\mathcal{I}^a, \label{eq:yangianindentification}$$ hence, the Yangian constitutes a symmetry of the HSM [@Haldane-92; @Hikami95npb]. The comultiplication defines the action of the operators (\[eq:yangianindentification\]) on two-particle states, and being a homomorphism it is consistent on three-particle states. There is another realization of the Yangian, first given by Drinfel’d in 1988 [@Drinfeld88], which will be used to set the representation theory of the Yangian in the next section. It is based on the Cartan–Weyl basis [@Cornwell84vol2] of sl$_n$, which is for $n=2$ is given in terms of the spin operators by $$H_1=2S^z,\quad X^\pm_1=S^\pm=S^x\pm\mathrm{i} S^y,$$ whereas for $n=3$ we have explicitly $$\begin{aligned} &&H_1=2J^3,\quad H_2=\sqrt{3}J^8-J^3,\\ &&X^\pm_1=I^\pm=J^1\pm\mathrm{i} J^2,\quad X^\pm_2=U^\pm=J^6\pm\mathrm{i} J^7,\quad V^\pm=J^4\pm\mathrm{i} J^5.\end{aligned}$$ The operators $I^\pm$ and $U^\pm$ are sufficient to span sl$_3$ as we can re-express the non-simple root as $V^\pm=I^\pm+U^\pm$. With the identifications $H_{i,0}=H_i$ and $X^\pm_{i,0}=X^\pm_i$ the relations of the Yangian Y(sl$_n)$ read as follows [@Drinfeld88; @ChariPressley98] ($1\le i\le n-1$, $k\in\mathbb{N}_0$): $$\begin{aligned} &&\hspace{-20mm} {\left[H_{i,k}^{\phantom{\pm}},H_{j,l}^{\phantom{\pm}}\right]}=0,\quad {\left[H_{i,0}^{\phantom{\pm}},X^\pm_{j,k}\right]}=\pm B_{ij} X^\pm_{j,k},\quad {\left[X^+_{i,k},X^-_{j,l}\right]}=\delta_{ij}H_{i,k+l}^{\phantom{\pm}},\\ &&\hspace{-20mm} {\left[H_{i,k+1}^{\phantom{\pm}},X^\pm_{j,l}\right]}- {\left[H^{\phantom{\pm}}_{i,k},X^\pm_{j,l+1}\right]}= \pm\frac{1}{2}B_{ij}\left(H_{i,k}^{\phantom{\pm}}X^\pm_{j,l}+ X^\pm_{j,l}H_{i,k}^{\phantom{\pm}}\right)\!,\\ &&\hspace{-20mm} {\left[X^\pm_{i,k+1},X^\pm_{j,l}\right]}-{\left[X^\pm_{i,k},X^\pm_{j,l+1}\right]}= \pm\frac{1}{2}B_{ij} \left(X^\pm_{i,k}X^\pm_{j,l}+X^\pm_{j,l}X^\pm_{i,k}\right)\!,\\ &&\hspace{-20mm} \Bigl[X^\pm_{i,k_1},{\left[X^\pm_{i,k_2},X^\pm_{j,l}\right]}\Bigr]+ \Bigl[X^\pm_{i,k_2},{\left[X^\pm_{i,k_1},X^\pm_{j,l}\right]}\Bigr]=0, \qquad\mathrm{for}\quad i=\pm(j+1),\\ &&\hspace{-20mm} {\left[X^\pm_{i,k},X^\pm_{j,l}\right]}=0, \qquad\mathrm{for}\quad i\neq j,\pm(j+1),\end{aligned}$$ where $B_{ii}=2$, $B_{i,i+1}=B_{i,i-1}=-1$, and $B_{ij}=0$ otherwise. For simplicity we state the isomorphism between the two realizations of Y(sl$_n$) only for the diagonal operators in the cases $n=2$ and $n=3$. For Y(sl$_2$) it is given by $$S^z \mapsto \frac{1}{2} H_{1,0},\quad \Lambda^z\mapsto \frac{1}{2} H_{1,1}+ \frac{1}{4}\left(S^+S^-+S^-S^+-H_{1,0}^2\right)\!, \label{eq:su2isomorpihsm}$$ while for Y(sl$_3$) it reads $$\begin{aligned} &&\hspace{-20mm} J^3\mapsto \frac{1}{2} H_{1,0},\quad J^8\mapsto \frac{1}{2\sqrt{3}}H_{1,0}+\frac{1}{\sqrt{3}}H_{2,0}, \label{eq:cwbj8}\\ &&\hspace{-20mm} \Lambda^3\mapsto \frac{1}{2} H_{1,1}-\frac{1}{4}H_{1,0}^2 +\frac{1}{4}\!\left(I^+I^-\!+\!I^-I^+\right) -\frac{1}{8}\!\left(U^+U^-\!+\!U^-U^+\!-\!V^+V^-\!-\!V^-V^+\right)\!, \label{eq:cwblambda3}\\ &&\hspace{-20mm} \Lambda^8\mapsto \frac{1}{2\sqrt{3}}H_{1,1}+\frac{1}{\sqrt{3}}H_{2,1} -\frac{1}{4\sqrt{3}}H_{1,0}^2-\frac{1}{2\sqrt{3}}H_{2,0}^2\nonumber\\ &&\hspace{-10mm} +\frac{\sqrt{3}}{8}\left(U^+U^-+U^-U^++V^+V^-+V^-V^+\right)\!. \label{eq:cwblambda8}\end{aligned}$$ Here, we have used the short-hand notations $S^\pm=X_{1,0}^\pm$ as well as $I^\pm=X_{1,0}^\pm$, $U^\pm=X_{2,0}^\pm$, and $V^\pm=X_{1,0}^\pm+X_{2,0}^\pm$. Representations of Yangians {#sec:repofyangians} =========================== The representation theory of Y(sl$_n$) [@ChariPressley90; @ChariPressley96; @ChariPressley98; @Molev02] is based on the existence of the evaluation homomorphism, which connects Y(sl$_n$) with the universal enveloping algebra U(sl$_n$) of sl$_n$. For all $\xi\in\mathbb{C}$, $\mathrm{ev_\xi}:\mathrm{Y(sl}_n)\to\mathrm{U(sl}_n)$ is given by $$I^a\mapsto I^a,\quad \mathcal{I}^a\mapsto\xi I^a+\frac{1}{4}\sum_{b,c=1}^{n^2-1}\mathrm{tr} \Bigl(I^a\left(I^bI^c+I^cI^b\right)\Bigr)\,I^bI^c, \label{eq:yangianevdef3}$$ where the trace is computed by representing the $I^a$ as $n\times n$ matrices. In general, given a representation of sl$_n$ one obtains a one-parameter family of Y(sl$_n$) representations via the pull-back of the evaluation homomorphism. Explicitly, if $\phi$ is a representation of sl$_n$ on $V$, $\phi_\xi=\phi\circ\mathrm{ev}_\xi$ is a representation of Y(sl$_n$) on $V$, which is called the evaluation representation with spectral parameter $\xi$. A representation $V$ of the Yangian Y(sl$_n$) is said to be highest weight, if there exists a vector $v\in V$ such that $V=\mathrm{Y(sl}_n)\,v$ and $$X_{i,k}^+ v=0, \qquad H_{i,k}^{\phantom{+}}v = d_{i,k}^{\phantom{+}}v, \label{eq:defhighetweight}$$ with an array of complex numbers $(d_{i,k})$. In this case, $v$ is called the Yangian highest weight state (YHWS) of $V$ and $(d_{i,k})$ its highest weight. As it was shown by Drinfel’d [@Drinfeld88], the irreducible highest weight representation $V$ of Y(sl$_n)$ with highest weight $(d_{i,k})$ is finite dimensional if and only if there exist monic polynomials $P_i\in\mathbb{C}[u]$, $1\le i\le n-1$ such that $$\frac{P_i(u+1)}{P_i(u)}=1+\sum_{k=0}^\infty \frac{d_{i,k}}{u^{k+1}}, \label{eq:defpolynomial}$$ in the sense that the right-hand side is a Laurent expansion of the left-hand side about $u=\infty$ [@ChariPressley96]. Hence, the Drinfel’d polynomials $P_i(u)$ classify the finite-dimensional irreducible representations of the Yangian. The $i$th fundamental representation of Y(sl$_n)$ with spectral parameter $\xi\in\mathbb{C}$ is defined as the irreducible highest weight representation with Drinfel’d polynomials $$P_i(u)=u-\xi,\qquad P_j(u)=1\quad\mathrm{for}\; j\neq i.$$ We will denote the fundamental representation of Y(sl$_2$) with Drinfel’d polynomial $P(u)=u-\xi$ by $V\left(\boldsymbol{\frac{1}{2}},\xi\right)$. It can be constructed explicitly as the pull-back of the sl$_2$ representation $\boldsymbol{\frac{1}{2}}$ under ev$_\xi$. For Y(sl$_3$) we denote by $V(\boldsymbol{3},\xi)$ and $V(\boldsymbol{\bar{3}},\xi)$ the three-dimensional representations with Drinfel’d polynomials $P_1(u)=u-\xi,\;P_2(u)=1$ and $P_1(u)=1,\;P_2(u)=u-\xi$, respectively. If $V(\boldsymbol{3},\xi)$ and $V(\boldsymbol{\bar{3}},\xi)$ are realised as evaluation representations, we obtain an additional shift in the spectral parameter due to the trace on the right-hand side of (\[eq:yangianevdef3\]). For example, $V(\boldsymbol{\bar{3}},\xi)$ is obtained as evaluation representation of the sl$_3$ representation $\boldsymbol{\bar{3}}$ with evaluation parameter $\xi+2/3$ (see \[sec:yangianappev\]). Representation theory of $Y(sl_2)$ {#sec:yangianrepsl2} ---------------------------------- In the following we denote the evaluation representation of the $(m+1)$-dimensional sl$_2$ representation $\boldsymbol{\frac{m}{2}}$ with spectral parameter $\xi\in\mathbb{C}$ by $V\left(\boldsymbol{\frac{m}{2}},\xi\right)$. The Drinfel’d polynomial of $V\left(\boldsymbol{\frac{m}{2}},\xi\right)$ is given by [@ChariPressley90] $$P(u)=\left(u-\xi+\frac{m-1}{2}\right)\left(u-\xi+\frac{m-3}{2}\right)\cdots \left(u-\xi-\frac{m-1}{2}\right)\!. \label{eq:yangpolyrep2}$$ Now, let $V_1$ and $V_2$ be two representations of Y(sl$_2$). The action of $X\in\mathrm{Y(sl}_2)$ on the tensor product $V_1\otimes V_2$ is given by $\Delta(X)$, where $\Delta$ is the comultiplication (\[eq:comultiplication\]). We stress that due to the last term of $\Delta(\mathcal{I}^a)$ $V_1$ and $V_2$ are intertwined. In particular, as $\Delta$ is not commutative, $V_1\otimes V_2$ and $V_2\otimes V_1$ are not isomorphic in general. In all cases, however, if $v_1^+$ and $v_2^+$ are the YHWS’s of $V_1$ and $V_2$, respectively, the vector $v_1^+\otimes v_2^+$ will be the YHWS of $V_1\otimes V_2$. The action of Y(sl$_2$) on $r$-fold tensor products is defined by repeated application of $\Delta$. The central theorem on the tensor product $V=V\left(\boldsymbol{\frac{m_1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{m_2}{2}},\xi_2\right)$ is due to Chari and Pressley [@ChariPressley90; @Tarasov85]: 1. If $\|\xi_1-\xi_2\|\neq\frac{m_1+m_2}{2}-p+1$ for all $p\in\mathbb{N}$ with $0<p\le\mathrm{min}(m_1,m_2)$, then $V$ is irreducible as Y(sl$_2$) representation. 2. If $\xi_2-\xi_1=\frac{m_1+m_2}{2}-p+1$ for some $p\in\mathbb{N}$ with $0<p\le\mathrm{min}(m_1,m_2)$, then $V$ has a unique proper Y(sl$_2)$ subrepresentation $W$ generated by the highest weight vector of $V$. Explicitly, we have $$\textstyle W\cong V\left(\boldsymbol{\frac{p-1}{2}},\xi_1+\frac{m_1-p+1}{2}\right)\otimes V\left(\boldsymbol{\frac{m_1+m_2-p+1}{2}},\xi_2-\frac{m_1-p+1}{2}\right)$$ and, as representation of sl$_2$, $W\cong \boldsymbol{\frac{m_1+m_2}{2}}\oplus\cdots\oplus \boldsymbol{\frac{m_1+m_2-2p+2}{2}}$. The third case, $\xi_1-\xi_2=\frac{m_1+m_2}{2}-p+1$ for some $p\in\mathbb{N}$ with $0<p\le\mathrm{min}(m_1,m_2)$, was also discussed in [@ChariPressley90] but will not be used in the study of the HSM. In order to illustrate the two different situations, we consider the simplest non-trivial case $V=V\left(\boldsymbol{\frac{1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_2\right)$ for $\xi_1<\xi_2$. Clearly, regarded as representation of sl$_2$ $V$ it decomposes as $V\cong\boldsymbol{1}\oplus\boldsymbol{0}$. If $\xi_2-\xi_1\neq 1$, then $V$ is irreducible as Y(sl$_2$) representation. One can always find an operator in Y(sl$_2$) which yields a singlet state when acting on a triplet state and vice versa. If $\xi_2-\xi_1=1$, however, $V$ contains a proper Y(sl$_2$) subrepresentation $W\cong\boldsymbol{1}$ generated by the YHWS. In particular, there exists no operator in Y(sl$_2$) which yields the singlet state when acting on a triplet state. However, it is possible to obtain a triplet state when acting on the spin singlet state with an appropriate operator. Hence, $V$ is not a direct sum of irreducible Y(sl$_2$) representations. The highest weight of $V=V\left(\boldsymbol{\frac{m_1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{m_2}{2}},\xi_2\right)$ is obtained from its Drinfel’d polynomial, which is in the irreducible case simply given by the product of the original polynomials [@ChariPressley90]. In the reducible case the highest weight is determined by the highest weight component of $V$ using (\[eq:yangpolyrep2\]). Representation theory of $Y(sl_3)$ {#sec:yangianrepsl3} ---------------------------------- The representation theory of Y(sl$_3$) is not known in the same detail as it is for Y(sl$_2$). Although there exist irreducibility criteria for tensor products of evaluation representations of Y(sl$_n$) [@Molev02], an explicit description of the irreducible subrepresentation of such tensor products including its spectral parameter is missing. We will restrict ourselves here to tensor products of two fundamental representations of Y(sl$_3$). There are three different situations [@ChariPressley96]: 1. $V=V(\boldsymbol{3},\xi_1)\otimes V(\boldsymbol{3},\xi_2)$ is reducible as Y(sl$_3$) representation if and only if $\xi_1-\xi_2=\pm 1$. If $\xi_2-\xi_1=1$, then $V$ has a proper Y(sl$_3$) subrepresentation $W$ generated by the highest weight vector of $V$ and, as representation of sl$_3$, $W\cong\boldsymbol{6}$. 2. $V=V(\boldsymbol{3},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_2)$ (or $V=V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{3},\xi_2)$) is reducible as Y(sl$_3$) representation if and only if $\xi_1-\xi_2=\pm 3/2$. If $\xi_2-\xi_1=3/2$, then $V$ has a proper Y(sl$_3$) subrepresentation $W$ generated by the highest weight vector of $V$ and, as representation of sl$_3$, $W\cong\boldsymbol{8}$. 3. $V=V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_2)$ is reducible as Y(sl$_3$) representation if and only if $\xi_1-\xi_2=\pm1$. If $\xi_1-\xi_2=1$, then $V$ has a proper Y(sl$_3$) subrepresentation $W$ not containing the highest weight vector of $V$ and, as representation of sl$_3$, $W\cong\boldsymbol{3}$. If $\xi_2-\xi_1=1$, then $V$ has a proper Y(sl$_3$) subrepresentation $W$ generated by the highest weight vector of $V$ and, as representation of sl$_3$, $W\cong\boldsymbol{\bar{6}}$. If the tensor product $V=V_1\otimes V_2$ is irreducible, the Drinfel’d polynomials of $V$ are given by the products of the polynomials of $V_1$ and $V_2$ [@ChariPressley98]. Furthermore, we show in \[sec:yangianappsc\] that the proper Y(sl$_3$) subrepresentation $W$ of $V(\boldsymbol{\bar{3}},\xi)\otimes V(\boldsymbol{\bar{3}},\xi-1)$ is given by $W\cong V\left(\boldsymbol{3},\xi-\frac{1}{2}\right)$. Spinons and representations of Y(sl$_{\boldsymbol{2}}$) {#sec:yangiansu2} ======================================================= As mentioned above it is well-known [@Haldane-92; @Hikami95npb] that the SU(2) HSM possesses the Yangian symmetry Y(sl$_2$) and therefore that its Hilbert space decomposes into irreducible representations of Y(sl$_2$). It is also known [@GS07] how to built up the Hilbert space of the HSM by non-interacting spinon excitations. In this section we will study the relation between these many-spinon states and representations of Y(sl$_2$). One-spinon states ----------------- We first derive the relation between the one-spinon states and the fundamental representations of Y(sl$_2$). Consider a chain with an odd number of sites $N$. The individual one-spinon momenta are given by [@Haldane91prl1; @Bernevig-01prb] $$p = \frac{\pi}{2}-\frac{2\pi}{N}\left(\mu+\frac{1}{4}\right)\!, \quad 0\le \mu\le\frac{N-1}{2}, \label{eq:yangianreponespinonmomentum}$$ where we have assumed $N-1$ to be divisible by four, and thereby restricted the momentum to $-\pi/2\le p\le\pi/2$. The up-spin spinons are YHWS’s (they are annihilated by $S^+,\Lambda^+,\ldots$), their spin currents are given by [@Bernevig-01prb] $$\Lambda^z{\left\|{\uparrow}\right\rangle}=\left(\frac{N-1}{4}-\mu\right){\left\|{\uparrow}\right\rangle}\!. \label{eq:yangiansc1}$$ Here ${\left\|{\uparrow}\right\rangle}$ denotes the state with one up-spin spinon. On the other hand the one-spinon states are represented by tableaux of the form (25,3)(0,0.5) (0,1)(1,0)[10]{}[(0,1)[2]{}]{} (0,2)(0,1)[2]{}[(1,0)[9]{}]{} (0,1)[(1,0)[3]{}]{} (4,1)[(1,0)[5]{}]{} (3.5,1.5) (3,0)[(1,1)[$a$]{}]{} (12,1.6)[$\displaystyle a=N-2\mu,\quad 0\le \mu\le\frac{N-1}{2}$,]{} where we omit the superfluous numbers in the boxes of the tableaux. Note that (\[eq:yangianreponespinonmomentum\]) is recovered using (\[eq:singlespinonmom\])–(\[eq:pzeroezero\]). Now, (\[eq:su2isomorpihsm\]) together with (\[eq:yangiansc1\]) yields $$\hspace{-10mm} H_{1,1}{\left\|{\uparrow}\right\rangle}=\left(2\Lambda^z- \frac{1}{2}\Bigl[S^+S^-+S^-S^+-4(S^z)^2\Bigr]\right){\left\|{\uparrow}\right\rangle} =\left(a-\frac{N+1}{2}\right){\left\|{\uparrow}\right\rangle}\!,$$ where the term in squared brackets vanishes as the spinon possesses spin $S=1/2$. Hence, individual spinons transform under the Y(sl$_2$) representation $V\left(\boldsymbol{\frac{1}{2}},\xi\right)$, where the spectral parameter $\xi$ is in terms of the SMN given by $$\xi=a-\frac{N+1}{2},\quad -\frac{N-1}{2}\le\xi\le\frac{N-1}{2}. \label{eq:yangianrepxiasu2}$$ Two-spinon states {#sec:yangiantwospinons} ----------------- Let us consider two spin-polarised spinons represented by the tableau (25,4.5)(0,-0.5) (0,1)(1,0)[10]{}[(0,1)[2]{}]{} (0,2)(0,1)[2]{}[(1,0)[9]{}]{} (0,1)[(1,0)[3]{}]{} (4,1)[(1,0)[2]{}]{} (7,1)[(1,0)[2]{}]{} (3.5,1.5)(3,0)[2]{} (3.1,-0.2)[(1,1)[$a_1$]{}]{} (4.6,-0.2)[(1,1)[$<$]{}]{} (6.1,-0.2)[(1,1)[$a_2$]{}]{} (12,2.7)[$a_1=N-2\mu-1,\quad a_2=N-2\nu,$]{} (12,0)[$\displaystyle 0\le\nu\le\mu\le\frac{N-2}{2}$.]{} Note that, as $N$ is even, $a_1$ is always odd and $a_2$ is always even. There are two fundamentally different cases. If $a_2-a_1>1$, there exists a two-spinon singlet with the same SMN’s (and hence the same energy), whereas for $a_2-a_1=1$ there exists no corresponding singlet. Graphically we have (28,3.5) (6,1)(1,0)[10]{}[(0,1)[2]{}]{} (6,2)(0,1)[2]{}[(1,0)[9]{}]{} (6,1)[(1,0)[3]{}]{} (10,1)[(1,0)[2]{}]{} (13,1)[(1,0)[2]{}]{} (9.5,1.5)(3,0)[2]{} (0,1.5)[(3,1)[$a_2-a_1>1:$]{}]{} (16,1.5)[(2,1)[and]{}]{} (19,1)(1,0)[10]{}[(0,1)[2]{}]{} (19,3)[(1,0)[6]{}]{} (26,3)[(1,0)[2]{}]{} (19,2)[(1,0)[9]{}]{} (19,1)[(1,0)[3]{}]{} (23,1)[(1,0)[5]{}]{} (22.5,1.5)(3,1)[2]{} (28,3)(0,0.5) (6,1)(1,0)[4]{}[(0,1)[2]{}]{} (11,1)(1,0)[5]{}[(0,1)[2]{}]{} (10,2)[(0,1)[1]{}]{} (6,2)(0,1)[2]{}[(1,0)[9]{}]{} (6,1)[(1,0)[3]{}]{} (11,1)[(1,0)[4]{}]{} (9.5,1.5)(1,0)[2]{} (0,1.5)[(3,1)[$a_2-a_1=1:$]{}]{} (15.8,1.5)[(3,1)[only.]{}]{} This can be understood by applying the representation theory of Y(sl$_2$). In both cases the two spinons transform under the product representation $V= V\left(\boldsymbol{\frac{1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_2\right)$, where the spectral parameters are given by $\xi_{1,2}=a_{1,2}-(N+1)/2$, respectively. In the first case we have $\xi_2-\xi_1>1$, hence by Sec. \[sec:yangianrepsl2\].i $V$ is irreducible as Y(sl$_2$) representation. As sl$_2$ representation we have $V\cong\boldsymbol{1}\oplus\boldsymbol{0}$, the triplet and singlet represented by the two tableaux shown above. $V$ is generated by its YHWS, which is the spin-polarised two-spinon state ${\left\|{\uparrow}{\uparrow}\right\rangle}={\left\|{\uparrow}\right\rangle}\otimes{\left\|{\uparrow}\right\rangle}$. In particular, the operator $\Lambda^zS^-\in\mathrm{Y(sl}_2)$ yields the two-spinon singlet state when acting on the YHWS, $\Lambda^zS^-{\left\|{\uparrow}{\uparrow}\right\rangle}\propto{\left\|{\uparrow}{\downarrow}\right\rangle}-{\left\|{\downarrow}{\uparrow}\right\rangle}$, while leaving the individual spinon momenta and hence the energy unchanged. In the second case we have $\xi_2-\xi_1=1$, and by Sec. \[sec:yangianrepsl2\].ii $V$ is reducible. The YHWS of $V$, which is again ${\left\|{\uparrow}{\uparrow}\right\rangle}$, generates the proper Y(sl$_2$) subrepresentation $W\cong\boldsymbol{1}$, only the triplet states are generated by ${\left\|{\uparrow}{\uparrow}\right\rangle}$ and, in particular, it is $\Lambda^zS^-{\left\|{\uparrow}{\uparrow}\right\rangle}=0$. This is reflected by the existence of only one tableau if the SMN’s fulfil $a_2-a_1=1$, and is consistent with results drawn from the asymptotic Bethe Ansatz for the HSM [@Haldane-92; @Bernard-93] as well as conformal field theory spectra [@Bernard-94]. The loss of the singlet, its non-existence in the spinon Hilbert space, is a manifestation of the fractional statistics of the spinons. The momentum spacings for two spinons with individual momenta $p_1$ and $p_2$ ($p_2>p_1$) are in general given by $p_2-p_1=2\pi(1/2+\ell)/N$, $\ell\in\mathbb{N}_0$ [@GS05prb]. When the spinons occupy adjacent momenta, $p_2-p_1=\pi/N$, only the triplet exists, which is mathematically implemented by the requirement of irreducibility under Y(sl$_2$) transformations. In analogy, the Pauli principle enforces two electrons with identical momenta to form a spin singlet, whereas otherwise a spin triplet exists as well. The difference between electrons and spinons is, however, that the wave function of free electrons factorises in a spin part, transforming under SU(2), as well as a momentum part, transforming under (lattice) translations; the product of both has to be antisymmetric under permutations. In contrast we cannot factorise spin and momentum of the spinons, as both are incorporated in the representations of Y(sl$_2$) (the lattice translations are implemented as shifts of the spectral parameter $\xi$). It seems that this entanglement of spin and momentum makes it impossible to implement the fractional statistics by the requirement of a definite transformation law under permutations of the spinons. In fact, this requirement is replaced by the postulate of irreducibility under Yangian transformations. The spin current of the polarised two-spinon state ${\left\|{\uparrow}{\uparrow}\right\rangle}$ is easily obtained from Drinfel’d polynomial of the irreducible subrepresentation of $V\left(\boldsymbol{\frac{1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_2\right)$, which is given by $P(u)=(u-\xi_1)(u-\xi_2)$. Hence, with (\[eq:defhighetweight\]) and (\[eq:defpolynomial\]) we find $$H_{1,1}{\left\|{\uparrow}{\uparrow}\right\rangle}=(\xi_1+\xi_2+1){\left\|{\uparrow}{\uparrow}\right\rangle}=(N-2\mu-2\nu-1){\left\|{\uparrow}{\uparrow}\right\rangle}\!,$$ and with (\[eq:su2isomorpihsm\]) we obtain the physical spin current $$\Lambda^z{\left\|{\uparrow}{\uparrow}\right\rangle}=\left(\frac{N-2}{2}-\mu-\nu\right){\left\|{\uparrow}{\uparrow}\right\rangle}\!,$$ which equals the result obtained using explicit wave functions [@Bernevig-01prb]. Many-spinon states {#sec:yangianmanyspinons} ------------------ If three spinons are present, there are three different cases, which are graphically represented by (28,4) (0,1)(1,0)[6]{}[(0,1)[2]{}]{} (6,2)[(0,1)[1]{}]{} (0,2)(0,1)[2]{}[(1,0)[6]{}]{} (0,1)(2,0)[3]{}[(1,0)[1]{}]{} (1.5,1.5)(2,0)[3]{} (1.1,-0.2)[(1,1)[$a_1$]{}]{} (3.1,-0.2)[(1,1)[$a_2$]{}]{} (5.1,-0.2)[(1,1)[$a_3$]{}]{} (-1.5,2.5)[(1,1)[(i)]{}]{} (11,1)(1,0)[2]{}[(0,1)[2]{}]{} (14,1)(1,0)[3]{}[(0,1)[2]{}]{} (13,2)(4,0)[2]{}[(0,1)[1]{}]{} (11,2)(0,1)[2]{}[(1,0)[6]{}]{} (11,1)[(1,0)[1]{}]{} (14,1)[(1,0)[2]{}]{} (12.5,1.5)(1,0)[2]{} (16.5,1.5) (12.1,-0.2)[(1,1)[$a_1$]{}]{} (13.2,-0.2)[(1,1)[$a_2$]{}]{} (16.1,-0.2)[(1,1)[$a_3$]{}]{} (9.35,2.5)[(1,1)[(ii)]{}]{} (22,1)(1,0)[2]{}[(0,1)[2]{}]{} (26,1)(1,0)[3]{}[(0,1)[2]{}]{} (24,2)(1,0)[2]{}[(0,1)[1]{}]{} (22,2)(0,1)[2]{}[(1,0)[6]{}]{} (22,1)[(1,0)[1]{}]{} (26,1)[(1,0)[2]{}]{} (23.5,1.5)(1,0)[3]{} (23.1,-0.2)[(1,1)[$a_1$]{}]{} (24.1,-0.2)[(1,1)[$a_2$]{}]{} (25.2,-0.2)[(1,1)[$a_3$]{}]{} (20.2,2.5)[(1,1)[(iii)]{}]{} In all three cases we have to determine the irreducible subrepresentation of $V=V\left(\boldsymbol{\frac{1}{2}},\xi_1\right)\linebreak \otimes V\left(\boldsymbol{\frac{1}{2}},\xi_2\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_3\right)$, where the spectral parameters are given by $\xi_i=a_i-(N+1)/2$. In the first case $V$ is irreducible and generated by its YHWS ${\left\|{\uparrow}{\uparrow}{\uparrow}\right\rangle}$. As sl$_2$ representation we find $\boldsymbol{\frac{3}{2}}\oplus\boldsymbol{\frac{1}{2}} \oplus\boldsymbol{\frac{1}{2}}$, which is the complete eight-dimensional space $\boldsymbol{\frac{1}{2}}\otimes\boldsymbol{\frac{1}{2}} \otimes\boldsymbol{\frac{1}{2}}$. The $\boldsymbol{\frac{3}{2}}$ is given by the tableau (i) above; the tableaux representing the two $\boldsymbol{\frac{1}{2}}$’s are obtained from (i) by moving either the second or the third spinon to the first row. In the second case we have $\xi_2-\xi_1=1$ and deduce using Sec. \[sec:yangianrepsl2\].ii that the irreducible subrepresentation of $V\left(\boldsymbol{\frac{1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_2\right)$ is $V\left(\boldsymbol{1},\xi_1+\frac{1}{2}\right)$. The remaining tensor product $V\left(\boldsymbol{1},\xi_1+\frac{1}{2}\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_3\right)$ is irreducible, and as sl$_2$ representation we obtain $\boldsymbol{\frac{3}{2}}\oplus\boldsymbol{\frac{1}{2}}$, which is only six-dimensional. The loss of one $\boldsymbol{\frac{1}{2}}$ is reflected by the fact that the second spinon in the tableau (ii) is fixed to the lower row. Note that this result is not affected if $a_2$ and $a_3$ were adjacent instead of $a_1$ and $a_2$, although the specific values of the spectral parameters will change. In the third case the irreducible subrepresentation of $V\left(\boldsymbol{\frac{1}{2}},\xi_1\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\xi_2\right)$ is again given by $V\left(\boldsymbol{1},\xi_1+\frac{1}{2}\right)$, however, this time the remaining tensor product is reducible as well; and its irreducible subrepresentation is given by $V\left(\boldsymbol{\frac{3}{2}},\xi_1+1\right)$. As sl$_2$ representation we only have $\boldsymbol{\frac{3}{2}}$ which is represented by the tableau (iii). To give a more general examples let us first consider a six-site chain and the four-spinon tableau (6,2)(0,1) (0,2)(1,0)[6]{}[(0,1)[1]{}]{} (2,1)(1,0)[2]{}[(0,1)[1]{}]{} (0,2)(0,1)[2]{}[(1,0)[5]{}]{} (2,1)[(1,0)[1]{}]{} (0.5,1.5)(1,0)[2]{} (3.5,1.5)(1,0)[2]{} The spin-polarised state in this multiplet, ${\left\|{\uparrow}{\uparrow}{\uparrow}{\uparrow}\right\rangle}$, generates the irreducible subrepresentation of $$\textstyle V\left(\boldsymbol{\frac{1}{2}},-\frac{5}{2}\right)\otimes V\left(\boldsymbol{\frac{1}{2}},-\frac{3}{2}\right) \otimes V\left(\boldsymbol{\frac{1}{2}},\frac{3}{2}\right)\otimes V\left(\boldsymbol{\frac{1}{2}},\frac{5}{2}\right),$$ which is given by $V(\boldsymbol{1},-2)\otimes V(\boldsymbol{1},2)$. As sl$_2$ representation this is given by $\boldsymbol{2}\oplus\boldsymbol{1}\oplus\boldsymbol{0}$, which is represented by the tableaux (25,2.5)(0,1.2) (0,2)(1,0)[6]{}[(0,1)[1]{}]{} (2,1)(1,0)[2]{}[(0,1)[1]{}]{} (0,2)(0,1)[2]{}[(1,0)[5]{}]{} (2,1)[(1,0)[1]{}]{} (0.5,1.5)(1,0)[2]{} (3.5,1.5)(1,0)[2]{} (9,2)(1,0)[6]{}[(0,1)[1]{}]{} (11,1)(1,0)[3]{}[(0,1)[1]{}]{} (9,2)[(1,0)[5]{}]{} (9,3)[(1,0)[3]{}]{} (13,3)[(1,0)[1]{}]{} (11,1)[(1,0)[2]{}]{} (9.5,1.5)(1,0)[2]{} (12.5,2.5)(1,-1)[2]{} (18,2)(1,0)[4]{}[(0,1)[1]{}]{} (20,1)(1,0)[4]{}[(0,1)[1]{}]{} (18,2)[(1,0)[5]{}]{} (18,3)(2,-2)[2]{}[(1,0)[3]{}]{} (18.5,1.5)(1,0)[2]{} (21.5,2.5)(1,0)[2]{} In the same way we can analyse the space generated by the YHWS of the seven-spinon tableau (15,2)(0,1.2) (0,2)(1,0)[13]{}[(0,1)[1]{}]{} (0,2)(0,1)[2]{}[(1,0)[12]{}]{} (0,1)(1,0)[2]{}[(0,1)[1]{}]{} (3,1)(1,0)[2]{}[(0,1)[1]{}]{} (7,1)(1,0)[3]{}[(0,1)[1]{}]{} (10,1)(1,0)[2]{}[(0,1)[1]{}]{} (0,1)[(1,0)[1]{}]{} (3,1)[(1,0)[1]{}]{} (7,1)[(1,0)[2]{}]{} (10,1)[(1,0)[1]{}]{} (1.5,1.5)(1,0)[2]{} (4.5,1.5)(1,0)[3]{} (9.5,1.5)(2,0)[2]{} where $N=17$. We couple adjacent spinons according to Sec. \[sec:yangianrepsl2\].ii, and find the irreducible subrepresentation to be $$\textstyle V\left(\boldsymbol{1},-\frac{11}{2}\right)\otimes V\left(\boldsymbol{\frac{3}{2}},-1\right)\otimes V\left(\boldsymbol{\frac{1}{2}},5\right)\otimes V\left(\boldsymbol{\frac{1}{2}},8\right),$$ which as sl$_2$ representation reads $$\textstyle \boldsymbol{1}\otimes\boldsymbol{\frac{3}{2}}\otimes \boldsymbol{\frac{1}{2}}\otimes\boldsymbol{\frac{1}{2}} \;=\; \boldsymbol{\frac{7}{2}}\oplus\boldsymbol{\frac{5}{2}} \oplus\boldsymbol{\frac{5}{2}} \oplus\boldsymbol{\frac{5}{2}}\oplus\boldsymbol{\frac{3}{2}} \oplus\boldsymbol{\frac{3}{2}} \oplus\boldsymbol{\frac{3}{2}}\oplus\boldsymbol{\frac{3}{2}} \oplus\boldsymbol{\frac{1}{2}} \oplus\boldsymbol{\frac{1}{2}}\oplus\boldsymbol{\frac{1}{2}}.$$ The corresponding tableaux are easily constructed using that the first, second, and fifth spinon are fixed to the lower row, and the fourth spinon can only move to the upper row if the third one does. The general scheme works as follows. Any spin-polarised $m$-spinon state is represented by a tableau with all spinons in the second row. The individual spinon momenta are given in terms of the SMN’s $a_i$. The space generated by this YHWS under the action of Y(sl$_2$) is the irreducible subrepresentation $W$ of the tensor product $$\textstyle V=\,\bigotimes_{i=1}^m V\left(\boldsymbol{\frac{1}{2}},\xi_i\right),\quad \displaystyle\xi_i=a_i-\frac{N+1}{2}, \label{eq:yangiangeneralsu2product}$$ where the $\xi_i$’s have ascending order. In order to construct $W$ one first determines the irreducible subrepresentations of all partial products in (\[eq:yangiangeneralsu2product\]) which have consecutive spectral parameters $\xi_{i+1}-\xi_i=1$ using Sec. \[sec:yangianrepsl2\].ii. (Note that we can without loss of generality begin with these products as the comultiplication (\[eq:comultiplication\]) is associative.) The remaining tensor product is then irreducible by repeated application of Sec. \[sec:yangianrepsl2\].i (for a proof see Ref. ). The sl$_2$ contents is determined by straightforward calculation. To sum up, spinons in the HSM transform under the Y(sl$_2$) representation $V\left(\boldsymbol{\frac{1}{2}},\xi\right)$, where the spectral parameter $\xi$ is via (\[eq:yangianrepxiasu2\]) and (\[eq:singlespinonmom\]) directly connected to the spinon momentum. All $m$-spinon states with given individual momenta $p_1,\dots,p_m$ are generated by the YHWS of (\[eq:yangiangeneralsu2product\]), meaning that they span the irreducible subrepresentation $W$. The complete Hilbert space is the direct sum of these subspaces. From a mathematical point of view the tableau formalism [@GS07] hence provides an algorithm to determine the sl$_2$ content of the irreducible subrepresentation of a tensor product of fundamental Y(sl$_2$) representations (\[eq:yangiangeneralsu2product\]) with increasing spectral parameters (the restriction to integer or half-integer spectral parameters $\xi_i$ is no limitation, since all $\xi_i$’s can be shifted simultaneously and tensor products where the spacings $\xi_j-\xi_i$ are not integers are irreducible [@ChariPressley90]). Colorons and representations of Y(sl$_{\boldsymbol{3}}$) {#sec:yangiansu3} ======================================================== In this section we will investigate the relation between the Y(sl$_3$) symmetry of the SU(3) HSM and its coloron excitations. One-coloron states ------------------ Consider a chain with $N=3M-1$, $M\in\mathbb{N}$, sites. Then the one-coloron momenta are given by [@SG05epl] $$p=\frac{4\pi}{3} -\frac{2\pi}{N}\left(\mu+\frac{1}{3}\right)\!, \quad 0\le \mu\le (N-2)/3. \label{eq:yangiancoloronmomentum}$$ The SU(3) spin (or colour) currents of a yellow coloron ${\left\|\y\right\rangle}$, which is a Yangian lowest weight state, are $$\frac{1}{\sqrt{3}}\Lambda^3{\left\|\y\right\rangle}= \Lambda^8{\left\|\y\right\rangle}=-\frac{\sqrt{3}}{2}\left(\frac{N-2}{6}-\mu\right){\left\|\y\right\rangle}\!.$$ In order to apply the representation theory of Y(sl$_3$) it will be appropriate to work with YHWS’s, that is magenta colorons ${\left\|\m\right\rangle}$, instead. As the fundamental representation $V(\boldsymbol{\bar{3}},\xi)$ of Y(sl$_3$) can be explicitly realized as evaluation representation (see \[sec:yangianappev\]), we obtain the spin currents of ${\left\|\m\right\rangle}$ to be $$\Lambda^3{\left\|\m\right\rangle}=0,\quad \Lambda^8{\left\|\m\right\rangle}=\sqrt{3}\left(\frac{N-2}{6}-\mu\right){\left\|\m\right\rangle}\!.$$ On the other hand a single coloron is represented by the tableau (25,3.5)(0,1) (0,1)(1,0)[10]{}[(0,1)[3]{}]{} (0,2)(0,1)[3]{}[(1,0)[9]{}]{} (0,1)[(1,0)[3]{}]{} (4,1)[(1,0)[5]{}]{} (3.5,1.5) (3,0)[(1,1)[$a$]{}]{} (12,2)[$\displaystyle a=N-3\mu-\frac{1}{2},\quad 0\le\mu\le\frac{N-2}{3}$.]{} The spectral parameter of $V(\boldsymbol{\bar{3}},\xi)$ is determined from the eigenvalue of $H_{2,1}$ when acting on the YHWS ${\left\|\m\right\rangle}$. Using (\[eq:cwblambda8\]) together with $H_{1,1}{\left\|\m\right\rangle}=0$ we find $H_{2,1}{\left\|\m\right\rangle}=(\sqrt{3}\Lambda^8-1/4){\left\|\m\right\rangle}=(a-(2N+3)/4){\left\|\m\right\rangle}$. Hence, colorons transform under the Y(sl$_3$) representation $V(\boldsymbol{\bar{3}},\xi)$ with spectral parameter $$\xi=a-\frac{2N+3}{4},\quad -\frac{2N-3}{4}\le\xi\le\frac{2N-5}{4}. \label{eq:yangianxidefinitionsu3}$$ Note that although the allowed values for $\xi$ are not symmetrically distributed around zero, the eigenvalues of the physical spin current $\Lambda^8$ are. Two-coloron states {#sec:yangiantwocolorons} ------------------ Compared to the many-spinon states discussed above, the effect of the fractional statistics on many-coloron states is rather complicated. We will discuss in this and the next sections how the requirement of irreducibility under Y(sl$_3$) transformations yields several restrictions on the allowed SU(3) representations for many-coloron states. Let us first consider two colorons with identical colours like ${\left\|\m\m\right\rangle}={\left\|\m\right\rangle}\otimes{\left\|\m\right\rangle}$. The individual coloron momenta $p_1$ and $p_2$ with $p_2>p_1$ are spaced according to $p_2-p_1=2\pi(2/3+\ell)/N$, $\ell\in\mathbb{N}_0$ (see (\[eq:twospinonmomenta\])). Furthermore, for all pairs of momenta satisfying this condition the SU(3) spin takes the values $\boldsymbol{\bar{3}}\otimes\boldsymbol{\bar{3}}= \boldsymbol{\bar{6}}\oplus\boldsymbol{3}$, which is graphically reflected by the two tableaux (40,4)(0,0.5) (2,1)(1,0)[5]{}[(0,1)[3]{}]{} (7,2)[(0,1)[2]{}]{} (2,2)(0,1)[3]{}[(1,0)[5]{}]{} (2,1)[(1,0)[1]{}]{} (4,1)[(1,0)[2]{}]{} (3.5,1.5)(3,0)[2]{} (3.1,-0.2)[(1,1)[$a_1$]{}]{} (4.6,-0.2)[(1,1)[$<$]{}]{} (6.1,-0.2)[(1,1)[$a_2$]{}]{} (10,2)[(2,1)[and always]{}]{} (15,1)(1,0)[5]{}[(0,1)[3]{}]{} (20,1)(0,2)[2]{}[(0,1)[1]{}]{} (15,2)(0,1)[3]{}[(1,0)[5]{}]{} (15,1)[(1,0)[1]{}]{} (17,1)[(1,0)[3]{}]{} (16.5,1.5)(3,1)[2]{} (22,3.5)[$\textstyle a_1=N-3\mu-\frac{5}{2},\quad a_2=N-3\nu-\frac{1}{2},$]{} (22,0.8)[$\displaystyle 0\le\nu\le\mu\le\frac{N-4}{3}$.]{} We note that $a_2-a_1\ge 2$ even if the colorons occupy adjacent columns, and that the YHWS ${\left\|\m\m\right\rangle}$ belongs to the left tableau. In order to determine the space generated by ${\left\|\m\m\right\rangle}$ we have to investigate the tensor product $V=V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_2)$, where $\xi_{1,2}=a_{1,2}-(2N+3)/4$, respectively. By application of Sec. \[sec:yangianrepsl3\].iii, $V$ is irreducible. As sl$_3$ representation we find $V\cong\boldsymbol{\bar{6}}\oplus\boldsymbol{3}$, where the $\boldsymbol{\bar{6}}$ is represented by the left tableau above and the $\boldsymbol{3}$ by the right tableau. The spin currents of ${\left\|\m\m\right\rangle}$ are obtained from the Drinfel’d polynomials of $V$, $P_1(u)=1$ and $P_2(u)=(u-\xi_1)(u-\xi_2)$, they equal the results derived using explicit wave functions [@SG05epl]. There are fundamentally different two-coloron states, namely the ones represented by tableaux like (22,4.5)(0,0) (0,1)(1,0)[6]{}[(0,1)[3]{}]{} (0,3)(0,1)[2]{}[(1,0)[5]{}]{} (0,1)(0,1)[2]{}[(1,0)[1]{}]{} (2,1)(0,1)[2]{}[(1,0)[3]{}]{} (1.5,1.5)(0,1)[2]{} (0.5,0)[$a_1, a_2$]{} (9,3.3)[$a_1=N-3\mu+\frac{1}{2},\;a_2=N-3\mu-\frac{1}{2}$,]{} (9,0.5)[$\displaystyle 0\le \mu\le\frac{N-1}{3}$.]{} We have chosen the SMN’s to satisfy $a_1=a_2+1$, which is supported by the following consideration: We rewrite the momentum spacing as $\Delta p=p_2-p_1=2\pi(g+\ell)/N$, $\ell\in\mathbb{N}_0$, where $g$ denotes the statistical parameter of the colorons. With the assignment $a_1=a_2+1$ we obtain (we keep the relations $p_i\leftrightarrow a_i$) $g=-1/3$ for the preceding tableau. Moreover, the momentum spacings for the left two-coloron tableau above are also given by $p_2-p_1=2\pi(-1/3+\ell)/N$ if $\ell$ takes the values $\ell\ge 1$. Hence, we obtain $g=-1/3$ for all two-coloron states where the SU(3) spins of the colorons are coupled antisymmetrically, all states represented by tableaux where the two colorons occupy different rows. The finding $g=2/3$ for colour-polarised colorons (in general symmetrically coupled) and $g=-1/3$ for colorons with different colours (in general antisymmetrically coupled) is also consistent with what we find by naive state counting. A negative mutual exclusion statistics was also deduced from the dynamical spin susceptibility of the SU(3) HSM calculated by Yamamoto [@Yamamoto-00prl], and observed in conformal field theory spectra analysed by Schoutens [@Schoutens97]. As a consequence, two colorons occupying the same column transform under the Y(sl$_3$) representation $V=V(\boldsymbol{\bar{3}},\xi)\otimes V(\boldsymbol{\bar{3}},\xi-1)$ with $\xi=a_1-(2N+3)/4$. By Sec. \[sec:yangianrepsl3\].iii, $V$ is reducible and the irreducible subrepresentation $W$ does not contain the YHWS of $V$ (which is ${\left\|\m\m\right\rangle}$). As sl$_3$ representation we have $W\cong\boldsymbol{3}$, the colorons are coupled antisymmetrically. Hence, if the individual coloron momenta satisfy $\|p_2-p_1\|=2\pi/3N$, we deduce with the choice $a_1=a_2+1$ and the requirement of irreducibility under Y(sl$_3$) transformations that only the sl$_3$ representation $\boldsymbol{3}$ exists in the spectrum. This was also found heuristically in the numerical study of the spectrum of the HSM [@GS07], and is consistent with similar results for conformal field theories [@BouwknegtSchoutens96]. Furthermore, it is shown in \[sec:yangianappsc\] that the proper Y(sl$_3$) subrepresentation $W$ of $V(\boldsymbol{\bar{3}},\xi)\otimes V(\boldsymbol{\bar{3}},\xi-1)$ is explicitly given by $$\textstyle W=V\left(\boldsymbol{3},\xi-\frac{1}{2}\right). \label{eq:yangianreptwocoloronsanti}$$ This means that $W$ is a highest weight representation with YHWS ${\left\|\b\right\rangle}\propto{\left\|\m\c\right\rangle}-{\left\|\c\m\right\rangle}$ (see Fig. \[fig:weightdiagrams\]). We will see below that (\[eq:yangianreptwocoloronsanti\]) is necessary and sufficient to built up the complete Hilbert space of the SU(3) HSM with many-coloron states and the restrictions imposed by the fractional statistics through the requirement of irreducibility under Y(sl$_3$) transformations. At this point we wish to underline that fractional statistics in SU($n$) spin chains cannot be implemented by the requirement of a definite transformation law under permutations of the spinons (a one-dimensional representation of the symmetric group), as in specific situations only the antisymmetric spin representations exist in the spinon Hilbert space, whereas in other situations only the symmetric representations exist. Three-coloron states {#sec:yangianthreecolorons} -------------------- If three colorons are present, there are three different cases to be investigated. In the first case the SMN’s $a_{1,2,3}$ satisfy $a_j-a_i\ge 2$, $i<j$. The tensor product $V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_2)\otimes V(\boldsymbol{\bar{3}},\xi_3)$ is irreducible, hence we find as sl$_3$ representation $\boldsymbol{\bar{3}}\otimes\boldsymbol{\bar{3}}\otimes\boldsymbol{\bar{3}}= \boldsymbol{\bar{10}}\oplus\boldsymbol{8}\oplus\boldsymbol{8} \oplus\boldsymbol{1}$, which is graphically represented by the corresponding tableaux: (21,3.5)(0,1) (0,2)(1,0)[4]{}[(0,1)[2]{}]{} (0,2)(0,1)[3]{}[(1,0)[3]{}]{} (0.5,1.5)(1,0)[3]{} (4,2)[(1,1)[$\oplus$]{}]{} (6,2)(1,0)[2]{}[(0,1)[2]{}]{} (8,1)[(0,1)[3]{}]{} (9,1)(0,2)[2]{}[(0,1)[1]{}]{} (6,2)(0,1)[3]{}[(1,0)[3]{}]{} (8,1)[(1,0)[1]{}]{} (6.5,1.5)(1,0)[2]{} (8.5,2.5) (10,2)[(1,1)[$\oplus$]{}]{} (12,2)(3,0)[2]{}[(0,1)[2]{}]{} (13,1)(1,0)[2]{}[(0,1)[3]{}]{} (12,2)(0,1)[3]{}[(1,0)[3]{}]{} (13,1)[(1,0)[1]{}]{} (12.5,1.5)(2,0)[2]{} (13.5,2.5) (16,2)[(1,1)[$\oplus$]{}]{} (18,2)(3,-1)[2]{}[(0,1)[2]{}]{} (19,1)(1,0)[2]{}[(0,1)[3]{}]{} (18,2)(0,1)[2]{}[(1,0)[3]{}]{} (19,1)(-1,3)[2]{}[(1,0)[2]{}]{} (18.5,1.5)(1,1)[3]{} We have drawn the tableaux for the smallest possible system with $N=6$, the situation is unchanged if the colorons do not occupy adjacent columns. The YHWS ${\left\|\m\m\m\right\rangle}$ belongs to the representation $\boldsymbol{\bar{10}}$ (the left-most tableau). In the second case two of the colorons occupy the same column, whereas the third coloron is separated by at least one column. Graphically we have (12,3.5)(0,1) (3,2)[(0,1)[2]{}]{} (1,1)(1,0)[2]{}[(0,1)[3]{}]{} (0,3)[(0,1)[1]{}]{} (0,3)(0,1)[2]{}[(1,0)[3]{}]{} (1,2)[(1,0)[2]{}]{} (1,1)[(1,0)[1]{}]{} (0.5,1.5)(0,1)[2]{} (2.5,1.5) (5,2)[(1,1)[or]{}]{} (8,2)[(0,1)[2]{}]{} (9,1)(1,0)[2]{}[(0,1)[3]{}]{} (11,3)[(0,1)[1]{}]{} (8,3)(0,1)[2]{}[(1,0)[3]{}]{} (8,2)[(1,0)[2]{}]{} (9,1)[(1,0)[1]{}]{} (10.5,1.5)(0,1)[2]{} (8.5,1.5) The left tableau stands for the sl$_3$ representation containing the YHWS of the tensor product $V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_1-1)\otimes V(\boldsymbol{\bar{3}},\xi_2)$, where $\xi_2-\xi_1\ge 4$. Using (\[eq:yangianreptwocoloronsanti\]) the irreducible subrepresentation of $V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_1-1)$ is $V\left(\boldsymbol{3},\xi_1-\frac{1}{2}\right)$, hence the remaining tensor product $V\left(\boldsymbol{3},\xi_1-\frac{1}{2}\right)\otimes V(\boldsymbol{\bar{3}},\xi_2)$ is irreducible by Sec. \[sec:yangianrepsl3\].ii. As sl$_3$ representation we find $\boldsymbol{8}\oplus\boldsymbol{1}$. The similar result is obtained for the right tableau. The sl$_3$ representations $\boldsymbol{8}$ are given by the tableaux above, the corresponding singlets are represented by (12,3.5)(0,1) (3,1)[(0,1)[2]{}]{} (1,1)(1,0)[2]{}[(0,1)[3]{}]{} (0,3)[(0,1)[1]{}]{} (1,1)(0,1)[2]{}[(1,0)[2]{}]{} (0,3)[(1,0)[3]{}]{} (0,4)[(1,0)[2]{}]{} (0.5,1.5)(0,1)[2]{} (2.5,3.5) (5,2)[(1,1)[and]{}]{} (8,2)[(0,1)[2]{}]{} (9,1)(1,0)[2]{}[(0,1)[3]{}]{} (11,1)[(0,1)[1]{}]{} (8,3)(0,1)[2]{}[(1,0)[2]{}]{} (8,2)[(1,0)[3]{}]{} (9,1)[(1,0)[2]{}]{} (10.5,2.5)(0,1)[2]{} (8.5,1.5) In the third case all three colorons are close together, and two of them occupy the same column. Graphically we have (12,3.5)(0,1) (1,3)[(0,1)[1]{}]{} (2,2)(1,0)[2]{}[(0,1)[2]{}]{} (1,3)(0,1)[2]{}[(1,0)[2]{}]{} (2,2)[(1,0)[1]{}]{} (1.5,1.5)(0,1)[2]{} (2.5,1.5) (5,2)[(1,1)[or]{}]{} (8,2)(1,0)[2]{}[(0,1)[2]{}]{} (10,3)[(0,1)[1]{}]{} (8,3)(0,1)[2]{}[(1,0)[2]{}]{} (8,2)[(1,0)[1]{}]{} (9.5,1.5)(0,1)[2]{} (8.5,1.5) For example, the left tableau is translated into the tensor product $V(\boldsymbol{\bar{3}},\xi)\otimes V(\boldsymbol{\bar{3}},\xi-1)\otimes V(\boldsymbol{\bar{3}},\xi+1)$. As in the second case we first construct the irreducible subrepresentation of the first two factors, which is given by $V\left(\boldsymbol{3},\xi-\frac{1}{2}\right)$. The remaining tensor product $V\left(\boldsymbol{3},\xi-\frac{1}{2}\right)\otimes V(\boldsymbol{\bar{3}},\xi+1)$ is reducible by Sec. \[sec:yangianrepsl3\].ii, its irreducible subrepresentation is as sl$_3$ representation given by $\boldsymbol{8}$. This is reflected in the fact that no singlet tableau with the same SMN’s exists. The loss of the singlet in this case was also observed in conformal field theory spectra [@BouwknegtSchoutens96]. Many-coloron states ------------------- Let us first consider four colorons forming two antisymmetrically coupled pairs. Hence we have to investigate the tensor product $V=V(\boldsymbol{3},\xi_1)\otimes V(\boldsymbol{3},\xi_2)$, where the results of Sec. \[sec:yangianrepsl3\].i apply. If $\xi_2-\xi_1>1$, then $V$ is irreducible and $V\cong\boldsymbol{6}\oplus\boldsymbol{\bar{3}}$. If, however, $\xi_2-\xi_1=1$, then $V$ is reducible and its irreducible subrepresentation is, as sl$_3$ representation, given by $\boldsymbol{6}$. These two situations are graphically represented by the tableaux (28,4) (6,1)(1,0)[4]{}[(0,1)[3]{}]{} (10,3)[(0,1)[1]{}]{} (6,3)(0,1)[2]{}[(1,0)[4]{}]{} (6,2)(2,0)[2]{}[(1,0)[1]{}]{} (6,1)(2,0)[2]{}[(1,0)[1]{}]{} (7.5,1.5)(2,0)[2]{} (7.5,2.5)(2,0)[2]{} (0,2)[(3,1)[$\xi_2-\xi_1>1:$]{}]{} (12,2)[(2,1)[and]{}]{} (16,1)(1,0)[4]{}[(0,1)[3]{}]{} (20,2)[(0,1)[1]{}]{} (16,4)[(1,0)[3]{}]{} (16,3)[(1,0)[4]{}]{} (19,2)[(1,0)[1]{}]{} (16,2)(2,0)[2]{}[(1,0)[1]{}]{} (16,1)(2,0)[2]{}[(1,0)[1]{}]{} (17.5,1.5)(2,0)[2]{} (17.5,2.5)(2,1)[2]{} (28,3)(0,1) (6,1)(1,0)[2]{}[(0,1)[3]{}]{} (9,1)(1,0)[2]{}[(0,1)[3]{}]{} (8,3)[(0,1)[1]{}]{} (6,3)(0,1)[2]{}[(1,0)[4]{}]{} (6,2)(3,0)[2]{}[(1,0)[1]{}]{} (6,1)(3,0)[2]{}[(1,0)[1]{}]{} (7.5,1.5)(1,0)[2]{} (7.5,2.5)(1,0)[2]{} (0,2)[(3,1)[$\xi_2-\xi_1=1:$]{}]{} (11.8,2)[(3,1)[only.]{}]{} If more than four colorons are present, the corresponding product representation of Y(sl$_3$) has to contain more than two fundamental representations. As the representation theory for Y(sl$_3$) is not known in the same detail as that of Y(sl$_2$), we have to restrict ourselves to some illuminating examples. Let us start with the highest-weight tableau (25,3.5)(0,0) (3,2)(0,1)[2]{}[(1,0)[4]{}]{} (5,1)[(1,0)[1]{}]{} (4,1)[(1,0)[1]{}]{} (3,2)(1,0)[5]{}[(0,1)[1]{}]{} (4,1)(1,0)[3]{}[(0,1)[1]{}]{} (3.5,1.5)(3,0)[2]{} (3.5,0.5)(1,0)[4]{} (10,2.3)[SMN’s:]{} (14.5,2.3)[$a_1=\frac{3}{2}$, $a_2=\frac{1}{2}$, $a_3=\frac{5}{2}$,]{} (14.5,0.3)[$a_4=\frac{9}{2}$, $a_5=\frac{13}{2}$, $a_6=\frac{11}{2}$,]{} which stands for the tensor product (we have coupled colorons in the same column already) $$\textstyle V\left(\boldsymbol{3},-\frac{11}{4}\right)\otimes V\left(\boldsymbol{\bar{3}},-\frac{5}{4}\right) \otimes V\left(\boldsymbol{\bar{3}},\frac{3}{4}\right)\otimes V\left(\boldsymbol{3},\frac{9}{4}\right).$$ Using Sec. \[sec:yangianrepsl3\].ii the first as well as the third tensor product is reducible, its irreducible subrepresentations are $V(\boldsymbol{8},\zeta_1)$ and $V(\boldsymbol{8},\zeta_2)$, respectively. We have not determined the spectral parameters explicitly, but expect them to satisfy $-11/4<\zeta_1<-5/4$ and $3/4<\zeta_2<9/4$ (in analogy to the Y(sl$_2$) case [@ChariPressley90]). Thus we have $\zeta_2-\zeta_1>2$, which causes the irreducibility of $V=V(\boldsymbol{8},\zeta_1)\otimes V(\boldsymbol{8},\zeta_2)$ [@Molev02]. As sl$_3$ representation we find $$\boldsymbol{8}\otimes\boldsymbol{8}\;=\;\boldsymbol{27}\oplus \boldsymbol{10}\oplus\boldsymbol{\bar{10}} \oplus\boldsymbol{8}\oplus\boldsymbol{8}\oplus\boldsymbol{1}. \label{eq:yangianhelp2}$$ The irreducibility of $V$ is confirmed by inspection of the allowed tableaux with six colorons and the SMN’s given above, which are (34,3.5)(-6,0) (-6,2)(0,1)[2]{}[(1,0)[4]{}]{} (-5,1)[(1,0)[1]{}]{} (-4,1)[(1,0)[1]{}]{} (-6,2)(1,0)[5]{}[(0,1)[1]{}]{} (-5,1)(1,0)[3]{}[(0,1)[1]{}]{} (-5.5,1.5)(3,0)[2]{} (-5.5,0.5)(1,0)[4]{} (0,2)(0,1)[2]{}[(1,0)[4]{}]{} (1,1)[(1,0)[2]{}]{} (2,0)[(1,0)[1]{}]{} (0,2)(1,0)[5]{}[(0,1)[1]{}]{} (1,1)(1,0)[2]{}[(0,1)[1]{}]{} (2,0)(1,0)[2]{}[(0,1)[1]{}]{} (0.5,0.5)(0,1)[2]{} (3.5,0.5)(0,1)[2]{} (1.5,0.5)(1,1)[2]{} (6,2)[(1,0)[4]{}]{} (6,3)(1,-2)[2]{}[(1,0)[3]{}]{} (6,2)(1,0)[4]{}[(0,1)[1]{}]{} (7,1)(1,0)[4]{}[(0,1)[1]{}]{} (6.5,0.5)(0,1)[2]{} (9.5,0.5)(0,2)[2]{} (7.5,0.5)(1,0)[2]{} (12,2)[(1,0)[4]{}]{} (12,3)(1,-2)[2]{}[(1,0)[2]{}]{} (15,3)(-1,-3)[2]{}[(1,0)[1]{}]{} (12,2)(4,0)[2]{}[(0,1)[1]{}]{} (14,0)(1,0)[2]{}[(0,1)[3]{}]{} (13,1)[(0,1)[2]{}]{} (12.5,0.5)(0,1)[2]{} (15.5,0.5)(0,1)[2]{} (13.5,0.5)(1,2)[2]{} (18,2)[(1,0)[4]{}]{} (18,3)(1,-2)[2]{}[(1,0)[3]{}]{} (20,0)[(1,0)[1]{}]{} (18,2)(4,-1)[2]{}[(0,1)[1]{}]{} (20,0)(1,0)[2]{}[(0,1)[3]{}]{} (19,1)[(0,1)[2]{}]{} (18.5,0.5)(0,1)[2]{} (21.5,0.5)(0,2)[2]{} (19.5,0.5)(1,1)[2]{} (24,2)(1,-1)[2]{}[(1,0)[3]{}]{} (24,3)(2,-3)[2]{}[(1,0)[2]{}]{} (24,2)(4,-2)[2]{}[(0,1)[1]{}]{} (25,1)(2,-1)[2]{}[(0,1)[2]{}]{} (26,0)[(0,1)[3]{}]{} (24.5,0.5)(0,1)[2]{} (27.5,1.5)(0,1)[2]{} (25.5,0.5)(1,2)[2]{} A similar example is obtained from the tableau (25,3.5)(0,0) (3,2)(0,1)[2]{}[(1,0)[4]{}]{} (6,1)[(1,0)[1]{}]{} (4,1)[(1,0)[1]{}]{} (4,1)(1,0)[4]{}[(0,1)[2]{}]{} (3,2)[(0,1)[1]{}]{} (3.5,1.5)(2,0)[2]{} (3.5,0.5)(1,0)[4]{} (10,2.3)[SMN’s:]{} (14.5,2.3)[$a_1=\frac{3}{2}$, $a_2=\frac{1}{2}$, $a_3=\frac{5}{2}$,]{} (14.5,0.3)[$a_4=\frac{9}{2}$, $a_5=\frac{7}{2}$, $a_6=\frac{11}{2}$]{} We have to determine the irreducible subrepresentation of the tensor product $$\textstyle V\left(\boldsymbol{3},-\frac{11}{4}\right)\otimes V\left(\boldsymbol{\bar{3}},-\frac{5}{4}\right) \otimes V\left(\boldsymbol{3},\frac{1}{4}\right)\otimes V\left(\boldsymbol{\bar{3}},\frac{7}{4}\right).$$ As before, we obtain as intermediate result $V=V(\boldsymbol{8},\zeta_1)\otimes V(\boldsymbol{8},\zeta_2)$, but this time the spectral parameters satisfy $-11/4<\zeta_1<-5/4$ and $1/4<\zeta_2<7/4$. In particular, they are separated by $1/2$ less than in the preceding example. Inspection of the allowed tableaux with six colorons and the given SMN’s, which are (22,3.5)(-6,0) (-6,2)(0,1)[2]{}[(1,0)[4]{}]{} (-5,1)[(1,0)[1]{}]{} (-3,1)[(1,0)[1]{}]{} (-5,1)(1,0)[4]{}[(0,1)[2]{}]{} (-6,2)[(0,1)[1]{}]{} (-5.5,1.5)(2,0)[2]{} (-5.5,0.5)(1,0)[4]{} (0,2)(0,1)[2]{}[(1,0)[4]{}]{} (1,1)[(1,0)[1]{}]{} (3,0)(0,1)[2]{}[(1,0)[1]{}]{} (0,2)(1,0)[5]{}[(0,1)[1]{}]{} (1,1)(1,0)[2]{}[(0,1)[1]{}]{} (3,0)(1,0)[2]{}[(0,1)[1]{}]{} (0.5,1.5) (0.5,0.5)(1,0)[3]{} (2.5,1.5)(1,0)[2]{} (6,2)[(1,0)[4]{}]{} (7,1)[(1,0)[3]{}]{} (6,3)[(1,0)[2]{}]{} (9,3)[(1,0)[1]{}]{} (6,2)(1,0)[5]{}[(0,1)[1]{}]{} (7,1)(1,0)[4]{}[(0,1)[1]{}]{} (6.5,0.5)(0,1)[2]{} (8.5,0.5)(0,2)[2]{} (7.5,0.5)(2,0)[2]{} (12,2)[(1,0)[4]{}]{} (12,3)(1,-2)[2]{}[(1,0)[2]{}]{} (15,3)(0,-1)[4]{}[(1,0)[1]{}]{} (12,2)(4,0)[2]{}[(0,1)[1]{}]{} (16,0)[(0,1)[1]{}]{} (15,0)[(0,1)[3]{}]{} (13,1)(1,0)[2]{}[(0,1)[2]{}]{} (12.5,0.5)(0,1)[2]{} (14.5,0.5)(1,1)[2]{} (13.5,0.5)(1,2)[2]{} shows that the irreducible subrepresentation of $V$ is, as representation of sl$_3$, given by $\boldsymbol{27}\oplus\boldsymbol{10}\oplus\boldsymbol{\bar{10}} \oplus\boldsymbol{8}$. The difference to (\[eq:yangianhelp2\]) implies that $V$ is reducible. Physically, the fractional statistics of the colorons restricts the allowed SU(3) representations more than above, as the individual coloron momenta are closer together. As final example consider the six-coloron states with SMN’s $a_1=3/2$, $a_2=7/2$, $a_3=5/2$, $a_4=9/2$, $a_5=7/2$, and $a_6=11/2$. For this set of SMN’s there exist only the two tableaux (12,3)(0,0.2) (0,2)(0,1)[2]{}[(1,0)[4]{}]{} (0,1)(3,0)[2]{}[(1,0)[1]{}]{} (0,2)(1,0)[5]{}[(0,1)[1]{}]{} (0,1)(1,0)[2]{}[(0,1)[1]{}]{} (3,1)(1,0)[2]{}[(0,1)[1]{}]{} (1.5,1.5)(1,0)[2]{} (0.5,0.5)(1,0)[4]{} (8,2)(0,1)[2]{}[(1,0)[4]{}]{} (8,1)[(1,0)[1]{}]{} (11,0)(0,1)[2]{}[(1,0)[1]{}]{} (8,2)(1,0)[5]{}[(0,1)[1]{}]{} (8,1)(1,0)[2]{}[(0,1)[1]{}]{} (11,0)(1,0)[2]{}[(0,1)[1]{}]{} (9.5,1.5)(1,0)[3]{} (8.5,0.5)(1,0)[3]{} the irreducible subrepresentation of $$\textstyle V\left(\boldsymbol{\bar{3}},-\frac{9}{4}\right)\otimes V\left(\boldsymbol{3},-\frac{3}{4}\right) \otimes V\left(\boldsymbol{3},\frac{1}{4}\right)\otimes V\left(\boldsymbol{\bar{3}},\frac{7}{4}\right), \label{eq:yangianmanycoloronsproduct3}$$ should be given by $\boldsymbol{27}\oplus\boldsymbol{10}$. The general scheme for SU(3) works as follows. An $m$-coloron YHWS is represented by a tableau in which the colorons sit at the bottom of the columns. First, we couple the colorons in the same column, we construct the representations $V(\boldsymbol{3},\zeta)$, where $\zeta$ is determined using (\[eq:yangianxidefinitionsu3\]) and (\[eq:yangianreptwocoloronsanti\]). The remaining colorons transform under $V(\boldsymbol{\bar{3}},\xi)$ with $\xi$ given by (\[eq:yangianxidefinitionsu3\]). The space generated under the action of Y(sl$_3$) by the YHWS is the irreducible subrepresentation $W$ of the tensor product $$V=\,\bigotimes_{i=1}^{m'} V\left(\boldsymbol{x}_i,\xi_i\right),\quad \xi_1<\xi_2<\ldots<\xi_{m'}, \label{eq:yangiangeneralsu3product}$$ where $\boldsymbol{x}_i$ denotes either $\boldsymbol{3}$ or $\boldsymbol{\bar{3}}$, and $m'$ is the number of occupied columns in the tableau (the number of isolated colorons plus the number of coloron pairs). As sl$_3$ representation, $W$ is given by all tableaux with $m$ colorons possessing the corresponding SMN’s. To sum up, colorons transform under the Y(sl$_3$) representation $V(\boldsymbol{\bar{3}},\xi)$, where the spectral parameter $\xi$ is directly connected to the individual coloron momentum. The space of $m$ colorons with momenta $p_1,\dots,p_m$ is generated by the YHWS of (\[eq:yangiangeneralsu3product\]) as explained above. The restrictions on the SU(3) content of this space are due to the fractional statistics of the colorons. From a mathematical point of view the tableau formalism provides an algorithm to derive the sl$_3$ content of the irreducible subrepresentation of a tensor product of fundamental Y(sl$_3$) representations (\[eq:yangiangeneralsu3product\]) with increasing spectral parameters. As a by-product, this yields an irreducibility criterion for tensor products of the form (\[eq:yangiangeneralsu3product\]). Conclusion ========== In conclusion, we have investigated the relation between the spinon excitations of the Haldane–Shastry model and its Yangian symmetry. Each individual spinon transforms under the fundamental representation of the Yangian. The associated spectral parameter is directly proportional to its momentum. We have obtained a generalised Pauli principle which states that the spinon Hilbert space is built up by the irreducible subrepresentations of tensor products of these fundamental representations. This enabled us to derive several restrictions on the total spin of many-spinon states. Although the fractional statistics of spinons can be implemented using the representation theory of the Yangian only for spinons in the Haldane–Shastry model, we expect the rules governing the allowed values of the total spin of many-spinon states to be valid for interacting spinons in general spin chains as well. Acknowledgments {#acknowledgments .unnumbered} =============== I would like thank Frank Göhmann and Martin Greiter for valuable discussions. This work was mainly carried out at the Institut für Theorie der Kondensierten Materie, Universität Karlsruhe. This work was supported by the German Research Foundation (DFG) through GK 284, the Center for Functional Nanostructures (CFN) Karlsruhe and the Deutsche Akademie der Naturforscher Leopoldina under grant no BMBF-LPD 9901/8-145. Realization of $\boldsymbol{V(\bar{3},\xi)}$ as evaluation representation {#sec:yangianappev} ========================================================================= Consider the representation $V(\boldsymbol{\bar{3}},\xi)$ with Drinfel’d polynomials $P_1(u)=1$ and $P_2(u)=u-\xi$, and denote by ${\left\|\m\right\rangle}$ its YHWS. Then we have $$H_{1,0}{\left\|\m\right\rangle}=H_{1,1}{\left\|\m\right\rangle}=0,\quad H_{2,0}{\left\|\m\right\rangle}={\left\|\m\right\rangle}\!, \quad H_{2,1}{\left\|\m\right\rangle}=\xi{\left\|\m\right\rangle}\!,$$ and with (\[eq:cwblambda8\]) we deduce $$\Lambda^8{\left\|\m\right\rangle}=\frac{1}{\sqrt{3}}\left(\xi+\frac{1}{4}\right){\left\|\m\right\rangle}\!. \label{eq:appyangian1}$$ On the other hand we find with (\[eq:yangianevdef3\]) that $$\begin{aligned} \mathrm{ev}_\zeta(\Lambda^8)&=&\;\zeta J^8+ \frac{1}{2\sqrt{3}}\left(J^3J^3-J^8J^8\right)+ \frac{1}{4\sqrt{3}}\left(I^+I^-+I^-I^+\right)\nonumber\\ & &-\frac{1}{8\sqrt{3}}\left(U^+U^-+U^-U^++V^+V^-+V^-V^+\right)\!,\end{aligned}$$ and hence for the action of $\Lambda^8$ on ${\left\|\m\right\rangle}$ in the evaluation representation $\phi_\zeta$ $$\Lambda^8{\left\|\m\right\rangle}=\frac{1}{\sqrt{3}}\left(\zeta-\frac{5}{12}\right){\left\|\m\right\rangle}\!. \label{eq:appyangian2}$$ Comparison of (\[eq:appyangian1\]) and (\[eq:appyangian2\]) yields $\zeta=\xi+2/3$. Irreducible subrepresentation of $\boldsymbol{V(\bar{3},\xi)\otimes V(\bar{3},\xi-1)}$ {#sec:yangianappsc} ====================================================================================== Consider the tensor product $V=V(\boldsymbol{\bar{3}},\xi_1)\otimes V(\boldsymbol{\bar{3}},\xi_2)$. $V$ contains a proper Y(sl$_3$) subrepresentation $W$ isomorphic to $\boldsymbol{3}$ as sl$_3$ representation if and only if [@ChariPressley96] $\xi_1-\xi_2=1$. We wish to determine the Drinfel’d polynomials of $W$. For that we have to evaluate the actions of $H_{1,1}$ and $H_{2,1}$ on the YHWS ${\left\|\b\right\rangle}={\left\|\m\right\rangle}\otimes{\left\|\c\right\rangle}-{\left\|\c\right\rangle}\otimes{\left\|\m\right\rangle}$, where ${\left\|\b\right\rangle}\in\boldsymbol{3}$ and ${\left\|\m\right\rangle},{\left\|\c\right\rangle}\in\boldsymbol{\bar{3}}$. First, we obtain from (\[eq:cwblambda3\]) $$\Lambda^3{\left\|\b\right\rangle}=\frac{1}{2}H_{1,1}{\left\|\b\right\rangle}+\frac{1}{8}{\left\|\b\right\rangle}\!. \label{eq:appyangian3}$$ On the other hand we find the action of $\Lambda^3$ on $V$ to be $$\begin{aligned} \Delta(\Lambda^3)&=&\Lambda^3\otimes 1+1\otimes\Lambda^3-f^{3ab}J^a\otimes J^b\nonumber\\ &=&\Lambda^3\otimes 1+1\otimes\Lambda^3+\frac{1}{2}\left(I^+\otimes I^--I^-\otimes I^+\right)\nonumber\\ & &-\frac{1}{4}\left(U^+\otimes U^--U^-\otimes U^-- V^+\otimes V^-+V^-\otimes V^+\right)\!.\end{aligned}$$ On each factor of $V$ the action of $\Lambda^3$ is by (\[eq:yangianevdef3\]) $$\hspace{-10mm} \mathrm{ev}_{\xi_{1,2}}(\Lambda^3)=\xi_{1,2} J^3+ \frac{1}{\sqrt{3}}\,J^3J^8 -\frac{1}{8}\left(U^+U^-+U^-U^+-V^+V^--V^-V^+\right)\!,$$ especially we get $$\Lambda^3{\left\|\m\right\rangle}=0,\quad\Lambda^3{\left\|\c\right\rangle}= \left(\frac{\xi_{1,2}}{2}+\frac{1}{8}\right){\left\|\c\right\rangle}\!.$$ Hence, we find $$\hspace{-10mm} \Delta(\Lambda^3){\left\|\b\right\rangle}= \left(\frac{\xi_2}{2}+\frac{3}{8}\right){\left\|\m\right\rangle}\otimes{\left\|\c\right\rangle}- \left(\frac{\xi_1}{2}-\frac{1}{8}\right){\left\|\c\right\rangle}\otimes{\left\|\m\right\rangle}= \left(\frac{\xi_1}{2}-\frac{1}{8}\right){\left\|\b\right\rangle}\!.\label{eq:appyangian4}$$ The last equality is valid if and only if $\xi_1-\xi_2=1$, when the Y(sl$_3$) subrepresentation $W$ is indeed isomorphic to $\boldsymbol{3}$. 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--- abstract: \| A famous open problem asks whether the asymptotic dimension of a CAT(0) group is necessarily finite. For hyperbolic $G$, it is known that $\operatorname{asdim}G$ is bounded above by $\dim\partial G+1$, which is known to be finite. For CAT(0) $G$, the latter quantity is also known to be finite, so one approach is to try proving a similar inequality. So far those efforts have failed. Motivated by these questions we work toward understanding the relationship between large scale dimension of CAT(0) groups and small scale dimension of the group’s boundary by shifting attention to the linearly controlled dimension of the boundary. To do that, one must choose appropriate metrics for the boundaries. In this paper, we suggest two candidates and develop some basic properties. Under one choice, we show that linearly controlled dimension of the boundary remains finite; under another choice, we prove that macroscopic dimension of the group is bounded above by $2\cdot\ell$-$\dim \partial G+1$. Other useful results are established, some basic examples are analyzed, and a variety of open questions are posed. author: - 'Molly A. Moran' bibliography: - 'Biblio.bib' title: 'Metrics on Visual Boundaries of CAT(0) Spaces' --- [^1] Introduction ============ In [@Mo14] and [@GuMo15], it was shown that coarse (large-scale) dimension properties of a space $X$ can impose restrictions on the classical (small-scale) dimension of boundaries attached to $X$. A natural question to ask is if the converse is true. For example, one might hope to use the finite-dimensionality of $\partial G$, proved first in [@Swe99] and following as a corollary of Theorem A in [@Mo14], to attack the following well-known open question: Does every CAT(0) group have finite asymptotic dimension? This question provides motivation for much of the work in what follows. Although we do not answer Question 1.0.1, a framework is developed that we expect will lead to future progress. Along the way, we prove some results that we hope are of independent interest; one such result is a partial solution to Question 1.0.1 that captures the spirit of our approach. As is often the case with questions about CAT(0) groups, Question 1.0.1 is rooted in known facts about hyperbolic groups. Gromov observed that all hyperbolic groups have finite asymptotic dimension. A more precise bound on the asymptotic dimension, which helps to establish our point of view, is the following: [@BuSc07; @BuLe07] For a hyperbolic group, $\emph{asdim}G=\emph{dim}\partial G+1=\ell\emph{-dim}\partial G+1<\infty$. In this theorem ‘$\operatorname{asdim}$’ denotes asymptotic dimension, ‘$\dim$’ denotes covering dimension, and ‘$\ell$-$\dim$’ denotes linearly controlled dimension. All of these terms will be explained in Section 2.3. For now, we note that linearly controlled dimension is similar to, but stronger than, covering dimension; both are small-scale invariants defined using fine open covers. The difference is that $\ell$-$\dim$ is a metric invariant, requiring a linear relationship between the mesh and the Lebesgue numbers of the covers used. Implicit in the statement of Theorem 1.0.2 is that $\partial G$ be endowed with a visual metric. There is a family of naturally occurring visual metrics on $\partial G$, but all are quasi-symmetric to one-another. That is enough to make $\ell$-$\dim\partial G$ well-defined. This also will be explained shortly. We can now summarize the content of this paper. We begin by reviewing a number of key definitions and properties from CAT(0) geometry. Next, we recall definitions of quasi-isometry and quasi-symmetry, and then we discuss variations, both small- and large-scale, on the notion of dimension. To bring the utility of linearly controlled dimension to CAT(0) spaces, it is necessary to have specific metrics on their visual boundaries. Although CAT(0) boundaries are important, well-understood, and metrizable, specific metrics have seldom been used in a significant way. In Sections 3 and 4, we develop two natural families of metrics for CAT(0) boundaries and verify a number of their basic properties. One of these families $\left\{ d_{A,x_{0}}\right\} _{x_{0}\in X}^{A>0}$ was discussed in [@Ka07], where B. Kleiner asked whether the induced action on $\partial X$ of a geometric action on a proper CAT(0) space $X$ is “nice”. After first showing that all metrics in the family $\left\{ d_{A,x_{0}}\right\} _{x_{0}\in X}^{A>0}$ are quasi-symmetric in Section 3.1, we provide an affirmative answer to Kleiner’s question with the following: Suppose $G$ acts geometrically on a proper CAT(0) space $X$, $x_{0}\in X$ and $A>0$. Then the induced action of $G$ on $\left( \partial X,d_{x_{0},A}\right) $ is by quasi-symmetries. In Section 3.2, we look to prove analogs of Theorem 1.0.2 for CAT(0) spaces. The question of whether $\ell$-dimension of a CAT(0) group boundary agrees with its covering dimension (under either of our metrics) is still open, but we can prove: If $G$ is a CAT(0) group, then $\left( \partial G,d_{A,x_{0}}\right) $ has finite $\ell$-dimension. As for the equality in Theorem 1.0.2, we are thus far unable to use the $\ell $-dimension of $\left( \partial X,d_{A,x_{0}}\right) $ to make conclusions about the asymptotic dimension of $X$. Instead we turn to our other family of metrics $\left\{ \overline{d}_{x_{0}}\right\} $. In some sense, these boundary metrics retain more information about the interior space $X$. That additional information allows us to prove the following theorem, which we view as a weak solution to Question 1.0.1. It is our primary application of the $\overline{d}_{x_{0}}$ metrics. Suppose $X$ is a geodesically complete CAT(0) space and, when endowed with the $\overline{d}_{x_0}$ metric for $x_0\in X$, $\ell$-$\dim\partial X\leq n$. Then the macroscopic dimension of $X$ is at most $2n+1$. In Section 5, we compare the $d_{A,x_{0}}$ and $\overline{d}_{x_{0}}$ metrics to each other by applying them to some simple examples. We also compare them to the established visual metrics when we have a space that is both CAT(0) and hyperbolic. Much work remains in this area and thus we conclude with a list of open questions. The author would like to thank Craig Guilbault for his guidance and suggestions during the course of this project. Preliminaries ============= Before discussing the possible metrics and their properties, we first review CAT(0) spaces and the visual boundary, quasi-symmetries, and the various dimension theories that will be discussed. The study of metrics on the boundary begins in Section 3. CAT(0) Spaces and their Geometry -------------------------------- In this section, we review the definition of CAT(0) spaces, some basic properties of these spaces, and the visual boundary. For a more thorough treatment of CAT(0) spaces, see [@BH99]. A geodesic metric space $(X,d)$ is a *CAT(0) space* if all of its geodesic triangles are no fatter than their corresponding Euclidean comparison triangles. That is, if $\Delta(p,q,r)$ is any geodesic triangle in $X$ and $\overline{\Delta}(\overline{p},\overline{q},\overline{r})$ is its comparison triangle in ${\mathbb{E}}^2$, then for any $x,y\in\Delta$ and the comparison points $\overline{x},\overline{y}$, then $d(x,y)\leq d_{{\mathbb{E}}}(\overline{x},\overline{y})$. A few important properties worth mentioning are that proper CAT(0) spaces are contractible, uniquely geodesic, balls in the space are convex, and the distance function is convex. Furthermore, we now record a very simple geometric property that will be used repeatedly throughout the rest of the paper. Let $(X,d)$ be a proper CAT(0) space and suppose $\alpha,\beta:[0,\infty)\to X$ are two geodesic rays based at the same point $x_0\in X$. Then for $0<s\leq t<\infty$, $d(\alpha(s),\beta(s))\leq \frac{s}{t}d(\alpha(s),\beta(t))$. Let $p=\alpha(t)$, $q=\beta(t)$, $x=\alpha(s)$, and $y=\beta(s)$. Consider the geodesic triangle $\Delta(x_0, p, q)$ in X and its comparison triangle $\overline{\Delta}(\overline{x_0},\overline{p},\overline{q})$ in ${\mathbb{E}}^2$. Let $\overline{x},\overline{y}$ be the corresponding points to $x,y$ on $\overline{\Delta}$. (See picture below.) ![image](similartriangles3.pdf) By similar triangles in ${\mathbb{E}}^2$, $$\frac{d_{{\mathbb{E}}}(\overline{p},\overline{q})}{d_{{\mathbb{E}}}(\overline{x},\overline{y})}=\frac{d_{{\mathbb{E}}}(\overline{x_0},\overline{p})}{d_{{\mathbb{E}}}(\overline{x_0},\overline{x})}=\frac{t}{s}$$ Thus, $d_{{\mathbb{E}}}(\overline{x},\overline{y})=\frac{s}{t}d_{{\mathbb{E}}}(\overline{p},\overline{q})=\frac{s}{t}d(p,q)$ Applying the CAT(0)-inequality, we obtain the desired inequality: $$d(x,y)\leq \left(\frac{s}{t}\right)d(p,q)$$ We now review the definition of the boundary of CAT(0) spaces: The *boundary* of a proper CAT(0) space $X$, denoted $\partial X$, is the set of equivalence classes of rays, where two rays are equivalent if and only if they are asymptotic. We say that two geodesic rays $\alpha, \alpha':[0,\infty)\to X$ are *asymptotic* if there is some constant $k$ such that $d(\alpha(t),\alpha'(t))\leq k$ for every $t\geq 0$. Once a base point is fixed, there is a a unique representative geodesic ray from each equivalence class by the following: If $X$ is a complete CAT(0) space and $\gamma:[0,\infty)\to X$ is a geodesic ray with $\gamma(0)=x$, then for every $x'\in X$, there is a unique geodesic ray $\gamma':[0,\infty)\to X$ asymptotic to $\gamma$ and with $\gamma'(0)=x'$. In the construction of the asymptotic ray for Proposition 2.1.4, it is easy to verify that $d(\gamma(t), \gamma '(t))\leq d(x,x')$ for all $t\geq 0$. We may endow $\overline{X}=X\cup \partial X$, with the cone topology, described below, which makes $\partial X$ a closed subspace of $\overline{X}$ and $\overline{X}$ compact (as long as $X$ is proper). With the topology on $\partial X$ induced by the cone topology on $\overline{X}$, the boundary is often called the visual boundary. In what follows, the term ‘boundary’ will always mean ‘visual boundary’. Furthermore, we will slightly abuse terminology and call the cone topology restricted to $\partial X$ simply the cone topology if it is clear that we are only interested in the topology on $\partial X$. One way in which to describe the cone topology on $\overline {X}$, denoted $\mathscr{T}(x_0)$ for $x_0\in X$, is by giving a basis. A basic neighborhood of a point at infinity has the following form: given a geodesic ray $c$ and positive numbers $r>0$, $\epsilon>0$, let $$U(c,r, \epsilon) = \{x \in X \| d(x, c(0)) > r, d(p_r(x), c(r)) < \epsilon\}$$ where $p_r$ is the natural projection of $\overline{X} $ onto $\overline{B}(c(0),r)$. Then a basis for the topology, $\mathscr{T}(x_0)$, on $\overline{X}$ consists of the set of all open balls $B(x,r) \subset X$, together with the collection of all sets of the form $U(c,r, \epsilon)$, where $c$ is a geodesic ray with $c(0) = x_0$. For all $x_0,x_0'\in X$, $\mathscr{T}(x_0)$ and $\mathscr{T}(x_0')$ are equivalent [@BH99 Proposition 8.8]. Quasi-Symmetries ---------------- As we are interested in both large-scale and small-scale properties of metric spaces, we briefly discuss two different types of maps that may be used to capture the particular scale we care about. The first type of map is a quasi-isometry. A map $f:(X,d_X)\to(Y,d_Y)$ between metric spaces is a *quasi-isometric embedding* if there exists constants $A,B>0$ such that for every $x,y\in X$, $\frac{1}{A}d_X(x,y)-B\leq d_Y(f(x),f(y))\leq Ad_X(x,y)+B$. Moreover, if there exists a $C>0$ such that for every $z\in Y$, there is some $x\in X$ such that $d_Y(f(x),z)\leq C$, then we call $f$ a *quasi-isometry. Quasi-isometries capture the large-scale geometry of a metric space, but ignore the small scale-behavior. Thus, they are ideal when studying large scale notions of dimension, which we will discuss briefly in the next section. Since small-scale behavior is ignored, all compact metric spaces turn out to be quasi-isometric because they are all quasi-isometric to a point. Thus, quasi-isometries are not particularly useful when studying compact metric spaces. When interested in compact metric spaces and small-scale behavior, we can turn to a second type of map: quasi-symmetry. Quasi-symmetric maps were defined to extend the notion of quasi-conformality. Since these maps care about local behavior, they are ideal when studying small scale notions of dimension, in particular, linearly controlled dimension. Quasi-symmetric maps have also played a large role in the the study of hyperbolic group boundaries. For example, it has been shown that all visual metrics on the boundary are quasi-symmetric. We review the definition and properties that will be needed in later sections. For more information, see [@TuVa80] or [@He01]. A map $f:X\to Y$ between metric spaces is said to be quasi-symmetric* if it is not constant and there is a homeomorphism $\eta:[0,\infty)\to[0,\infty)$ such that for any three points $x,y,z\in X$ satisfying $d(x,z)\leq td(y,z)$, it follows that $d(f(x), f(z))\leq\eta(t)d(f(y),f(z))$ for all $t\geq 0$. The function $\eta$ is often called a *control function* of $f$. A *quasi-symmetry* is a quasi-symmetric homeomorphism. [@He01 Proposition 10.6] If $f:X\to Y$ is $\eta$-quasi-symmetric, then $f^{-1}:f(X)\to X$ is $\eta'$-quasi-symmetric where $\eta'(t)=1/\eta^{-1}(t^{-1})$ for $t>0$. Moreover, if $f:X\to Y$ and $g:Y\to Z$ are $\eta_f$ and $\eta_g$ quasi-symmetric, respectively, then $g\circ f:X\to Z$ is $\eta_g\circ\eta_f$ quasi-symmetric. [@He01 Theorem 11.3]A quasi-symmetric embedding $f$ of a uniformly perfect space $X$ is $\eta$-quasi-symmetric with $\eta$ of the form $\eta(t)=c\emph{max}\{t^{\delta},t^{1/\delta}\}$ where $c\geq 1$ and $\delta\in(0,1]$ depends only on $f$ and $X$. We say that a metric space $X$ is uniformly perfect* if there exists a $c>1$ such that for all $x\in X$ and for all $r>0$, the set $B(x,r)-B(x,\frac{r}{c})\neq \emptyset$ whenever $X-B(x,r)\neq\emptyset$. Some examples of uniformly perfect spaces include connected spaces and the Cantor ternary set. Being uniformly perfect is a quasi-symmetry invariant [@He01]. A Review of Various Dimension Theories -------------------------------------- Recall that the covering dimension of a space $X$ is at most $n$, denoted dim$X\leq n$, if every open cover of $X$ has an open refinement of order at most $n+1$. The covering dimension can be studied for any topological space, in particular, spaces need not be metrizable. However, if $X$ is a compact metric space, we may use the following to show finite covering dimension. For a compact metric space $X$, dim$X\leq n$ if, for every $\epsilon>0$, there is a cover of $X$ with mesh smaller than $\epsilon$ and order at most $n+1$. In the preceding lemma, we use the terms ‘mesh’ and ‘order’. We now define this terminology, along with a few other terms needed for the other dimension theories. Given a cover $\mathscr{U}$ of a metric space $X$, we define mesh$(\mathscr{U})=\sup\{\text{diam}(U)\|U\in\mathscr{U}\}$. We say that the cover $\mathscr{U}$ is uniformly bounded if there exists some $D>0$ such that mesh$(\mathscr{U})\leq D$. The order of $\mathscr{U}$ is the smallest integer $n$ for which each element $x\in X$ is contained in at most $n$ elements of $\mathscr{U}$. The Lebesgue number of $\mathscr{U}$, denoted $\mathscr{L(U)}$, is defined as $\mathscr{L(U)}=\text{inf}_{x\in X}\mathscr{L}(\mathscr{U},x)$, where $\mathscr{L}(\mathscr{U},x)=\text{sup}\{d(x, X-U)\|U\in\mathscr{U}\}$ for each $x\in X$. One reason for pointing out the alternate characterization of covering dimension for compact metric spaces is that the other dimension theories that we discuss here are restricted to metric spaces. These restrictions are due to the need for control of Lebesgue numbers as well as the mesh of covers. In particular, we record two properties for covers that will be used to characterize the different notions of dimension. Let $\mathscr{U}$ be a uniformly bounded open cover of a metric space $X$. We say that $\mathscr{U}$ has - Property $\mathscr{P}_\lambda^n$ if $\mathscr{L(U)}\geq\lambda$ and order$(\mathscr{U})\leq n+1$. - Property $\mathscr{P}_{\lambda, c}^n$ if $\mathscr{L(U)}\geq\lambda$, mesh$(\mathscr{U)}\leq c\lambda$, and order$(\mathscr{U})\leq n+1$ This second property requires not only a given Lebesgue number, but also a linear relationship between the mesh of the cover and the Lebesgue number. These two properties capture key requirements in the remaining dimension theories, which we now describe, organized in terms of large-scale and small-scale properties. Let $X$ be a metric space. 1. The *macroscopic dimension* of $X$ is at most $n$, denoted $\dim_{\emph{mc}}X\leq n$, if there exists a single uniformly bounded open cover of $X$ with order $n+1$. 2. The *asymptotic dimension* of $X$ is at most $n$, denoted asdim$X\leq n$, if for every $\lambda>0$, there exists a cover $\mathscr{U}$ with Property $\mathscr{P_\lambda}^n$. 3. The *linearly-controlled asymptotic dimension* of $X$ is at most $n$, denoted $\ell\emph{-asdim}X\leq n$, if there exists $c\geq 1$ and $\lambda_0>0$ such that for all $\lambda\geq \lambda_0$, there is a cover $\mathscr{U}$ with Property $\mathscr{P}_{\lambda,c}^n$. 4. The *Assouad-Nagata dimension* of $X$ is at most $n$, denoted ANdim$X\leq n$, if there exists $c\geq 1$, such that for all $\lambda>0$, there is a cover $\mathscr{U}$ with Property $\mathscr{P}_{\lambda,c}^n$. 5. The *linearly-controlled dimension* of $X$ is at most $n$, denoted\ $\ell\emph{-dim}X\leq n$, if there exists $c\geq 1$ and $\lambda_0>0$ such that for all $0<\lambda\leq\lambda_0$, there is a cover $\mathscr{U}$ with Property $\mathscr{P}_{\lambda,c}^n$. We wish to record a few facts about the various dimension theories, as well as some relationships between them: 1. Asymptotic dimension and linearly-controlled asymptotic dimension are quasi-isometry invariants of a metric space. For a nice survey of asymptotic dimension, see [@BDr07]. It has become widely studied due in part to its relationship to the Novikov Conjecture. 2. Assouad-Nagata dimension is a quasi-symmetry invariant [@LS05]. Since $\ell\text{-dim}X= \text{ANdim}X$ for bounded metric spaces, linearly-controlled dimension is a quasi-symmetry invariant for bounded metric spaces 3. In fact, linearly-controlled metric dimension is a quasi-symmetry invariant of a larger class of metric spaces: uniformly perfect metric spaces [@BuSc07]. 4. For a metric space $X$, we have the following comparisons: $$\text{mdim}X\leq\text{dim}X\leq\ell\text{-dim}X\leq \text{ANdim}X$$ $$\text{mdim}X\leq\text{asdim}X\leq\ell\text{-asdim}X\leq\text{ANdim}X$$ For more on the above dimension theories, see [@BuSc07] The $d_{A,x_0}$ metrics ======================== We are now ready to define the first family of metrics on the visual boundary of a CAT(0) space: the $d_{A,x_0}$ metrics. Fix a base point $x_0\in X$ and choose $A>0$. For $[\alpha],[\beta]\in\partial X$, let $\alpha:[0,\infty)\to X$ and $\beta:[0,\infty)\to X$ be the geodesic rays based at $x_0$ and asymptotic to $[\alpha] $ and $[\beta]$, respectively. Let $a\in(0,\infty)$ be such that $d(\alpha(a),\beta(a))=A$. If such an $a$ does not exist, set $a=\infty$. Then, define $d_{A,x_0}:\partial X\times\partial X\to {\mathbb{R}}$ by $$d_{A,x_0}([\alpha],[\beta])=\frac{1}{a}$$ Basic Properties of the $d_{A,x_0}$ metrics ------------------------------------------- Before discussing any properties of the $d_{A,x_0}$ metrics, we must first show that each member of the family is indeed a metric and induces the cone topology on $\partial X$. If $(X,d)$ is a CAT(0) space and $x_0\in X$, then $d_{A,x_0}$ for any $A>0$ is a metric on $\partial X$. Fix a base point $x_0\in X$ and choose $A>0$. Let $[\alpha],[\beta],[\gamma]\in\partial X$ and $\alpha, \beta,\gamma: [0,\infty)\to X$ be the geodesic rays based at $x_0$ and asymptotic to $[\alpha],[\beta],[\gamma]$, respectively. Clearly, $d_{A,x_0}([\alpha],[\alpha])=0$ since $d(\alpha(t),\alpha(t))=0$ for every $t\geq 0$ and hence $a=\infty$. If $d_{A,x_0}([\alpha],[\beta])=0$, then there is no $a\in (0,\infty)$ such that $d(\alpha(a),\beta(a))=A$. By convexity of CAT(0) metric, this means $d(\alpha(t),\beta(t))=0$ for every $t\geq 0$. Hence, $\alpha=\beta$, which means $[\alpha]=[\beta]$. Also, $d_A([\alpha],[\beta])=d_A([\beta],[\alpha])$ since $d(\alpha(t),\beta(t))=d(\beta(t),\alpha(t))$. Finally, to verify the triangle inequality, suppose $a,b,c\in (0,\infty]$ satisfy $$d_{A,x_0}([\alpha],[\beta])=\frac{1}{a} \, , \, d_{A,x_0}([\beta], [\gamma])=\frac{1}{b} \, , \, d_{A,x_0}([\alpha],[\gamma])=\frac{1}{c}$$ If $c\geq a $ or $c\geq b$, then $$d_{A,x_0}([\alpha],[\gamma])=\frac{1}{c}\leq \frac{1}{a}\leq \frac{1}{a}+\frac{1}{b}=d_{A,x_0}([\alpha],[\beta])+d_{A,x_0}([\beta], [\gamma])$$ or $$d_{A,x_0}([\alpha],[\gamma])=\frac{1}{c}\leq \frac{1}{b}\leq \frac{1}{a}+\frac{1}{b}=d_{A,x_0}([\alpha],[\beta])+d_{A,x_0}([\beta], [\gamma])$$ Thus, the only interesting case is if $c<a$ and $c<b$. By Lemma 2.1.2 $$d(\alpha(c),\beta(c))\leq\frac{c}{a}A$$ and $$d(\beta(c), \gamma(c))\leq\frac{c}{b}A$$ Then, $$A=d(\alpha(c),\gamma(c))\leq d(\alpha(c), \beta(c))+d(\beta(c),\gamma(c))\leq \frac{c}{a}A+\frac{c}{b}A=Ac\left(\frac{a+b}{ab}\right)$$ Thus, $$c\geq \frac{ab}{a+b}$$ which proves: $$d_{A,x_0}([\alpha],[\gamma])=\frac{1}{c}\leq\frac{a+b}{ab}=\frac{1}{a}+\frac{1}{b}=d_{A,x_0}([\alpha],[\beta])+d_{A,x_0}([\beta], [\gamma])$$ The topology induced by the $d_{A,x_0}$ metric on $\partial X$ is equivalent to the cone topology on $\partial X$. Fix $A>0$ and $x_0\in X$. Since the base point is fixed, we will simplify $d_{A,x_0}$ to $d_A$. Consider the basic open set $B_{d_A}([\alpha],\epsilon)$ for $[\alpha]\in\partial X$ and $\epsilon>0$ and let $[\beta]\in B_{d_A}([\alpha],\epsilon)$. Let $\alpha,\beta:[0,\infty)\to X$ be the unique geodesic rays based at $x_0$ corresponding to $[\alpha]$ and $[\beta]$, respectively. Choose $\delta>0$ such that $B_{d_A}([\beta],\delta)\subset B_{d_A}([\alpha],\epsilon)$ and consider the basic open set in the cone topology $U(\beta,\frac{1}{\delta},A)\cap\partial X$. Let $[\gamma]\in U(\beta,\frac{1}{\delta},A)\cap\partial X$. Then $d(\beta(\frac{1}{\delta}),\gamma(\frac{1}{\delta}))<A$. If $a>0$ is the point such that $d(\beta(a),\gamma(a))=A$, then $a>\frac{1}{\delta}$. Thus, $d_A([\beta],[\gamma])=\frac{1}{a}<\delta$. Thus, $[\gamma]\in B_{d_A}([\beta],\delta)\subset B_{d_A}([\alpha],\epsilon)$, proving $[\beta]\in U(\beta,r,A)\cap \partial X \subset B_{d_A}([\alpha],\epsilon)$. Now consider a basic open set $U(\alpha, r, \epsilon)\cap\partial X$ in the cone topology where $r>0$,\ $A>\epsilon>0$ and $\alpha:[0,\infty)\to X$ is a geodesic ray based at $x_0$ . Let $[\beta]\in U(\alpha, r, \epsilon)\cap\partial X$. Choose $\delta>0$ such that $B_d(\beta(r),\delta)\cap S(x_0,r)\subset B_d(\alpha(r),\epsilon)\cap S(x_0,r)$ and consider the basic open set in the metric topology $B_{d_A}([\beta],\frac{\delta}{Ar})$. Let $[\gamma]\in B_{d_A}([\beta],\frac{\delta}{Ar})$. Then $d_A([\beta],[\gamma])=\frac{1}{a}<\frac{\delta}{Ar}$ where $a>0$ is such that $d(\beta(a),\gamma(a))=A$, which means $a>r$ since $A>\epsilon\geq \delta$. By Lemma 2.1.2, $d(\gamma(r),\beta(r))\leq \frac{r}{a}A<r\frac{\delta}{Ar}A=\delta$. Thus, $\gamma(r)\in B_d(\beta(r),\delta)\cap S(x_0,r)\subset B_d(\alpha(r),\epsilon)\cap S(x_0,r)$, proving $[\gamma]\in U(\alpha,r,\epsilon)$. Thus $[\beta]\in B_{d_A}([\beta],\frac{\delta}{Ar})\subset U(\alpha,r,\epsilon)$. Recall that the cone topology is defined on $\overline{X}=X\cup\partial X$. However, the preceding lemma restricts the cone topology to the boundary since there is not a natural extension of $d_{A,x_0}$ to $\overline{X}$. We now answer two important questions: what happens if we change $A$ and what happens if we move the base point? It turns out that in both cases, the metrics are quasi-symmetric. Thus, by transitivity, all members of the $d_{A,x_0}$ family are quasi-symmetric. Let $X$ be a proper CAT(0)-space. For all $A, A'>0$, $ id_{\partial X}:(\partial X, d_{A,x_0})\to(\partial X, d_{{A',x_0}})$ is a quasi-symmetry. Fix a base point $x_0\in X$ and suppose, without loss of generality, that $A<A'$. Clearly the identity map is a homeomorphism, so we need only verify that $id_{\partial X}$ is a quasi-symmetric map. Let $\eta(t)=\frac{A'}{A}t$; we will show this a control function for $id_{\partial X}$. Suppose that $[\alpha],[\beta],[\gamma]\in \partial X$ with $d_{A,x_0}([\alpha],[\gamma])\leq d_{A,x_0}([\beta],[\gamma])$ for $t>0$. Let $\alpha, \beta,\gamma:[0,\infty)\to X$ be geodesic rays based at $x_0$ that are asymptotic to $[\alpha],[\beta],[\gamma]$, respectively. Let $a,b,a',b'>0$ be such that $$d_{A,x_0}([\alpha],[\gamma])=\frac{1}{a} \, , \, d_{A,x_0}([\beta],[\gamma])=\frac{1}{b}$$ $$d_{A',x_0}([\alpha],[\gamma])=\frac{1}{a'}\, , \, d_{A',x_0}([\beta],[\gamma])=\frac{1}{b'}$$ By convexity of CAT(0) metric and since $A'>A$, then $a\leq a'$ and $b\leq b'$. Furthermore, applying Lemma 2.1.2, $$A=d(\beta(b),\gamma(b))\leq d_{{\mathbb{E}}}(\overline{\beta(b)},\overline{\gamma(b)})=\frac{A'b}{b'}$$ Thus, $\frac{Ab'}{A'}\leq b$. Applying the above, we obtain the following inequalities: $$d_{A',x_0}([\alpha],[\gamma])=\frac{1}{a'}\leq\frac{1}{a}=d_{A,x_0}([\alpha],[\gamma])\leq td_{A,x_0}([\beta],[\gamma])=t\frac{1}{b}\leq t\frac{A'}{A}\frac{1}{b'}=\eta(t)d_{A',x_0}([\beta],[\gamma])$$ Suppose $X$ is a complete CAT(0) space. For all $x_0,x_0'\in X$,\ $id_{\partial X}:(\partial X, d_{A,x_0})\to(\partial X, d_{A,x_0'})$ is a quasi-symmetry. Let $x_0,x_0'\in X$ with $x_0\neq x_0'$. We begin by assuming $A>2d(x_0,x_0')$. We show that $\eta(t)=\left(\frac{A}{A-2d(x_0,x_0')}\right)^2t$ is a control function for $id_{\partial X}$. Suppose that $[\alpha],[\beta],[\gamma]\in\partial X$ and satisfy the inequality $d_{A,x_0}([\alpha],[\gamma])\leq td_{A,x_0}([\beta],[\gamma])$ for $t>0$. Let $\alpha,\beta,\gamma:[0,\infty)\to X$ be geodesic rays based at $x_0$ and asymptotic to the corresponding points in $\partial X$. Let $a,b\in (0,\infty)$ be such that $d_{A,x_0}(\alpha(a),\gamma(a))=A$ and $d_{A,x_0}(\beta(b),\gamma(b))=A$. Since $X$ is a complete CAT(0) space, there exists unique geodesic rays $\alpha',\beta',\gamma'$ in $X$ based at $x_0'$ and asymptotic to $\alpha, \beta, \gamma$, respectively. Let $a',b'\in(0,\infty)$ be such that $d_{A,x_0'}(\alpha'(a'),\gamma'(a'))=A$ and $d_{A,x_0'}(\beta'(b'),\gamma'(b'))=A$. There are four cases to consider: : $a'\geq a$ and $b\geq b'$. Then $$d_{A,x_0'}([\alpha],[\gamma])=\frac{1}{a'}\leq\frac{1}{a}=d_{A,x_0}([\alpha],[\gamma])\leq td_{A,x_0}([\beta], [\gamma])=t\frac{1}{b}$$$$\leq t\frac{1}{b'}=td_{A,x_0'}([\beta],[\gamma])\leq\eta (t)d_{A,x_0'}([\beta],[\gamma])$$ : $a'\geq a$ and $b<b'$. Applying Lemma 2.1.2, $d(\beta'(b),\gamma'(b))\leq\frac{Ab}{b'}$. Thus, $\frac{b'}{A}d(\beta'(b),\gamma'(b))\leq b$. Furthermore, by Remark 1, $$A=d(\beta(b),\gamma(b))\leq d(\beta(b),\beta'(b))+d(\beta'(b),\gamma'(b))+d(\gamma'(b),\gamma(b))\leq 2d(x_0,x_0')+d(\beta'(b),\gamma'(b))$$ Thus, $A-2d(x_0,x_0')\leq d(\beta'(b),\gamma'(b))$ Applying all of the above, $$d_{A,x_0'}([\alpha],[\gamma])=\frac{1}{a'}\leq \frac{1}{a}=d_{A,x_0}([\alpha],[\gamma])\leq td_{A,x_0}([\beta], [\gamma])=t\frac{1}{b}$$$$\leq t\frac{A}{d(\beta'(b),\gamma'(b))}\frac{1}{b'}\leq t\frac{A}{A-2d(x_0,x_0')}d_{A,x_0'}([\beta],[\gamma])\leq \eta(t)d_{A,x_0'}([\beta],[\gamma])$$ : $a'<a$ and $b\geq b'$ Using Lemma 2.1.2, $d(\alpha(a'),\gamma(a'))\leq \frac{Aa'}{a}$. Furthermore, by Remark 1, $$A=d(\alpha'(a'),\gamma'(a'))\leq d(\alpha'(a'),\alpha(a'))+d(\alpha(a'),\gamma(a'))+d(\gamma(a'),\gamma'(a'))\leq 2d(x_0,x_0')+d(\alpha(a'),\gamma(a'))$$ Applying the above, $$d_{A,x_0'}([\alpha],[\gamma])=\frac{1}{a'}\leq \frac{A}{d(\alpha(a'),\gamma(a'))}\frac{1}{a}\leq \frac{A}{A-2d(x_0,x_0')}\frac{1}{a}=\frac{A}{A-2d(x_0,x_0')}d_{A,x_0}([\alpha],[\gamma])$$$$\leq \frac{A}{A-2d(x_0,x_0')}td_{A,x_0}([\beta], [\gamma])=\frac{A}{A-2d(x_0,x_0')}t\frac{1}{b}\leq \frac{A}{A-2d(x_0,x_0')}t\frac{1}{b'}$$$$=\frac{A}{A-2d(x_0,x_0')}td_{A,x_0'}([\beta], [\gamma])\leq \eta(t)d_{A,x_0'}([\beta],[\gamma])$$ : $a'<a$ and $b<b'$. Using the computations in Cases 2 and 3: $$d_{A,x_0'}([\alpha],[\gamma])=\frac{1}{a'}\leq \frac{A}{A-2d(x_0,x_0')}d_{A,x_0}([\alpha],[\gamma])$$$$\leq \frac{A}{A-2d(x_0,x_0')}td_{A,x_0}([\beta], [\gamma])=\frac{A}{A-2d(x_0,x_0')}t\frac{1}{b}\leq t\left(\frac{A}{A-2d(x_0,x_0')}\right)^2\frac{1}{b'}$$$$=t\left(\frac{A}{A-2d(x_0,x_0')}\right)^2d_{A,x_0'}([\beta],[\gamma]) = \eta(t)d_{A,x_0'}([\beta],[\gamma])$$ Thus, $\eta(t)=\left(\frac{A}{A-2d(x_0,x_0')}\right)^2t$ is a control function for $id_{\partial X}$ for $A>2d(x_0,x_0')$. Now, suppose we are given any $A>0$. Since $X$ is a CAT(0) space, it is path connected. Let $\gamma:[0,d(x_0,x_0')]\to X$ be a geodesic segment connecting $x_0$ to $x_0'$. Let $\{y_0,y_1,...,y_{n-1},y_n\}$ be a partition of $[0,d(x_0,x_0')]$ where $\|x_k-x_{k-1}\|<\frac{A}{2}$ for $k=1,2,...n$ and set $x_k=\gamma(y_k)$ for $k=0,1,...,n-1$ and $x_0'=\gamma(y_n)$. From above, we know $id^k_{\partial X}:(\partial X,d_{A,x_k})\to(\partial X, d_{A,x_{k-1}})$ is a quasi-symmetry for each $k$. Theorem 2.2.3 guarantees that $id_{\partial X}=id^n_{\partial X}\circ...\circ id^1_{\partial X}:(\partial X, d_{A,x_0})\to(\partial X, d_{A,x_0'})$ is a quasi-symmetry. In the future, we will use $d_A$ to denote an arbitrary representative of the family of metrics $\{d_{A,x_0}\}$. When specific calculations are to be done, $A>0$ should be fixed and a base point $x_0$ should be chosen. In problem 46 of [@Ka07], B. Kleiner asked whether the group of isometries of a CAT(0) space acts in a “nice” way on the boundary. The following theorem provides one answer. Suppose $G$ is a finitely generated group that acts by isometries on a complete CAT(0) space $X$. Then the induced action of $G$ on $(\partial X, d_{A,x_0})$ is a quasi-symmetry. In other words, $G$ acts by quasi-symmetries on $\partial X$. Fix a base point $x_0\in X$ and $A>0$. Notice that proving this theorem relies on knowing that changing base point is a quasi-symmetry, since if $\alpha,\beta,\gamma:[0,\infty)\to X$ are geodesic rays based at $x_0$, then $$d_{A,x_0}([\alpha],[\gamma])=d_{A,gx_0}([g\alpha],[g\gamma])$$ $$d_{A,x_0}([\beta],[\gamma])=d_{A,gx_0}([g\beta],[g\gamma]).$$ This is a simple consequence of the action being by isometries. Hence, to obtain the desired inequality for a quasi-symmetric map, all we need to do is find the distances of the translated rays with respect to the base point $x_0$ rather than $gx_0$. A simple application of Theorem 3.1.4 proves $g$ is a quasi-symmetry. Dimension Results Using the $d_A$ metric ---------------------------------------- In [@BuLe07], it is shown that the linearly controlled dimension of every compact locally self-similar metric space $X$ is finite and $\ell\hbox{-dim}X=\hbox{dim}X$. Since hyperbolic group boundaries are compact and locally self-similar, we obtain the equality of linearly controlled dimension and covering dimension of hyperbolic group boundaries in Theorem 1.0.2. Swenson shows in [@Swe99] that the boundary of a proper CAT(0) space admitting a cocompact action by isometries has finite topological dimension. Since topological dimension can be defined for arbitrary topological spaces, there was no need for a metric on the boundary to prove this fact. Now that we have the $d_A$ family of metrics on the boundary, we can examine the linearly controlled metric dimension. We have been unable to show equality of the two dimensions, but we do show that linearly controlled dimension of a CAT(0) group boundary must be finite. This proof was motivated by previous work found in [@Mo14]. Suppose $G$ acts geometrically on a proper CAT(0)-space $X$. Then $\ell$-dim$(\partial X, d_A)<\infty$. This proof relies on the existence of a single cover with Property $\mathscr{P}_{R,4R}^n$ for some $R,n>0$. Suppose a group $G$ acts geometrically on a proper CAT(0) space $(X,d)$. Then for all sufficiently large $R$, there exists a finite order open cover $\mathscr{V}$ of $X$ with mesh$(\mathscr{V})\leq 4R$ and $\mathscr{L(V)}\geq R$. Let $C\subseteq X$ be a compact set with $GC=X$ and choose $R$ large enough so that $C\subseteq B(x_0,R)$ for some $x_0\in X$. Then $\mathscr{V}=\cup_{g\in G}B(gx_0,2R)$ is a finite order open cover of $X$ with mesh bounded above by $4R$. Notice that the order of $\mathscr{V}$ is finite since the action of $G$ is proper, that is only finitely many $G$-translates of any compact set $C$ can intersect $C$. Since the cover is obtained by this nice geometric action, it must look the same everywhere. Thus, the order of $\mathscr{V}$ is bounded above by the finite number of translates of $gB(x_0,2R)$ intersecting $B(x_0,2R)$. Furthermore, the Lebesgue number of $\mathscr{V}$ is at least $R$. For if we take $x\in X$ and let $g\in G$ such that $gx\in C\subseteq B(x_0,R)$. Then $d(gx, X-B(x_0,2R))\geq R$. As the action is by isometries: $$R\leq d(gx, X-B(x_0,2r))=d(x, g^{-1}(X-B(x_0,2R)))=d(x, X-g^{-1}(B(x_0, 2R)))$$$$=d(x, X-B(g^{-1}x_0,2R))$$ Since $B(g^{-1}x_0,2R)\in\mathscr{V}$, and $d(x,B(g^{-1}x_0,2R))\geq R$, then $\mathscr{L}(\mathscr{V})\geq R$. Lemma 3.2.2 proves that $\dim_{\emph{mc}} X<\infty$ for a CAT(0) space admitting a geometric action. \[Proof of Theorem 3.2.1\] Fix $A>0$. By Lemma 3.2.2, we may choose a sufficiently large $R>A$ so that there is a finite order open cover $\mathscr{V}$ of $X$ with mesh$(\mathscr{V})\leq 4R$ and $\mathscr{L(V)}\geq R$. Set $n= $order$(\mathscr{V})$. Set $t_{\lambda}=\frac{1}{\lambda}$ for each $\lambda\in(0, \infty)$, and for each $V\in\mathscr{V}$, define $$U_V=\{[\gamma] \| \gamma \text{ is a geodesic ray based at } x_0 \text{ with } \gamma(t_{\lambda})\in V\}$$ We will show that $\mathscr{U}=\cup_{V\in\mathscr{V}}U_V$ is an open cover of $\partial X$ with order bounded above by $n$, Lebesgue number at least $\lambda$ and mesh at most $\frac{4R}{A}\lambda$. Clearly $\mathscr{U}$ is an open cover since $\mathscr{V}$ is an open cover of $X$. Furthermore, since $\gamma(t_{\lambda})$ can be in at most $n$-elements of $\mathscr{V}$, then $[\gamma]$ can be in at most $n$ elements of $\mathscr{U}$. We now show the Lebesgue number must be at least $\lambda$. Let $[\gamma]\in\partial X$ and $\gamma$ a geodesic ray in $X$ based at $x_0$ and asymptotic to $[\gamma]$. Since $\mathscr{L(V)}\geq R$, there is some $V\in\mathscr{V}$ such that $d(\gamma(t_{\lambda}), X-V)\geq R$. Consider then $d_A([\gamma],\partial X-U_V)$. If $[\beta]\in \partial X-U_V$, then $\beta(t_{\lambda})\notin V$ and hence $d(\gamma(t_{\lambda}),\beta(t_{\lambda}))\geq R$. Letting $a\in (0,\infty)$ be such that $d(\gamma(a), \beta(a))=A$, then $a\leq t_{\lambda}$ since $R\geq A$. Hence, $$d_A([\gamma],[\beta])=\frac{1}{a}\geq \frac{1}{t_{\lambda}}=\lambda$$ Hence, $d_A([\gamma],\partial X-U_V)\geq \lambda$, so $\mathscr{L}(\mathscr{V})\geq \lambda$. Lastly, we show mesh$(\mathscr{U})\leq \frac{4R}{A}\lambda$. Let $[\alpha],[\beta]\in U_V$ for some $U_V\in\mathscr{U}$. Let $\alpha,\beta$ be geodesic rays in $X$ based at $x_0$ and asymptotic to $[\alpha]$ and $[\beta]$, respectively. Let $a\in(0,\infty)$ be such that $d(\alpha(a), \beta(a))=A$. Since $\alpha(t_{\lambda}),\beta(t_{\lambda})\in V$, then $d(\alpha(t_{\lambda}),\beta(t_{\lambda}))\leq 4R$. There are then two cases to consider: : $d(\alpha(t_{\lambda}),\beta(t_{\lambda}))\leq A$. Then $a\geq t_{\lambda}$, so $$d_A([\alpha],[\beta])=\frac{1}{a}\leq\frac{1}{t_{\lambda}}=\lambda\leq \frac{4R}{A}\lambda$$ : $A\leq d(\alpha(t_{\lambda}),\beta(t_{\lambda}))\leq 4R$. Then $a\leq t_{\lambda}$, and by Lemma 2.1.2, $d(\alpha(a),\beta(a))\leq \frac{a}{t_{\lambda}}d(\alpha(t_{\lambda}),\beta(t_{\lambda}))$. Thus, $$A=d(\alpha(a),\beta(a))\leq \frac{a}{t_{\lambda}}d(\alpha(t_{\lambda}),\beta(t_{\lambda}))\leq \frac{a}{t_{\lambda}}(4R)$$ Rearranging, we obtain that $a\geq \frac{At_{\lambda}}{4R}$, and thus: $$d_A([\alpha],[\beta])=\frac{1}{a}\leq\frac{4R}{At_{\lambda}}=\frac{4R}{A}\lambda$$ Thus, there exists a $c\geq 1$ such that for every $\lambda>0$, there is an open cover $\mathscr{U}$ of $\partial X$ with order$(\mathscr{U})\leq n$, $\mathscr{L}(\mathscr{U})\geq\lambda$ and mesh$(\mathscr{U})\leq c\lambda$, proving $\ell$-dim$(\partial X, d_A)<\infty$. The above proof really only required the existence of a single finite order uniformly bounded open cover with large Lebesgue number. Thus, if we know a proper CAT(0) space has finite asymptotic dimension, we do not need a group action to provide such a cover. We point out that there are some CAT(0) spaces that are known to have finite asymptotic dimension: ${\mathbb{R}}^n$ for all $n\geq 0$, Gromov hyperbolic CAT(0) spaces, and CAT(0) cube complexes [@WR12]. Thus, there are spaces for which the following proposition will apply. Suppose $(X,d)$ is a proper CAT(0) space with finite asymptotic dimension. Then $\ell$-dim$(\partial X,d_A)\leq$ asdim$X$. Fix $A>0$. Since asdim$X\leq n$ for some $n>0$, there exists a uniformly bounded cover $\mathscr{V}$ with order $\mathscr{V}\leq n+1$ and $\mathscr{L(V)}\geq R$ for some $R\geq A$. We may assume that this cover is also open, because if it is not, we can simply choose a larger $R$, “push in” the cover $\mathscr{V}$ using the Lebesgue number, and obtain a smaller open cover with the desired properties. Repeat the same argument as in the proof of Theorem 3.2.2 to obtain an open cover $\mathscr{U}$ of $\partial X$ with order at most $n+1$, $\mathscr{L(U)}\geq \lambda$ and mesh$\mathscr{U}\leq \frac{\text{mesh}\mathscr{V}}{A}\lambda$. The $\overline{d}_{x_0}$-metrics ================================ To define the second family of metrics on $\partial X$, fix a base point $x_0\in X$. For $[\alpha],[\beta]\in\partial X$, let $\alpha:[0,\infty)\to X$ and $\beta:[0,\infty)\to X$ be the unique representatives of $[\alpha]$ and $[\beta]$ based at $x_0$. Define $\overline{d}_{x_0}:\partial X\times\partial X\to {\mathbb{R}}$ by $$\overline{d}_{x_0}([\alpha],[\beta])=\int_0^{\infty}\frac{d(\alpha(r),\beta(r))}{e^r}\, dr$$ This family of metrics, unlike the $d_{A}$ metrics, takes into account the entire timespan of the geodesic rays. Due to this fact, it can naturally be extended to $\overline{X}=X\cup\partial X$. To do so, consider $x,y\in X$. Let $c_x:[0,d(x_0,x)]\to X$ be the geodesic from $x_0$ to $x$ and $c_y:[0,d(x_0,y)]\to X$ the geodesic segment from $x_0$ to $y$. Extend $c_x$ to $c'_x:[0,\infty)\to X$ by letting $c'_x(r)=x$ for all $r>d(x_0,x)$ and $c'(r)=c(r)$ otherwise. Extend $c_y$ to $c'_y:[0,\infty)\to X$ in a similar fashion. Then $$\overline{d}_{x_0}(x,y)=\int_0^\infty\frac{d(c'_x(r),c'_y(r))}{e^r}\, dr$$ Basic Properties of the $\overline{d}_{x_0}$ metrics ---------------------------------------------------- The following lemma that $\overline{d}_{x_0}$ is a metric is trivial. If $(X,d)$ is a proper CAT(0) space and $x_0\in X$, then $\overline{d}_{x_0}$ is a metric on $\partial X$. The topology induced on $\overline{X}=X\cup \partial X$ by the $\overline{d}_{x_0}$ metric is equivalent to the cone topology on $\overline{X}$. Fix $x_0\in X$. We will denote $\overline{d}_{x_0}$ by $\overline{d}$. We first show the cone topology is finer than the metric topology by considering points in $X$ and $\partial X$, respectively. Let $y\in X$ and $B_{\overline{d}}(x,\epsilon)$ be a basic open set in $\overline{X}$ containing $y$ for some $\epsilon>0$ and $x\in \overline{X}$. Choose $\delta>0$ such that $B_{\overline{d}}(y,\delta)\subset B_{\overline{d}}(x,\epsilon)$ and $B_{\overline{d}}(y,\delta)\cap \partial X=\emptyset$. Consider the basic open set $B_d(y,\delta)$ in the cone topology. Clearly, $y\in B_d(y,\delta)$ and if $z\in B_d(y,\delta)$, then $z\in B_{\overline{d}}(y,\delta)$ since $\overline{d}(y,z)<d(y,z)$. Thus, $$y\in B_d(y,\delta)\subset B_{\overline{d}}(y,\delta)\subset B_{\overline{d}}(x,\epsilon)$$ Now, let $[\beta]\in\partial X$, and consider the basic open set $B_{\overline{d}}(x,\epsilon)$ for $\epsilon>0$ and $x\in\overline{X}$. Choose $\delta>0$ such that $B_{\overline{d}}([\beta],\delta)\subset B_{\overline{d}}(x,\epsilon)$. Let $t>0$ be such that $e^{-t}<\delta/4$ and consider the basic open set $U(\beta,t,\frac{\delta}{2})$ in the cone topology. Clearly $[\beta]\in U(\beta,t,\frac{\delta}{2})$, so if $[\gamma]\in U(\beta,t,\frac{\delta}{2})\cap\partial X$, then $$\overline{d}([\beta],[\gamma])=\int_0^t\frac{d(\beta(r),\gamma(r))}{e^r}\, dr+\int_t^\infty\frac{d(\beta(r),\gamma(r))}{e^r}\, dr$$ $$\leq \int_0^t\frac{d(\gamma(t),\beta(t))}{e^r}\, dr+\int_t^\infty\frac{2(r-t)+d(\gamma(t),\beta(t))}{e^r}\, dr$$ $$=d(\gamma(t),\beta(t))+\frac{2}{e^t}$$ $$<\frac{\delta}{2}+\frac{\delta}{2}=\delta$$ Moreover, if $y\in U(\beta,t,\frac{\delta}{2})\cap X$ and $c_y:[0,d(x_0,y)]\to X$ is the geodesic from $x_0$ to $y$, then $$\overline{d}([\beta], x)=\int_0^t\frac{d(c_y(r),\beta(r))}{e^r}\, dr+\int_t^\infty\frac{d(c_y(r),\beta(r))}{e^r}\, dr$$ $$<\int_0^t\frac{d(c_y(t),\beta(t))}{e^r}\, dr+\int_t^\infty\frac{(r-t)+d(c_y(t),\beta(t))}{e^r}\, dr$$ $$<\frac{3\delta}{4}<\delta$$ These two calculations show $U(\beta,t,\frac{\delta}{2})\subset B_{\overline{d}}([\beta,\delta])$ and thus, $$[\beta]\in U(\beta,t,\frac{\delta}{2})\subset B_{\overline{d}}([\beta,\delta])\subset B_{\overline{d}}(x,\epsilon)$$ Now, we show the metric topology is finer than the cone topology, again by considering points in $X$ and $\partial X$. Let $y\in X$ and $B$ a basic open set in the cone topology. Choose $\delta>0$ such that $B_d(y,\delta)\subset B\cap X$. Consider the basic open set $B_{\overline{d}}(y,R)$ where $R=\frac{\delta}{e^{d(x_0,y)}}$ (if necessary, choose $R$ smaller so that $B_{\overline{d}}(y,R)\subset X$). Let $z\in B_{\overline{d}}(y,R)$ and $c_y$ and $c_z$ the geodesics connecting $x_0$ to $y$ and $z$, respectively. Set $t=\hbox{max}\{d(x_0,y),d(x_0,z)\}$. Then $$\overline{d}(y,z)>\int_t^\infty\frac{d(c_z(r),c_y(r))}{e^r}\, dr$$ $$=\int_t^\infty\frac{d(y,z)}{e^r}\, dr$$ $$=\frac{d(y,z)}{e^t}$$ $$\geq \frac{d(y,z)}{e^{d(x_0,y)}}$$ Since $\overline{d}(y,z)<\frac{\delta}{e^{d(x_0,y)}}$, by the above calculation, $z\in B_d(y,\delta)$ proving $$y\in B_{\overline{d}}(y,R)\subset B_d(y,\delta)\subset B$$ For a boundary point $[\beta]\in \partial X$, let $U(\alpha,t,\epsilon)$ be a basic open set containing $[\beta]$ for $t,\epsilon>0$ and $\alpha$ a geodesic ray based at $x_0$. Choose $1>\delta>0$ so that\ $B_d(\beta(t), \delta)\cap S(x_0, t)\subset B_d(\alpha(t),\epsilon)\cap S(x_0,t)$. Consider the basic open set $B_{\overline{d}}([\beta],\frac{\delta}{e^t})$. If $[\gamma]\in B_{\overline{d}}([\beta],\frac{\delta}{e^t})\cap\partial X$, then $d(\beta(t),\gamma(t))<\delta$. Otherwise, $$\overline{d}([\gamma],[\beta])\geq\int_t^\infty\frac{\delta}{e^r}\, dr=\frac{\delta}{e^t}$$ Thus, $d(\gamma(t), \beta(t))<\delta<\epsilon$, so $[\gamma]\in U([\alpha],t,\epsilon)$. If $x\in B_{\overline{d}}([\beta],\frac{\delta}{e^t})\cap X$, we first notice that $\overline{d}(x,[\beta])\geq \overline{d}([\beta],\beta(d(x_0,x)))=e^{-d(x_0,x)}$. Thus, $d(x_0, x)\geq t$, otherwise $x\notin B_{\overline{d}}([\beta],\frac{\delta}{e^t})$. By the same argument just given for a boundary point, we see that $d(c_x(t),\beta(t))<\delta$ proving $x\in U([\alpha],t,\epsilon)$. Thus, $$[\beta]\in B_{\overline{d}}\left([\beta],\frac{\delta}{e^t}\right)\subset U([\alpha],t,\epsilon)$$ Thus far, we have been unable to prove analogs of Lemma 3.1.4 and Theorem 3.1.5 for this family of metrics. However, we will see that there are some significant advantages in using $\overline{d}_{x_0}$ for comparing dimension properties of $\partial X$ and $X$. In particular, we use the $\overline{d}_{x_0}$ metric to obtain a weak solution to Question 1.0.1 (which we have been unable to accomplish using the $d_A$ metrics). Dimension Results Using the $\overline{d}_{x_0}$ Metrics -------------------------------------------------------- Suppose $X$ is a geodesically complete CAT(0) space and $\ell$-dim$\partial X\leq n$, where $\partial X$ is endowed with the $\overline{d}_{x_0}$ metric. Then the macroscopic dimension of $X$ is bounded above by $2n+1$. The proof “pushes in” covers of the boundary obtained by knowing finite linearly controlled metric dimension of the boundary to create covers of the entire space. We will show that there exists a uniformly bounded cover $\mathscr{V}$ of $X$ with order$\mathscr{V}\leq 2n+1$. Fix a base point $x_0\in X$. Since $\ell-$dim$\partial X\leq n$, there exists constants $\lambda_0\in(0,1)$ and $c\geq 1$ and $n+1$-colored coverings (by a single coloring set $A$) $\mathscr{U}_k$ of $\partial X$ with - mesh$\mathscr{U}_k\leq c\lambda_k$ - $\mathscr{L}(\mathscr{U}_k)\geq\lambda_k/2$ - $\mathscr{U}_k^a$ is $\lambda_k/2$-disjoint for each $a\in A$. where $\lambda_k\leq\lambda_0$. Such a cover is guaranteed by [@BuSc07 Lemma 11.1.3]. Choose $R>0$ so that $\frac{4}{e^R}<\lambda_0$ and set $\lambda_k=\frac{4}{e^{kR}}$. Let $B_k=\{x\in X\|(k+\frac{1}{2})R\leq d(x,x_0)\leq (k+\frac{3}{2})R$ be an the annulus centered at $x_0$ for each $k=1,2,3,...$. We will cover each of these $B_k$ by “pushing in” the cover $\mathscr{U}_k$ of the boundary. To do so, let $$V_{U_k}=\{\gamma(kR,(k+2)R)\|\gamma \text{ is a geodesic ray with } [\gamma]\in U_k\}$$ and $\mathscr{V_k}=\cup_{U_k\in \mathscr{U}_k}V_{U_k}$. Clearly $\mathscr{V}_k$ is a cover of $B_k$. ![image](macdim.pdf) $\mathscr{V}_k$ is $(n+1)$-colored by the same set $A$. That is, $\mathscr{V}_k^a$ is a disjoint collection of sets for each $a\in A$. Suppose otherwise. That is, that there exists $V_U, V_{U'}\in \mathscr{V}_k^a$ with $V_U\cap V_{U'}\neq \emptyset$. If $x\in V_U\cap V_{U'}$ then there exists geodesic rays $\alpha$ and $\beta$ passing through $x$ with $[\alpha]\in U$ and $[\beta]\in U'$. Since $U,U'\in\mathscr{U}_k^a$, then $\overline{d}([\alpha],[\beta])\geq \lambda_k/2$. Thus, $$\frac{\lambda_k}{2}\leq \overline{d}([\alpha],[\beta])=\int_0^\infty\frac{d(\alpha(r),\beta(r))}{e^r}\, dr$$ $$=\int_{d(x,x_0}^\infty \frac{d(\alpha(r),\beta(r))}{e^r}\, dr$$ $$\leq \int_{d(x,x_0}^\infty \frac{2(r-d(x,x_0)}{e^r}\, dr$$ $$=\frac{2}{e^{d(x,x_0)}}$$ $$<\frac{2}{e^{kR}}=\frac{\lambda_k}{2}$$ The last line provides the required contradiction. Thus, order$(\mathscr{V}_k)\leq n$ for each $k$. For every $x,y\in V_{U_k}\in\mathscr{V}_k$ with $d(x_0,x)=(k+2)R=d(x_0,y)$, then $d(x,y)\leq 4ce^{2R}$. To show this, suppose otherwise. Choose $x,y\in \mathscr{V}_k$ with $d(x_0,x)=(k+2)R=d(x_0,y)$ and $d(x,y)> 4ce^{2R}$. Let $\gamma_x$ and $\gamma_y$ be geodesic rays based at $x_0$ with $[\gamma_x],[\gamma_y]\in U_k$ and such that $\gamma_x((k+2)R)=x$ and $\gamma_y((k+2)R)=y$. Thus, $$\overline{d}([\gamma_x],[\gamma_y])\geq \int_{(k+2)R}^\infty\frac{d(\gamma_x(r),\gamma_y(r))}{e^r}\, dr$$ $$>\int_{(k+2)R}^\infty\frac{4ce^{2R}}{e^r}\, dr$$ $$=\frac{4c}{e^{kR}}=c\lambda_k$$ Since $[\gamma_x],[\gamma_y]\in U_k$ and mesh$\mathscr{U}_k\leq c\lambda_k$, we obtain the desired contradiction. mesh$\mathscr{V}_k\leq 4ce^{2R}+2R$. Let $x,y\in V_{U_k}\in\mathscr{V}_k$. Let $\gamma_x$ and $\gamma_y$ be geodesic rays based at $x_0$ passing through $x$ and $y$, respectively. Suppose $\gamma_x(t)=x$ and $\gamma_y(s)=y$ for $t,s\in (kR,(k+2)R)$. Without loss of generality, suppose $s\leq t$. Then $$d(x,y)\leq d(x,\gamma_x(s))+d(\gamma_x(s),\gamma_y(s))$$ $$=(t-s)+d(\gamma_x(s),\gamma_y(s))$$ $$\leq 2R+d(\gamma_x((k+2)R),\gamma_y((k+2)R))$$ $$\leq 2R+4ce^{2R}$$ Thus, we have shown that mesh$\mathscr{V}_k\leq 4ce^{2R}+2R$ and order$\mathscr{V}_k\leq n$ for every $k$. Since $\mathscr{V}_k\cap\mathscr{V}_{k-1}=\emptyset$, then $\cup \mathscr{V}_k$ is a uniformly bounded cover of $X-B(x_0, \frac{3}{2}R)$ with order bounded above by $2n$. Letting $\mathscr{V}=\cup\mathscr{V}_k\cup B(x_0,2R)$ we obtain our desired cover. The missing piece in the above argument that would prove finite asymptotic dimension is having arbitrarily large Lebesgue numbers for the cover. Thus, this argument is a potential step in finally answering the open asymptotic dimension question. Examples ======== The previous sections highlight important properties and results that can be obtained using the $d_A$ and $\overline{d}$ metrics. Many of the results we obtained with the given techniques worked for one metric, but not the other. That is of course not to say that the same results cannot be obtained using different methods with the other metric. However, the different results do provide interesting comparisons between the two metrics and some insight into each ones strengths or weaknesses. In this section, we highlight some other differences by showing calculations done on $T_4$, the four valent tree. In this example, we show that $\overline{d}_{x_0}$ is a visual metric on $\partial T_4$, but $d_A$ is not a visual metric on $T_4$. Recall that a metric $d$ on the boundary of a hyperbolic space is called a visual metric with parameter $a>1$ if there exists constants $k_1,k_2>0$ such that $$k_1a^{-(\zeta,\zeta')_p}\leq d(\zeta,\zeta')\leq k_2a^{-(\zeta,\zeta')_p}$$ for all $\zeta,\zeta'\in\partial X$. \[Here $(\zeta,\zeta')_p$ is the extended Gromov product based at $p\in X$. See [@BH99] for more information on visual metrics.\] Fix a base point $x_0\in X$ and $A>0$. Let $[\alpha],[\beta]\in\partial T_4$ and let $\alpha, \beta:[0,\infty)\to T_4$ be the corresponding geodesic rays based at $x_0$. Set $t=\emph{max}\{r\|d(\alpha(r),\beta(r))=0\}$. Then $d(\alpha(r),\beta(r))=2(r-t)$ for all $r\geq t$. A simple computation shows: $$\overline{d}_{x_0}([\alpha],[\beta])=\int_t^\infty\frac{2(r-t)}{e^r}\, dr=\frac{2}{e^t}$$ Furthermore, since $([\alpha],[\beta])_{x_0}=t$, we see that $\overline{d}_{x_0}$ is a visual metric on $T_4$ with parameter $e$. Now, suppose, by way of contradiction, that $d_A$ is visual with parameter $a>1$. Then there exists $k_1,k_2>0$ such that $k_1a^{-(\zeta,\zeta')_{x_0}}\leq d_A(\zeta,\zeta')\leq k_2a^{-(\zeta,\zeta')_{x_0}}$ for all $\zeta,\zeta'\in\partial X$. Choose $n\in {\mathbb{Z}}^+$ large enough such that $\frac{a^n}{n+1}>k_2a^{A/2}$, which is possible since\ $\lim_{n\to\infty}\frac{a^n}{n+1}=\infty$. Let $\alpha,\beta:[0,\infty)\to X$ be any two proper geodesic rays based at $x_0$ with the property that $\alpha(t)=\beta(t)$ for all $t\leq \left \lceil{n-\frac{A}{2}}\right \rceil$ and $\alpha(t)\neq\beta(t)$ for all $t>\left \lceil{n-\frac{A}{2}}\right \rceil$ (that is, $\alpha$ and $\beta$ are two rays that branch at time $t=\left \lceil{n-\frac{A}{2}}\right \rceil$. Notice then that $$d_A([\alpha],[\beta])=\frac{1}{\left \lceil{n-\frac{A}{2}}\right \rceil+\frac{A}{2}} \, \text{ and } \, ([\alpha],[\beta])_{x_0}=\left \lceil{n-\frac{A}{2}}\right \rceil$$ By the visibility assumption, $$d_A([\alpha],[\beta])\leq k_2a^{-([\alpha],[\beta])_{x_0}}$$ and thus, $$\frac{1}{\left \lceil{n-\frac{A}{2}}\right \rceil+\frac{A}{2}}\leq k_2a^{-([\alpha],[\beta])_{x_0}}$$ Since $\left \lceil{n-\frac{A}{2}}\right \rceil\geq n-\frac{A}{2}$ and $\left \lceil{n-\frac{A}{2}}\right \rceil\leq n-\frac{A}{2}+1$, we obtain the following inequality: $$\frac{1}{n+1}\leq\frac{1}{\left \lceil{n-\frac{A}{2}}\right \rceil+\frac{A}{2}}\leq k_2a^{-([\alpha],[\beta])_{x_0}}=k_2a^{-\left \lceil{n-\frac{A}{2}}\right \rceil}\leq k_2a^{-(n-\frac{A}{2})}$$ Rearranging, we see that $$\frac{a^n}{n+1}\leq k_2a^{A/2},$$ a contradiction to the choice of $n$. $id_{\partial X}:(\partial X,d_A)\to (\partial X,\overline{d})$ is not a quasi-symmetry. We prove this proposition by showing it in the case that $X=T_4$. For this, we need the following lemma. $(\partial T_4,d_A)$ is uniformly perfect. Fix a base point $x_0\in T_4$. It suffices to show $(\partial T_4, d_1)$ is uniformly perfect since $(\partial T_4, d_A)$ is quasi-symmetric to $(\partial T_4, d_1)$ for every $A>0$ by Lemma 3.1.3. Let $[\alpha]\in\partial T_4$ and $\alpha:[0,\infty)\to T_4$ the ray asymptotic to $[\alpha]$ based at $x_0$. Since diam$(T_4,d_1)=2$, we show that $B([\alpha],r)-B([\alpha],\frac{r}{4})\neq\emptyset$ for all $0<r<2$. Consider the geodesic ray $\beta:[0,\infty)\to T_4$ based at $x_0$ with $\alpha(t)=\beta(t)$ for all $t\leq \lceil \frac{1}{r}\rceil$ and $\alpha(t)\neq\beta(t)$ for all $t>\lceil\frac{1}{r}\rceil$. Then, $d_1([\alpha],[\beta])=\frac{1}{\lceil1/r\rceil+1/2}$ which means $d_1([\alpha],[\beta])<r$. Moreover, $\lceil\frac{1}{r}\rceil +\frac{1}{2}\leq \frac{1}{r}+1+\frac{1}{2}<\frac{1}{r}+\frac{3}{r}$, so $d_1([\alpha],[\beta])>\frac{r}{4}$. This proves $[\beta]\in B([\alpha],r)-B([\alpha],\frac{r}{4})$. Let $X=T_4$. We will show that $id:(\partial T_4,d_A)\to(\partial T_4,\overline{d})$ is not a quasi-symmetry for $A=1$ and then refer to Proposition 3.1.3 for the full claim. Fix a base point $x_0\in T_4$ and suppose, by way of contradiction, that $id:(\partial T_4,d_1)\to(\partial T_4,\overline{d})$ is a quasi-symmetry. By Theorem 2.2.4 and Lemma 5.0.4, $\eta$ must be of the form $\eta(t)=c\, \text{max}\{t^{\delta},t^{1/\delta}\}$ where $c\geq 1$ and $\delta\in(0,1]$ depends only on $f$ and $X$. Let $\alpha,\gamma:[0,\infty)\to T_4$ be two proper geodesic rays such that $\alpha(t)\neq\gamma(t)$ for all $t>0$. Then $$d_1([\alpha],[\gamma])=\frac{1}{1/2}=2$$ $$\overline{d}([\alpha],[\gamma])=\int_0^{\infty}\frac{2r}{e^r}\, dr =2$$ Choose $n\in{\mathbb{Z}}^+$ large enough such that $n-\frac{1}{\delta}\ln(2n+1)>\ln(c)$, which is possible since $\lim_{n\to\infty} n-\frac{1}{\delta}\ln(2n+1)=\infty$. Let $\beta:[0,\infty)\to T_4$ be a proper geodesic ray with the property that $\beta(t)=\gamma(t)$ for all $t\leq n$ and $\beta(t)\neq\gamma(t)$ for all $t>n$. Then $$d_1([\beta],[\gamma])=\frac{1}{n+1/2}=\frac{2}{2n+1}$$ $$\overline{d}([\beta],[\gamma])=\int_n^{\infty}\frac{2(r-n)}{e^r}\, dr =\frac{2}{e^n}$$ Set $t=\frac{d_1([\alpha],[\gamma])}{d_1([\beta],[\gamma])}=2n+1$. By the quasi-symmetry assumption, $$\overline{d}([\alpha],[\gamma])\leq \eta(t)\overline{d}([\beta],[\gamma])$$ and thus, $$2\leq\eta(2n+1)\frac{2}{e^n}$$ $$\Rightarrow e^n\leq\eta(2n+1)=c\, \text{max}\{(2n+1)^{\delta},(2n+1)^{1/\delta}\}$$ $$\Rightarrow e^n\leq c(2n+1)^{1/\delta}$$ $$\Rightarrow n\leq \ln(c)+\frac{1}{\delta}\ln(2n+1)$$ This last inequality contradicts the choice of $n$, proving our claim. Open Questions ============== Since metrics on visual boundaries of CAT(0) spaces have not been widely studied, there is still much work to be done in this area. We hope that the results here show the development of these metrics is worthwhile and provides the opportunity to study CAT(0) boundaries from a different point of view, which may of course lead to answering interesting unanswered questions about these boundaries. We end with a list of open questions. Is there an extension of $d_A$ to $\overline{X}$ that is equivalent to the cone topology on $\overline{X}$? In the proof of Theorem 3.1.5, a different control function is used for each $g\in G$. Is there a single control function for the entire group? Are all of the members of the $\overline{d}_{x_0}$ family of metrics quasi-symmetric? The answer to this question is yes in the extreme cases that $X$ is ${\mathbb{R}}^2$ or the four-valent tree by simple calculations. If it can be shown that the answer is yes for any CAT(0) space $X$, then we could easily show that the group of isometries of a CAT(0) space acts by quasi-symmetries on the boundary as in Theorem 3.1.5. Is the linearly controlled dimension of CAT(0) group boundaries finite when the boundary is endowed with the $\overline{d}_{x_0}$ metric? Furthermore, if the answer to Question 6.0.7 is no, can a CAT(0) boundary with two different metrics from the same family $\{\overline{d}_{x_0}\}$ have different linearly controlled dimension? For a hyperbolic group $G$, $\ell\emph{-dim}\partial X=\emph{dim}\partial X$. Can the same be said for CAT(0) group boundaries? In particular, can it be shown for a CAT(0) group $G$, $\ell\emph{-dim}\partial X\leq \emph{dim}\partial X$ with respect to either the $d_A$ metric or $\overline{d}$ metric? In Example 1, we showed that $\overline{d}_{x_0}$ is a visual metric on $\partial T_4$. Is $\overline{d}_{x_0}$ a visual metric on the boundary of any $\delta$-hyperbolic space? [^1]: The contents of this paper constitutes part of the author’s dissertation for the degree of Doctor of Philosophy at the University of Wisconsin-Milwaukee under the direction of Professor Craig Guilbault.
--- abstract: 'The RXTE observations of GRS 1915+105 have given a new impulse to the study of spectral and timing properties of the X-ray emission of black hole candidates. At variance with any other known source, GRS 1915+105 shows dramatic changes on time scales as short as a tenth of a second. These changes are associated to marked spectral changes which have been interpreted as changes of the observable inner radius of the accretion disk. I review the existing results and discuss the current evidence for such disk oscillations. Independently of the precise theoretical models, the detailed study of these fast variations can provide an extremely valuable insight on the accretion processes onto black holes. Making use of these results, I compare the properties of GRS 1915+105 with those of other black hole candidates.' author: - Tomaso date: 'October 26, 2000' title: Inner disk oscillations --- Introduction: black hole candidates =================================== In the recent years, a classification scheme for the X-ray emission of black hole candidates has emerged. From the timing and spectral properties, four separate states (plus the quiescent state for transient systems) have been identified, probably in dependence of the accretion rate level (see van der Klis 1995). Unfortunately, all known persistent sources, Cyg X-1, GX 339-4, LMC X-1 and LMC X-3, show state transitions very rarely, making the accumulation of extensive datasets on different states difficult. Transient systems are more promising, but their transient nature limits the number of states and state transitions that can be observed from one system, and the comparison between different systems is always problematic. The spectral properties of black hole candidates are usually characterized in terms of a double-component model. The first component is thermal and (when present) contributes mostly to the flux below 10 keV. It is commonly interpreted as emission from an optically thick accretion disk (Mitsuda et al. 1984). The second component is much harder and extends above 10 keV. In its simple form, it can be approximated as a power law, although more sophisticated models are often required (see e.g. Frontera et al. 2000). An attractive feature of the optically thick disk model, the so-called [disk-blackbody]{}, is that one of its parameters is the value of the inner radius of the accretion disk. Therefore, the application of this model can in principle yield the measurement of this fundamental parameter, although a precise value depends on the distance and the inclination of the system. This model became rather popular when a number of black hole systems showed a rather constant inner disk radius, around 20-30 km, despite large changes in accretion rate (see Tanaka & Lewin 1995). Although this model is simplified and leads to an underestimate of the values of the radii, it can detect large variations in the inner radius of the disk (see Merloni, Fabian & Ross 2000). GRS 1915+105: inner disk oscillations ===================================== The galactic Microquasar GRS 1915+105 is known to exhibit dramatic variability in the soft (1-20 keV) energy band (see Greiner, Morgan & Remillard 1996; Belloni et al. 2000) and since its appearance in 1992 it has remained active up to the time of writing. This variability is accompanied by strong spectral changes (Greiner, Morgan & Remillard 1996; Belloni et al. 1997a,b; Muno, Morgan & Remillard 1999), providing an ideal system to study spectral transitions in a black-hole candidates: the source is always observable and significant spectral variability is present on very short time scales. At the same time, a complex behavior is observed in the power density spectra of the source (see Rao, this volume). Right: corresponding ![Count rate, inner disk temperature and radius as a function of time of the 1997 Jun 18th of GRS 1915+105 (from Belloni et al. 1997b).](belloni_TIII-01_fig1.ps){width="18pc"} ![Relation between disk re-fill time and inner disk radius for the same observation shown in Figure 1 (from Belloni et al. 1997b).](belloni_TIII-01_fig2.ps){width="18pc"} Belloni et al. (1997a,b) proposed a model for the observed spectral variations, based on fits to RXTE/PCA data with the approximated spectral shape described above. They interpret the observations as the onset of a thermal-viscous instability, during which the innermost part of the accretion disk becomes unobservable and is slowly refilled from the outer parts. This interpretation is based on the measurement of a variable inner disk radius from energy spectra (see Figure 1) and is supported by the observation of the expected dependence between size of the unstable region and refill time (Figure 2). The observed radius variations are too large to be attributed to spurious effects due to the approximate form of the model. This is what I will refer to as ‘inner disk oscillations’. The modeling by these authors was rather qualitative (with the exception of the estimate of the refill time scale), but other authors have developed more accurate models for this process (Szuszkiewicz & Miller 1998, Nayakshin, Rappaport & Melia 2000, Janiuk et al. 2000). Spectral states of GRS 1915+105 =============================== Belloni et al. (2000) analyzed a large number of RXTE/PCA observations of GRS 1915+105 and obtained a subdivision of the observed phenomenology into 12 separate classes. From this classification, they identified three basic states, the alternation of which cause all of the observed variability. The three classes, called A, B and C, are shown schematically in Figure 3, which represents a color-color diagram (both colors increase with increasing hardness of the spectrum). These states correspond to the three states already identified by Markwardt, Swank & Taam (1999). Class B correspond to the “normal” state of a black hole candidate at high accretion rate (the very high state), with an optically thick accretion disk extending down to the last stable orbit and a steep power-law component. State C corresponds to the instability periods: the accretion disk stops at a larger radius than in state B, while the power law component is harder. State A is a new state, not recognized earlier. It corresponds to an accretion disk like in state B, but with a lower temperature, and therefore lower local accretion rate. The relatively fast transition time between A and B is consistent with being the viscous time scale at the innermost stable orbit around the black hole. The observed variability consists of transitions between these three states: all possible transitions between two of the states are observed, with the exception of C to B transitions, of which no example has been found. There are intervals of time, even as long as a month, when the source is found only in state C. These are the ‘plateau’ intervals observed in the radio band (see Fender 1999). These are consistent with being long instability intervals, when a large missing inner disk is slowly refilled at a lower external accretion rate (Belloni et al. 2000). On the other hand, there are very fast variations as well. During some observations, variability of a factor of five in 0.1 seconds has been observed, corresponding to fast A–B/B–A transitions (Belloni et al. 2000). Timing properties ================= It is important to connect what is observed in the time domain and the state transitions described above. As a first step, I consider the three types of QPOs observed in GRS 1915+105: - 1-10 Hz QPO: (see Rao, this volume). This QPO is only observed during state C. Its central frequency varies systematically with time, count rate and hardness. Since state C is the only state when systematic changes in the inner disk radius are observed, it is natural to associate this oscillation to the inner radius, although often the missing part of the disk is so large that no disk component is observed directly, indicating that the QPO must be related to the power law component. - 67 Hz QPO: This QPO appears only during state B. Its high and rather constant frequency and its association with a state with small (and possibly constant) inner disk radius also point to a connection to the inner disk radius. It is interesting to note that this QPO is not observed in all state-B intervals, indicating that there must be another parameter involved in its production. - Low-$\nu$ QPO: These oscillations, in the range 10-100 seconds (see Morgan, Remillard & Greiner 1997) [are]{} the regular transitions between states that are often observed. The instability model provides an interpretation for the variability, but does not cast light on why the light curves are so regular. There is another important point about the timing properties of the source. Looking at the light curves (see Belloni et al. 2000), one notices not only that their time structure is very complex, but also it often repeats in an almost undistinguishable way, so that all observations can be grouped in twelve classes. As mentioned above, there is no clear explanation for why the structure of the light curves is the observed one. More than this, sometimes even the finest structures of the light curves repeat at a distance of years (see Figure 4). The basic question that need to be answered are: why do the light curves have those complex and repeatable shapes, and why do the light curves have [only]{} those complex shapes, with only a few possibilities to choose from? Answering these questions could provide crucial information about the accretion phenomenon. ![Two PCA light curves of GRS 1915+105 separated by almost two years. The similarity between the two panels, even in some small details, is evident.](belloni_TIII-01_fig4.ps){width="18pc"} A unique source? ================ Another important question is: are there other sources that show the same phenomenology seen here? To this date, the answer is: the C-state events (the instability) are observed only in GRS 1915+105, but A–B transitions have been observed in other systems, namely 4U 1630-47 and GRO J1655-40 (Trudolyubov, Borozdin & Priedhorsky 2000, see Figure 5). Possibly, the dips and the so-called flip-flops seen earlier in GX 339-4 (Miyamoto et al. 1991) are also of the same nature. It seems clear that this type of fast (and often very regular) temperature oscillations are more common among bright black hole candidates, although their origin is still unknown. The question of why the C-state instability is observed only in this source is still basically unanswered. Possibly, this type of instability appears only at very high levels of accretion rate, which would then be reached only by GRS 1915+105. The observation of the same phenomenon in another source would greatly help to understand this issue. Relation to the canonical black hole states =========================================== An important point is the comparison between the A/B/C states of GRS 1915+105 and the four canonical black hole states mentioned above. It is tempting to associate state C with the low/intermediate state (hard spectrum flat-top noise and QPO in the power spectrum), state B with the very high state (strong disk component, weaker noise level), and the A state with the high state (cooler disk component, low noise level). However, as mentioned above, the instability related to the C state is not observed in other sources. It is more likely that the similarities between the properties of GRS 1915+105 and the canonical black hole states are not indicative of them being the same, but rather of them looking the same. In other words, the onset of the instability in GRS 1915+105 influences the accretion structure in a way that makes it mimic the properties of the canonical states observed in “normal” sources. ![Comparison between selected RXTE/PCA light curves of 4U 1630-47 (upper panel, GRO J1655-40 (middle panel) and GRS 1915+105 (lower panel). From Trudolyubov, Borozdin & Priedhorsky (2000).](belloni_TIII-01_fig5.ps){width="18pc"} Inner disk oscillations and radio jet ejection ============================================== The X-ray phenomenology described above has been positively linked to the variability observed in the radio band, and major events have been associated to the emergence of superluminal radio jets. This topic is treated by other authors in this volume. Here it is important to stress that the analysis of simultaneous X-ray/radio observations has shown that radio flares seem to be associated only to state C events, and therefore to the inner disk oscillations, while observations containing only states A and B do not correspond to significant radio detections of the source (see Klein-Wolt et al., this volume). However, the presence of this instability has not yet been observed in the other galactic sources showing jet ejection in the radio. Conclusions =========== GRS 1915+105 is the only source up to now which can provide a large number of spectral transitions, enabling us to study different states of one source without having to wait for years. The most important of these transitions do not involve the canonical states of black hole candidates, but are associated to an instability of the innermost region of an optically thick accretion disk, which causes its inner parts to be evacuated and refilled on the local viscous time scale. Other variations observed in this source can be linked to those observed in other sources, strengthening the connection between this unique system and more conventional ones. On the X-ray side, it is now important to examine the spectral/timing behavior of GRS 1915+105 in more detail (see Migliari, Vignarca & Belloni, this volume), in order to provide a more complete phenomenological picture for further theoretical work. Modeling of the X-ray properties of GRS 1915+105 is at the moment the most promising way to make significant progress in the understanding of accretion onto black holes. Belloni, T., et al., 1997a, ApJ, 479, L145 Belloni, T., et al., 1997a, ApJ, 488, L109 Belloni, T., et al., 2000, A&A, 355, 271 Fender, R.P., et al., 1999, MNRAS, 304, 865 Frontera, F., et al., 2000, ApJ, in press (astro-ph/0009160) Greiner, J., Morgan, E.H., Remillard, R.A., 1996, ApJ, 473, L107 Janiuk, A., et al., 2000, ApJ, in press (astro-ph/0008354) Markwardt, C.B., Swank, J.H., Taam, R.E., 1999, ApJ, 513, L37 Morgan, E.H., Remillard, R.A., Greiner, J., 1997, ApJ, 482, 993 Merloni, A., Fabian, A.C., Ross, R.R., 2000, MNRAS, 313, 193 Muno, M.P., Morgan, E.H., Remillard, R.A., 1999, ApJ, 527, 321 Mitsuda, K., et al., 1984, PASJ, 36, 741 Miyamoto, S., et al., 1991, ApJ, 383, 784 Nayakshin, S., Rappaport, S., Melia, F., 2000, ApJ, 535, 798 Szuszkiewicz, E., Miller, J.C., 1998, MNRAS, 298, 888 Tanaka, Y., Lewin, W.H.G, 1995, in “X-ray binaries”, Cambridge Univ. Press, p126 Trudolyubov, S.P., Borozdin, K.N., Priedhorsky W.C., 2000, MNRAS, in press (astro-ph/9911345) van der Klis, M., 1995, in “X-ray binaries”, Cambridge Univ. Press, p252
--- bibliography: - 'EarthquakesMoonSun.bib' --- [The Association of the Moon and the Sun with Large Earthquakes ]{}\ Lyndie Chiou$^{1}$,\ [1]{} ResearchPipeline.com, Union City, CA, USA\ $\ast$ E-mail: Corresponding lyndie@researchpipeline.com Abstract {#abstract .unnumbered} ======== The role of the moon in triggering earthquakes has been studied since the early 1900s. Theory states that as land tides swept by the moon cross fault lines, stress in the Earth’s plates intensifies, increasing the likelihood of small earthquakes. This paper studied the association of the moon and sun with larger magnitude earthquakes (magnitude 5 and greater) using a worldwide dataset from the USGS. Initially, the positions of the moon and sun were considered separately. The moon showed a reduction of 1.74% (95% confidence) in earthquakes when it was 10 hours behind a longitude on earth and a 1.62% increase when it was 6 hours behind. The sun revealed even weaker associations (<1%). Binning the data in 6 hours quadrants (matching natural tide cycles) reduced the associations further. However, combinations of moon-sun positions displayed significant associations. Cycling the moon and sun in all possible quadrant permutations showed a decrease in earthquakes when they were paired together on the East and West horizons of an earthquake longitude (4.57% and 2.31% reductions). When the moon and sun were on opposite sides of a longitude, there was often a small (about 1%) increase in earthquakes. Reducing the bin size from 6 hours to 1 hour produced noisy results. By examining the outliers in the data, a pattern emerged that was independent of earthquake longitude. The results showed a significant decrease (3.33% less than expected) in earthquakes when the sun was located near the moon. There was an increase (2.23%) when the moon and sun were on opposite sides of the Earth. The association with earthquakes independent of terrestrial longitude suggests that the combined moon-sun tidal forces act deep below the Earth’s crust where circumferential forces are weaker. Introduction {#introduction .unnumbered} ============ There are many proposed triggers for earthquakes. Examples include underground temperature differences, solar activity and water movements both below and above ground (see, for example, [@subsurfacetempEQs; @glacialearthquakes; @earthquakepredictionchapter; @earthquakeoverviewchapter; @GlobalSeismicAndSolar; @EarthPlanet8]). Tides are especially intriguing as a potential trigger because they are induced by the gravitational pull of the moon and the sun which follow calculable orbits. Tides occur both in water and, to a lesser extent, in land. Land tides are a measurable effect wherein the Earth’s crust rises and falls as a result of the moon’s and sun’s gravitational pull. The effect can be up to 20 cm near the Earth’s poles [@NYTimesLandTides]. Is it possible that some earthquakes could be triggered by land tides? It seems a reasonable hypothesis that the intersection of land tides with the Earth’s fault lines could serve as a trigger for an earthquake. The reverse has proven true: moonquakes can be triggered by land tides caused by the Earth’s gravitational influence on the moon [@NationalGeographicNews]. However, the notion that the moon can similarly cause earthquakes is controversial as the Earth has 81 times more mass than the moon [@NASAmoonSite] and any influence is assumed to be very weak at best with many concluding the effect is non-existent (for example, see references [@SpringerLinkNote; @JournalGeoResearch1]). Despite the controversy, papers continue to be published on both sides of the debate (illustrated by [@EarthPlanet5; @EarthPlanet6; @EarthPlanet7; @EarthPlanet1; @EarthPlanet2; @EarthPlanet3; @EarthPlanet4]), many with conclusions based on minimally-sized datasets corresponding to specific faults. One noticeable exception was reference [@BigAnalysis] which reported an analysis using 442,412 earthquakes from magnitude 2.5 to 9. The authors concluded that land tides induced by the moon do trigger earthquakes, albeit primarily shallow earthquakes of low magnitude (a 0.5% to 1.0% increase over expected). The moon’s greatest influence occurred when it was overhead, corresponding to a rise in the Earth’s crust. A short-coming of the analysis was that low magnitude data, particularly below magnitude 4, was not uniformly recorded in the data and potentially lead to a bias in the conclusions. This paper uses a different dataset than the previous study and not only looks at the role of the moon and sun in triggering earthquakes separately, but extends the analysis to examine the role of the sun and moon in conjunction. The analysis is restricted to larger earthquakes, magnitude 5 and greater, since these magnitudes were accurately and comprehensively recorded across the globe. Also, larger magnitude earthquakes can exert a high negative economic impact, making the results more interesting outside of the geological community. Methods {#methods .unnumbered} ======= The position of the moon and sun as a proxy for land tides {#the-position-of-the-moon-and-sun-as-a-proxy-for-land-tides .unnumbered} ---------------------------------------------------------- A suitable technique to deduce the relationship between land tides and earthquakes would require a dataset that allowed users to extract the land tide value for a given latitude and longitude on a particular date and time. Unfortunately, no such dataset exists. The ability to calculate land tides relies on a detailed map of subsurface properties such as ocean floor depths and underground plate tectonics [@OceanMotion]. For this reason, the longitudinal positions of the moon and sun with respect to earthquakes were used as a proxy for the position of land tides. This approach is not the most ideal since, as mentioned, the actual height of land tides varies not just according to the position of the moon and sun, but also due to localized geographical properties. The assumed correlation of the moon and sun versus land tide heights is an important source of error in the upcoming statistical analysis. This paper measures the occurrence of global earthquakes between 1973 to 2011 relative to the longitudinal positions of the sun and moon and compares the result to a simulation of randomly generated earthquakes. The data was downloaded from the United States Geological Survey (USGS) website [@USGSEQdata]. Earthquakes of magnitude 5 and greater were extracted from a date range of January 1, 1973 to July 29, 2011 (the range available when the project was started). As mentioned earlier, lesser magnitude earthquakes, particularly in the magnitudes below 4, were not globally recorded as accurately as those of higher magnitudes. Therefore the analysis was limited to magnitude 5 and above which encompasses larger earthquakes. The resulting dataset consisted of 66,724 earthquakes located around the world. Figure \[WorldEQs\] shows a visualization of each earthquake on a world map generated with the data. ![ [Earthquakes from January 1, 1973 to July 29, 2011, magnitude 5 and greater.]{} The color of the dots denotes magnitude, from blue (magnitude 5) to red (highest magnitude, 9.1). []{data-label="WorldEQs"}](WorldEarthquakes.png){width="4in"} The following sections describe the method used for establishing the relative difference in position between the moon and sun and each earthquake as well as the histogram method used for the analysis. One Hour Binning {#Methods_1HrBins .unnumbered} ---------------- To evaluate the possibility of the moon influencing earthquakes, it was necessary to calculate the relative difference in position between the moon and each earthquake. This was done by subtracting the right ascension (RA) coordinates (calculated using Mathematica’s ephemerides tables which relied on [@Ephemerides]) for the moon from the longitudinal coordinate of each earthquake, converted to units of time. $$d_{rel}= Longitude_{EQ}(hrs) - RA_{moon}(hrs)$$ The result yielded the relative longitudinal difference between the Earth position of each earthquake and the corresponding moon position (latitude offsets were considered less important since the moon moves primarily circumferentially around the Earth). The longitudinal offsets were then binned according to relative time differences in hours. The left hand side of figure \[RelHistoEQs\] shows the entire dataset of calculated longitudinal offsets in a traditional histogram format, binned by hour. The right hand side of figure \[RelHistoEQs\] is an alternate visualization. The 24 wedges represent the 24 hours of possible relative longitudinal distances around the globe from a given earthquake. The height of the wedge reflects the number of earthquakes occurring during that offset hour. The visualization shows the geometrical relationship between the earthquake, the moon’s RA coordinate and the number of earthquakes for each distance. ![ [Histograms of earthquake-moon longitudinal offsets binned by hour.]{} The left hand side shows the traditional histogram format of the entire moon-earthquake data. The right hand side shows the results referenced to an example earthquake longitude. Note that the results are relative to earthquake longitudes in both histograms, instead of referenced to a specific longitude. The expected mean number of earthquakes/hour was 2780+/-51.2. []{data-label="RelHistoEQs"}](Combined_IntroHistos.png){width="6in"} An examination of the histograms shows slight variations in the number of earthquakes versus the position of the moon. But does this imply a slight increase in the chance of an earthquake during those moon-earthquake offsets? In a noiseless dataset, the total number of earthquakes would be divided evenly among the 24 hours of possible longitudinal offsets. This would imply 2780 earthquakes per hour of relative offset with a standard deviation of 0 (during the dataset time period, 38.6 years). However, since the dataset is limited to 66,724 data points, it is necessary to run a simulation that generates the equivalent number of random earthquakes and compare the results to the real-world data to determine if the fluctuations bear any statistical significance. This simulation was performed by randomly generating longitudinal offsets occurring throughout the day across the same year range as the real-world data. The distribution along fault lines was implicit as the calculation only used relative distances. In that way we don’t need to know the positioning of the fault lines. As mentioned, this was a source of error since local geography was not taken into account. In all, 70 sets of 66,724 earthquakes were generated to achieve a stable group of datasets. The stability was measured by calculating the standard deviation of the hourly bins from the histograms generated from $n$ random datasets. Equations and show how the averaged standard deviation was computed. As more and more datasets were included, the averaged standard deviation converged. $$stdev_{bins} = {stdev(\sum_{1}^{n} bin_{hour})}\Bigl\lvert_{each\,hour} \label{hist1}$$ $$stdev_{avg} = \frac{\sum_{1}^{24}{stdev_{bins}}}{n} \label{hist2}$$ Figure \[moonEQBGstability\] shows the averaged standard deviation for the averaged hourly bins as more and more random datasets are included. ![ [Stabilization of the standard deviation of the data.]{} As the number of datasets of randomly generated earthquakes increases, the standard deviation of the height of the averaged bins stabilizes. Each dataset has a full 66,724 points. []{data-label="moonEQBGstability"}](MoonEQBGstability.png){width="3in"} The resulting simulation predicted a mean of 2780 earthquakes with a standard deviation of $\pm$ 51.2 earthquakes. Finally, figure \[moonEQPlusBG\] shows the error estimate overlaid on top of the actual measured offset data. The gray overlapping ring shows $\pm1.96\times$(standard deviation) which implies a 95% confidence level (the threshold used throughout the paper). A first examination shows that indeed, some moon-earthquake offsets show a slight increase (and decrease) compared to what would be expected from totally random earthquakes. A more detailed discussion follows in the Results section of this paper. ![ [Relative histograms between earthquake longitude and moon’s (left) and sun’s (right) longitude (mapped back to Earth).]{} The left shows the same histogram data as figure \[RelHistoEQs\] with an overlay of the expected variation (95% confidence) shown as a gray ring. The actual increase or decrease is listed as a percentage outside each wedge. The inner number shows the relative displacement in hours between the earthquake’s and the moon’s longitudes. The expected mean was 2,780 earthquakes +/- 51.2 []{data-label="moonEQPlusBG"}](SideBySide_MoonSun_v2.png){width="6in"} Six Hour Binning {#Methods_6Hrs .unnumbered} ---------------- In addition to hourly binning, these methods were applied to six-hour longitudinal bins. Oceanic tides cycle through two high and two low tides lasting six hours over a period of a day. The high tides correspond to the moon sitting nearly overhead and below the observational longitude, while the low tides correspond to the moon near the horizons which are six hours away, hence the choice of six hour bins. While there is about a 1 hour lag between the position of the moon and oceanic tides, there is no such lag for land tides [@SAONASAinPopularAstronomy]. This allows the data to be re-binned into the following six hours increments which capture the entire phase of a tide: 21 hours to 3 hours (which is equivalent to -3 hours to 3 hours), 3 hours to 9 hours, 9 hours to 15 hours and 15 hours to 21 hours (see figure \[fig:World4Quadrants\]). ![ [The Earth divided into 4 quadrants reflecting the 4 phases of the tide referenced to an earthquake’s longitude (shown with a red dot).]{} The longitude indicated is merely an example. Results are relative to all earthquake data longitudes rather than with respect to an absolute longitude. []{data-label="fig:World4Quadrants"}](GLobeQuadrants.png){width="2in"} Moon-Sun Position Combinations {#moon-sun-position-combinations .unnumbered} ------------------------------ Rather than limiting the analysis to just the moon and sun individually, the relative positions of both the moon and sun in conjunction proved to be important. This is the final analytical method used on the data and provided the most interesting set of results. Results {#sec:results .unnumbered} ======= Moon and Sun Considered Separately {#hourly_results .unnumbered} ---------------------------------- ### Hourly Bins {#hourly-bins .unnumbered} A priori, it might be guessed that the moon and sun have a low influence on larger magnitude earthquakes. This proved true. The statistical significance of hourly binning of earthquakes with respect to the moon’s position produced nothing extraordinary. The same was true for the sun’s relative position. Tables \[table:moonZTable\] and \[table:sunZTable\] at the end of the paper show the number of earthquakes and Z-values for the moon and sun across the full 24 hours of relative longitudinal distance (the Z-value is defined as how many standard deviations the result lies away from the mean). This data was also presented in figure \[moonEQPlusBG\] which shows the moon’s results on the left and the sun’s results on the right. The results show that the moon and sun have a tiny effect on larger terrestrial earthquakes. Over the course of 38.6 years, the moon showed an increase in the likeliness of earthquakes at 4, 6 and 8 hour longitudinal offsets. Specifically, the moon-earthquake displacements with the largest associations (at the 95% confidence level): - 4 hours $ \implies $ 0.741% greater earthquakes - 6 hours $ \implies $ 1.62% greater earthquakes - 8 hours $ \implies $ 0.501% greater earthquakes Dividing by the 38.6 year duration, this amounted to a combined total of 2.2 extra earthquakes per year, magnitude 5 and greater during those longitudinal offsets. Similar to the moon, the sun showed an effect on terrestrial earthquakes, although the association was even weaker. Earthquakes were more likely than expected when the sun was 17 hours behind a geographical longitude (Z=2.38, 0.746% increase, 95% confidence). On the other hand, certain longitudinal offsets for the sun and moon reduced the chance of an earthquake. In total, the moon was associated with a reduction of 1.8 earthquakes per year during lower risk offsets (9th and 10th hours). The sun was associated with 1.3 fewer earthquakes per year during the longitudinal offsets of 5, 7 and 11 hours. ### Six Hour Bins {#six-hour-bins .unnumbered} The associations were non-existent for six-hour bins, defined according to the tidal cycles (figure \[fig:World4Quadrants\]). Tables \[table:moonQuadZTable\] and \[table:sunQuadZTable\] show the associations of the sun and moon with the relative positions of earthquakes. In retrospect, the averaging out of signal in the six hour bins should have been obvious due to the haphazard nature of the increases and decreases in the hourly bins. Aside from the minimal influence, one issue in particular stood out in this analysis: there was no clear cycle present. Reference [@BigAnalysis] showed a daily cycle similar to the water tide, although the pattern was for shallow, low-magnitude earthquakes which have different origins than high magnitude earthquakes. The next section examines combined sun-moon positions. Surprisingly, this approach resulted in a significant changes in earthquakes for certain moon-sun pairings that were cyclical in nature. Moon (quad) EQs Z-value % extra EQs ------------- ------- --------- ------------- I 16676 -0.0441 0 II 16612 -0.552 0 III 16901 1.76 0 IV 16535 -1.17 0 : [The Z-value associated with moon-earthquake relative distances with 6-hour binning]{} The mean and standard deviation of the equivalent size dataset of randomly generated earthquakes was 16,681 $\pm$ 125. \[table:moonQuadZTable\] Sun (quad) EQs Z-value % extra EQs ------------ ------- --------- ------------- I 16818 1.10 0 II 16463 -1.74 0 III 16692 0.088 0 IV 16751 0.056 0 : [The Z-value associated with sun-earthquake relative distances with 6-hour binning]{} The mean and standard deviation of the equivalent size dataset of randomly generated earthquakes was 16,681 $\pm$ 125. The sun’s results produced low statistical fluctuations. \[table:sunQuadZTable\] Combined Moon-Sun Positions {#binningtogether .unnumbered} --------------------------- ### Six Hour Binning {#binningtogether6 .unnumbered} While looking at the moon and sun separately revealed little, looking at moon-sun combinations did result in a significant influence. Unlike the expected mean number of earthquakes for the moon and sun considered separately, the combined moon-sun ephemerides caused the mean earthquakes to vary by each bin. The effect was small (<1%) since the bin widths were relatively large, but this was still taken into account when calculating the mean expected values for each of the six-hour bins. Whenever the moon and sun paired together in the horizon, there was a significant decrease in earthquakes. Table \[table:moonsunOppositesTable\] lead to the following conclusions for the (moon, sun) quadrants with a 95% confidence level: - (II,II) $ \implies $ 4.57% fewer earthquakes - (IV,IV) $ \implies $ 2.31% fewer earthquakes One physical interpretation is that when the moon and sun are paired together on the horizons they act to horizontally pull the Earth’s crust together, helping to seal fault lines. This force behaves differently than when the moon and sun are overhead and pull both sides of a fault upward rather than sideways. Conversely, the statistical increases in earthquakes were shallow and occurred when the moon and sun were not paired in the same quadrant: - (II,IV) $ \implies $ 1.08% increase in earthquakes - (III,I) $ \implies $ 1.21% increase in earthquakes - (III,IV) $ \implies $ 0.345% increase in earthquakes It seems intuitive that when the moon and sun are pulling on opposite sides of a fault they separate the two halves just enough to trigger an earthquake. What about the other permutations? Table \[table:moonsunOppositesTable\] shows the measured earthquakes versus expected earthquakes for all possible moon and sun quadrant pairings. Moon Sun EQs expected EQs Z-value % extra EQs ------ ----- ------ -------------- --------- ------------- I I 4025 4136 -1.77 0 I II 4295 4185 1.74 0 I III 4190 4156 0.539 0 I IV 4166 4197 -0.489 0 II I 4196 4191 0.0848 0 II II 3848 4147 -4.76 -4.57 II III 4244 4195 0.781 0 II IV 4324 4155 2.7 1.08 III I 4316 4141 2.8 1.21 III II 4133 4194 -0.975 0 III III 4115 4133 -0.286 0 III IV 4337 4199 2.2 0.345 IV I 428I 4202 1.26 0 IV II 4187 4154 0.534 0 IV III 4143 4203 -0.95 0 IV IV 3924 4136 -3.41 -2.31 : [The Z-value associated with all possible moon and sun quadrant permutations]{} Based on randomly generated datasets of simulated earthquakes, the standard deviation was $\pm$62.7. The number of expected earthquakes varied by bin and was therefore listed in a separate column in the table. \[table:moonsunOppositesTable\] ### One Hour Binning {#binningtogether1 .unnumbered} The final step was to calculate the hourly combined position results. The mean expected earthquakes varied more with hourly binning, amounting to a 4.4% effect as opposed to <1% with six-hour bins. Taking this into account, a large table of 576 bins resulted, of which most combinations produced null or negligible increases. However, 32 positions (about 5.6% of the possible permutations) were statistically significant. An initial plot of the moon’s and sun’s combined positions versus the increase in expected earthquakes showed the apparently random data (left, figure \[EarthquakesWRTMoonSun3D\]). The right hand side of figure \[EarthquakesWRTMoonSun3D\] showed the results if only the statistically significant results were plotted. Table \[table:sunmoonZTable\] at the end of the paper lists the values for the non-negligible results (using $Z>2.5$ so the chart could fit on one page). A moment should be taken to examine the true significance of the results in the table and figures. Some of the results show very sharp increases and decreases over expected, right next to an hourly bin combination that had no significant difference. For instance, (moon,sun)=(3,4) hrs offset resulted in 13% fewer earthquakes compared to the mean expected number. However, (moon,sun)=(4,4) hrs offset was within the expected range of variation. Clearly, even though the noise instrinsic to the size of the dataset was taken into account, another source of noise must still be present. As mentioned at the start of the paper, the method of looking only at the relative distance between an earthquake and the celestial positions of the moon and sun ignored localized geological properties that affect the heights of land tides. It is likely that this influence added an extra significant source of noise, on top of the noise intrinsic to the dataset. A new approach was thus proposed: treat the statistically significant points listed in Table \[table:sunmoonZTable\] as outliers on top of a noisy dataset. Examine the outliers for a periodic trend. Periodicity in the result adds validity since the system is cyclical in nature. Then, based on the pattern identified, select the broadest possible bins in order to average out the influence of the extra noise while still preserving the earthquake “signal”. ![ [Earthquakes for various moon/sun offset positions.]{} A plot of the moon/sun positions vs. the number of earthquakes at those positions. The redder the dot, the larger the number of earthquakes, the bluer the dot, the fewer. The plot on the right shows the same data with only the 95% confidence results. The positions which lay within expected variation were removed. []{data-label="EarthquakesWRTMoonSun3D"}](combinedLatticeMoonSun.png){width="6in"} ### Outlier Analysis {#OutlierAnalysis .unnumbered} Ignoring the specific value of the outliers and plotting the fewer-than-expected earthquake bins as blue circles and the higher-than-expected earthquake bins as red squares, a cyclical pattern emerged. Figure \[significantEQs\] shows that whenever the sun was +2.5 or -3.5 hours away from the moon, irrespective of longitude, there was a trend toward a reduction of earthquakes. The axes in figure \[significantEQs\] were labeled such that: $$x = Longitude_{EQ}(hrs) - RA_{moon}(hrs)$$ $$y = Longitude_{EQ}(hrs) - RA_{sun}(hrs)$$ Thresholds were selected to identify the region of reduced earthquakes as three hours wide around the central, dotted line, $y=x-1/2$. This bin definition captured the most blue points and a minimum of red points. Labeled regions in the figure identify the bin definitions. Because the days cycle every 24 hours, some regions are extensions of others, reflected in the labeling. - Region 1 accounted for 67% of the fewer-than-expected earthquake moon-sun positions, and only 11% of the higher-than-expected earthquake combinations fell in between these thresholds. - Region 2 had only 13% of the lowered combinations and 34% of the higher combinations. - Region 3 had 13% lowered combinations and 37% higher combinations. - Region 4 had 7% lowered combinations and 17% higher combinations. ![ [Significant changes in earthquakes for various moon/sun offset longitudes.]{} On the left, a plot shows the moon/sun relative longitudes. The blue circles show where there were fewer than expected earthquakes and the red squares show where there were higher than expected earthquakes. Because the day cycles every 24 hours, certain regions are extensions of each other and have been labeled to show this relationship. The diagram on the right shows a histogram of the lower than expected earthquakes (blue) and higher than expected earthquakes (red) in the labeled regions. Region 1 (when the sun was +2.5 or -3.5 hours of the moon, independent of longitude) showed reduced earthquakes. Regions 3 and 4 showed increases in high earthquake moon-sun pairings. []{data-label="significantEQs"}](linear_rel_v7_reordered.png){width="6in"} Do the outliers predict real earthquake trends? The question is answered by summing the earthquakes data in the four regions and again comparing to the 70 randomly generated datasets. These 4 regions encompassed about 16,700 points each as opposed to the 116 points for the hourly bins which reduces random noise fluctuations by a factor of six. The mean expected earthquakes for each bin were recalculated for the four regions. Table \[table:regionsTable\] shows the results. Region EQs expected EQs Z-value % extra EQs -------- ------- -------------- --------- ------------- 1 15714 16509 -6.36 -3.33 2 16776 16832 -0.445 0 3 17169 16554 4.92 2.23 4 17065 16829 1.89 0 : [The Z-value associated with the four regions depicted in Figure \[significantEQs\]]{} Based on randomly generated datasets of simulated earthquakes, the standard deviation of earthquakes is +/- 125. \[table:regionsTable\] Indeed, the abundance of outlier lower-than-expected earthquakes correctly identified a region of generally reduced earthquakes. A reduction of 3.33% earthquakes (95% confidence) implied about 14.3 fewer large magnitude earthquakes per year when the sun was +2.5 or -3.5 hours away from the moon, independent of longitude. Conversely, Region 3 (the sun was 8.5 to 14.5 hours behind the moon) showed significantly higher than expected earthquakes, 2.23% at the 95% confidence level. This amounted to 9.6 extra earthquakes per year. Starting in Region 1 and stepping the 6-hour bin through the moon-sun earthquake space an hour at a time revealed a cyclical pattern, seen in figure \[fig:stepEQs\]. The minimum earthquake total occured for Region 1 and the maximum aligned with Region 3. The sinusoidal line shows the estimated mean background for each of the steps, a number which varied due to the ephemerides of the combined moon-sun positions. ![ [Cyclical pattern from summing earthquakes by stepping through offsets to y=x-1/2.]{} Stepping through hour-by-hour earthquakes binned +/-3 hours around the line y=x-1/2. For example, Region 1 corresponds to the 0 hours offset from y=x-1/2 (+/-3 hours) and Region 3 corresponds to the 11 hours offset (+/-3 hours). The right hand side shows the raw summed earthquakes plotted with estimated error bars. The sinusoidal line shows the estimated mean background for each of the positions, a number which varied due to the ephemerides of the combined moon-sun positions. []{data-label="fig:stepEQs"}](EQ_6-hr_cycle.png){width="6in"} Plotting the difference between the mean expected earthquakes and the measured earthquakes produced figure \[fig:Final\_Plot\]. The data for the plot was recorded in Table \[table:regionsTable\]. Fitting an equation to the data gave: $$EQ_{extra}=-15.3 Cos[\frac{2Pi}{24}offset_{y=x-1/2}]$$ This equation gives a simple relationship between the measured number of earthquakes over (or less than) the mean expected number of earthquakes which depends strictly on the relative distance between the moon and sun. It is also notable that these conclusions agree with the more narrow conclusions of 6-hour combination binning relative to a specific longitude (Table \[table:moonsunOppositesTable\]). ![ [Cyclical pattern from summing earthquakes by stepping through offsets to y=x-1/2.]{} Stepping through hour-by-hour earthquakes binned +/-3 hours around the line y=x-1/2. For example, the 0 hours offset corresponds to Region 1 and the 12 hour offset corresponds to Region 3. []{data-label="fig:Final_Plot"}](Final_Plot.png){width="3in"} Discussion {#Discussion .unnumbered} ========== The earthquake data, despite coming from a large, accurate record of events, was still noisy. The assumption in the paper was that the method of looking at relative offsets did not account for local properties that influenced land tide heights and, in turn, influenced the likeliness of an earthquake. The only way that local geography could possibly influence the results would be if specific faults or fault types were over-represented in the outlier moon-sun combinations. Then the local geography would exert an influence. If this was not the case, but the fault lines and types were randomly scattered in the moon-sun earthquake space then local geography could not be a noise source and something else must be influencing the results. A future step to research would be to examine the specific fault and fault type distributions in the data in more detail to verify this statement. Despite this, by examining outliers, a clear association of the positions of the moon and sun and earthquakes was identified. Surprisingly, the clearest trend was for a reduction of earthquakes when the moon and sun were close together in the sky, independent of the longitude of an earthquake. When the moon and sun were on opposite sides of the Earth (Region 3) there was a marked increase in large magnitude earthquakes. The fact that the location of the earthquake was irrelevant may be related to the depth of the earthquakes. Deeper fault lines, closer to the Earth’s core, may be less influenced by the circumferential effects of the moon and sun. It is interesting to note that although the association was based on a statistical analysis of a large dataset, the pattern holds for three of the four highest-magnitude earthquakes in the study, including: - December 2004 Northern Sumatra earthquake (magnitude 9.1, relative sun RA=0.227, relative moon RA=12.5) - March 2005 Indonesia earthquake (magnitude 8.6, relative sun RA=9.17, relative moon RA=18.8) and - February 2010 Chile earthquake (magnitude 8.8, relative sun RA=4.09, relative moon RA=15.0) As always, correlation does not necessarily imply causation. One alternate source for the results of the study could be an estimate for a correction to the moon/sun ephemerides. The appearance of an increase or decrease in earthquakes could imply an error in the calculated moon/sun positions at the time of the earthquakes. This paper used Mathematica to generate the RA coordinates which which employed the current standard, VSOP87 (Variations Séculaires des Orbites Planétaires). The precision of this standard is stated as 1" per 4000 years around the year 2000 (see reference [@Ephemerides]), making the possibility of the bias unlikely. Acknowledgments {#acknowledgments .unnumbered} =============== The author expresses thanks to the USGS for posting their earthquake data online and Mathematica for making this project possible in a finite amount of time. Tables {#tables .unnumbered} ====== Moon (hrs) EQs Z-value % extra EQs ------------ ------ --------- ------------- 0 2767 -0.257 0 1 2840 1.17 0 2 2863 1.62 0 3 2816 0.7 0 4 2902 2.38 0.741 5 2765 -0.296 0 6 2928 2.89 1.62 7 2834 1.05 0 8 2895 2.24 0.501 9 2659 -2.37 -0.784 10 2634 -2.86 -1.74 11 2715 -1.27 0 12 2755 -0.492 0 13 2836 1.09 0 14 2751 -0.57 0 15 2742 -0.746 0 16 2825 0.876 0 17 2772 -0.16 0 18 2734 -0.902 0 19 2740 -0.785 0 20 2698 -1.61 0 21 2748 -0.628 0 22 2716 -1.25 0 23 2789 0.173 0 : [The Z-value associated with moon-earthquake relative distances]{}* A positive Z-value reflects more earthquakes than expected while a negative Z-value implies less earthquakes than expected. A simulation was used to calculate the mean number of earthquakes and the mean standard deviation of the equivalent size dataset of randomly generated earthquakes. This produced the values 2780 $\pm$ 51.2. As can be seen in the table, the results had mostly no or low statistical significance. \[table:moonZTable\] Sun (hrs) EQs Z-value % extra EQs ----------- ------ --------- ------------- 0 2856 1.48 0 1 2758 -0.434 0 2 2823 0.838 0 3 2757 -0.453 0 4 2759 -0.414 0 5 2666 -2.23 -0.524 6 2839 1.15 0 7 2668 -2.19 -0.449 8 2774 -0.121 0 9 2850 1.37 0 10 2830 0.975 0 11 2655 -2.45 -0.94 12 2771 -0.179 0 13 2811 0.603 0 14 2775 -0.101 0 15 2801 0.408 0 16 2766 -0.277 0 17 2902 2.38 0.746 18 2747 -0.649 0 19 2725 -1.08 0 20 2810 0.584 0 21 2793 0.251 0 22 2796 0.31 0 23 2792 0.231 0 : [The Z-value associated with sun-earthquake relative distances]{} A simulation showed the mean number of earthquakes during the time-period to be 2780 $\pm$ 51.2. The sun produced even lower statistically significant results than the moon. \[table:sunZTable\] Moon (hrs) Sun (hrs) EQs Z-value % extra EQs ------------ ----------- ----- --------- ------------- 1 0 85 -2.85 -10.9 1 21 150 3.17 8.48 3 0 141 2.52 4.16 3 4 85 -3.02 -13.1 3 13 152 3.47 10.4 5 4 93 -2.3 -3.78 5 14 92 -2.22 -2.93 5 15 143 2.76 5.86 5 18 148 3.2 8.79 5 23 147 2.81 6.06 6 7 88 -2.25 -3.42 6 18 151 3.55 11.1 7 5 85 -2.78 -10.2 7 7 88 -2.46 -5.93 8 7 90 -2.34 -4.45 9 14 158 3.79 12.2 9 15 87 -2.9 -11.4 9 17 86 -2.91 -11.6 10 13 141 2.55 4.39 11 15 143 2.55 4.35 12 2 144 2.77 5.91 12 17 155 3.58 11. 13 20 154 3.33 9.31 14 2 140 2.38 3.17 14 14 89 -2.55 -6.91 16 4 146 3.05 7.84 17 21 148 2.92 6.84 18 14 91 -2.47 -5.94 18 15 91 -2.42 -5.36 21 6 150 3.22 8.81 21 13 86 -2.91 -11.6 23 18 94 -2.26 -3.3 : [The non-zero values for earthquakes exceeding (or less than) the expected variation in earthquakes at the 95% confidence level. The moon-sun-earthquake relative distances.]{} The mean of the equivalent size dataset of randomly generated earthquakes varied by position according to the moon and sun ephemerides (calculated via Mathematica). The means ranged from 111 to 120 earthquakes during the dataset time span (38.6 years). The standard deviation was $\pm$ 10.8. The large number of statistically significant results is attributed to noise introduced by the method of analysis (relying on the position of the sun and moon to gauge the effect of land-tides on earthquake fault lines). \[table:sunmoonZTable\] Region EQs expected EQs Z-value % extra EQs extra EQ/yr -------- ------- -------------- --------- ------------- ------------- 0 15714 16509 -6.36 -3.33 -14.3 1 15758 16561 -6.43 -3.37 -14.5 2 16002 16638 -5.09 -2.35 -10.1 3 16330 16719 -3.11 -0.863 -3.74 4 16616 16783 -1.34 0 0 5 16609 16829 -1.76 0 0 6 16776 16832 -0.445 0 0 7 16969 16798 1.36 0 0 8 16979 16726 2.02 0.0477 0.207 9 17015 16647 2.95 0.74 3.19 10 17025 16589 3.49 1.15 4.94 11 17146 16548 4.78 2.13 9.15 12 17169 16554 4.92 2.23 9.59 13 16923 16587 2.69 0.55 2.37 14 16866 16659 1.65 0 0 15 16820 16741 0.63 0 0 16 17055 16800 2.04 0.0622 0.271 17 17110 16841 2.15 0.144 0.629 18 17065 16829 1.89 0 0 19 17074 16777 2.37 0.308 1.34 20 16877 16700 1.41 0 0 21 16559 16616 -0.46 0 0 22 16028 16552 -4.19 -1.69 -7.23 23 15859 16506 -5.17 -2.43 -10.4 : [The Z-value associated with the y=x-1/2 region stepped through all 24 hours, as depicted in Figure \[fig:stepEQs\]]{} Based on randomly generated datasets of simulated earthquakes, the standard deviation of earthquakes is +/- 125. \[table:regionsTable\]
--- author: - \| Daniel Kosiorowski\ Cracow University of Economics Zygmunt Zawadzki\ Cracow University of Economics bibliography: - 'literatura\_JSS.bib' title: ': An Package for Robust Exploration of Multidimensional Economic Phenomena' --- Introduction ============ The modern Economics crucially depend on advances in applications of multivariate statistics. We mean here for example theory and practice of the portfolio optimisation, a practice of credit scoring, evaluation of results of government aid programs, creation of a taxation system or assessment of attractiveness of candidates on a labour market. Unfortunately, in the Economics we very often cannot use powerful tools of the classical multivariate statistics basing on the mean vector, the covariance matrix and the normality assumptions. In a great part, the economic phenomena departure from normality. Usually our knowledge of the economic laws is not sufficient for a parametric modelling. Moreover, the today Economics significantly differs from a tomorrow Economics due to technological development and/or an appearance of new social phenomena. Additionally the data sets under our consideration consist of outliers and or inliers of various kind and/or we have to cope with a missing data phenomenon. Robust statistics aims at identifying a tendency represented by an influential majority of data and detecting observations departing from that tendency (see [@Marona:2006]). Nonparametric and robust statistical procedures are especially useful in the Economics where an activity of influential majority of agents determines behaviour of a market, closeness to a crash etc. From a conceptual point of view, robust statistics is closely tied with well known economic ideas like Pareto’s effectiveness or Nash equilibrium (see [@Mizera:2002]). The main aim of this paper is to present an package ([@R]) consisting of successful implementations of a selection of multivariate nonparametric and robust procedures belonging to so called Data Depth Concept (DDC), which are especially useful in exploration of socio-economic phenomena. The package is available under GPL-2 license on and .\ The rest of the paper is organized as follows: in Section 2, basic notions related to the data depth concept are briefly described. In Section 3, the procedures offered by the package are briefly presented. In Section 4, an illustrative example is presented. The paper ends with some conclusions and references. All data sets and examples considered within the paper are available after installing the package. In this paper we use the following notation and definitions borrowed from [@Dyck:2004]. ${S}^{d-1}$ is the $(d-1)$ dimensional unit sphere in ${\mathbb{R}}^{d}$ , ${S}^{d-1}=\left\{ x\in {{\mathbb{R}}^{d}}:\left\\| x \right\\|=1 \right\}$ . $\mathcal{B}^{d}$ denotes Borel $\sigma$ algebra in ${\mathbb{R}}^{d}$. The transpose of a vector $x\in \mathbb{R}^{d}$ is written by ${x}^\top$ . For a random variable $X$ we write ${Q}_{X}$ for the usual (lower) quantile function, ${Q}_{X}:(0,1)\to \mathbb{R}$ , ${{Q}_{X}}(p)=\min \{x\in {{\mathbb{R}}^{d}}:P(X\le x)\ge p\}$ , and ${{\bar{Q}}_{X}}$ for the upper quantile function $\bar{Q}_{X}:(0,1)\to \mathbb{R}$, ${{\bar{Q}}_{X}}(p)=\max \{x\in \mathbb{R}^{d}:P(X\ge x)\ge p\}$. A sample consisting of $n$ observations is denoted by ${X}^{n}=\{{x}_{1},...,{x}_{n}\}$ , $F$ denotes a probability distribution in $\mathbb{R}^{d}$ , and ${F}_{n}$ its empirical counterpart. Data depth concept ================== Data depth concept was originally introduced as a way to generalize the concepts of median and quantiles to the multivariate framework. A detailed presentation of the concept can be found in [@Liu:1999], [@Zuo:2000], [@Serfling:2004], [@Serfling:2006], and [@Mosler:2013]. Nowadays the DDC offers a variety of powerful techniques for exploration and inference on economic phenomena involving robust clustering and classification, robust quality control and streaming data analysis, robust multivariate location, scale, symmetry tests. Theoretical aspects of the concept could be found for example in [@Kong:2010] and in references therein, recent developments of the computational aspects presents for example [@Shao:2012]. Our package uses so called location depths and their derivatives, i.e., regression depth and Student depth. The implements also recently developed concept of local depth presented in [@Pain:2012] and [@Pain:2013]. A developer version of the package, which is available on servers, additionally consists of fast algorithms for calculating selected depths for functional data and weighted by the local depth nonparametric estimators of a predictive distribution. Basic definitions ----------------- Following [@Dyck:2004] we consider the depth of a point w.r.t. a probability distribution. Let $\mathcal{P}_{0}$ be the set of all probability measures on $(\mathbb{R}^{d},\mathcal{B}^{d})$ and $\mathcal{P}$ a subset $\mathcal{P}_{0}$. A depth assigns to each probability measure $F\in \mathcal{P}$ a real function $D(\cdot ,F):{\mathbb{R}^{d}}\to {{\mathbb{R}}_{+}}$ , the so-called depth function w.r.t. $F$. The set of all points that have depth at least $\alpha $ is called $\alpha -$ trimmed region. The $\alpha -$ trimmed region w.r.t. $F$ is denoted by ${D}_{\alpha }(F)$, i.e., $${D}_{\alpha }(F)=\left\{ z\in {\mathbb{R}}^{d}:D(z,F)\ge \alpha \right\}.$$ In a context of applications, the probability measure is the distribution ${F}^{X}$ of a d-variate random vector $X$. In this case we write shortly $D(z,X)$ instead of $D(z,{F}^{X})$ and ${D}_{\alpha }(X)$ instead of ${D}_{\alpha}({F}^{X})$. The data depth is then defined on the set $\mathcal{X}$ of all random vectors $X$ for which ${F}^{X}$ is in $\mathcal{P}$ .\ Formal definitions of the depth functions can be found in [@Liu:1999], [@Zuo:2000], [@Mosler:2013]. There is an agreement in the literature, that every concept of depth should satisfy some reasonable properties: - Affine invariance: For every regular $d\times d$ matrix $A$ and $b\in \mathbb{R}^{d}$ it holds $D(z,X)=D(az+b,AX+b)$ . - Vanishing at infinity: For each sequence ${{\{{{x}_{n}}\}}_{n\in \mathbb{N}}}$ with $\underset{n\to \infty }{\mathop{\lim }}\,\left\\| {{x}_{n}} \right\\|=\infty $ holds $\underset{n\to \infty }{\mathop{\lim }}\,D({{x}_{n}},X)=0$. - Upper semicontinuity: For each $\alpha >0$ the set ${{D}_{\alpha }}(X)$ is closed. - Monotone on rays: For each ${{x}_{0}}$ of maximal depth and each $r\in {{S}^{d-1}}$ , the function ${{\mathbb{R}}_{+}}\to \mathbb{R}$ , $\lambda \mapsto D({{x}_{0}}+\lambda r,X)$ is monotone decreasing. - Quasiconcavity: For every $\alpha \ge 0$ holds: If ${{z}_{1}},{{z}_{2}}$ are two points with a depth of at least $\alpha $ , then every point on the line segment joining ${{z}_{1}}$ and ${{z}_{2}}$ has depth of at least $\alpha $ , too. DEFINITION ([@Dyck:2004]): A mapping $D$ , that assigns to each random vector $X$ in a certain set $\mathcal{X}$ of random vectors a function $D(\cdot ,F):{{\mathbb{R}}^{d}}\to {{\mathbb{R}}_{+}}$ and that satisfies the properties T1, T2, T3 and T4 is called depth. A depth that satisfies $T4^{}$ is called convex depth Properties T1 to T4 are formulated in terms of the depth itself. It is very useful to notice however, that these properties can also be formulated in terms of the trimmed regions (what is useful for approximate depth calculation): - Affine equivariance: For every regular $d\times d$ matrix $A$ and $b\in {{\mathbb{R}}^{d}}$ it holds ${{D}_{\alpha }}(AX+b)=A{{D}_{\alpha }}(X)+b$ . - Boundedness: For every $\alpha >0$ the $\alpha -$ trimmed region ${{D}_{\alpha }}(X)$ is bounded. - Closedness: For every $\alpha >0$ the $\alpha -$ trimmed region ${{D}_{\alpha }}(X)$ is closed. - Starshapedness: If ${{x}_{0}}$ is contained in all nonempty trimmed regions, then the trimmed regions ${{D}_{\alpha }}(X)$ , $\alpha \ge 0$ , are starshaped w.r.t. ${{x}_{0}}$ . - Convexity: For every $\alpha >0$ the $\alpha -$ trimmed region ${{D}_{\alpha }}(X)$is convex. - Intersection property: For every $\alpha >0$ holds ${{D}_{\alpha }}(X)=\bigcap\nolimits_{\beta :\beta <\alpha }{{{D}_{\beta }}}(X)$ . The simplest example of the depth is the Euclidean depth* defined as (see \[fig11\]) $${D}_{EUK}(y,{X}^{n})=\frac{1}{1+{{\left\\| y-\bar{x} \right\\|}^{2}}},$$ where $\bar{x}$ denotes the mean vector calculated from a sample ${X}^{n}$. As a next example let us take the Mahalanobis depth (see Fig. \[fig3\]) $${D}_{MAH}(y,{X}^{n})=\frac{1}{1+{{(y-\bar{x})}^\top}{{S}^{-1}}(y-\bar{x})},$$ where $S$ denotes the sample covariance matrix ${X}^{n}$. A symmetric projection depth $D\left( x,X \right)$ of a point $x\in {{\mathbb{R}}^{d}}$, $d\ge 1$ is defined as $$D\left( x,X \right)_{PRO}={{\left[ 1+su{{p}_{\left\\| u \right\\|=1}}\frac{\left\| {{u}^{\top}}x-Med\left( {{u}^{\top}}X \right) \right\|}{MAD\left( {{u}^{\top}}X \right)} \right]}^{-1}},$$ where $Med$ denotes the univariate median, $MAD\left( Z \right)$ = $Med\left( \left\| Z-Med\left( Z \right) \right\| \right)$. Its sample version denoted by $D\left( x,{X}^{n} \right)$ or $D\left( x,{X}^{n} \right)$ is obtained by replacing $F$ by its empirical counterpart ${{F}_{n}}$ calculated from the sample ${X}^{n}$ (see Fig. \[fig1\]). This depth is affine invariant and $D(x,{F}_{n})$ converges uniformly and strongly to $D(x,F)$. The affine invariance ensures that our proposed inference methods are coordinate-free, and the convergence of $D(x,{X}^{n})$ to $D(x,X)$ allows us to approximate $D(x,F)$ by $D(x,{X}^{n})$ when $F$ is unknown. Induced by this depth, multivariate location and scatter estimators have high breakdown points and bounded Hampel’s influence function (for further details see [@Zuo:2003]). Next, very important depth is the weighted ${L}^{p}$ depth. The weighted ${L}^{p}$ depth $D(\mathbf{x},F)$ of a point $\mathbf{x}\in \mathbb{R}^{d}$, $d\ge 1$ generated by $d$ dimensional random vector $\mathbf{X}$ with distribution $F$, is defined as (see Fig. \[fig4\]) $$D(x,F)=\frac{1}{1+Ew({{\left\\| x-X \right\\|}_{p}})},$$ where $w$ is a suitable weight function on $[0,\infty )$ , and ${{\left\\| \cdot \right\\|}_{p}}$ stands for the ${L}^{p}$ norm (when $p=2$ we have usual Euclidean norm). We assume that $w$ is non-decreasing and continuous on $[0,\infty )$ with $w(\infty -)=\infty $, and for $a,b\in {{\mathbb{R}}^{d}}$ satisfying $w(\left\\| a+b \right\\|)\le w(\left\\| a \right\\|)+w(\left\\| b \right\\|)$. Examples of the weight functions are: $w(x)=a+bx$ , $a,b>0$ or $w(x)={x}^{\alpha }$. The empirical version of the weighted ${L}^{p}$ depth is obtained by replacing distribution $F$ of ${X}$ in $Ew({{\left\\| {x}-{X} \right\\|}_{p}})=\int{w({{\left\\| x-t \right\\|}_{p}})}dF(t)$ by its empirical counterpart. The weighted $L^p$ depth from sample $X^n=\{x_1,...,x_n\}$ is computed as follows: $$D(x,X^n)=\frac{1}{1+\frac{1}{n}\sum\limits_{i=1}^{n}{w\left( {{\left\\| x-{X}_{i} \right\\|}_{p}} \right)}},$$ The weighted ${L}^{p}$ depth function in a point, has the low breakdown point (BP) and unbounded influence function IF (see [@Marona:2006] for the BP and IF definitions). On the other hand, the weighted ${L}^{p}$ depth induced medians (multivariate location estimator) are globally robust with the highest BP for any reasonable estimator. The weighted ${L}^{p}$ medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted ${L}^{p}$ depth and medians are computationally feasible for on-line applications and easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted ${L}^{p}$ depth and medians, respectively. Theoretical properties of this depth can be found in [@Zuo:2004]. ![Tukey depth.[]{data-label="fig2"}](figure/PROC_1){width=".95\linewidth"} ![Tukey depth.[]{data-label="fig2"}](figure/PROC_2){width=".95\linewidth"} ![$L^2$ depth.[]{data-label="fig4"}](figure/PROC_3){width=".95\linewidth"} ![$L^2$ depth.[]{data-label="fig4"}](figure/PROC_4){width=".95\linewidth"} Next, very important depth is the halfspace depth (Tukey depth, see Fig. \[fig2\]) $$D(x,F)=\underset{H}{\mathop{\inf }}\,\,\left\{ P(H):x\in H\subset {{\mathbb{R}}^{d}},\text{ X is closed subspace} \right\}$$ A very useful for the economic applications example of depth, originating from the halfspace depth, is regression depth introduced in [@Hubert:1999] and intensively studied in [@Van:2000] and in [@Mizera:2002]. Let ${Z}^{n}=\left\{ ({x}_{1},{y}_{1}),...,({x}_{n},{y}_{n}) \right\}\subset \mathbb{R}^{d}$ denotes a sample considered from a following semiparametric model: $${{y}_{l}}={{a}_{0}}+{{a}_{1}}{{x}_{1l}}+...+{{a}_{(d-1)l}}{{x}_{(d-1)l}}+{{\varepsilon }_{l}}, l=1,...,n,$$ we calculate a depth of a fit $\alpha=(a_{0},...,a_{d-1})$ as $$RD(\alpha ,{{Z}^{n}})=\underset{u\ne 0}{\mathop{\min }}\,\sharp\left\{l: \frac{{{r}_{l}}(\alpha )}{{{u}^{\top}}{{x}_{l}}}<0,l=1,...,n \right\},$$ where $r(\cdot )$ denotes the regression residual, $\alpha=(a_{0},...,a_{d-1})$, ${u}^{\top}{x}_{l}\ne 0$. The deepest regression estimator $DR(\alpha,{{Z}^{n}})$ is defined as $$DR(\alpha ,{{Z}^{n}})=\underset{\alpha \ne 0}{\mathop{\arg \max }}\,RD(\alpha ,{{Z}^{n}})$$ Fig. \[fig5\] presents a comparison of least squares and DR estimators of simple regression. ![Depth trimmed regression.[]{data-label="fig6"}](figure/PROC_5){width=".95\linewidth"} ![Depth trimmed regression.[]{data-label="fig6"}](figure/PROC_6){width=".95\linewidth"} The regression depth has its local version thanks to its relation to the halfspace depth (see [@Pain:2013]). The local version of this depth may be easily calculated within package. Next depth, which is implemented within the package, is the Student depth originating from [@Mizera:2002] and which was proposed in [@Mizera:2004]. It is pointed out in [@Mizera:2002] that general halfspace depth can be defined as a measure of data-analytic admissibility of a fit. Depth of the fit $\theta $ is defined as proportion of the observations whose omission causes $\theta $ to become a nonfit, a fit that can be uniformly dominated by another one. For a sample ${X}^{n}=\{{x}_{1},...,{x}_{n}\}$ we consider a criterial function ${F}_{i}$, given a fit represented by $\alpha$ , the criterial function evaluates the lack of fit of $\alpha $ to the particular observation ${x}_{i}$ . It means ${\alpha }^{}$ fitting ${x}_{i}$ better than $\alpha $, if ${F}_{i}({\alpha }^{})<{F}_{i}(\alpha )$ . In [@Mizera:2002] more operational version – the tangent depth of a fit $\alpha $ is defined $$d(\alpha )=\mathop{\inf_{u}\ne \mathbf{0}} \left\{\sharp n:{{{u}}^{\top}}{{\nabla }_{\alpha }}{{F}_{i}}(\alpha )\ge 0\right\},$$ where $\sharp$ stands for the relative proportion in the index set - its cardinality divided by $n$ . In [@Mizera:2004] authors suggest assuming the location-scale model for the data and taking log-likelihood in a role of the criterial function. They suggest taking the criterial function $${F}_{i}(\mu ,\sigma )=-\log f\left( \frac{{{y}_{i}}-\mu }{\sigma } \right)+\log \sigma$$ Substituting into (12) into (11) we obtain a family of location-scale depths. The Student depth of $(\mu ,\sigma )\in \mathbb{R}\times [0,\infty )$ is obtained substituting into the above expression the density of the $t$ distribution with $v$ degrees of freedom $$d(\mu ,\sigma )=\mathop{\inf_{{{u}\ne {0}}}} \left\{\sharp i : ({u}_{1},{u}_{2}) \left( \begin{array}{c} {\tau }_{i} \\ \frac{v}{v+1}(\tau_{i}^{2}-1) \\ \end{array} \right) \ge 0 \right\} ,$$ where by the multiplication we mean the dot product,${\tau }_{i}$ is a shorthand for $({{y}_{i}}-\mu )/\sigma $, and we can absorb the constant $v/(v+1)$ into the ${u}$ term (see Fig. \[fig7\] - \[fig8\]) The Student Median (SM) is the maximum depth estimator induced by the Student depth. It is very interesting joint estimator of location and scale in a context of robust time series analysis. It is robust but not very robust – its BP is about 33% and hence is robust to a moderate fraction of outliers but is sensitive to a regime change of a time series at the same time. It is worth noticing, that by its definition, the SM is not affected by temporal dependence of the observations. ![Sample student depth contour plot, data from t(1).[]{data-label="fig8"}](figure/PROC_7){width=".95\linewidth"} ![Sample student depth contour plot, data from t(1).[]{data-label="fig8"}](figure/PROC_8){width=".95\linewidth"} Local depth ----------- In an opposition to the density function, the depth function has a global nature i.e., e.g., that it expresses a centrality of a point w.r.t. a whole sample. This property is an advantage of depth for some applications but may be treated as its disadvantage in the context od classification of objects or for k-nearest neighbour rule applications. Depth based classifier or depth based k-nearest density estimators need local version of depths. A successful concept of local depth was proposed in [@Pain:2012]. For defining a neighbourhood of a point, authors proposed using idea of symmetrisation of a distribution (a sample) with respect to a point in which depth is calculated. In their approach instead of a distribution ${P}^{X}$, a distribution ${{P}_{x}}=1/2{{P}^{X}}+1/2{{P}^{2x-X}}$ is used. For any $\beta \in (0,1]$, let us introduce the smallest depth region with probability bigger or equal to $\beta $, $${R}^{\beta }(F)=\bigcap\limits_{\alpha \in A(\beta )}{{{D}_{\alpha }}}(F),$$ where $A(\beta )=\left\{ \alpha \ge 0:P\left[ {{D}_{\alpha }}(F) \right]\ge \beta \right\}$. Then for a locality parameter $\beta \in (0,1]$ we can take a neighbourhood of a point $x$ as $R^{\beta }(P_{x})$ (see Fig. \[fig9\] - \[fig10\]). Formally, let $D(\cdot,P)$ be a depth function. Then the local depth with the locality parameter $\beta \in (0,1]$ and w.r.t. a point $x$ is defined as $$L{{D}^{\beta }}(z,P):z\to D(z,P_{x}^{\beta }),$$ where $P_{x}^{\beta }(\cdot )=P\left( \cdot \|R_{x}^{\beta }(P) \right)$ is cond. distr. of $P$ conditioned on $R_{x}^{\beta }(P)$. For $\beta=1$ the local depth reduces to its global counterpart (no localization). ![Local $L^2$ depth, locality = 20%.[]{data-label="fig10"}](figure/PROC_9){width=".95\linewidth"} ![Local $L^2$ depth, locality = 20%.[]{data-label="fig10"}](figure/PROC_10){width=".95\linewidth"} In a sample case ${{X}^{n}}=\{{{x}_{1}},...,{{x}_{n}}\}$, in a first step we calculate depth of a point $y$ by adding to the original observations ${{x}_{1}},...,{{x}_{n}}$ their reflections $2y-{{x}_{1}},...,2y-{{x}_{n}}$ w.r.t. $y$ – let us denote this combined sample $X_{n}^{y}$ and then calculating usual depth. Then we order observations from the original sample w.r.t. $D(\cdot ,X_{n}^{y})$ the sample depth calculated from the combined sample: $D({{x}_{(1)}},X_{n}^{y})\ge ...\ge D({{x}_{(n)}},X_{n}^{y})$. We choose the locality parameter $\beta \in (0,1]$ determining a size of depth based neighbourhood of the point $x$. Then we determine ${{n}_{\beta }}(X_{n}^{y})=\max \left\{ l=\left\lceil n\beta \right\rceil ,...,n \right\}:D({{x}_{(l)}},X_{n}^{y})=D({{x}_{(\left\lceil n\beta \right\rceil )}},X_{n}^{y})\}$ . Finally we calculate $L{{D}^{\beta }}(y,{{X}^{n}})=D(y,X_{n}^{y,\beta })$ , where $X_{n}^{y,\beta }$ denotes subsample ${{x}_{(1)}},...,{{x}_{(n\beta )}}$ of $X_{n}^{y}$. Further theoretical properties involving its weak continuity and almost sure consistency can be found in [@Pain:2012] and [@Pain:2013]. Approximate depth calculation ----------------------------- A direct calculation of many statistical depth functions is a very challenging computational issue. On the other hand a computational tractability of depths and induced by them procedures is especially important for online economy involving monitoring high frequency financial data, social networks or for shopping center management (see [@Kos:2014c]). Within the package we use approximate algorithm proposed in [@Dyck:2004] to calculation of a certain class of location depth functions (depths possessing so called strong projection property), we base also on a algorithm proposed by Rousseeuw i Hubert (1998) for deepest regression calculation and direct algorithm for Student depth calculation proposed in [@lsdepth]. For calculation of the local depths we use direct method described in [@Pain:2012]. Below we briefly present main ideas of the Dyckerhoff algorithm. DEFINITION ([@Dyck:2004]): Let $D$ be a depth on $\mathcal{X}$ . $D$ satisfies the (weak) projection property, if for each point $y\in \mathbb{R}^{d}$ and each random vector $X\in \mathcal{X}$ it holds: $$D(y,X)=\inf \left\{ D({{p}^{\top}}y,{{p}^{\top}}X):p\in {S}^{d-1} \right\}.$$ THEOREM 1 ([@Dyck:2004]): For each $X\in \mathcal{X}$ let ${{\left( {{Z}_{\alpha }}(X) \right)}_{\alpha \ge 0}}$ be a family of subsets of ${{\mathbb{R}}^{d}}$ that satisfy the properties Z1 to Z5. Further let ${{Z}_{0}}(X)={{\mathbb{R}}^{d}}$ for every $X\in \mathcal{X}$. If $D$ is defined by $D(z,X)=\sup \left\{ \alpha :{{Z}_{\alpha }}(X) \right\}$ , then $D$ is a depth on $\mathcal{X}$ and the sets ${{Z}_{\alpha }}(X)$ are trimmed regions of $D$. THEOREM 2 ([@Dyck:2004]): Let ${{D}^{1}}$ be a univariate depth. If $D$ is defined by $D(z,X)=\underset{p\in {{S}^{d-1}}}{\mathop{\inf }}\,{{D}^{1}}({{p}^{\top}}z,{{p}^{\top}}X)$ , then $D$ is a multivariate convex depth that satisfies the weak projection property. Theorem 2 shows how multivariate depths can be obtained from univariate depths via the projection property. In theorem 1 a depth was defined by the family of its trimmed regions. By combining these two results one arrives at a construction method of multivariate depths from univariate trimmed regions. For practical applications of the above approach it is of prior importance to replace $sup$ and $inf$ by means of $max$ and $min$, i.e., approximate multivariate depth by means of a finite number of projections. Theoretical background of the issue can be found in [@Cuesta:2008] and references therein. In the in order to decrease the computational burden related to sample depth calculation we use proposition 11 from [@Dyck:2004]. We use 1000 random projections from the uniform distribution on a sphere. We use following families of one-dimensional central regions: 1\. For Tukey depth ${{Z}_{\alpha }}(X)=\left[ {{Q}_{X}}(\alpha ),{{{\bar{Q}}}_{X}}(\alpha ) \right],$ 2\. For zonoid depth (see also [@Mosler:2013], [@Mosler:2014]) ${{Z}_{\alpha }}(X)=\left[ \frac{1}{\alpha }\int\limits_{0}^{\alpha }{{{Q}_{X}}(p)dp,\frac{1}{\alpha }\int\limits_{0}^{\alpha }{{{{\bar{Q}}}_{X}}(p)dp,}} \right],$ 3\. For a symmetric projection depth (see [@Zuo:2003]) ${{D}_{\alpha }}(X)=\left[ me{{d}_{X}}-c(\alpha )MA{{D}_{X}},me{{d}_{X}}+c(\alpha )MA{{D}_{X}} \right],$ where $c(\alpha )=(1-\alpha )/\alpha $. Existing software for depth calculation --------------------------------------- Currently there are four packages available which are directly dedicated for depth calculation: [@depthPack], [@depthTools], [@localdepth] and [@Mosler:2014]. Additionally, two packages [@fda.usc] and [@Ramsay:2009], consist of tools related to depths for functional data. The package allows for exact and approximate calculation of Tukey, Liu and Oja depths. It also provides tools for visualisation contour plots and perspective plots of depth functions, and function for depth median calculation. Note, that commands and which are available within the were patterned on these commands. The is focused on the Modified Band Depth (MBD) for functional data [@Lop:2009]. It provides scale curve, rank test based on MBD and two methods of supervised classification techniques, the DS and TAD methods. The package enables us for calculation of local version of “simplicial”, “ellipsoid”, “halfspace” (Tukey’s depth), “mahalanobis” and “hyperspheresimplicial” depth functions. The also has a function for depth-vs-depth plot, which differs from the function which is available within the . In the , the DDPlot is a plot of normalized localdepth versus normalized depth.We should note also that version of the local depth which is available within the differs from a more general version proposed in [@Pain:2013], which is available within the . The package concentrates around a new method for classification basing on the DD-plot prepared using the random Tukey depth and zonoid depth. Package description and illustrative examples ============================================= Our package comprises among other of the commands listed in a Table \[tab1\]. Command Short description ------------- ------------------------------------------------------------------ multivariate asymmetry functional depth based simple binning of 2D data $L^p$ depth weighted location and scatter estimator multivariate quantile-quantile normality plot deepest regression estimator for simple regression depth calculation depth contour plot depth weighted density estimator fast modified band depth calculation multivariate median calculation depth perspective plot local depth calculation Student median calculation bootstrap region for a multivariate median multivariate Wilcoxon test for location and/or scale differences multivariate scatter functional projection depth trimmed regression 2D : Main commands available within the .[]{data-label="tab1"} The and commands, dedicated correspondingly to nonparametric weighted by local depth conditional probability density estimator and for fast calculation of the modified band depth for functional data, are under development. These commands indicate a direction of a further development of the package however. Available depths functions -------------------------- A basic command for depth calculation is\ 0.5mm 0.5mm Arguments\ 0.5mm u: Numerical vector or matrix, which depth is to be calculated. A dimension has to be the same as that of the observations.\ X: The data as a matrix, a data frame or a list. If it is a matrix or data frame, then each row is treated as one multivariate observation. If it is a list, all components must be numerical vectors of equal length (coordinates of the observations).\ method: Character string determining the depth function. The method can be “Projection” (the default), “Mahalanobis”, “Euclidean”, “Tukey”, “LP” or “Local”.\ p: $L^p$ depth parameter.\ beta: locality parameter. Maximal depth estimators ------------------------ The enables for calculating multivariate medians induced by depth functions. 0.5mm 0.5mm Arguments: 0.5mm x: The data as a $k\times 2$ matrix or data frame.\ method: Character string determining the depth function. The method can be “Projection” (the default), “Mahalanobis”, “Euclidean”, “Tukey”, “LP” or “Local”.\ p: $L^p$ depth parameter.\ DepthContour ------------ Basic statistical plots offered by are the contour plot and the perspective plot (see Fig. \[fig11\] – \[fig12\]).\ 0.5mm Arguments\ 0.5mm x: The data as a $k\times 2$ matrix or data frame.\ 0.5mm 3d Plot – by default, plot from lattice is drawn. You can use plot\_method=“rgl”, but currently is not on “depends” - list. Note - can cause some problems with installation on clusters without OpenGL. 0.5mm\ 0.5mm Arguments\ 0.5mm x: The data as a $k\times 2$ matrix or data frame.\ ![Sample perspective plot.[]{data-label="fig12"}](figure/UN_4){width=".95\linewidth"} ![Sample perspective plot.[]{data-label="fig12"}](figure/UN_5){width=".95\linewidth"} DD-plots -------- For two probability distributions $F$ and $G$, both in $\mathbb{R}^{d}$, we can define depth vs. depth plot being very useful generalization of the one dimensional quantile-quantile plot: $$DD(F,G)=\left\{ \left( D({z},F),D({z},G) \right),{z}\in {{\mathbb{R}}^{d}} \right\}$$ Its sample counterpart calculated for two samples ${{{X}}^{n}}=\{{{X}_{1}},...,{{X}_{n}}\}$ from $F$, and ${{Y}^{m}}=\{{{Y}_{1}},...,{{Y}_{m}}\}$ from $G$ is defined as $$DD({{F}_{n}},{{G}_{m}})=\text{ }\left\{ \left( D({z},{{F}_{n}}),D({z},{{G}_{m}}) \right),{z}\in \text{ }\!\!\{\!\!\text{ }{{{X}}^{n}}\cup {{{Y}}^{m}}\} \right\}$$ A detailed presentation of the DD-plot can be found in [@Liu:1999]. Fig. \[fig13\] presents DD-plot with a heart-shaped pattern in case of differences in location between two samples, whereas Fig. \[fig14\] presents a moon-shaped pattern in case of scale differences between samples. Applications of DD-plot and theoretical properties of statistical procedures using this plot can be found in [@Li:2004], [@Liu:1993], [@Jure:2012], [@Zuo:2006], [@Kos:2014b]. In [@Mosler:2014] an application of the DD-plot for classification can be found.\ Within the we can use DD-plot in a following way: Arguments\ x: The first or only data sample for ddPlot.\ y: The second data sample. x and y must be of the same dimension.\ scale : If TRUE samples are centered using multivariate medians. and Arguments\ x: The data sample for DD plot.\ size: Size of theoretical set.\ robust: Logical. Default FALSE. If TRUE, robust measures are used to specify the parameters of theoretical distribution.\ alpha:cutoff point for robust measure of covariance. ![DD-plot, scatter differences.[]{data-label="fig14"}](figure/PROC_11){width=".95\linewidth"} ![DD-plot, scatter differences.[]{data-label="fig14"}](figure/PROC_12){width=".95\linewidth"} 0.5mm Multivariate Wilcoxon test -------------------------- Having two samples $\mathbf{X}^{n}$ and $\mathbf{Y}^{m}$ using any depth function, we can compute depth values in a combined sample $\mathbf{Z}^{n+m}$ = $\mathbf{X}^{n}\cup \mathbf{Y}^{m}$, assuming the empirical distribution calculated basing on all observations, or only on observations belonging to one of the samples $\mathbf{X}^{n}$ or $\mathbf{Y}^{m}.$ For example if we observe ${X}_{l}'s$ depths are more likely to cluster tightly around the center of the combined sample, while ${Y}_{l}'s$ depths are more likely to scatter outlying positions, then we conclude $\mathbf{Y}^{m}$ was drawn from a distribution with larger scale. Properties of the DD plot based statistics in the i.i.d setting were studied in [@Li:2004]. Authors proposed several DD-plot based statistics and presented bootstrap arguments for their consistency and good effectiveness in comparison to Hotelling $T^2$ and multivariate analogues of Ansari-Bradley and Tukey-Siegel statistics. Asymptotic distributions of depth based multivariate Wilcoxon rank-sum test statistic under the null and general alternative hypotheses were obtained in [@Zuo:2006]. Several properties of the depth based rang test involving its unbiasedness was critically discussed in [@Jure:2012]. Basing on DD-plot object, which is available within the it is possible to calculate several multivariate generalizations of one-dimensional rank and order statistics. These generalizations cover well known Wilcoxon rang-sum statistic.\ The depth based multivariate Wilcoxon rang sum test is especially useful for the multivariate scale changes detection and was introduced among other in [@Liu:1993]\ 0.5mm For the samples ${{\mathbf{X}}^{m}}=\{{{\mathbf{X}}_{1}},...,{{\mathbf{X}}_{m}}\}$ , ${{\mathbf{Y}}^{n}}=\{{{\mathbf{Y}}_{1}},...,{{\mathbf{Y}}_{n}}\}$, and a combined sample ${\mathbf{Z}}={{\mathbf{X}}^{n}}\cup {{\mathbf{Y}}^{m}}$ the Wilcoxon statistic is defined as $$S=\sum\limits_{i=1}^{m}{{{R}_{i}}}, \label{eq4}$$ where ${R}_{i}$ denotes the rang of the i-th observation, $i=1,...,m$ in the combined sample $R({{\mathbf{x}}_{l}})= \#\left\{ {{\mathbf{z}}_{j}}\in {{\mathbf{Z}}}:D({{\mathbf{z}}_{j}},{\mathbf{Z}})\le D({{\mathbf{x}}_{l}},{\mathbf{Z}}) \right\}, l=1,...,m.$ 0.5mm The distribution of $S$ is symmetric about $E(S)=1/2m\text{(}m\text{+}n\text{+1)}$, its variance is ${{D}^{2}}(S)={1}/{12}\;mn(m+n+1).$ For theoretical properties statistic see [@Li:2004] and [@Zuo:2006].\ Using DD-plot object it is easy to calculate other multivariate test statistics involving for example Haga or Kamat tests and apply them for robust monitoring of multivariate time series (see [@Kos:2014b]). 0.5mm 0.5mm Arguments\ 0.5mm x,y: data matrices or data frames of the same dimension\ alternative:\ Character string determining the alternative, as in one-dimensional Wilcoxon test\ method: Character string determining the depth function. method can be “Projection” (the default), “Mahalanobis”, “Euclidean”, “Tukey”, “LP” or “Local”.\ 0.5mm EXAMPLE >require(MASS) >x = mvrnorm(100, c(0,0), diag(2)) >y = mvrnorm(100, c(0,0), diag(2)\1.4) >mWilcoxonTest(x,y) Wilcoxon rank sum test with continuity correction data: dep\_x and dep\_y W = 5103, p-value = 0.4011 alternative hypothesis: true location shift is greater than 0 Scale and asymmetry curves --------------------------- For sample depth function $D({x},{{{Z}}^{n}})$, ${x}\in {{\mathbb{R}}^{d}}$, $d\ge 2$, ${Z}^{n}=\{{{{z}}_{1}},...,{{{z}}_{n}}\}\subset {{\mathbb{R}}^{d}}$, ${{D}_{\alpha }}({{{Z}}^{n}})$ denoting $\alpha-$ central region, we can define the scale curve* (see Fig. \[fig15\]) $$SC(\alpha )=\left( \alpha ,vol({{D}_{\alpha }}({{{Z}}^{n}}) \right)\subset {{\mathbb{R}}^{2}},\hskip2mm for \hskip2mm \alpha \in [0,1],$$ and the asymmetry curve (see Fig. \[fig16\]) $$AC(\alpha )=\left( \alpha ,\left\\| {{c}^{-1}}(\{{\bar{z}}-med\|{{D}_{\alpha }}({{{Z}}^{n}})\}) \right\\| \right)\subset {{\mathbb{R}}^{2}}, \hskip2mm for \hskip2mm \alpha \in [0,1]$$ being nonparametric scale and asymmetry functional correspondingly, where $c-$denotes constant, ${\bar{z}}-$denotes mean vector, denotes multivariate median induced by depth function and $vol-$ denotes a volume. Further information on the scale curve and the asymmetry curve can be found in [@Liu:1999], [@Serfling:2006], [@Serfling:2004], [@Serfling:2006a]. Arguments\ x: a matrix consisting data.\ y: additional data matrix.\ alpha: a vector of central regions indices.\ method: character string which determines the depth function used, method can be “Projection” (the default), “Mahalanobis”, “Euclidean”, “Tukey” or “LP”.\ plot: Logical. Default TRUE produces scale curve plot; otherwise, returns a data frame containing the central areas and their volume. 0.5mm Arguments\ 0.5mm movingmedian: Logical. For default FALSE only one depth median is used to compute asymmetry norm. If TRUE – for every central area, a new depth median will be used - this approach needs much more computation time. ![Asymmetry curves.[]{data-label="fig16"}](figure/PROC_13){width=".95\linewidth"} ![Asymmetry curves.[]{data-label="fig16"}](figure/PROC_14){width=".95\linewidth"} EXAMPLE >x = mvrnorm(1000, c(0,0),diag(2)) >s1 = scaleCurve(x,name = “Curve 1”) >s2 = scaleCurve(x\2,x\3,name = “Curve 2”, name\_y = “Curve 3”, plot = FALSE) >w = getPlot(s1 >w + theme(text = element\_text(size = 25)) >xx = mvrnorm(1000, c(0,0),diag(2)) >yy = mvrnorm(1000, c(0,0),diag(2)) >p = asymmetryCurve(xx,yy, plot = FALSE) >getPlot(p)+ggtitle(“Plot”) >xx = mvrnorm(1000, c(0,0),diag(2)) >yy = mvrnorm(1000, c(0,0),diag(2)) >p = asymmetryCurve(xx,yy, plot = FALSE) >getPlot(p)+ggtitle(“Plot”) Simple robust regressions ------------------------- Within the package two simple (two dimensional) robust regressions are available: the deepest regression and projection depth trimmed regression – see Fig. \[fig17\].\ \ 0.5mm Arguments 0.5mm x,y: data vectors alpha: trimming parameter ![Simple regressions.[]{data-label="fig17"}](figure/PROC_15){width=".95\linewidth"} EXAMPLE >plot(starsCYG,cex=1.4) >deepreg = deepReg2d(starsCYG$log.Te, starsCYG$log.light) >trimreg = trimProjReg2d(starsCYG$log.Te, starsCYG$log.light) >least.sq = lm(starsCYG$log.Te~starsCYG$log.light) >abline(deepreg, lwd = 3, col = “red”) >abline(trimreg, lwd = 3, col = “brown”) >abline(least.sq, lwd = 3, col = “blue”) coefficients: deepreg@coef -7.903043 2.913043 trimreg@coef -7.403531 2.802837 Weighted estimators of location and scatter ------------------------------------------- Using depth function one can define a depth-weighted multivariate location and scatter estimators possessing high breakdown points and which for several depths are computationally tractable (see [@Zuo:2005]). In case of location, the estimator is defined as $$L(F)={\int{{x}{{w}_{1}}(D({x},F))dF({x})}}/{{{w}_{1}}(D({x},F))dF({x})},$$ Subsequently, a depth-weighted scatter estimator is defined as $$S(F)=\frac{\int{({x}-L(F)){{({x}-L(F))}^{\top}}{{w}_{2}}(D({x},F))dF({x})}}{\int{{{w}_{2}}(D({x},F))dF({x})}},$$ where ${{w}_{2}}(\cdot )$ is a suitable weight function that can be different from ${{w}_{1}}(\cdot )$. 0.5mm The package offers these estimators in case of computationally feasible weighted ${L}^{p}$ depth. Note that$L(\cdot )$ and $S(\cdot )$ include multivariate versions of trimmed means and covariance matrices. Sample counterparts of (20) and (21) take the forms $${{T}_{WD}}({{{X}}^{n}})={\sum\limits_{i=1}^{n}{{w({d}_{i})}{{X}_{i}}}}/{\sum\limits_{i=1}^{n}{{w({d}_{i})}}} ,$$ $$DIS({{{X}}^{n}})=\frac{\sum\limits_{i=1}^{n}{{w({d}_{i})}\left( {{{X}}_{i}}-{{T}_{WD}}({{{X}}^{n}}) \right){{\left( {{{X}}_{i}}-{{T}_{WD}}({{{X}}^{n}}) \right)}^{T}}}}{\sum\limits_{i=1}^{n}{{w({d}_{i})}}},$$ where ${{d}_{i}}$ are sample depth weights, ${{w}_{1}}(x)={{w}_{2}}(x)=a\cdot x +b$, $a, b \in \mathbb{R}$.\ Computational complexity of the scatter estimator crucially depend on the complexity of the depth used. For the weighted ${L}^{p}$ depth we have $O({{d}^{2}}n+{{n}^{2}}d)$ complexity and good perspective for its distributed calculation [@Zuo:2004]. EXAMPLE >require(MASS) >Sigma1 <- matrix(c(10,3,3,2),2,2) >X1 = mvrnorm(n= 8500, mu= c(0,0),Sigma1) >Sigma2 <- matrix(c(10,0,0,2),2,2) >X2 = mvrnorm(n= 1500, mu= c(-10,6),Sigma2) >BALLOT<-rbind(X1,X2) >train <- sample(1:10000, 500) >data<-BALLOT\[train,\] >cov\_x = CovLP(data,1,1,1) >cov\_x Call: -> Method: Depth Weighted Estimator Robust Estimate of Location: \[1\] -1.6980 0.8844 Robust Estimate of Covariance: \[,1\] \[,2\] \[1,\] 15.249 -2.352 \[2,\] -2.352 4.863 Student and $L^{p}$ binning --------------------------- Let us recall, that binning is a popular method allowing for faster computation by reducing the continuous sample space to a discrete grid (see [@Hall:1996]). It is useful for example in case predictive distribution estimation by means of kernel methods. To bin a window of $n$ points ${W}_{i,n}=\left\{{X}_{i-n+1},...,{X}_{i} \right\}$ to a grid ${X}'_{1},...,{X}'_{m}$ we simply assign each sample point ${X}_{i}$ to the nearest grid point ${X}'_{j}$. When binning is completed, each grid point ${X}'_{j}$ has an associated number ${c}_{i}$, which is the sum of all the points that have been assigned to ${X}'_{j}$. This procedure replaces the data ${W}_{i,n}=\left\{ {X}_{i-n+1},...,{X}_{i} \right\}$ with the smaller set ${W}'_{j,m}=\left\{ {X}'_{j-m+1},...,{X}'_{j} \right\}$. Although simple binning can speed up the computation, it is criticized for a lack of a precise approximate control over the accuracy of the approximation. Robust binning however stresses properties of the majority of the data and decreases the computational complexity of the DSA at the same time. For a 1D window ${W}_{i,n}$, let ${Z}_{i,n-k}$ denote a 2D window created basing on ${W}_{i,n}$ and consisted of $n-k$ pairs of observations and the $k$ lagged observations ${Z}_{i,n-k}$=$\left\{ ({X}_{i-n-k},{X}_{i-n+1})\right\}$, $1\le i\le n-k$ . Robust 2D binning of the ${Z}_{i,n-p}$ is a very useful technique in a context of robust estimation of the predictive distribution of a time series (see [@Kosiorowski:2013b]) or robust monitoring of a data stream (see [@Kos:2014b]). Assume we analyze a data stream $\{{X}_{t}\}$ using a moving window of a fixed length $n$, i.e., ${W}_{i,n}$ and the derivative window ${Z}_{i,n-1}$. In a first step we calculate the weighted sample $L^p$ depth for ${W}_{i,n}$. Next we choose equally spaced grid of points ${l}_{1},...,{l}_{m}$ in this way that $[{{l}_{1}},{{l}_{m}}]\times [{{l}_{1}},{{l}_{m}}]$ covers fraction of the $\beta$ central points of ${Z}_{i,n-1}$ w.r.t. the calculated $L^p$ depth, i.e., it covers ${R}^{\beta }({Z}_{i,n-1})$ for certain prefixed threshold $\beta \in (0,1)$. For both ${X}_{t}$ and ${X}_{t-1}$ we perform a simple binning using following bins: $(-\infty ,{l}_{1})$, $({l}_{1},{l}_{2})$,..., $({l}_{m},\infty )$. For robust binning we reject “border” classes and further use only midpoints and binned frequencies for classes $({l}_{1},{l}_{2})$, $({l}_{2},{l}_{3})$,..., $({l}_{m-1},{l}_{m})$. Figures \[fig18\] – \[fig19\] present the idea of the simple $L^{p}$ binning in case of data generated from a mixture of two two-dimensional normal distributions. The midpoints are represented by triangles. ![The second step in $L^p$ depth binning.[]{data-label="fig19"}](figure/bin_1){width=".99\linewidth"} ![The second step in $L^p$ depth binning.[]{data-label="fig19"}](figure/bin_2){width=".99\linewidth"} EXAMPLE 1 >require(MASS) >Sigma1 = matrix(c(10,3,3,2),2,2) >X1 = mvrnorm(n= 8500, mu= c(0,0),Sigma1) >Sigma2 = matrix(c(10,0,0,2),2,2) >X2 = mvrnorm(n= 1500, mu= c(-10,6),Sigma2) >BALLOT = rbind(X1,X2) >train = sample(1:10000, 500) >data =BALLOT\[train,\] >plot(data) >b1=binningDepth2D(data, remove\_borders = FALSE, nbins = 12, k = 1 ) >b2=binningDepth2D(data, nbins = 12, k = 1,remove\_borders = TRUE ) >plot(b1) >plot(b2) EXAMPLE 2 >data(under5.mort) >data(maesles.imm) >data2011=cbind(under5.mort\[,22\],maesles.imm\[,22\]) >plot(binningDepth2D(data2011, nbins = 8, k = 0.5, remove\_borders = TRUE )) The package architecture ======================== Nomenclature conventions ------------------------ There is no agreed naming convention within project. In our package we use following coding style: - Class names start with an uppercase letter (e.g. ). - For methods and functions we use lower camel case convention (e.g. ) - All functions related to location-scale depth starts with ’lsd’ prefix (e.g. ). - Sometimes we depart from these rules whenever to preserve compatibility, with other packages (e.g. - it is a function from that follows naming convention). Dependencies ------------ Algorithms for depth functions were written in , and they are completely independent from . For matrix operations we use [@Armadillo], and library [@openmp2013] for parallel computing. The communication between and is performed by package [@RcppArmadillo]. For plotting we use graphic (contours plots), package [@lattice] (perspective plot), and [@ggplot2] (other plots). We also uses functions from [@rrcov], [@np], [@geometry] packages. Parallel computing ------------------ By default uses multi-threading and tries to utilize all available processors. User can control this behaviour with threads parameter: EXAMPLE: Tested on: Intel(R) Core(TM) i5-2500K CPU @ 3.30GHz >x = matrix(rnorm(200000), ncol = 5) >system.time(depth(x)) user system elapsed 1.484 0.060 0.420 EXAMPLE: only one thread (approximately 3 times slower): >system.time(depth(x, threads = 1)) user system elapsed 1.368 0.000 1.371 EXAMPLE: any value <1 means “use all possible cores” >system.time(depth(x, threads = -10)) user system elapsed 1.472 0.076 0.416 Classes ------- Below we describe only , , and classes in details, because only them have non standard behaviour. Other classes are very simple. is a class for function. It inherits behaviour from class from package. Description of this class can be found in [@rrcov]. UML diagrams and classes ------------------------ In this paper we exploit UML class diagrams to describe a behaviour of main structures. The UML abbreviation stands for Unified Modelling Language, a system of notation for describing object oriented programs. In the UML, class is denoted by a box with three compartments which contain the name, the attributes (slots) and operations (methods) of the class. Each attribute is followed by its type, and each method by its return value. Inheritance relation between classes are depicted by arrowhead pointing to the base class. Depth class ----------- \[ZZ1\] ![Object structure for classes related to depth functions.](figure/Depth "fig:"){width="50.00000%"} Fig. 20 shows an object structure for classes related to depth functions. Each depth class inherits Depth and standard Numeric. Through inheritance after Numeric these classes are treated as a standard vector, and one can use them with all functions that are appropriate for vectors (e.g. max, min). Depth class is mainly used in internal package operations, but it can be used for extracting depth median without recomputing depth values. This mechanism is show in following example: EXAMPLE: function for numeric vector >x = matrix(rnorm(1e5), ncol = 2) >dep = depth(x) >max(dep) \[1\] 0.9860889 EXAMPLE: function for raw matrix - all depths must be recomputed: >system.time(dx <- depthMedian(x)) user system elapsed 1.609 0.072 0.451 EXAMPLE: function for Depth class - result is immediate system.time(dm <- depthMedian(dep)) user system elapsed 0.000 0.000 0.001 In order to check the equality >all.equal(dm, dx) \[1\] TRUE DepthCurve and DDplot classes ----------------------------- The is a main class for storing results from and the functions, and describing their behaviour - see Fig. 20. The stores results from and functions. Both classes and can be converted into object for further appearance modifications via function.\ EXAMPLE: >x = matrix(rnorm(1e2), ncol = 2) >y = matrix(rnorm(1e2), ncol = 2) >ddplot = ddPlot(x,y) >p = getPlot(ddplot) In order to modify a title >p + ggtitle(“X vs Y”) >scplot = scaleCurve(x,y) >p = getPlot(scplot) In order to change a color palette: >p + scale\_color\_brewer(palette = “Set1”) Fig. 21 shows class structure for . Class is a container for storing multiple curves for charting them on one plot. It inherits behaviour from standard list, but it can be also converted into object with method. We introduced $\%+\%$ operator for combining into . This operator is presented in following example:\ EXAMPLE >data(under5.mort) >data(maesles.imm) >data2011=cbind(under5.mort\[,“2011”\],maesles.imm\[,“2011”\]) >data2000=cbind(under5.mort\[,“2000”\],maesles.imm\[,“2000”\]) >data1995=cbind(under5.mort\[,“1995”\],maesles.imm\[,“1995”\]) >sc2011 = scaleCurve(data2011, name = “2011”) >sc2000 = scaleCurve(data2000, name = “2000”) In order to create ScalueCurveList >sclist = sc2000 >sclist In order to add another Curve >sc1995 = scaleCurve(data1995, name = “1995”) >sclist \[ZZ2\] ![Class structure for .](figure/DepthCurve "fig:"){width="80.00000%"} EXAMPLE >n = 200 >mat\_list = replicate(n,matrix(rnorm(200),ncol = 2),simplify = FALSE) >scurves = lapply(mat\_list, scaleCurve) >scurves = Reduce(“>p = getPlot(scurves) >p + theme(legend.position=”none“) + >scale\_color\_manual(values = rep(”black",n)) Empirical example ================= For illustrating usefulness of the package in a socio-economic researches, let us consider an issue of a nonparametric evaluation of the Fourth Millennium Development Goal of The United Nations (4MG). Main aim of the goal was reducing by two–thirds, between 1990 – 2015, the under five months child mortality. Using selected multivariate techniques which are available within our package we answer a question, if during the period 1990 – 2015 differences between developed and developing countries really decreased.. In the study we jointly considered following variables: - [Children under 5 months mortality rate per 1,000 live births ($Y_1$)]{} - [Infant mortality rate (0–1 year) per 1,000 live births ($Y_2$)]{} - [Children 1 year old immunized against measles, percentage ($Y_3$)]{} Data sets were obtained from <http://mdgs.un.org/unsd/mdg/Data.aspx> and are available within the package. ![2011: $L^2$ depth contour plot $Y_2$ vs. $Y_3$](figure/E_1){width=".95\linewidth"} ![2011: $L^2$ depth contour plot $Y_2$ vs. $Y_3$](figure/E_2 "fig:"){width=".95\linewidth"} \[fig21\] ![2011: $L^2$ depth contour plot $Y_2$ vs. $Y_3$](figure/E_3){width=".95\linewidth"} ![2011: $L^2$ depth contour plot $Y_2$ vs. $Y_3$](figure/E_4 "fig:"){width=".95\linewidth"} \[fig23\] Fig. \[fig20\] shows weighted $L^2$ depth contour with locality parameter $\beta=0.5$ for countries in 1990 considered w.r.t. variables $Y_1$ and $Y_3$ whereas Fig. \[fig21\] presents the same issue but in 2011. Fig. \[fig22\] shows weighted $L^2$ depth contour with locality parameter $\beta=0.5$ for countries in 1990 considered w.r.t. variables $Y_2$ and $Y_3$ whereas Fig. \[fig23\] presents the same issue but in 2011. Although we can notice a socio-economic development between 1990 and 2011 – the clusters of developed and developing countries are still evident in 2011 as they were in 1990. For assessing changes in location of the centers and scatters of the data between 1990 and 2011 we calculated $L^2$ medians and $L^2$ weighted covariance matrices for $(Y_1,Y_2,Y_3)$ which are presented below\ MED(1990): (23.2; 19.6; 86.0) MED(1995): (18.5; 15.5; 90.0) MED(2000): (17.2; 14; 93.0) MED(2005): (18.4; 15.5; 94.0) MED(2010): (13.8; 11.7; 95.0) $$COV_{L^2}(1990)=\left( \begin{matrix} 3044 & 1786 & -488 \\ 1786 & 1086 & -287 \\ -488 & -287 & 247 \\ \end{matrix} \right)$$ $$COV_{L^2}(2010)=\left( \begin{matrix} 914 & 600 & -195 \\ 600 & 402 & -127 \\ -195 & -127 & 126 \\ \end{matrix} \right)$$ Fig. \[fig24\] presents DD-plot for inspecting location changes between 1990 and 2011 for countries considered w.r.t. variables $Y_1, Y_2, Y_3$ and Fig. \[fig25\] presents DD-plot for inspecting scale changes for the same data. ![DD plot for inspecting scale differences.[]{data-label="fig25"}](figure/E_5){width=".95\linewidth"} ![DD plot for inspecting scale differences.[]{data-label="fig25"}](figure/E_6){width=".95\linewidth"} We performed multivariate Wilcoxon test (using $L^2$ depth) for scale change detection for $(Y_1,Y_2,Y_3)$ in 1990 and in 2011 induced by projection depth and obtained: W=21150 and p-value=0.0046. We can conclude therefore that both the scale and the location changed. Fig. 28 presents scale curves for the countries considered in the period 1990–2011 jointly w.r.t. all variables whereas Fig. 29 presents Student depth contour plots for variable $Y_1$ in 1990–2011. ![Student depth contour plots - $Y_1$ in 1990–2011](figure/FIG_3 "fig:"){width=".95\linewidth"} \[fig26\] ![Student depth contour plots - $Y_1$ in 1990–2011](figure/FIG_5 "fig:"){width=".95\linewidth"} \[fig27\] The results of the analysis lead us to following conclusions: 1. [There are big chances for obtaining the 4MG. In the 2010 year, the decrease in the under five months child mortality was about 40% with robust estimates used.]{} 2. [For the considered variables, both multivariate as well as univariate, scatters decreased in 1990–2011.]{} 3. [The dispersion between countries considered jointly with respect to variables $(Y_1,Y_2,Y_3)$ significantly decreased in 1990–2011. The clusters of rich and poor countries are still easily distinguishable however.]{} 4. [A comparison of Student depth medians of Children under 5 months mortality rate per 1,000 live births in 1990–2011 indicates significant one-dimensional tendency for obtaining the 4MG.]{} 5. [Calculated simple deepest regressions for the variables and additional socio-economic variables show clear relations between the 4MG Indicators and with other economic variables representing economic devolvement (e.g., GDP per Capita).]{} 6. [The data depth concept offers a complex family of powerful and user-friendly tools for nonparametric and robust analysis of socio-economic multivariate data.]{} Further considerations related to the issue can be found in [@Kos:2014a]. Summary ======= This paper presents package which offers a selection of multivariate statistical methods originating from the DDC.\ Theory of the DDC is still developing by many authors. Recent findings presented in the DDC literature involve among other depths on infinite dimensional spaces, very fast algorithms for approximate depth calculation, new classification rules and new depths on functional spaces. The package consists of a selection of very powerful but simple and user friendly tools dedicated for a robust economic analysis.\ Our plans for a future development of the package concentrate around the concepts of local depth and and depth for functional data. We are going to incorporate these ideas into the Theory of Economics. Acknowledgement {#acknowledgement .unnumbered} =============== Daniel Kosiorowski thanks for the polish NCS financial support DEC-011/03/B/HS4/01138.
--- abstract: 'We study inflation in the Brans-Dicke gravity as a special model of the scalar-tensor gravity. We obtain the inflationary observables containing the scalar spectral index, the tensor-to-scalar ratio, the running of the scalar spectral index and the equilateral non-Gaussianity parameter in terms of the general form of the potential in the Jordan frame. Then, we compare the results for various inflationary potentials in light of the Planck 2015 data. Our study shows that in the Brans-Dicke gravity, the power-law, inverse power-law and exponential potentials are ruled out by the Planck 2015 data. But, the hilltop, Higgs, Coleman-Weinberg and natural potentials can be compatible with Planck 2015 TT,TE,EE+lowP data at 95% CL. Moreover, the D-brane, SB SUSY and displaced quadratic potentials can be in well agreement with the observational data since their results can lie inside the 68% CL region of Planck 2015 TT,TE,EE+lowP data.' address: \| $^1$Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan, Iran\ $^2$Department of Physics, University of Kurdistan, Pasdaran St., P.O. Box 66177-15175, Sanandaj, Iran author: - 'B. Tahmasebzadeh$^{1}$[^1], K. Rezazadeh$^{2}$[^2] and K. Karami$^{2}$[^3]' title: 'Brans-Dicke inflation in light of the Planck 2015 data' --- Introduction {#sec:int} ============ The model of Hot Big Bang cosmology has impressive successes such as explaining the light nucleosynthesis and the cosmic microwave background (CMB) radiation. Despite its considerable successes, it suffers from central problems such as the flatness problem, the horizon problem and also the magnetic monopole problem. Inflation theory was proposed to solve all of these problems [@Starobinsky1980; @Sato1981a; @Sato1981; @Guth1981; @Linde1982; @Albrecht1982; @Linde1983]. Inflation is not a replacement for the Hot Big Bang cosmology, but rather an extra add-on idea which supposes that a short period of rapid accelerated expansion has occurred before the radiation dominated era. In addition to solving the problems of the Hot Big Bang cosmology, inflation can provide a reasonable explanation for the anisotropy observed in the CMB radiation and also in the large-scale structure (LSS) of the universe [@Mukhanov1981; @Hawking1982; @Starobinsky1982; @Guth1982]. This fact makes it possible for us to contact the late time observations to the early stages of our universe. Important observational results are provided by the Planck satellite from probing of the CMB radiation anisotropies in both temperature and polarization [@Planck2015]. Using these observational results, we can distinguish viable inflationary models and also constrain them. The standard inflationary scenario is based on a canonical scalar field in the framework of Einstein gravity. Viability of different inflationary models in the framework of standard inflationary scenario in light of the observational results has been extensively investigated in the literature [@Martin2014a; @Martin2014b; @Rezazadeh2015; @Huang2015; @Okada2014]. So far, many inflationary models have been proposed. One important class of inflationary models are based on the extended theories of gravity. The well-known instance for this class of models is the Starobinsky $R^2$ inflation [@Starobinsky1980]. Despite the fact that this model is the first inflationary model, it is in well agreement with the observational results [@Planck2015; @Martin2014a; @Martin2014b; @Rezazadeh2015; @Huang2015]. Inflationary models on the extended theories of gravity have been extensively studied in the literature [@Felice2011a; @Tsujikawa2013; @Artymowski2014; @Artymowski2015; @Barrow1995; @Berman2009; @Rinaldi2015; @Sebastiani2015; @Myrzakulov2015; @Myrzakul2015; @Nashed2014; @Jamil2015; @Sharif2015; @Rezazadeh2016; @Bamba2014; @Bamba2014a; @Bamba2014b; @Bamba2015; @Myrzakulov2015a; @Sebastiani2014; @Cerioni2009; @Cerioni2010; @Finelli2008; @Tronconi2011; @Kamenshchik2011; @Kumar2016; @Kannike2015; @Kannike2016; @Marzola2016]. One important branch of the extended theories of gravity is the scalar-tensor gravity which is a general theory that includes the $f(R)$-gravity, the Brans-Dicke gravity and the dilatonic gravity [@Faraoni2004; @Fujii2004; @Felice2010]. In the present paper, we focus on the Brans-Dicke gravity and study inflation in this framework. In study of inflation, the scalar field is called “inflaton” that can provide a negative pressure needed to have an accelerated expansion. During inflation, the inflaton rolls slowly downward a potential and we can examine its evolution classically [@Lyth2009; @Baumann2009]. At the end of inflation, the inflaton begins to oscillate around the minimum of the potential that leads to particle production and provides for the universe to transit into the radiation dominated era. This period is known as the “reheating” process that its details are unknown to us so far. Also, we don’t know the shape of the inflationary potential that determines the dynamics of the inflaton. In order to understand the inflationary potential, we need more advances in both theory and observations. However, by examination of different potentials in light of the observational results, we can specify some features of the original inflationary potential. In order to relate the present time observations to the inflationary era, we note that besides the classical evolution, the inflaton scalar field has some quantum fluctuations during inflation that can lead to the primordial perturbations whose we can see the imprints on the anisotropies observed in the CMB radiation and in the LSS formation [@Lyth2009; @Baumann2009; @Mukhanov1992; @Mukhanov2005; @Weinberg2008; @Malik2009]. Note that from the energy scale of the primordial universe, it is believed that cosmological inflation has occurred in the regime of high energy physics. Also in one hand, the effective quantum field theory predicts that the high energy theory has fields with non-canonical kinetic terms [@Franche2010; @Franche2010a; @Tolley2010]. On the other hand, from the action of the Brans-Dicke gravity, we know that the kinetic term of this theory has a non-canonical form. This motivates us to investigate the cosmic inflation of the early universe within the framework of the Brans-Dicke gravity. In our work, we concentrate on the various inflationary potentials which have motivations from quantum field theory or string theory and check their viability in light of the Planck 2015 observational results. To do so, first we present a brief review on the scalar-tensor gravity that it will be done in sec. \[sec:st\]. Then, in sec. \[sec:BD\], we will apply the results of sec. \[sec:st\] for the Brans-Dicke gravity as a special case of the scalar-tensor gravity and find the relations of the inflationary observables. This makes it possible for us to examine various inflationary potentials in comparison with the observational results that we proceed to it in sec. \[sec:pot\]. Finally, in sec. \[sec:con\], we summarize our concluding remarks. A brief review on the scalar-tensor gravity {#sec:st} =========================================== At first, in 1950, Jordan applied a scalar field in the gravitational part of the action. Then, in 1961, Brans and Dicke [@Brans1961] introduced a formalism for gravity in which the metric field together with a scalar field have been invoked to describe the gravitational force. After the discovery of the present accelerated expansion of the universe in 1998, other models on the base of the scalar-tensor gravity were proposed to explain this phenomenon [@Amendola1999; @Chiba1999; @Uzan1999; @Perrotta1999; @Boisseau2000; @Esposito2001; @Torres2002]. In this class of models, a scalar field is considered to solve the cosmological constant problems. The scalar-tensor gravity relative to the other competitor theories, posses the advantage that it can involve the dark energy in the form of the energy-momentum tensor $T_{\mu \nu }^\varphi$ and involve the modified gravity in the form of the Einstein tensor $G_{\mu \nu }^\varphi$. Throughout this paper we take the Jordan frame as the physical frame. In the Jordan frame, the general action of the scalar-tensor models can be written in the form [@Faraoni2004; @Fujii2004; @Felice2010a; @Felice2010; @Abdolmaleki2014] $$S_{J} = \int {{d^4}} x\sqrt { - g} \left[ {\frac{1}{2}f(R,\varphi ) - \frac{1}{2}\omega (\varphi ){g^{\mu \nu }}{\partial _\mu }\varphi {\partial _\nu }\varphi - U(\varphi )} \right],\label{Sst}$$ where $f(R,\varphi )$ is a general function of the Ricci scalar $R$ and the scalar field $\varphi$ while the parameter $\omega$ and the potential $U$ are general functions of $\varphi$. Hereafter, we take the reduced Planck mass equal to unity, ${M_P} \equiv 1/\sqrt {8\pi G} = 1$. The above action for the scalar-tensor gravity includes the $f(R)$ models, the Brans-Dicke gravity and also the dilatonic models [@Faraoni2004; @Fujii2004; @Felice2010]. Now, we turn to examine the dynamics of background cosmology in the scalar-tensor gravity. The variation of the action (\[Sst\]) with respect to the metric tensor ${g_{\mu \nu }}$ leads to [@Felice2010; @Abdolmaleki2014] $$\begin{aligned} \nonumber & F{R_{\mu \nu }} - \frac{1}{2}f{g_{\mu \nu }} - {\nabla _\mu }{\nabla _\nu }F + {g_{\mu \nu }}{\nabla ^\alpha }{\nabla _\alpha }F =\\ \label{FRmunu} & \omega (\varphi )\left( {{\nabla _\mu }\varphi {\nabla _\nu }\varphi - \frac{1}{2}{g_{\mu \nu }}{\nabla ^\lambda }\varphi {\nabla _\lambda }\varphi } \right) - U(\varphi ){g_{\mu \nu }},\end{aligned}$$ where ${\nabla _\mu }$ indicates covariant derivative and the function $F$ is defined as $F \equiv \partial f/\partial R$. Variation of the action (\[Sst\]) relative to the scalar field $\varphi$ gives rise to $$\label{nnphi} {\nabla ^\alpha }{\nabla _\alpha }\varphi + \frac{1}{{2\omega (\varphi )}}\left( {{\omega _{,\varphi }}{\nabla ^\lambda }\varphi {\nabla _\lambda }\varphi - 2{U_{,\varphi }} + {f_{,\varphi }}} \right) = 0, ‎$$ where ${U_{,\varphi }} \equiv dU/d\varphi$. For a spatially flat Friedmann-Robertson-Walker (FRW) universe, Eqs. (\[FRmunu\]) and (\[nnphi\]) turn into [@Felice2010; @Abdolmaleki2014] $$\begin{aligned} \label{3HF} 3{H^2}F - \frac{1}{2}(RF - f) + 3H\dot F - \frac{1}{2}\omega (\varphi ){\dot \varphi ^2} - U(\varphi ) &=& 0, \\ \label{2FHdot} \ddot F - H\dot F + 2F\dot H + \omega (\varphi ){\dot \varphi ^2} &=& 0, \\ \label{varphiddot} \ddot \varphi + 3H\dot \varphi + \frac{1}{{2\omega (\varphi )}}\left( {{\omega _{,\varphi }}{{\dot \varphi }^2} + 2{U_{,\varphi }} - {f_{,\varphi }}} \right) &=& 0,\end{aligned}$$ where the dot denotes a derivative with respect to the cosmic time $t$. The Hubble parameter is denoted by $H \equiv \dot a/a$ where $a$ is the scale factor of the universe. Also, $R$ is the Ricci scalar which is given by $$\label{R} R = 6\left( {\frac{{\ddot a}}{a} + \frac{{{{\dot a}^2}}}{{{a^2}}}} \right) = 6\left( {2{H^2} + \dot H} \right).$$ In the following, we briefly review the cosmological perturbations in the scalar-tensor gravity (for more details about this subject see e.g. [@Hwang1990; @Hwang1991; @Hwang1997; @Hwang2001a; @Hwang2001; @Hwang2002; @Hwang2005; @Felice2010]). We consider a general perturbed metric about the flat FRW background as [@Felice2010] $$\label{FRWpert} d{s^2} = - (1 + 2\alpha )d{t^2} - 2a(t)({\partial _i}\beta - {S_i})dt\,d{x^i} + {a^2}(t)({\delta _{ij}} + 2\psi {\delta _{ij}} + 2{\partial _i}{\partial _j}\gamma + 2{\partial _j}{F_i} + {h_{ij}})d{x^i}d{x^j},$$ where $\alpha$, $\beta$, $\psi$ and $\gamma$ are scalar perturbations, $S_i$ and $F_i$ are vector perturbations, and $h_{ij}$ is tensor perturbations. Also the energy-momentum tensor of a perfect fluid with perturbations is given by [@Felice2010] $$\label{Tmunu} T_0^0 = - (\rho + \delta \rho ),\qquad T_i^0 = - (\rho + p){\partial _i}v,\qquad T_j^i = (p + \delta p)\delta _j^i,$$ where $\rho$ and $p$ are the energy density and pressure of the perfect fluid, respectively. Also, ${\partial _i}v$ characterizes the scalar part of the velocity potential of the fluid. Here, it is useful to introduce the momentum density $\delta {q^i} \equiv ( \rho + p){v^i}$. Applying the scalar-vector-tensor (SVT) decomposition, the momentum density $\delta {q_i}$ can be expressed in terms of the scalar and vector parts, $\delta {q_i} = {\partial _i}\delta q + \delta {\hat q_i}$, where the vector part is divergenceless, ${\partial ^i}\delta {\hat q_i} = 0$. As a results, Eq. (\[Tmunu\]) follows that the scalar part of the 3-momentum energy-momentum tensor $\delta T_i^0$ is equal to the scalar part of the momentum density ${\partial _i}\delta q$. The scalar part of the momentum density ${\partial_i}\delta q$ is used in definition of an important gauge-invariant quantity which is the curvature perturbation [@Felice2010] $$\label{calR} {\cal R} \equiv \psi + \frac{H}{{\rho + p}}\delta q.$$ Indeed, deviation from the homogeneous and isotropic FRW metric leads to perturbation in the constant-time spatial slices that this perturbation is specified by the curvature perturbation $\mathcal{R}$. The attractive feature of $\mathcal{R}$ is the fact that it remains constant outside the horizon. In particular, its amplitude is not affected by the unknown physical properties of the reheating process occurred at the end of inflation. It is the constancy of $\mathcal{R}$ outside the horizon that allows us to nevertheless predict cosmological observables. After inflation, the comoving horizon grows, so eventually all fluctuations will re-enter the horizon. After horizon re-entry, $\mathcal{R}$ determines the perturbations of the cosmic fluid resulting in the observed CMB anisotropies and the LSS [@Lyth2009; @Baumann2009; @Mukhanov1992; @Mukhanov2005; @Weinberg2008; @Malik2009]. In order to examine the evolution of $\mathcal{R}$ during inflation, first we need to write it in the form that remains invariant under coordinate transformation so that we can distinguish the physical perturbations from the nonphysical ones. Then, we should apply the Arnowitt-Deser-Misner (ADM) formalism based on the variation from the second order action to obtain the evolution equation for the curvature perturbation $\mathcal{R}$. Subsequently, we quantize the perturbations to find an initial condition for the evolution equation and then obtain its general solution for the quasi-de Sitter universe. In the next step, we evaluate the solution at the time of horizon exit and find the power spectrum of perturbations. In the standard inflationary model based on a minimally coupled scalar field in the Einstein gravity, the equation for the scalar perturbations is known as the “Mukhanov-Sasaki equation”. For reviews on cosmological perturbations theory in the standard inflationary scenario see e.g. [@Lyth2009; @Baumann2009; @Mukhanov1992; @Mukhanov2005; @Weinberg2008; @Malik2009]. Using the perturbed equations in the scalar-tensor gravity, the equation of motion for the curvature perturbation can be derived as [@Felice2010] $$‎\label{d2u} ‎{u''_k} - \left( {{k^2} - \frac{{z''}}{z}} \right){u_k} = 0, ‎$$ where the prime indicates a derivative with respect to the conformal time $\tau = \int {{a^{ - 1}}dt}$. The normalized variable $u$ is defined as $$‎\label{u} ‎u \equiv z \mathcal{R}.‎ ‎$$ For the standard inflationary scenario, the variable $z$ is defined as $z \equiv a\dot \varphi /H$, but for the scalar-tensor gravity, this variable is obtained as $$‎\label{z} ‎z=a\sqrt{Q_{s}},$$ where $$\label{Qs} {Q_s} \equiv \frac{{\omega (\varphi ){{\dot \varphi }^2} + \frac{{3{{\dot F}^2}}}{{2F}}}}{{{{\left( {H + \frac{{\dot F}}{{2F}}} \right)}^2}}}.$$ To obtain the power spectrum of the curvature perturbation, it is useful to introduce the slow-roll parameters [@Hwang2001a; @Felice2010] $$\label{eps} {\varepsilon _1} \equiv - \frac{{\dot H}}{H^2}, ‎\hspace{.3cm} {\varepsilon _2} \equiv \frac{{\ddot \varphi }}{{H\dot \varphi }}‎, ‎\hspace{.3cm} {\varepsilon _3} \equiv \frac{{\dot F}}{{2HF}}, ‎\hspace{.3cm} {\varepsilon _4} \equiv \frac{{\dot E}}{{2HE}}‎‎.$$ In the slow-roll approximation, we assume that the slow-roll parameters are much smaller than unity. In the above expressions, the parameter $E$ is defined as $$\label{E} E \equiv F\left[ {\omega (\varphi ) + \frac{{3{{\dot F}^2}}}{{2{{\dot \varphi }^2}F}}} \right].$$ Therefore, using Eqs. (\[Qs\]), (\[eps\]) and (\[E\]), we can rewrite $Q_s$ as $$‎\label{Qs2} ‎Q_{s}=\dot{\varphi}^{2} \frac{E}{FH^{2}(1+\varepsilon_{3})^{2}}‎. ‎$$ If the slow-roll parameters are constant, i.e. ${\dot \varepsilon _i} = 0\,\, (i = 1,2,3,4)$, then using Eqs. (\[z\]) and (\[Qs2\]), we will have $$‎\label{d2z} ‎\frac{{z''}}{z} = \frac{{\nu _{\cal R}^2 - 1/4}}{{{\tau ^2}}}, ‎$$ where $$‎\label{nuR} \nu _{\cal R}^2 = \frac{1}{4} + \frac{{\left( {1 + {\varepsilon _1} + {\varepsilon _2} - {\varepsilon _3} + {\varepsilon _4}} \right)\left( {2 + {\varepsilon _2} - {\varepsilon _3} + {\varepsilon _4}} \right)}}{{{{\left( {1 - {\varepsilon _1}} \right)}^2}}}.‎$$ In addition, the conformal time reads $$‎\label{tau} ‎\tau = - \frac{1}{{\left( {1 - {\varepsilon _1}} \right)aH}}.‎$$ Consequently, the solution of Eq. (\[d2u\]) can be expressed as a linear combination of the Hankel functions, $$‎\label{uktau} ‎{u_k}(\tau ) = \frac{{\sqrt {\pi \|\tau \|} }}{2}{e^{i(1 + 2{\nu _{\cal R}})\pi /4}}\left[ {{C_1}H_{{\nu _{\cal R}}}^{(1)}\left( {k\|\tau \|} \right) + {C_2}H_{{\nu _{\cal R}}}^{(2)}\left( {k\|\tau \|} \right)} \right], ‎$$ where the integration constants $C_1$ and $C_2$ are determined by imposing the suitable initial conditions. Finally, the acceptable solution for $u_{k}(\tau)$ is obtained as [@Felice2010] $$‎\label{uktau2} ‎{u_k}(\tau ) = \frac{{\sqrt {\pi \|\tau \|} }}{2}{e^{i(1 + 2{\nu _{\cal R}})\pi /4}}H_{{\nu _{\cal R}}}^{(1)}\left( {k\|\tau \|} \right)‎‎. ‎$$ The scalar power spectrum is defined as $$‎\label{Ps} {{\cal P}_{s}} \equiv \frac{{{k^3}}}{{2{\pi ^2}}}\|{\cal R}{\|^2}‎. ‎$$ Using Eqs. (\[u\]) and (\[uktau2\]) in the above definition, we get $$‎\label{PsGam} {{\cal P}_{s}} = \frac{1}{{{Q_s}}}{\left[ {\left( {1 - {\varepsilon _1}} \right)\frac{{\Gamma \left( {{\nu _{\cal R}}} \right)}}{{\Gamma \left( {3/2} \right)}}\frac{H}{{2\pi }}} \right]^2}{\left( {\frac{{\|k\tau \|}}{2}} \right)^{3 - 2{\nu _{\cal R}}}}‎, ‎$$ where $\Gamma$ is the Gamma function. The power spectrum of the curvature perturbation must be evaluated at the horizon crossing for which $k=aH$. In the slow-roll approximation, it takes the form $$\label{PsH}‎ ‎{{\cal P}_{s}} \simeq \frac{1}{{{Q_s}}}{\left( {\frac{H}{{2\pi }}} \right)^2}\Big\|_{k=aH}‎. ‎$$ The scale-dependence of the scalar power spectrum is specified by the scalar spectral index defined as $$‎\label{ns} {n_s} - 1 \equiv \frac{{d\ln {{\cal P}_{s}}}}{{d\ln k}}. ‎$$ With the help of Eq. (\[PsGam\]), the scalar spectral index (\[ns\]) reads $$\label{nsnuR}‎ ‎{n_s} - 1 = 3 - 2{\nu _{\cal R}}. ‎$$ In the slow-roll approximation, it therefore can be written as [@Felice2010] $$\label{nseps} {n_s} \simeq 1 - 4{\varepsilon _1} - 2{\varepsilon _2} + 2{\varepsilon _3} - 2{\varepsilon _4}‎.$$ Here, we concentrate on the tensor perturbations in the framework of the scalar-tensor gravity. The power spectrum of the tensor perturbations can be derived in a similar procedure to the one followed for the scalar perturbations and in the slow-roll regime it takes the form [@Felice2010] $$‎\label{Pt} {{\cal P}_t} \simeq \frac{2}{{{\pi ^2}}}\frac{{{H^2}}}{F}\Big\|_{k=aH}‎. ‎$$ To specify the scale-dependence of the tensor power spectrum, one can define the tensor spectral index $$\label{nt} {n_t} \equiv \frac{{d\ln {{\cal P}_t}}}{{d\ln k}}.$$ For the scalar-tensor gravity and in the slow-roll approximation, it can be obtained as $$‎\label{ntst} {n_t} \simeq - 2{\varepsilon _1} - 2{\varepsilon _3}.$$ An important inflationary observable is the tensor-to-scalar ratio which is defined as $$‎\label{r} ‎r \equiv \frac{{\cal P}_t}{{\cal P}_{s}}‎. ‎$$ Using Eqs. (\[PsH\]) and (\[Pt\]) in (\[r\]), the tensor-to-scalar ratio for the scalar-tensor gravity in the slow-roll approximation turns into $$\label{rst} ‎r \simeq 8\frac{{{Q_s}}}{F}.$$ So far, we have obtained the inflationary observables in the Jordan frame which is our physical frame in this paper. Applying the conformal transformations, we can go from the Jordan frame to the Einstein frame and calculate the inflationary observables in that frame too. The issue of the conformal transformations is an important subject in the context of modified theories of gravity. Also, implications of the Einstein and Jordan frames and physicalness of these frames has always been controversial [@Kaiser1995; @Kaiser1995a; @Faraoni1999; @Postma2014; @Banerjee2016]. The conformal transformations define the induced degrees of freedom as scalar fields in the extended theories of gravity and consequently these transformations are used to investigate the models with different couplings between matter-energy content and the geometry. Indeed, the conformal transformations indicate the mathematical equivalence between the scalar-tensor gravity and the Einstein general relativity. In the conformal transformations, the metric re-scaling which is dependent on the spacetime, is considered in the form $$‎\label{ct} {g_{\mu \nu }} \rightarrow {\tilde g_{\mu \nu }} = {\Omega ^2}{g_{\mu \nu }}, ‎$$ that we specify quantities in the Einstein frame by tilde. For the scalar-tensor gravities in which $ f(R,\phi ) =F(\phi)R$, the transformation parameter becomes $$\label{Omega}‎ \Omega^{2} =F\equiv \frac{\partial f}{\partial R}‎, ‎\hspace{.5cm} F>‎0.$$ As a result, the action (\[Sst\]) in the Einstein frame turns into $$\begin{aligned} \nonumber {S_E} = \int d {x^4}\sqrt { - \tilde g} \left[ {\frac{1}{2}\tilde R - \frac{1}{2}{{\tilde g}^{\mu \nu }}{\partial _\mu }\phi {\partial _\nu }\phi - V(\phi )} \right], \label{SstE}\end{aligned}$$ that now there exist no coupling between the Ricci scalar and the scalar field. In the above equation, $\phi$ and $V(\phi)$ are the scalar field and the potential in the Einstein frame, respectively. In order to the kinetic energy have the canonical form, we define the scalar field in the Einstein frame as $$‎\label{phivarphi} \phi = \int d \varphi \sqrt {\frac{3}{2}{{\left( {\frac{{{F_{,\varphi }}}}{F}} \right)}^2} + \frac{{\omega (\varphi )}}{F}}. ‎$$ Due to conformal transformation, the time and scale factor change as $$\label{taE}‎ ‎d\tilde{t}=\sqrt{F}dt‎, ‎\hspace{.5cm} \tilde{a}=\sqrt{F}a‎. ‎$$ Therefore, the Hubble parameter changes in the form $$‎\label{HE} \tilde H = \frac{1}{{\tilde a}}\frac{{d\tilde a}}{{d\tilde t}} = \frac{1}{{\sqrt F }}\left( {H + \frac{{\dot F}}{{2F}}} \right).$$ In addition, the potential in the Einstein frame is given by [@Faraoni2004; @Fujii2004] $$‎\label{VU} V(\phi ) = {\left. {\frac{{U(\varphi )}}{{{F^2}(\varphi )}}} \right\|_{\varphi = \varphi (\phi )}}.‎ ‎$$ If we have the scalar field and Hubble parameter in the Einstein frame, we can calculate the scalar power spectrum from [@Baumann2009] $$\label{PsE} {\tilde {\cal P}_s} = {\left( {\frac{{\tilde H}}{{2\pi }}} \right)^2}{\left( {\frac{{\tilde H}}{{\phi '}}} \right)^2},$$ where prime denotes derivative with respect to time in the Einstein frame. Also, if we obtain the potential in the Einstein frame, then we can simply calculate the observational parameters in terms of the potential slow-roll parameters which are expressed in terms of the potential and its derivatives as $$\begin{aligned} \label{epsV} & {\varepsilon _V} \equiv \frac{1}{2}{\left( {\frac{{{V_{,\phi }}}}{V}} \right)^2}, \\ \label{etaV} & {\eta _V} \equiv \frac{{{V_{,\phi \phi }}}}{V}.\end{aligned}$$ In the Einstein frame, we can express the scalar spectral index and tensor-to-scalar ratio in terms of the potential slow-roll parameters as [@Baumann2009] $$\begin{aligned} \label{nsE} & {\tilde{n}_s} \simeq 1 + 2{\eta _V} - 6{\varepsilon _V}, \\ \label{rE} & \tilde{r} \simeq 16 \varepsilon_{V},\end{aligned}$$ which are valid in the slow-roll approximation. Inflation in the Brans-Dicke gravity {#sec:BD} ==================================== In this section, we consider the Brans-Dicke gravity as a special model of the scalar-tensor gravity and derive the background field equations in this model. Then, we turn to study inflation in this model and using the relations expressed in the previous section, we obtain the observational quantities for the Brans-Dicke gravity in both the Jordan and Einstein frames. In the next section, we will use the results for the inflationary observables for different potentials which have motivations from quantum field theory or string theory. In this way, we will be able to compare behaviors of those potentials in the Brans-Dicke gravity versus their behaviors in the standard inflationary scenario based on the Einstein gravity. Furthermore, we will check viability of those inflationary potentials in light of the Planck 2015 observational data. Brans and Dicke [@Brans1961] proposed a specific form of the scalar-tensor gravity that it is founded on the Mach principle, which implies that the inertial mass of an object depends on the matter distribution in the universe and thus the gravitational constant should have time-dependence. This idea was in agreement with Dirac’s prediction about the time-dependence of the gravitational constant so that the quantities constructed from the fundamental constants, take the values of order of the elementary particles. In the Brans-Dicke theory a scalar field is invoked to describe the time-dependence of the gravitational constant. In order to the action (\[Sst\]) turn into the action of the Brans-Dicke gravity, we should consider $$\label{BD}‎ ‎f(R,\varphi)=\varphi R‎ , ‎\hspace{.5cm} \omega(\varphi)=\frac{\omega_{BD}}{\varphi}‎, ‎$$ where $\omega_{BD}$ is the Brans-Dicke parameter which is a constant. Therefore, the form of the Brans-Dicke action in the Jordan frame becomes $$‎\label{SBD} S_{J} = \int {{d^4}} x\sqrt { - g} \left[ {\frac{1}{2}\varphi R - \frac{1}{2}\frac{{{\omega _{BD}}}}{\varphi }{g^{\mu \nu }}{\partial _\mu }\varphi {\partial _\nu }\varphi - U(\varphi )} \right]. ‎$$ Hereafter, we drop out the subscript “BD” in the Brans-Dicke parameter and write it as $\omega$. It should be noted that the original Brans-Dicke theory does not contain the potential ($U(\varphi ) = 0$) [@Brans1961]. Using Eqs. (\[3HF\]) and (\[varphiddot\]) for the Brans-Dicke action (\[SBD\]), we obtain the evolution equations for a spatially flat FRW universe as $$\begin{aligned} \label{HU} & 3{\left( {H + \frac{{\dot \varphi }}{{2\varphi }}} \right)^2} - \frac{{\left( {2\omega + 3} \right)}}{4}{\left( {\frac{{\dot \varphi }}{\varphi }} \right)^2}-\frac{U}{\varphi } = 0, \\ \label{varphiddotBDwp} & \ddot \varphi + 3H\dot \varphi + \frac{2}{{\left( {2\omega + 3} \right)}}\left( {\varphi {U_{,\varphi }} - 2U} \right) = 0.\end{aligned}$$ Considering the slow-roll conditions $\left\| {\dot \varphi } \right\| \ll \left\| {H\varphi } \right\|$ and $\left\| {\ddot \varphi } \right\| \ll \left\| {3H\dot \varphi } \right\|$, Eqs. (\[HU\]) and (\[varphiddotBDwp\]) reduce to $$\begin{aligned} \label{HUsr} & 3{H^2}\varphi - U \simeq 0, \\ \label{varphidotUsr} & 3H\dot \varphi + \frac{2}{{\left( {2\omega + 3} \right)}}\left( {\varphi {U_{,\varphi }} - 2U} \right) \simeq 0.\end{aligned}$$ From Eqs. (\[HUsr\]) and (\[varphidotUsr\]), one can get $H$ and $\dot{\varphi}$ in terms of the potential $U(\varphi)$ in the slow-roll approximation. Here, we introduce the $e$-fold number which is used to determine the amount of inflation and is defined as $$\label{N} N \equiv \ln \left({\frac{{{a_e}}}{a}} \right),$$ where $a_e$ is the scale factor at the end of inflation. The above definition gives rise to $$\label{dN} dN = - H dt = - \frac{H}{{\dot \varphi }}d\varphi.$$ The anisotropies observed in the CMB correspond to the perturbations whose wavelengths crossed the Hubble radius around $N_* \approx 50 - 60$ before the end of inflation [@Liddle2003; @Dodelson2003]. This result can be obtained with the assumption that during inflationary era, a slow-roll inflation has occurred that it provides a quasi-de Sitter expansion with $H \approx {\rm{constant}}$ for the universe. In addition, the evolution of the universe after inflation is assumed to be determined by the standard model of cosmology. In this work, we have used these two assumptions and thus we can take the $e$-folds number of the horizon crossing as $N_* \approx 50 - 60$ from the end of inflation. Substituting $H$ and $\dot{\varphi}$ from Eqs. (\[HUsr\]) and (\[varphidotUsr\]), respectively, into Eq. (\[dN\]), we obtain $$\label{Nsr} N \simeq \frac{{\left( {2\omega + 3} \right)}}{2}\int_{{\varphi _e}}^\varphi {\frac{U}{{\varphi \left( {\varphi {U_{,\varphi }} - 2U} \right)}}} d\varphi,$$ where $\varphi_e$ is the scalar field at the end of inflation that to determine it, we use the relation ${\varepsilon _1} = 1$, because the slow-roll conditions are violated at the end of inflation. From Eqs. (\[E\]) and (\[BD\]), we see that the parameter $E$ for the Brans-Dicke gravity becomes a constant as $E = \omega + 3/2$, and therefore the fourth slow-roll parameter in Eq. (\[eps\]) vanishes ($\varepsilon_{4}=0$). Consequently, the scalar spectral index for this model results from Eq. (\[nseps\]) as $$\label{nsBD} n{_s} \simeq 1 - 4{\varepsilon _1} - 2{\varepsilon _2} + 2{\varepsilon _3}.$$ From the above equation, we calculate the running of the scalar spectral index for the Brans-Dicke gravity as $$\label{dnsBD} \frac{{d{n_s}}}{{d\ln k}} \simeq - 8\varepsilon _1^2 + 2\varepsilon _2^2 - 4\varepsilon _3^2 - 2{\varepsilon _1}{\varepsilon _2} + 4{\varepsilon _1}{\varepsilon _3},$$ that we have used the relation $k=a H$ which is valid at the horizon crossing. Within the framework of Brans-Dicke gravity, we get the parameter $Q_s$ from Eq. (\[Qs2\]) as $$\label{QsBD} {Q_s} = \frac{{{{\dot \varphi }^2}\left( {2\omega + 3} \right)}}{{{2H^2}\varphi {{\left( {1 + \frac{{\dot \varphi }}{{2H\varphi }}} \right)}^2}}}.$$ Substituting the above result into Eq. (\[rst\]), we obtain the tensor-to-scalar ratio for the Brans-Dicke gravity as $$\label{rBD} r \simeq 4\left( {2\omega + 3} \right)\frac{{{{\dot \varphi }^2}}}{{{H^2}{\varphi ^2}}}.$$ In the following, we try to find the inflationary observables in terms of the potential. To this aim, it is useful to find expressions of the slow-roll parameters in the slow-roll approximation. If we use Eqs. (\[HUsr\]) and (\[varphidotUsr\]) in (\[eps\]), we get the non-vanishing slow-roll parameters $$\begin{aligned} \label{eps1} & {\varepsilon _1} = \frac{{\left( {\varphi {U_{,\varphi }} - 2U} \right)\left( {\varphi {U_{,\varphi }} - U} \right)}}{{\left( {2\omega + 3} \right){U^2}}}, \\ \label{eps2} & {\varepsilon _2} = {\varepsilon _1} - \frac{{2\varphi \left( {\varphi {U_{,\varphi \varphi }} - {U_\varphi }} \right)}}{{\left( {2\omega + 3} \right)U}}, \\ \label{eps3} & {\varepsilon _3} = - \frac{{\left( {\varphi {U_{,\varphi }} - 2U} \right)}}{{\left( {2\omega + 3} \right)U}}.\end{aligned}$$ Consequently, if we use Eqs. (\[HUsr\]) and (\[varphidotUsr\]) in Eq. (\[QsBD\]) and then insert the result into Eq. (\[PsH\]), we obtain the scalar power spectrum in terms of the inflationary potential as $$\label{PsU} {{\cal P}_s} \simeq \frac{{\left( {2\omega + 3} \right){U^3}}}{{24{\pi ^2}{\varphi ^2}{{\left( {\varphi {U_{,\varphi }} - 2U} \right)}^2}}}.$$ In addition, substituting the slow-roll parameters (\[eps1\]), (\[eps2\]) and (\[eps3\]) into Eq. (\[nsBD\]), the scalar spectral index takes the form $$\label{nsU} {n_s} \simeq 1 + \frac{2}{{\left( {2\omega + 3} \right){U^2}}}\left[ {\varphi \left( {6U{U_{,\varphi }} + 2\varphi U{U_{,\varphi \varphi }} - 3\varphi U_\varphi ^2} \right) - 4{U^2}} \right].$$ Moreover, using Eqs. (\[HUsr\]) and (\[varphidotUsr\]), the tensor-to-scalar ratio (\[rBD\]) is obtained as $$\label{rU} r \simeq \frac{{16{{\left( {\varphi {U_{,\varphi }} - 2U} \right)}^2}}}{{\left( {2\omega + 3} \right){U^2}}}.$$ Another inflationary observable which can be used to discriminate between inflationary models, is the non-Gaussianity parameter (for review see e.g. [@Bartolo2004; @Chen2010]). Different inflationary models predict maximal signal for different shapes of non-Gaussianity. Therefore, the shape of non-Gaussianity is potentially a powerful probe of the mechanism that generate the primordial perturbations [@Babich2004; @Baumann2009]. For the single field inflationary models with non-canonical kinetic terms, the non-Gaussianity parameter has peak in the equilateral shape. Also, the squeezed shape is the dominant mode of models with multiple light fields during inflation. Furthermore, the folded non-Gaussianity becomes dominant in models with non-standard initial states. The subject of primordial non-Gaussianities in the Brans-Dicke theory has been investigated in details in [@Felice2011]. However, since in the present work, we deal with a single field inflation with a non-canonical kinetic term and standard initial states (such as the Bunch-Davies vacuum initial conditions for perturbations), therefore we focus on the non-Gaussianity parameter in the equilateral limit. The equilateral non-Gaussianity parameter for the Brans-Dicke gravity has been obtained in [@Artymowski2015] as $$\label{fNLequil} ‎f_{{\rm{NL}}}^{{\rm{equil}}} = - \frac{5}{4}{\varepsilon _2} + \frac{5}{6}{\varepsilon _3}.$$ We see that in the Brans-Dicke gravity, the equilateral non-Gaussianity is of order of the slow-roll parameters which are very smaller than unity in the slow-roll regime. On the other hand, the slow-roll conditions can be perfectly satisfied in the Brans-Dicke gravity. Therefore, the equilateral non-Gaussianity parameter in the Brans-Dicke gravity can be in agreement with the Planck 2015 prediction, $f_{{\rm{NL}}}^{{\rm{equil}}} = - 16 \pm 70$ (68% CL, Planck 2015 T-only), see [@Planck2015]. We will show this fact in the next section explicitly for different inflationary potentials. At the end of this section, we discus about equivalence of the results for the inflationary observables in the Jordan and Einstein frames. We saw before that via the conformal transformation $\tilde{g}=\Omega^{2}g$, we can go from the Jordan frame to the Einstein frame. For the Brans-Dicke gravity, $F=\varphi$, and thus from Eq. (\[taE\]) we see that the time and scale factor change under the conformal transformation as $$\label{taEBD} d\tilde t = \sqrt \varphi~ dt‎, ‎\hspace{.5cm} \tilde a = \sqrt \varphi~ a‎.$$ Also, from Eq. (\[HE\]) we conclude that the Hubble parameter transforms in the form $$\label{HEBD} \tilde H = \frac{H}{{\sqrt \varphi }}.$$ To find the relation between the scalar fields in the Einstein and Jordan frames in the Brans-Dicke gravity, we use Eq. (\[phivarphi\]) and get $$\label{phivarphiBD} \phi = \sqrt {\frac{{2\omega + 3}}{2}} \ln \varphi‎.$$ Also, from Eq. (\[VU\]), we see that the relation between potentials in the two frames is $$\label{VUBD} V(\phi ) = {\left. {\frac{{U(\varphi )}}{{{\varphi ^2}}}} \right\|_{\varphi = \varphi (\phi )}}.$$ Here, we want to know how the inflationary observables change under the conformal transformation from the Jordan frame to the Einstein frame in the Brans-Dicke gravity. First, we focus on the transformation of the scalar power spectrum. If we use Eqs. (\[taEBD\]), (\[HEBD\]) and (\[phivarphiBD\]) in Eq. (\[PsE\]), and compare the result with Eq. (\[PsU\]), we conclude that $$\label{PsEJ} {\tilde {\cal P}_s} \simeq {{\cal P}_s},$$ which implies that in the slow-roll approximation, the equations for the scalar power spectrum are same in both the Einstein and Jordan frames. In what follows, we proceed to find the transformations of the scalar spectral index and tensor-to-scalar ratio. To do so, we can use Eqs. (\[phivarphiBD\]) and (\[VUBD\]) in Eqs. (\[epsV\]) and (\[etaV\]), and obtain the potential slow-roll parameters in the Einstein frame in terms of the scalar field $\varphi$ and potential $U(\varphi)$ in the Jordan frame, as $$\begin{aligned} \label{epsVBD} & {\varepsilon _V} = \frac{{{{\left( {\varphi {U_\varphi } - 2U} \right)}^2}}}{{\left( {2\omega + 3} \right){U^2}}}, \\ \label{etaVBD} & {\eta _V} = {\varepsilon _V} + \frac{{2\varphi \left( {{U_{,\varphi \varphi }} - {U_\varphi }} \right)}}{{\left( {2\omega + 3} \right)U}} - \frac{{{\varphi ^2}U_{,\varphi }^2}}{{\left( {2\omega + 3} \right){U^2}}} + \frac{4}{{\left( {2\omega + 3} \right)}}.\end{aligned}$$ If we use these relations in Eqs. (\[nsE\]) and (\[rE\]), and compare the result with Eqs. (\[nsU\]) and (\[rU\]), then we see that $$\begin{aligned} \label{nsEJ} & \tilde{n}_{s} \simeq n_{s}‎, \\ \label{rEJ} ‎& \tilde{r} \simeq r,\end{aligned}$$ which means that in the Brans-Dicke gravity and in the slow-roll approximation, the relations for the scalar spectral index and tenor-to-scalar ratio are identical in the Einstein and Jordan frames. There are much discussion and challenge about the Jordan and Einstein frames and also about the results corresponding to the inflationary observables in these two frames [@Kaiser1995; @Kaiser1995a; @Faraoni1999; @Postma2014; @Banerjee2016; @Makino1991; @Fakir1990; @Salopek1989; @Tsujikawa2004; @Hinterbichler2012; @Chiba2008; @Gong2011; @Catena2007; @Chiba2013]. In fact, the conformal invariance of the scalar power spectrum from the Jordan frame to the Einstein frame, Eq. (\[PsEJ\]), is expected because of the conformal invariance of the curvature perturbation, $\tilde {\cal R} = {\cal R}$ (for more details, see [@Felice2010]). The conformal invariance of the amplitude of scalar perturbations was firstly shown in [@Makino1991], for $\lambda {\varphi ^4}$ chaotic inflation model with the non-minimal coupling $\xi {\varphi ^2}R$ in a more rigorous manner relative to the previous papers [@Fakir1990; @Salopek1989]. In [@Tsujikawa2004], the authors have generalized the conformal invariance of both the scalar and tensor power spectra for the model with non-minimal coupling term $F(\varphi)R$. In [@Kaiser1995], it has been clarified that the scalar spectral index for the new inflation model [@Linde1982; @Albrecht1982], is different in the two frames, but for the chaotic inflation model [@Linde1983] with various initial conditions, the results are identical in both frames. Furthermore, in [@Kaiser1995a], it has been discussed that if one applies the slow-roll approximation in obtaining the scalar power spectrum, then the scalar spectral index for both the new and chaotic inflation models with various initial conditions, are same in the two frames. In [@Felice2010], it was shown that the curvature perturbation and the tensor perturbations remain invariant under the conformal transformations, and hence the scalar and tensor power spectrum remain invariant in the two frames. Consequently, the tensor-to-scalar ratio is identical for the both frames. In the present paper, our study implies that for the Brans-Dicke gravity, the relations of the scalar power spectrum ${\cal P}_s$, the scalar spectral index ${n_s}$ and the tensor-to-scalar ratio $r$ are same in the two frames, only in the slow-roll approximation. It is worth mentioning that the conformal invariance holds for the adiabatic modes even beyond the slow-roll approximation [@Hinterbichler2012]. The conformal equivalence of the inflationary observables also holds at the non-linear level as shown in [@Chiba2008; @Gong2011]. Furthermore, it was pointed out that the other cosmological observables/relations besides the inflationary ones, such as redshift, luminosity distance, temperature anisotropies, cross sections, etc. are frame-independent [@Catena2007; @Chiba2013]. Study of various inflationary potentials in the Brans-Dicke gravity {#sec:pot} =================================================================== In the previous section, we obtained the relations of the inflationary observables in the Brans-Dicke gravity using the Jordan frame. Here, we apply the results of the previous section for various inflationary potentials and check their viability in light of the Planck 2015 observational results [@Planck2015]. We examine the potentials which have motivations from quantum field theory or string theory. Note that validity of these potentials in comparison with the observational data has been already investigated in [@Martin2014a; @Martin2014b; @Rezazadeh2015; @Huang2015; @Okada2014], but within the framework of standard inflationary scenario based on Einstein’s gravity. To examine each potential, first we use it in Eqs. (\[nsU\]) and (\[rU\]), and find the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ in terms of the inflaton scalar field $\varphi$. Then, we set ${\varepsilon _1} = 1$ in Eq. (\[eps1\]) to determine analytically the scalar field at the end of inflation, ${\varphi _e}$. Next, we use ${\varphi _e}$ in Eq. (\[Nsr\]) and apply a numerical method to obtain the inflaton scalar field at the horizon exit, $\varphi_$, that we take the $e$-fold number of the epoch of horizon exit as ${N_} = 50$ or $60$. In this way, we can evaluate $n_s$ and $r$ at the horizon exit and then plot the $r-n_s$ diagram for the model. Finally, comparing the result of the model in $r-n_s$ plane with the allowed region by the Planck 2015 data [@Planck2015], we are able to check viability of the considered inflationary potential in light of the observational results. Power-law potential {#subsec:pl} ------------------- We start with the simplest inflationary potential which is the power-law potential $$\label{Upl} U(\varphi ) = {U_0}{\varphi ^n},$$ where $U_0$ and $n>0$ are constant parameters. This class of potentials includes the simplest chaotic inflationary models introduced by [@Linde1983], in which inflation starts from large values for the inflaton, i.e. $\varphi > M_P$. In the standard inflationary scenario, this potential can be in agreement with Planck 2015 TT,TE,EE+lowP data [@Planck2015] at 95% CL, as it has been shown in [@Rezazadeh2015]. In the Brans-Dicke gravity setting, the power-law potential (\[Upl\]) with $n>2$ leads to the power-law inflation with the scale factor $a(t) \propto t^q$ where $q>1$ [@Artymowski2015]. Therefore, the slow-roll parameters (\[eps\]) turn to be constant and they become dependent on the parameters $n$ and $\omega$. As a result, the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ become constant as $$\begin{aligned} \label{nspl} & {n_s} = 1 - \frac{{2{{\left( {n - 2} \right)}^2}}}{{2\omega + 3}}, \\ \label{rpl} & r = \frac{{16{{\left( {n - 2} \right)}^2}}}{{2\omega + 3}}.\end{aligned}$$ From the two above equations, we see that for large values of the Brans-Dicke parameter $\omega$, the scalar spectral index $n_s$ approaches unity while the tensor-to-scalar ratio $r$ converges to zero. We see that the two above equations can be easily combined to give the linear relation $$\label{rnspl} r = 8\left( {1 - {n_s}} \right).$$ This relation implies that the prediction of the power-law potential (\[Upl\]) in $r-n_s$ plane is independent of the parameters $\omega$ and $n$. Using the above relation, we can draw the $r-n_s$ plot as shown by a black line in Fig. \[fig:pl\]. Moreover, in Fig. \[fig:pl\], the marginalized joint 68% and 95% confidence limit (CL) regions for Planck 2013, Planck 2015 TT+lowP and Planck 2015 TT,TE,EE+lowP data [@Planck2015] are specified by gray, red and blue, respectively. The figure shows that (i) the result of the power-law potential in the Brans-Dicke gravity in contrary to the standard model, lies outside the range allowed by the Planck 2015 data. (ii) The prediction of this potential takes place in the region 95% CL of Planck 2013 data. This is in good agreement with that obtained by [@Felice2011a] using the data of WMAP7. \[1\][![Prediction of power-law potential (\[Upl\]) in $r-n_s$ plane in the Brans-Dicke gravity (black line). The marginalized joint 68% and 95% CL regions of Planck 2013, Planck 2015 TT+lowP and Planck 2015 TT,TE,EE+lowP data [@Planck2015] are specified by gray, red and blue, respectively.[]{data-label="fig:pl"}](pl.eps "fig:")]{} Inverse power-law potential {#subsec:ipl} --------------------------- The next potential which we examine is the inverse power-law potential $$\label{Uipl} U(\varphi ) = {U_0}{\varphi ^{ - n}},$$ where $U_0$ and $n>0$ are two model parameters. This potential is a steep potential and in the standard inflationary setting, it gives rise to the intermediate inflation with the scale factor $a(t) \propto \exp [A{({M_P}t)^\lambda}]$ where $A>0$ and $0<\lambda<1$ [@Barrow1990; @Barrow2006; @Barrow2007], which is not consistent with the Planck 2015 observational results, as it has been discussed in [@Rezazadeh2015]. To obtain the equations of $n_s$ and $r$ for this potential in the Brans-Dicke theory, we can simply change $n \to - n$ in Eqs. (\[nspl\]) and (\[rpl\]). In this way, if we can combine the results, we again recover relation (\[rnspl\]) between $n_s$ and $r$. Therefore, the $r-n_s$ plot for the inverse power-law potential (\[Uipl\]) becomes like the one for the power-law potential (\[Upl\]) which it has been shown in Fig. \[fig:pl\]. Consequently, inflation with the inverse power-law potential in the Brans-Dicke gravity like the standard setting is ruled out by the Planck 2015 data. Exponential potential {#subsec:exp} --------------------- Another steep potential that we study, is the exponential potential $$\label{Uexp} U(\varphi ) = {U_0}{e^{ - \alpha \varphi }},$$ where $U_0$ and $\alpha >0$ are constant parameters. In the standard inflation model, this potential provides the power-law inflation with the scale factor $a(t) \propto t^q$ where $q>1$ [@Lucchin1985; @Halliwell1987; @Yokoyama1988], that cannot be compatible with the Planck 2015 results, as it has been demonstrated in [@Rezazadeh2015; @Rezazadeh2016]. Within the framework of Brans-Dicke gravity, the observables $n_s$ and $r$ for the exponential potential (\[Uexp\]) become independent of the parameters $U_0$ and $\alpha$. We can evaluate $n_s$ and $r$ for different values of the Brans-Dicke parameter $\omega$ and the horizon exit $e$-fold number $N_$. Our examination shows that with $N_=50$ and $N_=60$, the tensor-to-scalar ratio for different values of $\omega$, varies in the ranges $r \ge 0.687$ and $r \ge 0.580$, respectively. These results for $r$ are not consistent with the upper bound $r< 0.149$ (95% CL) deduced from Planck 2015 TT,TE,EE+lowP data [@Planck2015]. Therefore, the exponential potential (\[Uexp\]) in the Brans-Dicke gravity like the standard scenario is disfavored by the observational data. Hilltop potential {#subsec:hil} ----------------- A potential which has a remarkable importance in study of inflation is the hilltop potential $$\label{Uhil} U(\varphi ) = {U_0}\left( {1 - \frac{{{\varphi ^p}}}{{{\mu ^p}}} + ...} \right),$$ where $U_0$, $\mu$ and $p>0$ are constant parameters of the model [@Boubekeur2005]. In this interesting class of potentials, the inflaton rolls away from an unstable equilibrium as in the first new inflationary models [@Linde1982; @Albrecht1982]. This potential in the standard inflationary scenario can be in excellent agreement with the Planck 2015 results, because its prediction can lie inside the region 68% CL of Planck 2015 TT,TE,EE+lowP data [@Planck2015]. Study of this potential in the Brans-Dicke gravity shows that the quantities $n_s$ and $r$ do not depend on $U_0$ and $\mu$. Furthermore, we conclude that for $p = 1,\,2,\,3,\,4,\,6$, results of this potential can be placed inside the region 95% CL of Planck 2015 TT,TE,EE+lowP data, if we increase the parameter $\omega$ sufficiently. In Fig. \[fig:hil\], the $r-n_s$ plot for the hilltop potential (\[Uhil\]) with $p=4$ is illustrated in comparison with the observational data. In the figure, the results of the model with $N_=50$ and $N_=60$ are shown by the dashed and solid black lines, respectively. Here, it is worthwhile to mention that in this figure, the result of the model lies inside the region 95% CL for $\omega \gtrsim {10^3}$. In this model, a large values of the parameter $\omega$ gives larger values for $n_s$ and $r$. For very large values of $\omega$ relative to unity, the observables $n_s$ and $r$ approach $0.9708$ ($0.9757$) and $0.08$ ($0.07$), respectively, if we take the $e$-fold number of horizon crossing as $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the hilltop potential (\[Uhil\]). The result of the potential for $N_=50$ and $N_=60$ are shown by the dashed and solid black lines, respectively.[]{data-label="fig:hil"}](hil.eps "fig:")]{} D-brane potential {#subsec:Db} ----------------- Another inflationary potential which has motivations from the physical theories with extra dimensions is the D-brane potential $$\label{UDb} U(\varphi ) = {U_0}\left( {1 - \frac{{{\mu ^p}}}{{{\varphi ^p}}} + ...} \right),$$ where $U_0$, $\mu$ and $p>0$ are constant parameters. Two important cases of this potential correspond to $p=2$ [@Garcia-Bellido2002] and $p=4$ [@Dvali2001; @Kachru2003] are compatible with the Planck 2015 data in the standard inflationary scenario, as mentioned in [@Planck2015]. In the inflationary scenario based on the Brans-Dicke gravity, the observables $n_s$ and $r$ for the D-brane potential (\[UDb\]) depend only on $\omega$ and $N_$. For $N_=50$ and $N_=60$, the results of this potential with $p=2,\, 4$ lie inside the 68% CL region of Planck 2015 TT,TE,EE+lowP data. We see this fact for the case $p=4$ in Fig. \[fig:Db\], that the dashed and solid black lines correspond to $N_=50$ and $N_=60$, respectively. In this figure, as the parameter $\omega$ increases, the observables $n_s$ and $r$ grow and converge respectively to $0.9701$ ($0.9751$) and $0.08$ ($0.07$), for the $e$-fold number of horizon exit $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the D-brane potential (\[UDb\]). The results for $N_=50$ and $N_=60$ are shown by the dashed and solid black lines, respectively.[]{data-label="fig:Db"}](Db.eps "fig:")]{} Higgs potential {#subsec:Hig} --------------- Now, we investigate the Higgs potential $$\label{UHig} U(\varphi ) = {U_0}{\left[ {1 - {{\left( {\frac{\varphi }{\mu }} \right)}^2}} \right]^2},$$ where $U_0$ and $\mu>0$ are model parameters [@Kallosh2007; @Okada2014]. This potential leads to a mechanism of symmetry breaking, where the field rolls off an unstable equilibrium toward a displaced vacuum [@Baumann2009]. The Higgs potential (\[UHig\]) in the standard inflationary scenario behaves like a small-field potential when $\varphi<\mu$, and like a large-field potential when $\varphi>\mu$. In [@Rezazadeh2015] it has been shown that the result of this potential in the standard inflationary framework can lie within the joint 68% CL region of Planck 2015 TT,TE,EE+lowP data. Within the framework of Brans-Dicke gravity, our results for the Higgs potential (\[UHig\]) show that the observables $n_s$ and $r$ does not depend on the parameters $U_0$ and $\mu$. Surprisingly, we found that the result of this potential for $n_s$ and $r$ are completely identical for both sides of the potential minimum, i.e. for $\varphi<\mu$ and $\varphi>\mu$. In order to explain this unexpected result, we note that to know whether the result of a potential in $r-n_s$ plane is the same for the both sides of its minimum in the Jordan frame, we should go to the Einstein frame via the conformal transformation. If the shape of the potential is symmetric around its minimum in the Einstein frame, then its prediction in $r-n_s$ plane will be the same for the both sides of its minimum, and otherwise the results will be different. For the Higgs potential (\[UHig\]), using Eqs. (\[phivarphiBD\]) and (\[VUBD\]), it changes under the conformal transformation as $$\label{VHig} V(\phi ) = \frac{{{U_0}}}{\mu }{\left( {{e^{\sqrt {\frac{2}{{2\omega + 3}}}~\phi }} - {\mu ^2}{e^{ - \sqrt {\frac{2}{{2\omega + 3}}}~\phi }}} \right)^2}.$$ The above potential is completely symmetric around its minimum, $\tilde \mu = \sqrt {\frac{{2\omega + 3}}{2}} \ln \mu $. Consequently, although the Higgs potential (\[UHig\]) is not symmetric around its minimum in the Jordan frame, the transformed potential (\[VHig\]) is completely symmetric around its minimum in the Einstein frame. In addition, we should note that the values of $\phi < \tilde \mu $ and $\phi > \tilde \mu $ in the Einstein frame are related respectively to the values $\varphi < \mu $ and $\varphi > \mu $, in the Jordan frame. As a result, we conclude that the results of Higgs potential (\[UHig\]) for $n_s$ and $r$ should be same for the both regions $\varphi<\mu$ and $\varphi>\mu$. The $r-n_s$ diagram for this potential is shown in Fig. \[fig:Hig\], and as we see in the figure, the result of the Higgs potential (\[UHig\]) for $N_=60$ can be placed inside the 95% CL region of Planck 2015 TT,TE,EE+lowP data. For this model, as $\omega$ becomes larger, the scalar spectral index $n_s$ increases and approaches $0.9604$ ($0.9669$) for $N_=50$ ($N_=60$). Moreover, the greater $\omega$ leads to the smaller values for the tensor-to-scalar ratio $r$ and it finally approaches $0.16$ ($0.13$) for $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the Higgs potential (\[UHig\]). The results for $N_=50$ and $N_=60$ are shown by the dashed and solid black lines, respectively.[]{data-label="fig:Hig"}](Hig.eps "fig:")]{} Coleman-Weinberg potential {#subsec:CW} -------------------------- A famous inflationary potential which has ideas from quantum field theory, is the Coleman-Weinberg potential $$\label{UCW} U(\varphi ) = {U_0}\left[ {{{\left( {\frac{\varphi }{\mu }} \right)}^4}\left( {\ln \left( {\frac{\varphi }{\mu }} \right) - \frac{1}{4}} \right) + \frac{1}{4}} \right],$$ with constants $U_0$ and $\mu>0$ [@Martin2014a; @Martin2014b; @Okada2014; @Barenboim2014]. This potential is historically famous since it was applied in the original papers of the new inflation model [@Linde1982; @Albrecht1982]. In [@Rezazadeh2015], the authors have examined this potential in the standard inflationary framework and shown that it can be consistent with 68% CL region of Planck 2015 TT,TE,EE+lowP data. The result of Coleman-Weinberg potential (\[UCW\]) in the Brans-Dicke gravity is plotted in Fig. \[fig:CW\]. In contrast with the Higgs potential (\[UHig\]), the prediction of this potential for the two regimes $\varphi<\mu$ and $\varphi>\mu$ are completely different as shown in Fig. \[fig:CW\] by black and orange colors, respectively. For the values $\omega \gg 1$, the results of the two regimes approach to a common point in $r-n_s$ plane, that we see this behavior in Fig. \[fig:CW\]. Also, from the figure we see that the prediction of the Coleman-Weinberg potential (\[UCW\]) for the both regimes $\varphi<\mu$ and $\varphi>\mu$ can place within the joint 95% CL region of Planck 2015 TT,TE,EE+lowP data, if we take the horizon exit $e$-fold number as $N_=60$. For this potential, for the both cases $\varphi<\mu$ and $\varphi>\mu$, for large values of $\omega$, $n_s$ and $r$ converge respectively to $0.9604$ ($0.9669$) and $0.16$ ($0.13$), if we take the $e$-fold number of horizon exit as $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the Coleman-Weinberg potential (\[UCW\]). The predictions of the model for $N_=50$ and $N_=60$ are shown by the dashed and solid black lines, respectively. In addition, the results of this potential for the two ranges $\varphi<\mu$ and $\varphi>\mu$ are specified by black and orange colors, respectively.[]{data-label="fig:CW"}](CW.eps "fig:")]{} Natural inflation {#subsec:nat} ----------------- In what follows, we concentrate on one of the most elegant inflationary models which is natural inflation given by the periodic potential [@Freese1990; @Adams1993; @Freese2014] $$\label{Unat} U(\varphi ) = {U_0}\left[ {1 + \cos \left( {\frac{\varphi }{f}} \right)} \right],$$ where $f>0$ is the scale which determines the curvature of the potential. This potential has motivations from string theory and it often arises if the inflaton field is taken to be a pseudo-Nambu-Goldstone boson, i.e. an axion, [@Freese1990]. This potential behaves like a small-field potential for $2\pi f < {M_P}$, and like a large-field potential for $2\pi f > {M_P}$ [@Baumann2009]. The result of this potential in the standard inflationary scenario is in agreement with the Planck 2015 observational data at 95% CL, as demonstrated in [@Planck2015]. Although the result of the natural potential (\[Unat\]) in the standard inflationary scenario is same for the both ranges $0 < \varphi /f < \pi $ and $\pi < \varphi /f < 2\pi $ [@Freese1990], their results are different in the Brans-Dicke theory. To account for this fact, we note that using Eqs. (\[phivarphiBD\]) and (\[VUBD\]), the potential (\[Unat\]) changes under the conformal transformation to the Einstein frame as $$\label{Vnat} V(\phi ) = {U_0}{e^{ - 2\sqrt {\frac{2}{{2\omega + 3}}}~\phi }}\left[ {1 + \cos \left( {\frac{1}{f}{e^{\sqrt {\frac{2}{{2\omega + 3}}}~\phi }}} \right)} \right].$$ Consequently, although the natural potential (\[Unat\]) is symmetric in the Jordan frame, the transformed potential (\[Vnat\]) is not symmetric in the Einstein frame. Indeed, in the Einstein frame the shape of the potential is not the same for the both sides around the minimum of potential, i.e. for the ranges $\phi < \sqrt {2\omega + 3} \ln \left( {\pi f} \right)$ and $\phi > \sqrt {2\omega + 3} \ln \left( {\pi f} \right)$. Additionally, these two ranges correspond respectively to the ranges $0 < \varphi /f < \pi $ and $\pi < \varphi /f < 2\pi $, in the Jordan frame. Putting all of these notes together, we conclude that the results of natural potential (\[Unat\]) are not the same for the two sides of its minimum in the Jordan frame. The results of this potential for the two ranges $0 < \varphi /f < \pi $ and $\pi < \varphi /f < 2\pi $ are shown in Fig. \[fig:nat\], by black and orange colors, respectively. We see in the figure that the result of the range $0 < \varphi /f < \pi $ is outside the region allowed by Planck 2015 TT,TE,EE+lowP data for $N_=50$, but if we take $N_=60$, then its result can enter the 95% CL region of the same data. It is evident from the figure that the result of the range $\pi < \varphi /f < 2\pi$ for both $N_=50$ and $N_=60$ can be lied inside the marginalized joint 95% CL region of Planck 2015 TT,TE,EE+lowP data. It is worth mentioning that in this model, for the case $0 < \varphi /f < \pi $, a larger value of $\omega$ gives a lower value for $r$, while for the case $\pi < \varphi /f < 2\pi$, the greater $\omega$ leads to the smaller $r$. But for the both cases, $r$ approaches $0.16$ ($0.13$) for large values of $\omega$, if we take $N_=50$ ($N_=60$). Also, in the both cases, $n_s$ approaches $0.9605$ ($0.9670$), for $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the natural potential (\[Unat\]). The predictions of this potential for $N_=50$ and $N_=60$ are shown by the dashed and solid black lines, respectively. Furthermore, the results for the two ranges $0 < \varphi /f < \pi $ and $\pi < \varphi /f < 2\pi$ are specified by black and orange colors, respectively.[]{data-label="fig:nat"}](nat.eps "fig:")]{} Spontaneously broken supersymmetry (SB SUSY) potential {#subsec:SBS} ------------------------------------------------------ Here, we proceed to investigate inflation with the spontaneously broken supersymmetry (SB SUSY) potential $$\label{USBS} U(\varphi ) = {U_0}\left( {1 + b\ln \varphi } \right),$$ where $b>0$ is a dimensionless parameter. This potential has wide usage in the hybrid models to provide $n_s<1$ [@Mazumdar2011]. However, the result of this potential in the standard inflationary model cannot be compatible with Planck 2015 TT,TE,EE+lowP data [@Planck2015]. In the inflationary framework based on the Brans-Dicke gravity, the result of the SUSY breaking potential (\[USBS\]) for the observational quantities $n_s$ and $r$ depend on the Brans-Dicke parameter and the horizon exit $e$-fold number $N_$. The result of the potential in $r-n_s$ plane is presented in Fig. \[fig:SBS\]. It shows that the prediction of the SUSY breaking potential (\[USBS\]) within the framework of Brans-Dicke gravity in contrary to the standard setting, can lie inside the 68% CL region of Planck 2015 TT,TE,EE+lowP data [@Planck2015]. For this potential, a larger value of the Brans-Dicke parameter $\omega$ gives rise to larger value for both $n_s$ and $r$. But finally, as $\omega$ increases, $n_s$ and $r$ converge respectively to the values $0.9701$ ($0.9751$) and $0.08$ ($0.07$), if we consider $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the SB SUSY potential (\[USBS\]). The results for $N_=50$ and $N_=60$ are demonstrated by the dashed and solid black lines, respectively.[]{data-label="fig:SBS"}](SBS.eps "fig:")]{} Displaced quadratic potential {#subsec:dq} ----------------------------- The last inflationary potential that we investigate in the Brans-Dicke scenario, is the quadratic potential with displaced minimum $$\label{Udq} U(\varphi ) = {U_0}{\left( {1 - \frac{\varphi }{\mu }} \right)^2},$$ where $U_0$ and $\mu>0$ are constant parameters. By use of the transformation relations (\[phivarphiBD\]) and (\[VUBD\]), one can show that for the vanishing Brans-Dicke parameter ($\omega=0$), the above potential changes into the potential corresponding to the Starobinsky $R^2$ inflation in the Einstein frame. Therefore, we can consider the inflationary model with the above potential in the Brans-Dicke gravity as a generalized version of the Starobinsky $R^2$ inflation. Note that the consistency of the potential (\[Udq\]) with the Planck 2013 data in the Brans-Dicke gravity has been already investigated by [@Tsujikawa2013] using the Einstein frame. But in the present work, we study this potential in the Jordan frame which is our physical frame. Furthermore, we check comparability of this potential in comparison with the Planck 2015 observational data. We show the $r-n_s$ plot of the displaced quadratic potential (\[Udq\]) in Fig. \[fig:dq\] in comparison with the observational results. The results of the potential for the ranges $ \varphi < \mu $ and $\varphi > \mu $ are specified by black and orange colors, respectively. As it is obvious from the figure, result of the range $ \varphi < \mu $ is not consistent with Planck 2015 TT,TE,EE+lowP data [@Planck2015] for $N_=50$. But for $N_=60$, its result can be placed inside the 95% CL region of the same data. Also, it is clear from the figure that for the range $\varphi > \mu $, the potential can be in well agreement with the observation such that its prediction can lie inside the 68% CL region of Planck 2015 TT,TE,EE+lowP data [@Planck2015] for both $N_=50$ and $N_=60$. We see in Fig. \[fig:dq\] that the result of the range $\varphi > \mu $ approaches to the Starobinsky $R^2$ inflation that its prediction has been specified by a green line. In the figure, for the case $\varphi < \mu $, the larger value of $\omega$ gives rise to a larger value for $n_s$, but a smaller value for $r$. It should be noted that for the both cases $\varphi < \mu $ and $\varphi > \mu $, if we take $\omega$ very larger than unity, then the prediction of the potential (\[Udq\]) for $n_s$ and $r$ converges respectively to the values $0.9604$ ($0.9669$) and $0.16$ ($0.13$), for $N_=50$ ($N_=60$). \[1\][![Same as Fig. \[fig:pl\] but for the displaced quadratic potential (\[Udq\]). The results for $N_=50$ and $N_=60$ are shown by the dashed and solid lines, respectively. Furthermore, the predictions of the model for the two ranges $\varphi < \mu $ and $\varphi > \mu $ are specified by black and orange colors, respectively. The result of the Starobinsky $R^2$ inflation for $50 < {N_} < 60$ is shown by the green line, while the smaller and larger green points demonstrate the results corresponding to ${N_} = 50$ and ${N_} = 60$, respectively.[]{data-label="fig:dq"}](dq.eps "fig:")]{} So far, we tested the predictions of various potentials in $r-n_s$ plane relative to the Planck 2015 observational results. In Table \[tab:tab1\], we summarize the results of the examined inflationary potentials. To specify the viable inflationary potentials in light of the observational results, it is further needed to check consistency of their predictions for other inflationary observables such as the running of the scalar spectral index $d{n_s}/d\ln k$, Eq. (\[dnsBD\]), and the equilateral non-Gaussianity parameter $f_{{\rm{NL}}}^{{\rm{equil}}}$, Eq. (\[fNLequil\]). We evaluate these two observable parameters for the potentials which are successful in the $r-n_s$ test. Subsequently, we compare our results for different potentials with the results deduced from the Planck 2015 data implying $d{n_s}/d\ln k = - {\rm{0}}{\rm{.0085}} \pm {\rm{0}}{\rm{.0076}}$ (68% CL, Planck 2015 TT,TE,EE+lowP) and $f_{{\rm{NL}}}^{{\rm{equil}}} = - 16 \pm 70$ (68% CL, Planck 2015 T-only) [@Planck2015]. In Table \[tab:tab2\], we summarize the predictions of only viable potentials for $d{n_s}/d\ln k$ and $f_{{\rm{NL}}}^{{\rm{equil}}}$ with the allowed ranges for the Brans-Dicke parameter $\omega$, which are compatible with the Planck 2015 results. Here, we should notice that the values of Brans-Dicke parameter $\omega$ in several models do not seem to be compatible with the Solar System constraint, $\omega \gtrsim 10^5$. Of course, if the Brans-Dicke scalar field decays after inflation, the subsequent cosmology coincides with Einstein’s general relativity and today’s bound does not apply. Conclusions {#sec:con} =========== We studied inflation in the framework of Brans-Dicke gravity. For this purpose, first we presented a brief review on the scalar-tensor theories of gravity and expressed the equations governing the background cosmology. We also, reviewed briefly the cosmological perturbations in the scalar-tensor gravity and obtained the scalar and tensor power spectra for this general class of models. Applying the scalar and tensor power spectra, we found relations of the inflationary observables for the model that it makes possible for us to connect theory with observation. In the next step, we considered the Brans-Dicke gravity as a special case of the scalar-tensor gravity and provided a brief review on this theory of gravity. The Brans-Dicke gravity is based on Mach’s principle implying that the inertial mass of an object depends on the matter distribution in the universe so that the gravitational constant should have time-dependence and is usually described by a scalar field. Using the results of the scalar-tensor gravity, we obtained the equations governing the background cosmology in the Brans-Dicke gravity. Then, we considered the slow-roll approximation to simplify the background equations. We further obtained relations of the inflationary observables for the Brans-Dicke gravity, in the slow-roll approximation. Subsequently, we discussed about the conformal transformations from the Jordan frame to the Einstein frame. Although in this paper, we considered the Jordan frame as our physical frame, however our analysis shows explicitly that in the slow-roll approximation the relations of the inflationary observables including the scalar power spectrum ${{\cal P}_s}$, the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$, are identical in both the Jordan and Einstein frames. In addition, we checked viability of different inflationary potentials in the framework of Brans-Dicke gravity. We chose the potentials that have wide usage in study of inflation and they have motivations from quantum field theory or string theory. Our study shows that in the Brans-Dicke gravity, results of the power-law, inverse power-law and exponential potentials lie completely outside the region allowed by the Planck 2015 data, and therefore these inflationary potentials are ruled out. The hilltop, Higgs, Coleman-Weinberg and natural potentials can be compatible with Planck 2015 TT,TE,EE+lowP data at 95% CL. Moreover, the D-brane and SB SUSY potentials can be in well agreement with the observational data since their results can lie inside the 68% CL region of Planck 2015 TT,TE,EE+lowP data. Another inflationary potential that we examined in the Brans-Dicke gravity, was the quadratic potential with displaced minimum. This potential for the zero Brans-Dicke parameter ($\omega=0$) leads to the Starobinsky $R^2$ inflation. The result of the quadratic potential with displaced minimum can be placed within the 68% CL region of Planck 2015 results. We also examined the other inflationary observables including the running of the scalar spectral index $d{n_s}/d\ln k$ and the equilateral non-Gaussianity parameter $f_{{\rm{NL}}}^{{\rm{equil}}}$ for those potentials whose results in $r-n_s$ plane were consistent with the Planck 2015 data. We concluded that results of those potentials for $d{n_s}/d\ln k$ and $f_{{\rm{NL}}}^{{\rm{equil}}}$ are compatible with the Planck 2015 results too. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors thank the referee for his/her valuable comments. 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[^1]: b.tahmasebzadeh@iasbs.ac.ir [^2]: rezazadeh86@gmail.com [^3]: kkarami@uok.ac.ir
--- abstract: 'We investigate the transport dynamics of partons in proton-proton collisions at the Large Hadron Collider using a Boltzmann transport approach, the parton cascade model. The calculations include semi-hard pQCD interaction of partons populating the nucleons and provide a space-time description of the collision in terms of cascading partons undergoing scatterings and fragmentations. Parton production and number of collisions rise rapidly with increase in center of mass energy of the collision. For a given center of mass energy, the number of parton interactions is seen to rise stronger than linear with decreasing impact parameter before saturating for very central collisions. The strangeness enhancement factor $\gamma_s$ for the semi-hard processes is found to rise rapidly and saturate towards the highest collision energies. Overall, our study indicates a significant amount of partonic interactions in proton-proton collisions, which supports the observation of fluid-like behavior for high multiplicity proton-proton collisions observed in the experiments.' author: - 'Dinesh K. Srivastava' - Rupa Chatterjee - 'Steffen A. Bass' title: Transport Dynamics of Parton Interactions in pp Collisions at LHC Energies --- Introduction ============ Relativistic heavy-ion collisions have been used with great success to probe the properties of hot and dense QCD matter, the quark-gluon-plasma (QGP) [@Arsene:2004fa; @Adcox:2004mh; @Back:2004je; @Adams:2005dq; @Gyulassy:2004zy; @Muller:2006ee; @Muller:2012zq]. Proton-proton collisions at a given center of mass energy per nucleon have been thought to provide a baseline measurement without the creation of a QGP when extrapolated to the corresponding nucleus-nucleus case using simple geometric models [@Miller:2007ri]. However, recently this canonical picture has undergone a considerable change as several experimental “indications” of formation of a medium, e.g., flow and enhanced production of strangeness have also been seen in proton-proton collisions, albeit only when triggering on high multiplicity events [@ALICE:2017jyt]. One should note that on the theory side the notion of possible QGP formation in proton-proton collisions dates back several decades: hydrodynamics has been used for a long time while exploring $pp$ collisions, with several theoretical studies assuming formation of QGP (see, e.g., Ref. [@Shuryak:1978ij; @VonGersdorff:1986tqh; @McLerran:1986nc]). It was suggested that the particle spectra for 1.8 TeV proton-antiproton collisions showed evidence of flow [@Levai:1991be] which indicated formation of quark-gluon plasma in such collisions. One may recall though, that it was argued [@Wang:1991vx] that the increase in $\langle p_T^2 \rangle$ could be attributed to events simply having a large multiplicity, i.e. increased minijet activity, without the formation of a deconfined medium. Subsequently, additional data for the same system, along with measurements of HBT radii, were used [@Alexopoulos:2002eh] to claim evidence for a deconfining phase transition in these collisions. However, only through the advent of recent high quality data have these calculations and analyses gained renewed traction. By now a large body of data for $pp$ collisions at RHIC and ever-increasing LHC energies has been accumulated, which provides enough indications for flow and enhanced production of strangeness in events having large multiplicity [@ALICE:2017jyt], even though non-QGP based interpretations of the data have remained viable [@Blok:2017pui; @Greif:2017dua; @Aichelin:2016vpr]. In the present work we aim at quantifying the amount of parton interactions and rescattering present in proton-proton collisions and whether the amount of interactions observed may lend credence to the notion of collectivity in these collision-systems and the application of hydrodynamic models. For this purpose we use a microscopic Boltzmann transport approach, the parton cascade model [@Geiger:1991nj] as implemented in VNI/BMS [@Bass:2002fh] and extended recently to include heavy quark production [@Younus:2013rja; @Srivastava:2017bcm] to explore the emergence of semi-hard multi-partonic collisions and parton multiplications in $pp$ collisions using pQCD matrix elements. This treatment [@Bass:2002fh] has several inherent advantages. First of all, all parton scatterings leading to $p_T \ge p_T^\text{cut-off}$ are treated within (lowest order) perturbative QCD, avoiding any arbitrariness, except for the dependence of the results on the momentum cut-off, introduced to avoid singular cross-sections for mass-less partons at lower momentum transfers. However, it is expected that spectra etc. for larger $p_T$ should be reasonable. It should be noted, however, that the limitation to pQCD matrix elements with a momentum cut-off implies that our approach does not describe the dynamics of thermalized degrees of freedom. We thus will only be able to assess whether the conditions necessary for the formation of (equilibrated) QCD matter are met, but will not be able to describe the development and evolution of a QGP itself. The tracking of the hard collision dynamics and all the partons involved in these interactions allows us to perform calculations at several levels of complexity: in a first step, we only allow the primary partons from the projectile nucleon to collide with primary partons from the target nucleon. Next we consider scattering among primary and secondary partons. This corresponds [@Bass:2002fh] most closely to minijet calculations [@Eichten:1984eu]. Finally we perform calculations which account for fragmentation of final state partons following semi-hard scatterings. These radiative processes are included following the original PCM implementation [@Geiger:1991nj] in the leading-logarithmic approximation (LLA). We do not consider hadronization of either the partons which have undergone interaction or the un-interacted partons, and thus our findings relate only to the partons produced in the semi-hard processes. We report our results for minimum bias collisions of protons at center of mass energies of 0.2, 2.76, 5.02, 7.00, and 14 TeV for the three implementations discussed above and study the evolution of parton production and multiple collisions with the increase in collision energy. Subsequently we explore the collision of protons at a center of mass energy of 7.00 TeV as a function of impact parameter and the $p_T^{\text {cut-off}}$ used for regularizing the infra-red divergences for pQCD cross-sections. Finally, we study some of these systematics as a function of number of quarks (charged particles). ![(Color online) \[Upper panel\] Production of light quarks, strange quarks, charm quarks and gluons as a function of centre of mass energy for semi-hard partonic collisions in $pp$ system for calculations involving scattering only between primary partons (filled circles), multiple scatterings [with-out]{} fragmentation of scattered partons (hollow squares) and multiple scatterings [with]{} fragmentations of scattered partons (filled diamonds). \[Lower panel\] Number of collisions and number of fragmentations as a function of centre of mass energy.[]{data-label="sqrts"}](sqrt_new.eps){width="8.3"} ![(Color online) \[Upper panel\] Production of light quarks, strange quarks, charm quarks and gluons as a function of centre of mass energy for semi-hard partonic collisions in $pp$ system for calculations involving scattering only between primary partons (filled circles), multiple scatterings [with-out]{} fragmentation of scattered partons (hollow squares) and multiple scatterings [with]{} fragmentations of scattered partons (filled diamonds). \[Lower panel\] Number of collisions and number of fragmentations as a function of centre of mass energy.[]{data-label="sqrts"}](coll_frag_sqrt.eps){width="8.3"} Formulation =========== The Monte Carlo implementation of the parton cascade model has been discussed in [@Geiger:1991nj; @Bass:2002fh] for production of light quarks, gluons, and heavy quarks. The $2\rightarrow 2$ scatterings involving light quarks and gluons included in VNI/BMS are: $$\begin{aligned} q_i q_j &\rightarrow& q_i q_j , \, q_i \bar{q}_i \rightarrow q_j \bar{q}_j \, , \nonumber\\ q_i\bar{q}_i &\rightarrow & gg , \, q_i \bar{q}_i \rightarrow g \gamma \, ,\nonumber\\ q_i \bar{q}_i &\rightarrow & \gamma \gamma , \, q_i g \rightarrow q_i g \, , \nonumber\\ q_i g & \rightarrow & q_i \gamma , \, gg \rightarrow q_i \bar{q}_i , \, \nonumber\\ gg &\rightarrow & gg ~.\end{aligned}$$ The heavy quark production is included [@Younus:2013rja; @Srivastava:2017bcm] via, $$q \bar{q} \rightarrow Q \bar{Q} , \, \, \, \, gg \rightarrow Q \bar{Q} \nonumber\\$$ processes, while their scatterings with light quarks or gluons are included via the $$q Q \rightarrow qQ , \, \, \, g Q \rightarrow g Q ~,$$ processes, where $g$ stands for gluons, $q$ stands for light quarks, and $Q$ stands for heavy quarks. The $2\rightarrow 3$ reactions are included via time-like branchings of the final-state partons (see Ref. [@Bass:2002fh]) : $$\begin{aligned} g^{} &\rightarrow& q_i \bar{q}_i \, , \, {q_i}^ \rightarrow q_i g \, , \nonumber\\ g^{} &\rightarrow& gg \, , \, {q_i}^ \rightarrow q_i \gamma~,\end{aligned}$$ following the well tested procedure adopted in PYTHIA. ![(Color online) Rapidity density of partons produced in $pp$ interactions due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different centre of mass energies.[]{data-label="dndy-sqrts"}](dndy-prim-prim.eps){width="8.0"} ![(Color online) Rapidity density of partons produced in $pp$ interactions due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different centre of mass energies.[]{data-label="dndy-sqrts"}](dndy-mult.eps){width="8.0"} ![(Color online) Rapidity density of partons produced in $pp$ interactions due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different centre of mass energies.[]{data-label="dndy-sqrts"}](dndy-full.eps){width="8.0"} ![(Color online) Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different centre of mass energies.[]{data-label="spec-sqrts"}](spec_prim-prim.eps){width="8.0"} ![(Color online) Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different centre of mass energies.[]{data-label="spec-sqrts"}](spec_mult.eps){width="8.0"} ![(Color online) Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different centre of mass energies.[]{data-label="spec-sqrts"}](spec_full.eps){width="8.0"} We add that the IR-singularities in these pQCD cross-sections are avoided in PCM by introducing a lower cut-off on the momentum transfer $p_T^\text{cut-off} \approx $ 2 GeV. We also add that, as discussed in Ref. [@Srivastava:2017bcm], the processes $gQ \rightarrow gQ$ and $qQ \rightarrow qQ$ have been explicitly excluded in these calculations when the heavy quark belongs to the sea, in order to account for the strong suppression of these interactions when NLO terms are included. ![(Color online) The ratio of strange quarks and light quarks produced in semi-hard processes for the three set of calculations discussed here.[]{data-label="gamma-sqrts"}](gamma.eps){width="8.0"} The $2 \rightarrow 3$ processes are included by inclusion of radiative processes for the final state partons in a leading logarithmic approximation. The collinear sigularities have been regularized by terminating the time-line branchings, once the virtuality of the parton drops to $Q_0^2=m_i^2+\mu_0^2$, where $m_i$ is the current mass of the parton (zero for gluons, current mass for quarks) and $\mu_0 $ has been kept fixed as 1 GeV. We have included $g \rightarrow gg$, $q \rightarrow q g$, $g \rightarrow q \bar{q}$, and $q \rightarrow q \gamma$ branchings for which the relevant branching functions $P_{a\rightarrow bc}$ are taken from Altarelli and Parisi [@Altarelli:1977zs]. ![(Color online) A comparison of our calculations with prompt charm production measured by ALICE experiment in $pp$ collisions at 2.76 TeV [@Abelev:2012vra].[]{data-label="2.76_data"}](plot_2.76.eps){width="8.0"} ![(Color online) A comparison of our calculations with prompt charm production measured by ALICE experiment in $pp$ collisions at 7.00 TeV [@ALICE:2011aa].[]{data-label="7.00_data"}](plot_7.00.eps){width="8.0"} The initial state of the nucleons has been set up in terms of partons whose momentum distributions are described by the parton distribution functions initialized at the scale of $Q_\text{ini}^2$ = 4 GeV$^2$. We have used GRV-HO function for our studies even though more modern functions are now available in the literature, primarily as we are more interested in the evolution of the multi-parton interactions when the centre of mass energy or the impact parameter or the lower momentum cut-off for parton scattering is altered. The partons are distributed around the centres of nucleons according to the distribution, $$h_N(\vec{r})= \frac{1}{4\pi} \frac{\nu^3}{8\pi}\exp(-\nu r) \label{nucleon}$$ with $\nu$ chosen to give the root mean square radius $R_N^\text{ms}\equiv\sqrt{12/\nu}= $ 0.81 fm. Unless otherwise stated, we have kept $p_T^\text{cut-off}$ fixed at 2 GeV. Most of our studies using PCM at RHIC energies used a more modest value of $p_T^\text{cut-off} \approx$ 0.78 GeV. For the corresponding results at $\sqrt{s}$ =0.2 TeV, the reader is referred to Ref. [@Bass:2002fh; @Bass:2002vm; @Bass:2003mk; @Chang:2004eha; @Renk:2005yg]. Our calculations do not consider hadronization and subsequent interactions among the hadrons. Analysis Setup ============== As mentioned earlier, we perform three sets of calculations to investigate the essential features of the evolution of the partonic cascade in $pp$ collisions. We define primary partons as partons which constitute the nucleon and which have not undergone any interaction. The secondary partons are those which are produced in collisions or fragmentation of scattered partons. The first set of calculations look at the system which would be formed if only primary-primary collisions are included in the calculations. The second set of calculations look at the system when primary-primary, primary-secondary, and secondary-secondary collisions are permitted but fragmentation of final state partons is not permitted, thus effectively blocking parton multiplication. The final set of calculations describe the system when all possible multiple scatterings among partons are tracked and when the final state partons fragment, leading to a substantial increase in number of collisions and parton production from semi-hard processes. We discuss our results in terms of partons produced in these semi-hard interactions and number of collisions as well as number of fragmentations (when applicable). We also give our results for relative abundance of strange quarks with respect to light quarks which are produced by semi-hard interactions considered here, defined by: $$\gamma_s^{\rm{semi-hard}} = \frac {2( N_s + N_{\bar s})} {N_u + N_{\bar u} + N_d + N_{\bar d}} \, . \label{gamma}$$ ![(Color online) \[Upper panel\] Production of light quarks, strange quarks, charm quarks and gluons as a function of impact parameter for semi-hard partonic collisions in $pp$ system for calculations involving scattering only between primary partons (filled circles), multiple scatterings [with-out]{} fragmentation of scattered partons (hollow squares) and multiple scatterings [with]{} fragmentation of scattered partons (filled diamonds). \[Lower panel\] Number of collisions and number of fragmentations as a function of impact parameter.[]{data-label="impact"}](b_new.eps){width="8.3"} ![(Color online) \[Upper panel\] Production of light quarks, strange quarks, charm quarks and gluons as a function of impact parameter for semi-hard partonic collisions in $pp$ system for calculations involving scattering only between primary partons (filled circles), multiple scatterings [with-out]{} fragmentation of scattered partons (hollow squares) and multiple scatterings [with]{} fragmentation of scattered partons (filled diamonds). \[Lower panel\] Number of collisions and number of fragmentations as a function of impact parameter.[]{data-label="impact"}](coll_frag_b.eps){width="8.3"} We re-emphasize that the multi-parton interactions included in these calculations take place only if the momentum-transfer is larger than the $p_T^\text{cut-off} \approx$ 2 GeV. This necessarily provides that quite a large part of the initial state partons continue without interaction. These include valence/sea light quarks, sea strange quarks and gluons, which move with the momenta with which they were initialized. Many more partons would interact if $p_T^\text{cut-off}$ is lowered or if a suitably screened interaction, e.g, by including Debye screening is considered[^1]. The lack of interactions below the $p_T^\text{cut-off}$ implies that we cannot directly study the possible formation of a thermalized medium, which would require abundant interactions at scales below the $p_T^\text{cut-off}$. However, we can ascertain whether a sufficient number of interacting partons is deposited into the system that could potentially lead to the formation of a thermalized medium. ![(Color online) Rapidity density of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different impact parameters.[]{data-label="dndy-b"}](dndy_prim_b.eps){width="8.0"} ![(Color online) Rapidity density of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different impact parameters.[]{data-label="dndy-b"}](dndy_mult_b.eps){width="8.0"} ![(Color online) Rapidity density of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different impact parameters.[]{data-label="dndy-b"}](dndy_full_b.eps){width="8.0"} ![(Color online) Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions at $\sqrt{s}$ = 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different impact parameters.[]{data-label="spec-b"}](spec_prim_b.eps){width="8.0"} ![(Color online) Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions at $\sqrt{s}$ = 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different impact parameters.[]{data-label="spec-b"}](spec_mult_b.eps){width="8.0"} ![(Color online) Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions at $\sqrt{s}$ = 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentations of final state partons (lower panel) at different impact parameters.[]{data-label="spec-b"}](spec_full_b.eps){width="8.0"} Evolution of multi-parton interactions with centre of mass energy ================================================================= We first look at partons produced in minimum bias collisions of protons at varying center of mass energies (Fig. \[sqrts\]). We see only a marginal difference between the results for the calculations involving primary-primary collisions and multiple collisions (without fragmentation). The production of gluons, light quarks, and strange quarks is seen to rise monotonically along with the number of semi-hard collisions with the center of mass energy of the $pp$ collision, but overall the number of partons involved in these interactions remains small, even at the highest beam energies. These results imply that semi-hard $2 \rightarrow 2$ interactions, without fragmentation of final state partons, are too few in $pp$ collisions even at the highest energy considered to lead to a hot and dense interacting medium. The corresponding results for the calculations with fragmentation of final state partons using the procedure indicated earlier, show a rapid multiplication of partons and a sharp rise in the number of collisions compared to the cases discussed above. This increase is clearly driven by the number of fragmentations, We also see a sharp increase in production of strange and charm quarks due to the multiple interactions as gluons multiply and interact. The increase could also be partly due to the opening up of processes like $g^{} \rightarrow Q \overline{Q}$ as the centre of mass energy increases. We note that already at $\sqrt{s}_{NN}=200$ GeV more than 100 interacting partons are deposited into the system, which can participate in the partonic collisions having momentum transfers of more than 2 GeV. This number grows to about 500 at 2.76 TeV. These should be sufficient to lead towards a thermalized system of partons with signs of collectivity. This would become even more likely once softer collisions are accounted for. The calculations clearly indicate that parton multiplication following initial scattering among primary partons drives and is driven by substantially increased multiple scatterings. The system thus created has a large number of partons undergoing semi-hard multiple collisions and the multi-parton interactions rise rapidly with increase in the center of mass energy. The corresponding rapidity density distributions of partons produced for the three sets of calculations are shown in Fig. \[dndy-sqrts\]. Once again we see that semi-hard collisions among partons [without]{} radiative processes do not lead to a substantial rise in the production of partons in $pp$ collisions. The radiative processes following multiple collisions lead to an increase in parton production by a factor of 10–20 at LHC energies. We also note that the greatly increased multiple collisions can lead to a substantial production of strange and charm quarks as the centre of mass energy increases. One should note, however, that the average $p_T$ of partons included in the analysis of the upper two frames is considerably higher due to the momentum cut-off of the scattering cross section than in the lowest frame, which includes fragmented partons that can carry a significantly lower $p_T$. The transverse momentum spectra for these calculations (for $p_T \geq$ 2 GeV) are shown in Fig. \[spec-sqrts\] and reveal a power-law behavior as expected. The ratio of the number of strange and light quarks produced in such collisions, $\gamma_s$ (Eq.\[gamma\]), is often used as a measure of strangeness or chemical equilibration. The results for $\gamma_s$ as a function of center of mass energy for the three sets of the calculations for semi-hard processes considered here are shown in Fig. \[gamma-sqrts\]. We immediately see that fragmentations play an important role in increasing the value of $\gamma_s$ which for the highest energy is seen to saturate at a value of about 0.9, suggesting that the fragmentations and enhanced multiple scatterings may push the system towards equilibration of strangeness even in $pp$ collisions at higher energies. Let us pause here to understand this large opening up of multi-partonic interactions in $pp$ collisions as the energy of the collision increases. This has several origins. First of all, we recall that the center of mass energy in a collision of primary partons - $\widehat{s}$ is equal to $x_1 x_2 \sqrt{s}$, where $x_i$’s stand for the fractions of nucleon momenta carried by the partons. The lower cut-off on the transverse momentum for the collision requires (see e.g., Ref. [@Wang:1991hta]) that $x_1 x_2 \sqrt{s} \geq 2 p_T^\text{cut-off}$. As $0\le x_i \leq 1$, the partons must have $x_i \geq 2 p_T^\text{cut-off}/\sqrt{s}$ to be able to participate in the semi-hard partonic collisions considered here. The structure functions for gluons and sea quarks increase with decreasing $x$ and this will bring in many more partons which can participate in the semi-hard collisions as the center of mass energy increases. Secondly the parton-parton cross-sections rise as the available center of mass energy increases. And lastly with the increase in the center of mass energy, many more collisions will have large momentum transfers making it possible for the fragmentation processes to contribute to partons multiplications. In Figs. \[2.76\_data\] and \[7.00\_data\] we give a comparison of our calculations with the prompt charm production measured by the ALICE experiment [@Abelev:2012vra; @ALICE:2011aa] at 2.76 and 7.00 TeV respectively. The experimental values for $D^0$ have been divided by 0.565- the fraction for fragmentation of $c$ quarks into $D^0$. We have limited the comparison to $p_T >$ 2 GeV in view of the $p_T^\text{cut-off}$ used in our calculations. A fair agreement is seen, though we note a definite tendency of the calculations to give a larger production of charm as the $p_T$ decreases, especially at the higher incident energy. This is under investigation. ![(Color online) The ratio of strange quarks and light quarks produced in semi-hard processes for the three set of calculations discussed here as a function of impact parameter.[]{data-label="gamma-b"}](gamma_b.eps){width="8.0"} ![(Color online) The frequency distribution of number of collisions (upper panel) and number of radiations (lower panel) suffered by partons for different impact parameters at $\sqrt{s}$ of 7 TeV for $pp$ collisions. The multiple collisions among the partons and radiative processes are included.[]{data-label="coll_rad"}](ncoll.eps){width="8.0"} ![(Color online) The frequency distribution of number of collisions (upper panel) and number of radiations (lower panel) suffered by partons for different impact parameters at $\sqrt{s}$ of 7 TeV for $pp$ collisions. The multiple collisions among the partons and radiative processes are included.[]{data-label="coll_rad"}](nrad.eps){width="8.0"} Evolution of partonic cascades as a function of impact parameter ($b$) and $p_T^\text{cut-off}$ =============================================================================================== Our calculations provide an opportunity to study the evolution of the partonic cascade as a function of impact parameter as the number of partons in the region of over-lap changes with the impact parameter. Thus for example in a collision of protons at the center of mass energy of 7 TeV (a case which we study in greater detail here), each proton is populated by about 270 partons, which include the up and down valence quarks, up, down and strange sea quarks, and gluons distributed according to the function given by Eq. \[nucleon\] given earlier. This immediately provides for a larger possibility for multiple collisions for smaller impact parameters. We acknowledge that the identification of the impact parameter in a $pp$ collision is experimentally rather challenging, yet we proceed under the assumption that some measure of centrality can be identified that allows experimental data to be mapped to our systematic study as a function of the impact parameter (over which we have full control in our calculation). We note that our calculation contains a large number of uninteracted partons which will subsequently hadronize. It is quite likely that the hadrons arising from these uninteracted partons will have lower transverse momenta while those resulting from the partons which have undergone semi-hard collisions will have larger transverse momenta. We give results of our calculations for impact parameter, $b$, equal to 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 fm. The $p_T^\text{cut-off}$ for these calculations has been fixed at 2 GeV. Looking at the total number of light quarks, strange quarks, charm quarks, gluons and number of collisions and fragmentations (when applicable) (see Fig \[impact\]), we see a very clear increase in the number of collisions as the impact parameter decreases. These variations are large enough to provide a distinctive classification of events with large semi-hard partonic collisions for the more realistic calculation of partonic collisions along with radiative processes. ![(Color online) \[Upper panel\] Effect of changing the lower $p_T^\text{cut-off}$ on production of light quarks, strange quarks, charm quarks and gluons as a function of $p_T^{\rm{cut-off}}$ for semi-hard partonic collisions in $pp$ collisions at $\sqrt{s}=$ 7 TeV for calculations involving scattering only between primary partons (filled circles), multiple scatterings [with-out]{} fragmentation of scattered partons (hollow squares) and multiple scatterings [with]{} fragmentation of scattered partons (filled diamonds). \[Lower panel\] Number of collisions and number of fragmentations as a function of $p_T^{\rm{cut-off}}$ set of calculations.[]{data-label="ptcut"}](pt_new.eps){width="8.3"} ![(Color online) \[Upper panel\] Effect of changing the lower $p_T^\text{cut-off}$ on production of light quarks, strange quarks, charm quarks and gluons as a function of $p_T^{\rm{cut-off}}$ for semi-hard partonic collisions in $pp$ collisions at $\sqrt{s}=$ 7 TeV for calculations involving scattering only between primary partons (filled circles), multiple scatterings [with-out]{} fragmentation of scattered partons (hollow squares) and multiple scatterings [with]{} fragmentation of scattered partons (filled diamonds). \[Lower panel\] Number of collisions and number of fragmentations as a function of $p_T^{\rm{cut-off}}$ set of calculations.[]{data-label="ptcut"}](coll_frag_pt.eps){width="8.3"} How are the rapidity density and transverse momentum distributions of partons produced in these semi-hard processes affected by variation in impact parameter? We give these results for the rapidity densities for the three set of calculations in Fig.\[dndy-b\]. ![(Color online) Rapidity density of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentation of final state partons (lower panel) at different cut-offs for transverse momentum in semi-hard collisions.[]{data-label="dndy-pt"}](dndy_prim_pt.eps){width="8.0"} ![(Color online) Rapidity density of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentation of final state partons (lower panel) at different cut-offs for transverse momentum in semi-hard collisions.[]{data-label="dndy-pt"}](dndy_mult_pt.eps){width="8.0"} ![(Color online) Rapidity density of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentation of final state partons (lower panel) at different cut-offs for transverse momentum in semi-hard collisions.[]{data-label="dndy-pt"}](dndy_full_pt.eps){width="8.0"} ![(Color online)Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentation of final state partons (lower panel) at different cut-offs for transverse momentum in semi-hard collisions.[]{data-label="spec-pt"}](spec_prim_pt.eps){width="8.0"} ![(Color online)Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentation of final state partons (lower panel) at different cut-offs for transverse momentum in semi-hard collisions.[]{data-label="spec-pt"}](spec_mult_pt.eps){width="8.0"} ![(Color online)Rapidity integrated $p_T$ spectra of partons produced in $pp$ interactions at $\sqrt{s}=$ 7 TeV due to semi-hard collisions among primary partons (upper panel), multiple collisions [without]{} fragmentations of final state partons (middle panel) and multiple collisions [with]{} fragmentation of final state partons (lower panel) at different cut-offs for transverse momentum in semi-hard collisions.[]{data-label="spec-pt"}](spec_full_pt.eps){width="8.0"} We see once again that multiple collisions along with parton fragmentations lead to a large production of partons due to semi-hard processes. This production is seen to rise with decrease in impact parameter which leads to a larger overlap of partonic clouds. The production is seen to be largest at central rapidities. (The structures seen in the $dN/dy$ distributions for the calculations invoking only primary-primary or multiple collisions without fragmentations arise due to cut-offs on $p_T$.) Fig.\[spec-b\] shows the respective transverse momentum spectra: we find that the distributions are quite similar for different impact parameters but differ in magnitude. This may indicate that in high energy $pp$ interactions even though the number of partonic collisions increases as we reduce the impact parameter, the number of semi-hard collisions suffered by individual partons may not increase very substantially - as that could alter the shape of these momentum distributions. We shall come back to this point again. The results for the strangeness enhancement factor $\gamma_s$ in semi-hard processes are given in Fig. \[gamma-b\]. We find that $\gamma_s$ is only marginally dependent on the impact parameter. Note, however, that even the lowest transverse momentum that we have considered is much larger than the mass of strange quarks. The behavior of the strangeness enhancement parameter can be understood by noting that the frequency of number of collisions and number of radiations suffered by the parton indicate the possibility of creating an interacting medium of partons that would allow for a measure of chemical equilibration as well. Thus, for the most central collisions in our most realistic calculation (the scenario that includes primary and secondary scattering as well as fragmentation) we find on average in excess of 170 parton-parton scatterings and more than 300 fragmentations, even in the absence of low momentum interactions below the $p_T$-cutoff (see Fig. \[coll\_rad\]). This figure shows the frequency of number of collisions and number of radiations suffered by individual partons at three impact parameter: a substantial number of partons interact multiple times. The number of collisions as well as the parton rescattering that we observe indicate the formation of an interacting medium and a system that could potentially thermalize, as is hinted by experimental observations. Stronger statements regarding thermalization are hampered by the presence of the $p_T$-cutoff that is used to regularize the interaction cross sections in our model. In Fig. \[ptcut\], we vary the value of $p_T^\text{cut-off}$ (for a minimum bias sample of p+p collisions) to investigate these quantities within a reasonable range of cut-off values. The observed trends (strongly rising collision and fragmentation numbers with reduced values of the cut-off) are certainly favorable for the formation of a thermalized medium. The effect of the variation of the cut-off on the rapidity density distribution and rapidity integrated transverse momentum spectra is shown in figures \[dndy-pt\] and \[spec-pt\] respectively. We again see a rapid rise in the parton production as the $p_T^\text{cut-off}$ is reduced leading to more collisions and fragmentations. We have shown only a selected set of results for the parton spectra to avoid severe over-crowding. The results for other values of the cut-off parameter lie between appropriate curves given here. We see a near identity of $p_T$ spectra beyond the largest $p_T^\text{cut-off}$, as expected, i.e. while the low $p_T$ results of our calculation are significantly affected by the cut-off, the high momentum results remain stable. ![(Color online) The ratio of strange quarks and light quarks produced in semi-hard processes for the three set of calculations discussed here as a function of $p_T^\text{cut-off}$ in $pp$ collisions at $\sqrt{s}$ = 7 TeV.[]{data-label="gamma-pt"}](gamma_pt.eps){width="8.0"} ![(Color online) Number of strange and charm quarks produced in semi-hard interactions vs. average number of charge particles produced (upper panel). The lower panel depicts the variation of number of strange quarks, its square and cube.[]{data-label="str_vs_charge"}](charge_vs_str_charm.eps){width="8.0"} ![(Color online) Number of strange and charm quarks produced in semi-hard interactions vs. average number of charge particles produced (upper panel). The lower panel depicts the variation of number of strange quarks, its square and cube.[]{data-label="str_vs_charge"}](ns123.eps){width="8.0"} ![(Color online) The ratio of strange quarks and light quarks produced in semi-hard processes as a function of charged particles (quarks) produced. []{data-label="charge_gammas"}](gammas.eps){width="8.0"} Finally the dependence of the strangeness enhancement factor $\gamma_s$ on the transverse momentum cut-off is shown in Fig. \[gamma-pt\]. We see a mild rise in $\gamma_s$ as the $p_T^\text{cut-off}$ is decreased for all the calculations reported here as these lead to increased number of partonic collisions. Evolution of strangeness for events with large multiplicity =========================================================== In the previous section we saw that production of partons rises with decrease in impact parameter. While the determination of impact parameter for $pp$ collisions may be non-trivial, one can easily compare the results for minimum bias events to those with high multiplicity. In Fig. \[str\_vs\_charge\] we show the increase in production of strangeness and charm in collisions with large multiplicity, for the calculations allowing multiple scattering among partons and the fragmentation of final state partons. A rapid increase in strangeness and charm multiplicity is seen as the number of partons rises. The lower frame of Fig. \[str\_vs\_charge\] shows the variation of the ratio of strange quarks vs. the number of quarks (charged particles) produced against the multiplicity of the charged particles produced in semi-hard interactions. We find it to be essentially constant, basically as the $p_T^{\text{cut-off}}$ of 2 GeV chosen in these studies is much larger than the mass of strange quark taken in the calculations. The behavior of the square and the cube of the number of strange quarks is also shown. Taken together, these results indicate an increasing production of hadrons with one, two, and three strange quarks as we look at events with larger multiplicity, as indeed observed in recent experiments at LHC. In Fig. \[charge\_gammas\] we have plotted our result for strangeness enhancement factor as a function of charged particles. We see that as the number of quarks (charged particles) produced rises in the event, the strangeness tends to equilibrate, as indeed indicated in results from the ALICE experiment. Summary and conclusions ======================= We have studied the formation of a partonic medium produced in $pp$ collisions using parton cascade model. The production of light quarks, strange quarks, charm quarks, and gluons in calculations permitting multiple collisions and fragmentation of scattered partons is seen to rise rapidly with the energy due to semi-hard interactions treated using pQCD. The multi-parton interactions (collisions and radiations) are seen to rise with decrease in impact parameter and decrease in the lower cut-off on the transverse momenta at a given centre of mass energy. The strangeness enhance factor for the semi-hard processes at higher energies is seen to be close to unity and only marginally dependent on the impact parameter. A detailed analysis of number of collisions and radiations suffered by partons reveals that while many partons undergo only one collision and one fragmentation per event, the chances of the same parton undergoing multiple collisions or several radiations is still substantial. This indicates the emergence of an interacting medium. The strangeness and charm production is also seen to rise as a function of charged particles as indicated in recent experiments. The strangeness is seen to equilibrate in high multiplicity events and as the centre of mass energy increases. Overall we deem the amount of partons produced and the multiple interactions among these partons favorable for the formation of an interaction medium that may give rise to collective effects. Acknowledgments {#acknowledgments .unnumbered} =============== DKS gratefully acknowledges the support by the Department of Atomic Energy. This research was supported in part by the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany. DKS also acknowledges valuable comments by Helmut Satz and Dirk Rischke. SAB acknowledges support by US Department of Energy grant DE-FG02-05ER41367. [10]{} I. Arsene [et al.]{} \[BRAHMS Collaboration\], Nucl. Phys. [A757]{}, 1 (2005). K. 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--- abstract: \| For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots, k\}$. A class $\mathcal C$ of graphs is [balanced $k$-partitionable]{} and [unbalanced $k$-partitionable]{} if there exists a nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots, D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple has length $k$. A set $X$ of cycles is a [cycle obstruction set]{} of a class $\mathcal C$ of planar graphs if every planar graph containing none of the cycles in $X$ as a subgraph belongs to $\mathcal C$. This paper characterizes all cycle obstruction sets of planar graphs to be balanced $k$-partitionable and unbalanced $k$-partitionable for all $k$; namely, we identify all inclusion-wise minimal cycle obstruction sets for all $k$. author: - 'Ilkyoo Choi[^1]' - 'Chun-Hung Liu[^2]' - 'Sang-il Oum[^3]' title: Characterization of cycle obstruction sets for improper coloring planar graphs --- Introduction ============ All graphs in this paper are finite and simple, which means no loops and no parallel edges. Let $C_k$ denote a $k$-cycle, which is a cycle of length $k$. A set $X$ of cycles is a [cycle obstruction set]{} of a class $\mathcal C$ of planar graphs if every planar graph containing none of the cycles in $X$ as a subgraph belongs to $\mathcal C$. A graph is [$k$-colorable]{} if its vertex set can be partitioned into $k$ color classes so that each color class is an independent set. The celebrated Four Color Theorem [@1977ApHa; @1977ApHaKo] (later reproved in [@1997RoSaSeTh]) states that every planar graph is $4$-colorable. Since there are planar graphs that are not $3$-colorable, finding sufficient conditions for a planar graph to be $3$-colorable has been an active area of research; many of these conditions can be translated into the language of obstruction sets. Perhaps the most well-known result is the following theorem, known as Grötzsch’s Theorem [@1959Gr]: Planar graphs with no $3$-cycles are $3$-colorable. In the language of obstruction sets, Grötzsch’s Theorem states that $\{C_3\}$ is an obstruction set of $3$-colorable planar graphs. There is also a vast literature regarding forbidding various cycle lengths to guarantee a planar graph to be $3$-colorable; see Table \[tab:history\] for a summary of some of these results. \[tab:history\] year reference $ 3$ $ 4$ $ 5$ $ 6$ $ 7$ $ 8$ $ 9$ authors ------ --------------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ------------------- 1959 [@1959Gr] $\times$ Grötzsch 2005 [@2005ZhWu] $\times$ $\times$ $\times$ $\times$ Zhang–Wu 2006 [@2006Xu] $\times$ $\times$ $\times$ Xu 2010 [@2010WaLuCh] $\times$ $\times$ $\times$ $\times$ Wang–Lu–Chen 2007 [@2007ChRaWa] $\times$ $\times$ $\times$ $\times$ Chen–Raspaud–Wang 2007 [@2007WaCh] $\times$ $\times$ $\times$ Wang–Chen 2011 [@2011WaWuSh] $\times$ $\times$ $\times$ $\times$ Wang–Wu–Shen : Forbidding various cycle lengths to guarantee 3-colorability of planar graphs Each result in the aforementioned theorem reveals a new obstruction set of $3$-colorable planar graphs. The interest in forbidding various cycle lengths stems from Steinberg’s Conjecture [@1993St], which states that planar graphs with neither $4$-cycles nor $5$-cycles are $3$-colorable. There was almost no progress after the conjecture was first proposed in 1976, but many partial results were produced after 1991, which is when Erdős [@1993St] proposed the following approach towards Steinberg’s Conjecture: find the minimum $k$ such that planar graphs with no cycle lengths in $\{4, \ldots, k\}$ are $3$-colorable. After 40 years of effort by the coloring community to try to prove Steinberg’s Conjecture, only recently it was disproved via a clever construction by Cohen-Addad et al. [@2017CoHeKrLiSa]. Yet, the question of whether planar graphs with no cycle lengths in $\{4, 5, 6\}$ are $3$-colorable or not remains open. Recently, the following relaxation of proper coloring, also known as [improper coloring]{}, has attracted much attention: for nonnegative integers $k, d_1, \ldots, d_k$, a graph is [$(d_1, \ldots, d_k)$-colorable]{} if its vertex set can be partitioned into $k$ color classes $V_1, \ldots, V_k$ so that $V_i$ induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots, k\}$. This relaxation allows some prescribed defects in each color class, where defects are measured in terms of the maximum degree of the graph induced by the vertices of a color class. We say a class $\mathcal C$ of graphs is [balanced $k$-partitionable]{} and [unbalanced $k$-partitionable]{} if there exists a nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots, D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple has length $k$. There is a vast literature in improper coloring planar graphs. By the Four Color Theorem, planar graphs are $4$-colorable, which is equivalent to $(0, 0, 0, 0)$-colorable, and Cowen et al. [@1986CoCoWo] proved that planar graphs are $(2, 2, 2)$-colorable. This is best possible in the sense that for any given nonnegative integers $d_1$ and $d_2$, there exists a planar graph that is not $(1, d_1, d_2)$-colorable; for one such construction see [@unpub_ChEs]. Therefore, the question of partitioning planar graphs with no extra conditions into at least three subgraphs of bounded maximum degrees is completely solved. It is often useful to consider girth conditions along with planarity to obtain positive results. Regarding partitioning planar graphs into two parts, for any given nonnegative integers $d_1$ and $d_2$, a planar graph with girth $4$ that is not $(d_1, d_2)$-colorable is constructed in [@2015MoOc]. Yet, Choi et al. [@2016ChChJeSu], Borodin and Kostochka [@2014BoKo], Choi and Raspaud [@2015ChRa], and Škrekovski [@2000Sk] proved that planar graphs with girth at least $5$ are $(1, 10)$-, $(2, 6)$-, $(3, 5)$-, and $(4, 4)$-colorable, respectively. Also, given a nonnegative integer $d$, a planar graph with girth $6$ that is not $(0, d)$-colorable is constructed in [@2010BoIvMoOcRa]. On the other hand, it is known that every planar graph with girth at least $7$ is $(0, 4)$-colorable [@2014BoKo]. For other papers regarding improper coloring sparse (not necessarily planar) graphs, see [@2013BoKoYa; @2013EsMoOcPi; @2006HaSe; @2014KiKoZh; @2016KiKoZh]. The previous paragraph concerns girth conditions enforced on planar graphs to obtain positive results. Instead of forbidding all short cycles, we are interested in finding the minimal sets of obstacles in partitioning planar graphs into parts with bounded maximum degrees. We succeed in identifying which cycle lengths are essential obstructions when it comes to partitioning planar graphs in a balanced and unbalanced way. In other words, this paper characterizes all cycle obstruction sets of balanced $k$-partitionable and unbalanced $k$-partitionable planar graphs for all $k$; namely, we identify all the inclusion-wise minimal cycle obstruction sets. By the Four Color Theorem, the empty set is the (only) inclusion-wise minimal cycle obstruction set of both balanced $k$-partitionable and unbalanced $k$-partitionable planar graphs when $k\geq 4$. The empty set is also the (only) inclusion-wise minimal cycle obstruction set of balanced $3$-partitionable planar graphs, since Cowen et al. [@1986CoCoWo] proved that planar graphs are $(2, 2, 2)$-colorable. For the remaining cases, we characterize the inclusion-wise minimal obstruction sets, and for each case there are exactly two. Our main results are the following three theorems: [ A set $S$ of cycles is an inclusion-wise minimal cycle obstruction set of balanced $2$-partitionable planar graphs if and only if $S=\{C_4\}$ or $S$ is the set of all odd cycles. ]{} [ A set $S$ of cycles is an inclusion-wise minimal cycle obstruction set of unbalanced $2$-partitionable planar graphs if and only if $S=\{C_3,C_4,C_6\}$ or $S$ is the set of all odd cycles. ]{} [ A set $S$ of cycles is an inclusion-wise minimal cycle obstruction set of unbalanced $3$-partitionable planar graphs if and only if $S=\{C_3\}$ or $S=\{C_4\}$. ]{} Theorem \[thmm:bal2\] and Theorem \[thmm:unbal2\] state that for planar graphs to be balanced $2$-partitionable and unbalanced $2$-partitionable, respectively, there is only one inclusion-wise minimal cycle obstruction set other than the set of all odd cycles. Since forbidding all odd cycles makes the graph bipartite, and thus $2$-colorable, which is equivalent to $(0, 0)$-colorable, the minimal cycle obstructions for planar graphs to be balanced $2$-partitionable and unbalanced $2$-partitionable is a $4$-cycle and all of $3$-, $4$-, $6$-cycles, respectively. Note that previous results by Škrekovski [@2000Sk] and Borodin and Kostochka [@2014BoKo] imply that planar graphs are balanced $2$-partitionable and unbalanced $2$-partitionable when the forbidden cycle lengths are $3, 4$ and $3, 4, 5, 6$, respectively. Theorem \[thmm:unbal3\] states that other than a $3$-cycle, there is only one other inclusion-wise minimal cycle obstruction set of unbalanced $3$-partitionable planar graphs. Since Grötzsch’s Theorem says that forbidding a $3$-cycle in planar graphs guarantees that it is $3$-colorable, which is equivalent to $(0, 0, 0)$-colorable, the minimal cycle obstruction for non-3-colorable planar graphs to be unbalanced $3$-partitionable is a $4$-cycle. Note that for both balanced $1$-partitioning and unbalanced $1$-partitioning, cycle obstruction sets simply do not exist because of planar graphs with arbitrarily large maximum degree. See Table \[tab:results\] for a complete list of cycle obstruction sets of both balanced $k$-partitionable and unbalanced $k$-partitionable planar graphs. $ k$ balanced unbalanced --------------------------- ------------------------------------ ------------------------------------------------ ${4^+}$-[partitionable]{} [$\emptyset$]{} [$\emptyset$]{} $ 3$-[partitionable]{} [$\emptyset$]{} $\{ C_3\}, \{ C_4\}$ $ 2$-[partitionable]{} $\{C_{2i+1}:i\geq 1\}$, $\{ C_4\}$ $\{C_{2i+1}:i\geq 1\}$, $\{ C_3, C_4, C_6\}$ $ 1$-[partitionable]{} does not exist! does not exist! : Characterization of inclusion-wise minimal cycle obstruction sets[]{data-label="tab:results"} In Section \[sec:bal2\], Section \[sec:unbal2\], and Section \[sec:unbal3\], we prove Theorem \[thmm:bal2\], Theorem \[thmm:unbal2\], and Theorem \[thmm:unbal3\], respectively. The constants in all of our main results are probably improvable with some effort. Yet, we focused on simplifying the proofs and using the minimum number of reducible configurations and basic discharging rules in order to improve the readability of the paper. We end this section by posing three questions and some definitions that will be used in the next sections. What is the minimum $D$ such that every planar graph with no $4$-cycles is $(D, D)$-colorable? What is the minimum $D$ such that every planar graph with no $3$-, $4$-, $6$-cycles is $(0, D)$-colorable? What is the minimum $D$ such that every planar graph with no $4$-cycles is $(0, 0, D)$-colorable? The [degree]{} of a vertex $v$, denoted by $d(v)$, is the number of edges incident with it. A [$k$-vertex]{}, [$k^+$-vertex]{}, and [$k^-$-vertex]{} is a vertex of degree exactly $k$, at least $k$, and at most $k$, respectively. Given any embedding of a connected planar graph $G$ on at least two vertices on the plane, for every face $f$, we say that a boundary walk $W_f$ of $f$ is [canonical]{} if it traces the edges incident with $f$ according to one of the two obvious cyclic orderings of those edges. The [degree]{} of a face $f$, denoted by $d(f)$, is the length of $W_f$; note that cut edges are counted twice. A [$k$-face]{}, [$k^+$-face]{}, and [$k^-$-face]{} is a face of degree exactly $k$, at least $k$, and at most $k$, respectively. For each face $f$ and each vertex $v$ of $G$, we define $k_{f,v}$ to be the number of triples $(e,v,e')$ such that $e,e' \in E(G)$ and $eve'$ is a subwalk of $W_f$. It is well-known that the degree of $f$ and $k_{f,v}$ is independent of the choice of $W_f$. Clearly, the degree of $f$ equals $\sum_{v \in V(G)} k_{f,v}$. Balanced $2$-partitions {#sec:bal2} ======================= In this section, we prove the following theorem: \[thmm:bal2\] [ A set $S$ of cycles is an inclusion-wise minimal cycle obstruction set of balanced $2$-partitionable planar graphs if and only if $S=\{C_4\}$ or $S$ is the set of all odd cycles. ]{} We will first show a necessary condition for cycle obstruction sets, and then show that it is sufficient afterwards. \[lem:bal2:nec\] If a set $S$ of cycles is an obstruction set of balanced $2$-partitionable planar graphs, then either $C_4 \in S$ or $S$ contains all odd cycles. Given a nonnegative integer $D$ and two vertices $x$ and $y$, let $H_2(D; x, y)$ be the graph consisting of $2D+1$ internally disjoint $x, y$-paths of length $2$. For a positive integer $l$ and a vertex $v_1$, let $H_1(D, l; v_1)$ be the graph obtained from a cycle with vertices $v_1, \ldots, v_{l+1}$ and replacing each edge $v_iv_{i+1}$ with a copy of $H_2(D; v_i, v_{i+1})$ where $i\in\{1, \ldots, l\}$. Finally, let $H(D, l)$ be the graph obtained from $D+1$ pairwise disjoint copies of $H_1(D, l; v^j_1)$ and identifying all of $v^j_1$ for $j\in\{1, \ldots, D+1\}$. Now in any $(D, D)$-coloring of $H_2(D; x, y)$, it is easy to see that $x$ and $y$ must receive the same color. This implies that the cut-vertex of $H(D, l)$ has $D+1$ neighbors of the same color, which shows that $H(D, l)$ is not $(D, D)$-colorable. It is not hard to see that the cycles in $H(D, l)$ have length either $4$ or $2l+1$. Therefore the obstruction set of balanced $2$-partitionable planar graphs contains either $C_4$ or all odd cycles. See Figure \[fig:bal2-tight\] for an illustration of $H_2(D; x, y)$ and $H(D, 2)$. ![Graphs that are not $(D, D)$-colorable[]{data-label="fig:bal2-tight"}](fig-bal2-tight.pdf) If a planar graph does not contain any odd cycles, then it is bipartite, and thus it is $(0, 0)$-colorable, and hence it is balanced $2$-partitionable. The remaining of this section proves that planar graphs with no $4$-cycles are balanced $2$-partitionable. Note that Lemma \[lem:bal2:nec\] and Theorem \[thm:bal2\] imply Theorem \[thmm:bal2\]. \[thm:bal2\] A planar graph with no $4$-cycles is $(5, 5)$-colorable. In the rest of this section, let $G$ be a counterexample to Theorem \[thm:bal2\] with the minimum number of $3^+$-vertices, and subject to that choose one with the minimum number of edges. Also, fix a plane embedding of $G$. It is easy to see that $G$ is connected and has no $1$-vertices. From now on, given a (partially) $(5, 5)$-colored graph, we will let $a$ and $b$ be the two colors, and we say a vertex with a color is [saturated]{} if it already has five neighbors of the same color. Structural lemmas ----------------- \[lem:bal2:edge\] Every edge $xy$ of $G$ has an endpoint with degree at least $7$. Suppose to the contrary that $x$ and $y$ are both $6^-$-vertices. Since $G\setminus xy$ is a graph with fewer edges than $G$ and the number of $3^+$-vertices did not increase, there is a $(5, 5)$-coloring $\varphi:V(G)\rightarrow\{a, b\}$ of $G\setminus xy$. If $\varphi$ is not a $(5, 5)$-coloring of $G$, then $\varphi(x)=\varphi(y)$, and either $x$ or $y$ is saturated in $G\setminus xy$. For each saturated vertex $z$ in $\{x, y\}$, we may recolor it with the color in $\{a, b\}\setminus\{\varphi(z)\}$ since all of its neighbors have color $\varphi(z)$ in $G\setminus xy$. We end up with a $(5, 5)$-coloring of $G$, which is a contradiction. \[lem:bal2:3-vx\] There are no $3$-vertices in $G$. Suppose to the contrary that $v$ is a $3$-vertex of $G$ with neighbors $v_1, v_2, v_3$. By Lemma \[lem:bal2:edge\], we know that $v_1, v_2, v_3$ are $7^+$-vertices. Obtain a graph $H$ from $G-v$ by adding paths $v_1u_1v_2, v_2u_2v_3, v_3u_3v_1$, where $u_1,u_2,u_3$ are three distinct vertices not in $G$. Note that $H$ is planar and has no $4$-cycles since the pairwise distance between $v_1, v_2, v_3$ did not change. See Figure \[fig:bal2-3-vx\] for an illustration. Since $H$ has fewer $3^+$-vertices than $G$, there is a $(5, 5)$-coloring $\varphi:V(H)\rightarrow\{a, b\}$ of $H$. If $\varphi(v_1)=\varphi(v_2)=\varphi(v_3)$, then we may extend $\varphi$ to $G$ by using the color in $\{a, b\}\setminus\{\varphi(v_1)\}$ on $v$. Otherwise, without loss of generality we may assume $\varphi(v_1)=a$ and $\varphi(v_2)=\varphi(v_3)=b$. If $a\in\{\varphi(u_1), \varphi(u_3)\}$, then we may extend $\varphi$ to $G$ by using $a$ on $v$. Otherwise, $\varphi(u_1)=\varphi(u_3)=b$, so we may extend $\varphi$ to $G$ by using $b$ on $v$. In all cases we end up with a $(5, 5)$-coloring of $G$, which is a contradiction. ![Obtaining $H$ from $G$ in Lemma \[lem:bal2:3-vx\][]{data-label="fig:bal2-3-vx"}](fig-bal2-3-vx.pdf) A $3$-face is [terrible]{} if it is incident with a $2$-vertex. \[lem:bal2:terrible\] A $7^+$-vertex $v$ is incident with at most $\min\{\lfloor{d(v)\over 2}\rfloor, d(v)-6\}$ terrible $3$-faces. Since $G$ has no $4$-cycles, two $3$-faces cannot share an edge, and thus $v$ is incident with at most $\lfloor{d(v)\over 2}\rfloor$ terrible 3-faces. Since $\lfloor{d(v)\over 2}\rfloor\leq d(v)-6$ when $d(v)\geq 11$, we may assume $d(v)\leq 10$. Suppose to the contrary that $v$ is incident with $t$ terrible 3-faces, where $t \geq d(v)-5$. Let $w$ be a $2$-vertex of a terrible 3-face $wvu$; note that $u$ is also a $7^+$-vertex by Lemma \[lem:bal2:edge\]. Since $G-w$ is a graph with fewer edges than $G$ and the number of $3^+$-vertices did not increase, there is a $(5, 5)$-coloring $\varphi:V(G)\setminus\{w\}\rightarrow\{a, b\}$ of $G-w$. If $\varphi(u)=\varphi(v)$, then we may extend $\varphi$ to $G$ by using the color in $\{a, b\}\setminus\{\varphi(u)\}$ on $w$. Thus, we may assume $\varphi(u)=a$ and $\varphi(v)=b$. Since using the color $b$ on $w$ should not extend $\varphi$ to $G$, we know that $v$ must be saturated by $\varphi$. There are $d(v)-2t$ neighbors of $v$ in $G-w$ that are not in terrible 3-faces incident with $v$. Since $v$ has five neighbors with the color $b$, at least $5-(d(v)-2t)=5+2t-d(v)$ neighbors of $v$ in $G-w$ with the color $b$ are incident with a terrible 3-face incident with $v$. Since neither $w$ nor $u$ is colored with $b$, there are $t-1$ terrible 3-faces incident with $v$ that might have a vertex colored with $b$. Since $t\geq d(v)-5$ implies $5+2t-d(v)>t-1$, there exists a terrible 3-face $xyv$ where $x$ is a 2-vertex, other than $wuv$ with $\varphi(x)=\varphi(y)=b$. Now, we can extend $\varphi$ to $G$ by coloring $w$ with $b$ and recoloring $x$ with $a$, which contradicts the assumption that $G$ has no $(5, 5)$-coloring. Discharging ----------- We now define the initial charge at each vertex and each face. For every $v\in V(G)$, let $\mu(v)=2d(v)-6$ and for every face $f\in F(G)$, let $\mu(f)=d(f)-6$. The total initial charge is negative since $$\begin{aligned} \sum_{z\in V(G)\cup F(G)} \mu(z) =\sum_{v\in V(G)} (2d(v)-6)+\sum_{f\in F(G)} (d(f)-6) =-6\|V(G)\|+6\|E(G)\|-6\|F(G)\| =-12 <0.\end{aligned}$$ The last equality holds by Euler’s formula. Recall that a $3$-face is terrible if it is incident with a $2$-vertex. Here are the discharging rules: 1. Each $7^+$-vertex sends charge $1$ to each adjacent $2$-vertex. 2. Each $4$-, $5$-, $6$-vertex sends charge $1$ to each incident $3$-face. 3. Let $v$ be a $7^+$-vertex. 1. $v$ sends charge $3\over 2$ to each incident terrible 3-face. 2. $v$ sends charge $1$ to each incident $3$-face that is not terrible. 3. $v$ sends charge $1\over 2$ to each $5$-face $f$ that is incident with $v$ and incident with a neighbor of $v$ with degree at least $7$. See Figure \[fig:bal2-rules\] for an illustration of the discharging rules. ![Discharging rules[]{data-label="fig:bal2-rules"}](fig-bal2-rules2.pdf) We denote the final charge of $z$ by $\mu^(z)$ for each $z \in V(G) \cup F(G)$. The rest of this section will prove that $\mu^(z)$ is nonnegative for each $z\in V(G)\cup F(G)$. Every face has nonnegative final charge. Let $f$ be a face. It only receives charge and does not give out any charge. Thus if $f$ is a $6^+$-face, then $f$ has nonnegative final charge since $\mu^(f)=\mu(f)=d(f)-6\geq 0$. By Lemma \[lem:bal2:edge\], every $5$-face $f$ is incident with at least three $7^+$-vertices, and at least two of these are adjacent to each other. Thus, by rule (R3C), $f$ receives charge $1\over 2$ at least twice. Thus, $\mu^(f)\geq 5-6+2\cdot{1\over 2}=0$. Note that $f$ cannot be a $4$-face since $G$ has no $4$-cycles. Now assume $f$ is a $3$-face. By Lemma \[lem:bal2:edge\], $f$ is incident with at least two $7^+$-vertices, which must be pairwise adjacent to each other. If $f$ is incident with two $7^+$-vertices and the third vertex is a $4^+$-vertex, then $f$ is not a terrible face. Now $f$ receives either charge $1$ twice by rule (R3B) and charge $1$ once by rule (R2) or charge $1$ three times by rule (R3B). In either case, $\mu^(f)=3-6+3\cdot{1}=0$. Note that there are no $3$-vertices by Lemma \[lem:bal2:3-vx\]. If $f$ is incident with exactly two $7^+$-vertices, then the third vertex is a $2$-vertex, and $f$ is a terrible 3-face. Thus it receives charge ${3\over 2}$ twice by rule (R3A). Thus, $\mu^(f)=3-6+2\cdot{3\over 2}=0$. Each vertex has nonnegative final charge. Each neighbor of a $2$-vertex $v$ must be a $7^+$-vertex by Lemma \[lem:bal2:edge\]. Therefore $v$ receives charge $1$ twice by (R1). Thus, $\mu^(v)=2\cdot2-6+2\cdot1=0$. Note that there are no $3$-vertices by Lemma \[lem:bal2:3-vx\], and every vertex is incident with at most $\lfloor{d(v)\over 2}\rfloor$ $3$-faces since there are no $4$-cycles in $G$. If $v$ is a vertex with $d(v)\in\{4, 5, 6\}$, then $v$ sends charge $1$ at most $\lfloor{d(v)\over 2}\rfloor$ times by rule (R2). Thus, $\mu^(v)\geq 2d(v)-6-\lfloor{d(v)\over 2}\rfloor\geq 0$. Now assume $v$ is a $7^+$-vertex. We will show that $v$ has nonnegative final charge by a weighting argument. Let $u_1, \ldots, u_{d(v)}$ be the neighbors of $v$ in some cyclic order. First give all neighbors of $v$ a weight of $1$. If $u_i$ is not a $2$-vertex, then split the weight of $1$ it received from $v$, and transfer weight $1\over 2$ to each of the two faces that are incident with $vu_i$; if $vu_i$ is incident with only one face, then transfer the entire weight of $1$ to this face. Now, every neighbor of $v$ that is a $2$-vertex and every face incident with $v$ that is not a terrible 3-face have weight at least the charge that they should receive from $v$ by the discharging rules. Every terrible 3-face has weight at most $1$ short of the charge it should receive from $v$ by the discharging rules. Now give weight $1$ to each terrible 3-face incident with $v$. Since $v$ is incident with at most $d(v)-6$ terrible 3-faces by Lemma \[lem:bal2:terrible\], and each neighbor of $v$ received weight $1$ initially, the total weight spend is at most $2d(v)-6$, which is exactly the initial charge of $v$. Thus, the total weight sent is no more than the initial charge of $v$, which proves that the final charge of $v$ is nonnegative. Unbalanced $2$-partitions {#sec:unbal2} ========================= In this section, we prove the following theorem: \[thmm:unbal2\] [ A set $S$ of cycles is an inclusion-wise minimal cycle obstruction set of unbalanced $2$-partitionable planar graphs if and only if $S=\{C_3,C_4,C_6\}$ or $S$ is the set of all odd cycles. ]{} We will first show a necessary condition for cycle obstruction sets, and then show that it is sufficient afterwards. \[lem:unbal2:nec\] If a set $S$ of cycles is an obstruction set of unbalanced $2$-partitionable planar graphs, then either $\{C_3, C_4, C_6\} \subseteq S$ or $S$ contains all odd cycles. For a nonnegative integer $D$, a positive integer $l$, and a vertex $v$, recall that $H(D, l)$ from Section \[sec:bal2\] is not $(D, D)$-colorable and the only cycles in $H_1(D, l; v)$ have length either $4$ or $2l+1$. Therefore $H(D, l)$ is not $(0, D)$-colorable as well. Therefore $S$ contains either $C_4$ or all odd cycles. Given a nonnegative integer $D$ and two vertices $x$ and $y$, let $F_1(D; x, y)$ be the graph that consists of $2D+1$ internally disjoint $x, y$-paths of length $3$. See Figure \[fig:unbal2-tight\] for an illustration of $F_1(D;x,y)$. For an odd integer $l\geq 3$ and a vertex $v_1$, let $F_o(D, l; v_1)$ be the graph obtained from an odd cycle with vertices $v_1, \ldots, v_l$ by replacing each edge $v_iv_{i+1}$ with $F_1(D; v_i, v_{i+1})$ where $i$ is an odd integer at most $l$ (where $v_{l+1}$ is treated as $v_1$). Finally, obtain $F(D; l)$ from two disjoint copies of $F_o(D, l; v_1)$ and adding an edge between the two vertices that correspond to $v_1$. See Figure \[fig:unbal2-tight\] for an illustration of $F(D;5)$. Now in any $(0, D)$-coloring of $F_1(D; x, y)$, it is easy to see that $x$ and $y$ cannot both receive the color $2$. The two cutvertices of $F(D; l)$ cannot both receive the color $1$ in any $(0, D)$-coloring, thus at least one cutvertex $v$ receives the color $2$. In the copy that corresponds to $F_o(D, l; v)$, either there is an edge with both endpoints colored with the color $1$ or there is a copy of $F_1(D; x, y)$ where both $x$ and $y$ receive the color $2$. This shows that $F(D; l)$ is not $(0, D)$-colorable. It is not hard to see that the only cycles in $F(D; l)$ have length either $6$ or $2l+1$ where $l$ is an odd integer at least $3$. Therefore $S$ contains either $C_6$ or all cycles of lengths $4k+3$ where $k$ is a positive integer. For an odd integer $l\geq 3$ and a vertex $v_1$, let $F_e(D, l; v_1)$ be the graph obtained from an odd cycle with vertices $v_1, \ldots, v_l$ by replacing each edge $v_iv_{i+1}$ with $F_1(D; v_i, v_{i+1})$ where $i$ is an even integer at most $l$. Finally, let $F'(D; l)$ be the graph obtained from a star with $D+2$ vertices by attaching a copy of $F_e(D, l; v)$ to each vertex $v$ of the star. See Figure \[fig:unbal2-tight\] for an illustration of $F'(2;5)$. As above, $x$ and $y$ cannot both receive the color $2$ in any $(0,D)$-coloring of $F_1(D; x, y)$. This implies that every cutvertex of $F'(D; l)$ must be colored with color $2$ in a $(0, D)$-coloring. Yet, now there exists a cutvertex of that has $D+1$ neighbors colored with the color $2$, which implies that $F'(D; l)$ is not $(0, D)$-colorable. It is not hard to see that the cycles in $F'(D; l)$ have length either $6$ or $2l-1$ where $l$ is an odd integer at least $3$. Therefore $S$ contains either $C_6$ or all cycles of lengths $4k+1$ where $k$ is a positive integer. Let $T_0(D; x)$ be the graph obtained from $D+1$ pairwise disjoint $3$-cycles by identifying one vertex in each cycle into $x$. Now let $T(D)$ be the graph obtained from two copies of $T_0(D; x)$ and adding an edge between the two vertices corresponding to $x$. In any $(0, D)$-coloring of $T_0(D; x)$, the vertex $x$ must not receive color $2$ since it will have $D+1$ neighbors colored with $2$. Yet, in $T(D)$, one of the two cutvertices, which corresponds to $x$ in a copy of $T_0(D; x)$, will receive color $2$. This shows that $T(D)$ is not $(0, D)$-colorable, and it is easy to see that $T(D)$ contains only $3$-cycles. Hence $S$ contains $C_3$. To sum up, the obstruction set of unbalanced $2$-partitionable planar graphs must contain $C_3$, and contains either $\{C_4, C_6\}$ or all odd cycles of length at least five. In other words, $S$ contains either $\{C_3, C_4, C_6\}$ or all odd cycles. ![Graphs that are not $(0, D)$-colorable[]{data-label="fig:unbal2-tight"}](fig-unbal2-tight.pdf) If either $C_4$ or $C_6$ is not in an obstruction set $S$ of unbalanced $2$-partitionable planar graphs, then all odd cycles must be in $S$. This implies that the graph is bipartite and $(0, 0)$-colorable, and hence it is unbalanced $2$-partitionable. The following theorem shows that $\{C_3, C_4, C_6\}$ is an obstruction set of unbalanced $2$-partitionable planar graphs. Note that Lemma \[lem:unbal2:nec\] and Theorem \[thm:unbal2\] imply Theorem \[thmm:unbal2\]. \[thm:unbal2\] A planar graph with no $3$-, $4$-, $6$-cycles is $(0, 45)$-colorable. In this section, let $G$ be a counterexample to Theorem \[thm:unbal2\] with the minimum number of vertices. Also, fix a plane embedding of $G$. It is easy to see that $G$ is connected and has no $1$-vertices. From now on, given a (partially) $(0, 45)$-colored graph, we will let $a$ and $b$ be the two colors where $b$ is the color class allowed to have maximum degree at most $45$, and we say a vertex colored with $b$ is [saturated]{} if it already has 45 neighbors colored with $b$. Structural lemmas ----------------- \[lem:unbal2:vx-degree\] Any $46^-$-vertex is adjacent to a $47^+$-vertex. Suppose to the contrary that a $46^-$-vertex $v$ is adjacent to only $46^-$-vertices. Since $G-v$ is a graph with fewer vertices than $G$, there is a $(0, 45)$-coloring $\varphi$ of $G-v$; choose $\varphi$ that maximizes the number of neighbors of $v$ with the color $a$. At least one neighbor of $v$ has color $a$, since otherwise we can extend $\varphi$ to all of $G$ by coloring $v$ with color $a$. Also, every neighbor of $v$ colored $b$ has a neighbor in $G-v$ with the color $a$, otherwise it can be recolored by $a$ and violates the choice of $\varphi$. Since each neighbor $u$ of $v$ has at most $45$ neighbors in $G-v$, $u$ has at most $44$ neighbors with the color $b$ in $G-v$. So no neighbor of $v$ is saturated. Hence we can extend $\varphi$ to $G$ by coloring $v$ with color $b$. This contradicts that $G$ is a counterexample, and thus proves the claim. Since $G$ has no $3$-cycles and no $4$-cycles, every $5$-face is bounded by a cycle. A [bad face]{} is a $5$-face $f$ where the degrees of the vertices on a boundary walk is as in Figure \[fig:unbal2-bad\]. ![Bad faces[]{data-label="fig:unbal2-bad"}](fig-unbal2-bad.pdf) \[lem:unbal2:bad-faces\] Any $2$-vertex cannot be incident with two bad faces. Suppose to the contrary that a $2$-vertex $v$ is incident with two bad faces where $x, v,y,v_1,v_2$ and $x, v, y,u_1,u_2$ are vertices, in this order, of boundary walks of the two bad faces. If $v_1=u_2$ (or $v_2=u_1$), then $G$ contains a $3$-cycle $xv_1v_2$ (or $yv_1v_2$), which is a contradiction. If $v_1=u_1$ (or $v_2=u_2$), then $G$ has a $4$-cycle $v_1v_2xu_2$ (or $yv_1v_2u_1$), which is again a contradiction. Therefore, $\{v_1, v_2\}\cap\{u_1, u_2\}=\emptyset$, and this implies that $G$ contains a $6$-cycle with vertices $x, v_2, v_1, y, u_1, u_2$, which is a contradiction. \[lem:unbal2:bad-faces1\] Any $47^+$-vertex $v$ is incident with at most $\lfloor{d(v)\over 2}\rfloor$ bad faces. Suppose to the contrary that some $47^+$-vertex $v$ is incident with at least $\lfloor {d(v)\over2} \rfloor+1$ bad faces. Then some edge $e$ incident with $v$ is contained in two different bad faces. By the definition of bad faces, the end of $e$ other than $v$ has degree $2$. So this $2$-vertex is incident with two different bad faces, contradicting Lemma \[lem:unbal2:bad-faces\]. Discharging ----------- We now define the initial charge at each vertex and each face. For every $v\in V(G)$, let $\mu(v)=2d(v)-6$ and for every face $f\in F(G)$, let $\mu(f)=d(f)-6$. The total initial charge is negative since $$\begin{aligned} \sum_{z\in V(G)\cup F(G)} \mu(z) =\sum_{v\in V(G)} (2d(v)-6)+\sum_{f\in F(G)} (d(f)-6) =-6\|V(G)\|+6\|E(G)\|-6\|F(G)\| =-12 <0.\end{aligned}$$ The last equality holds by Euler’s formula. Recall that a bad face is a $5$-face and there are two non-adjacent $2$-vertices on that face. For each face $f$, let $W_f$ be a canonical boundary walk of $f$. Recall that for any face $f$ and vertex $v$, $k_{f,v}$ is the number of triples $(e,v,e')$ such that $e,e' \in E(G)$ and $eve'$ is a subwalk of $W_f$. Here are the discharging rules: 1. Let $v$ be a $47^+$-vertex. 1. $v$ sends charge $1$ to each adjacent vertex. 2. $v$ sends charge $1$ to each incident bad face. 3. $v$ sends charge $\frac{3}{4}k_{f,v}$ to each incident face $f$ that is not bad. 2. Let $v$ be a vertex where $d(v)\in\{3, \ldots, 46\}$. 1. $v$ sends charge $1\over 2$ to each adjacent $2$-vertex. 2. $v$ sends charge $t \over 2$ to each incident face $f$, where $t$ is the number of triples $(x,v,y)$ such that $x,y \in V(G)$, $xvy$ is a subpath in $W_f$, and either both $d(x),d(y)$ are at least $47$, or both $d(x),d(y) \in \{3, \ldots,46\}$. 3. $v$ sends charge $t \over 4$ to each incident face $f$, where $t$ is the number of triples $(x,v,y)$ such that $x,y \in V(G)$, $xvy$ is a subpath in $W_f$, $d(x)=2$ and $d(y) \in \{3, \ldots,46\}$. 3. Let $f$ be a face. 1. $f$ sends charge $\frac{1}{2} k_{f,v}$ to each incident $2$-vertex $v$ that is adjacent to another $2$-vertex. 2. $f$ sends charge $\frac{1}{4} k_{f,v}$ to each incident $2$-vertex $v$ that is adjacent to a vertex $y$ with $d(y)\in\{3, \ldots, 46\}$. The discharging rule (R1) shows how a $47^+$-vertex distributes its initial charge, (R2) shows how a vertex with degree in $\{3, \ldots, 46\}$ sends charge, and (R3) shows how a face sends its charge. Note that by Lemma \[lem:unbal2:vx-degree\], a face does not send charge to a $2$-vertex via both (R3A) and (R3B). See Figure \[fig:unbal2-rules\] for an illustration of the discharging rules. ![Discharging rules.[]{data-label="fig:unbal2-rules"}](fig-unbal2-rules.pdf) The rest of this section will prove that the final charge $\mu^(z)$ is nonnegative for each $z\in V(G)\cup F(G)$. Every vertex has nonnegative final charge. Assume $v$ is a $2$-vertex. If $v$ is adjacent to two $47^+$-vertices, then $v$ receives charge $1$ from each of its neighbors by (R1A). Thus, $\mu^(v)=-2+2\cdot{1}=0$. Note that $v$ is adjacent to at least one $47^+$-vertex by Lemma \[lem:unbal2:vx-degree\]. If $v$ is adjacent to another $2$-vertex, then $v$ receives charge $2 \cdot \frac{1}{2}$ from the faces incident with $v$ by (R3A). Thus, $\mu^(v)=-2+2\cdot{1\over 2}+1=0$. Otherwise, $v$ is adjacent to a vertex of degree from $3$ to $46$, which sends charge $1\over 2$ to $v$ by (R2A). Also, $v$ receives charge $2 \cdot \frac{1}{4}$ from the faces incident with $v$ by (R3B). Thus, $\mu^(v)=-2+2\cdot{1\over 4}+{1\over 2}+1=0$. Assume $v$ is a $47^+$-vertex. By (R1A), $v$ sends charge at most $d(v)$ to its adjacent vertices in total. By Lemma \[lem:unbal2:bad-faces1\], $v$ is incident with at most $\lfloor{d(v)\over 2}\rfloor$ bad faces. Since $v$ sends charge $1$ to each of its incident bad faces by (R1B) and sends charge $3\over 4$ to each of its incident faces that are not bad by (R1C), the final charge $\mu^(v)$ is at least $2d(v)-6-d(v)-\lfloor{d(v)\over 2}\rfloor-{3\over4}\cdot \lceil{d(v)\over 2}\rceil$, which is nonnegative since $d(v)\geq 47$. Assume $d(v)\in\{4, \ldots, 46\}$. We will show that $v$ has nonnegative final charge by using a weighting argument. Let $u_1, \ldots, u_{d(v)}$ be the neighbors of $v$ in some cyclic order. First give all neighbors of $v$ a weight of $1\over 2$. If $u_i$ is not a $2$-vertex, then split the weight of $1\over 2$ it received from $v$, and transfer weight $1\over 4$ to each of the two faces that are incident with $vu_i$ (if $vu_i$ is incident with only one face, then transfer weight $1 \over 2$ to this face). Now, every $2$-vertex adjacent to $v$ and every face that is incident with $v$ have weight equal to the charge sent from $v$ in the discharging rules. So the total charge sent from $v$ is at most the weight sent from $v$. Since $v$ has charge $2d(v)-6\geq {d(v)\over 2}$ when $d(v)\geq 4$, $v$ has nonnegative final charge. Assume $v$ is a $3$-vertex. If $v$ is adjacent to at least two $47^+$-vertices, which each sends charge $1$ to $v$ by (R1A), then $v$ is adjacent to at most one $2$-vertex. Thus, $\mu^(v)\geq 0+2-4\cdot{1\over 2}=0$. If $v$ is adjacent to exactly one $47^+$-vertex, then $v$ sends charge at most $1\over 2$ to at most twice according to the discharging rules. In either case, $\mu^(v)\geq 0+1-2\cdot{1\over 2}=0$. Each $7^+$-face $f$ has nonnegative final charge. We will show that $f$ has nonnegative final charge by using a weighting argument. Pull weight $\frac{3}{4}k_{f,v}$ from each $47^+$-vertex $v$ on $f$ (note that this corresponds to (R1C)), and transfer weight $\frac{3}{8}k_{f,v}$ to each $2$-vertex on $f$ that is adjacent to $v$. Each $2$-vertex on $f$ receives weight at least $\frac{3}{8}k_{f,v}$, since it must be adjacent to a $47^+$-vertex, which is on $f$, by Lemma \[lem:unbal2:vx-degree\]. Now if $f$ sends an additional weight of $\frac{1}{8}k_{f,v}$ to each $2$-vertex on $f$, then (R3) is satisfied. By Lemma \[lem:unbal2:vx-degree\], there cannot be three consecutive $2$-vertices on a boundary walk of $f$, so it follows that $\sum k_{f,v} \leq \lfloor \frac{2}{3}d(f) \rfloor$, where the sum is over all 2-vertices incident with $f$. Therefore, $\mu^(f) \geq d(f)-6-\frac{1}{8} \sum k_{f,v} \geq d(f)-6-\frac{1}{8} \lfloor \frac{2}{3}d(f) \rfloor>0$ when $d(f) \geq 7$, where the sum is over all 2-vertices incident with $f$. Note that there is no $6$-face since $G$ has no $1$-vertex and no $3$-, $4$-, $6$-cycles. Each $5$-face $f$ has nonnegative final charge. Since $G$ has no $1$-vertex, every 5-face is bounded by a cycle. Let $v_1, v_2, v_3, v_4, v_5$ be the vertices of $f$ in some cyclic order. Assume $f$ is incident with at most one $2$-vertex, and assume $v_1$ is the $2$-vertex, if any. Note that $f$ sends charge $1\over 4$ to $v_1$ by (R3B) if it is a $2$-vertex. If at least two of $v_2, \ldots, v_5$ are $47^+$-vertices, then $f$ receives charge $3\over 4$ from each one by (R1C), thus, $\mu^(f)\geq -1+2\cdot{3\over 4}-{1\over 4}>0$. If exactly one of $v_2, \ldots, v_5$ is a $47^+$-vertex, then without loss of generality we may assume it is $v_2$ by Lemma \[lem:unbal2:vx-degree\]. Now $v_4$ sends charge $1\over 2$ to $f$ by (R2B), thus, $\mu^(f)\geq -1+{3\over 4}+{1\over 2}-{1\over 4}=0$. If none of $v_2, \ldots, v_5$ is a $47^+$-vertex, then $v_1$ cannot be a $2$-vertex. Since both $v_3$ and $v_4$ send charge $1\over 2$ by (R2B), it follows that $\mu^(f)\geq -1+2\cdot{1\over 2}=0$. Assume $f$ is incident with at least two $2$-vertices where two of them, say $v_2$ and $v_3$, are adjacent to each other. Note that $f$ sends charge $1\over 2$ to each of $v_2$ and $v_3$ by (R3A). By Lemma \[lem:unbal2:vx-degree\], both $v_1$ and $v_4$ must be $47^+$-vertices. If $v_5$ is not a $2$-vertex, then $f$ is not a bad face, and $v_1, v_4, v_5$ send charge $3\over 4$, $3\over 4$, at least $1\over 2$, respectively, by (R1C) and (R2B). Thus, $\mu^(f)\geq -1+2\cdot{3\over 4}+{1\over 2}-2\cdot{1\over 2}=0$. If $v_5$ is a $2$-vertex, then $f$ is a bad face, and both $v_1$ and $v_4$ send charge $1$ each to $f$ by (R1B). Thus, $\mu^(f)\geq -1+2\cdot{1}-2\cdot{1\over 2}=0$. If $f$ is incident with at least two $2$-vertices and where no pair is nonadjacent, then $f$ is incident to exactly two $2$-vertices by Lemma \[lem:unbal2:vx-degree\]. Thus, the only remaining case is when $f$ is incident with exactly two nonadjacent $2$-vertices, say $v_1$ and $v_3$. Note that $f$ sends charge $1\over 4$ to each of $v_1$ and $v_3$ by (R3B). If $f$ is incident with at least two $47^+$-vertices, which each sends charge at least $3\over 4$ to $f$ by (R1), then $\mu^(f)\geq -1+2\cdot{3\over 4}-2\cdot{1\over 4}=0$. Now $f$ must be incident with exactly one $47^+$-vertex because $f$ is incident with a $2$-vertex, and by Lemma \[lem:unbal2:vx-degree\] we know that $v_2$ must be the $47^+$-vertex. It follows that $d(v_4), d(v_5)\in\{3, \ldots, 46\}$, and therefore $f$ is a bad face. Now, $v_2, v_4, v_5$ send charge $1, {1\over 4}, {1\over 4}$, respectively, to $f$ by (R1B) and (R2C). Thus, $\mu^(f)\geq -1+1+2\cdot{1\over 4}-2\cdot{1\over 4}=0$. Unbalanced $3$-partitions {#sec:unbal3} ========================= In this section, we prove the following theorem: \[thmm:unbal3\] [ A set $S$ of cycles is an inclusion-wise minimal cycle obstruction set of unbalanced $3$-partitionable planar graphs if and only if $S=\{C_3\}$ or $S=\{C_4\}$. ]{} We will first show a necessary condition for cycle obstruction sets, and then show that it is sufficient afterwards. \[lem:unbal3:nec\] If a set $S$ of cycles is an obstruction set of unbalanced $3$-partitionable planar graphs, then either $C_3\in S$ or $C_4\in S$. Let $X_0(D; v)$ be the graph that is obtained from starting with $D+1$ pairwise disjoint copies of $K_4$ and picking one vertex from each copy of $K_4$ and identifying them into $v$. Now let $X(D)$ be the graph obtained from three copies of $X_0(D; v)$ and adding three edges between the three vertices that correspond to $v$. See Figure \[fig:unbal3-tight\] for an illustration of $X_0(2; v)$ and $X(2)$. Now in any $(0, 0, D)$-coloring of $X_0(D; v)$, the vertex $v$ cannot receive the color $3$. This is because each copy of $K_4-v$ must contain a vertex colored with $3$, and since there are $D+1$ copies, $v$ has $D+1$ neighbors with the same color, which is a contradiction. However, in any $(0, 0, D)$-coloring of $X(D)$, one vertex $v$ of the three cutvertices must receive the color $3$, and this shows that $X(D)$ is not $(0, 0, D)$-colorable. It is not hard to see that the only cycles in $X(D)$ have length either $3$ or $4$. ![Graphs that are not $(0, 0, D)$-colorable[]{data-label="fig:unbal3-tight"}](fig-unbal3-tight.pdf) If a planar graph does not contain $3$-cycles, then it is $3$-colorable, which is equivalent to $(0, 0, 0)$-colorable, by Grötzsch’s Theorem [@1959Gr], and thus it is unbalanced $3$-partitionable. This means that $\{C_3\}$ is an inclusion-wise minimal obstruction set of unbalanced 3-partitionable planar graphs. The remaining of this section proves Theorem \[thm:unbal3\] below, which states that planar graphs with no $4$-cycles are unbalanced $3$-partitionable. Note that Lemma \[lem:unbal3:nec\] and Theorem \[thm:unbal3\] imply Theorem \[thmm:unbal3\]. \[thm:unbal3\] Any planar graph with no $4$-cycles is $(0, 0, 117)$-colorable. In this section, let $G$ be a counterexample to Theorem \[thm:unbal3\] with the minimum number of vertices. Also, fix a plane embedding of $G$. It is easy to see that $G$ is connected and there are no $2^-$-vertices in $G$. From now on, given a (partially) $(0, 0, 117)$-colored graph, we will let $a$, $b$, $c$ be the color of the color class that is allowed to have maximum degree at most $0$, $0$, $117$, respectively, and we say a vertex colored with $c$ is [saturated]{} if it already has $117$ neighbors colored with $c$. Structural lemmas ----------------- \[lem:unbal3:vx-degree\] A $119^-$-vertex is adjacent to a $120^+$-vertex. Suppose to the contrary that a $119^-$-vertex $v$ is adjacent to only $119^-$-vertices. Since $G-v$ is a graph with fewer vertices than $G$, there is a $(0, 0, 117)$-coloring $\varphi$ of $G-v$. We further assume that $\varphi$ minimizes the number of neighbors of $v$ colored with $c$. If there exists a neighbor $u$ of $v$ in $G$ such that $\varphi(u)=c$ and $u$ is saturated, then at most one neighbor of $u$ in $G-v$ has a color in $\{a, b\}$, so we can recolor $u$ to be a color in $\{a,b\}$ that does not appear in its neighborhood in $G-v$, contradicting the minimality of $\varphi$. Hence no neighbor $u$ of $v$ with color $c$ is saturated. If no neighbor of $v$ is colored with a color in $\{a, b\}$, then we can extend $\varphi$ to all of $G$ by coloring $v$ with a color in $\{a,b\}$ that does not appear in the neighborhood of $v$ in $G$, contradicting that $G$ is a counterexample. So both $a$ and $b$ appear in the neighborhood of $v$ in $G$, and thus there are at most $117$ neighbors of $v$ colored with $c$. Since no neighbor of $v$ with color $c$ is saturated, we can extend $\varphi$ to all of $G$ by coloring $v$ with color $c$, a contradiction. \[lem:unbal3:ext-coloring2\] Let $X$ be a set of $3$-vertices of $G$ such that the subgraph of $G$ induced on $X$ is a path $v_1v_2\ldots v_{k}$ where $k\geq 2$. If $x$ and $y$ are the neighbors of $v_k$ in $G-X$, then $c \in \{\varphi(x),\varphi(y)\}$ and $\varphi(x) \neq \varphi(y)$ for every $(0,0,117)$-coloring $\varphi$ of $G-X$. Moreover, the vertex in $\{x, y\}$ that receives the color $c$ must be a $116^+$-vertex. Let $\varphi$ be a $(0,0,117)$-coloring of $G-X$ and let $x'$ and $y'$ be the neighbors of $v_1$ in $G-X$. For each integer $i$ with $2 \leq i \leq k-1$, let $u_i$ be the vertex in $G-X$ adjacent to $v_i$. First we extend $\varphi$ to a $(0,0,117)$-coloring of $G-v_k$ by defining $\varphi(v_1) \in \{a,b,c\}\setminus\{\varphi(x'),\varphi(y')\}$ and $\varphi(v_i) \in \{a,b,c\}-\{\varphi(v_{i-1}),\varphi(u_i)\}$ for each $i\in\{2, \ldots, k-1\}$. If $\varphi(x)=\varphi(y)$, then we can extend $\varphi$ to be a $(0,0,117)$-coloring of $G$ by further defining $\varphi(v_k)$ to be an element in $\{a,b,c\}\setminus\{\varphi(v_{k-1}),\varphi(x)\}$. This proves $\varphi(x) \neq \varphi(y)$. If either $c \not \in \{\varphi(x),\varphi(y)\}$ or the vertex in $\{x, y\}$ with the color $c$ is a $115^-$-vertex, then by defining $\varphi(v_k)=c$, we extended $\varphi$ to a $(0,0,117)$-coloring of $G$ since the degree of $v_{k-1}$ is $3$. Therefore, $c \in \{\varphi(x),\varphi(y)\}$. \[lem:unbal3:ext-coloring3\] Let $X$ be a set of $3$-vertices of $G$ such that the subgraph of $G$ induced on $X$ is a path $v_1v_2\ldots v_{2k}$ on an even number of vertices. Let $u_i$ be a neighbor of $v_i$ in $G-X$ for each $i$ with $1 \leq i \leq 2k$. Let $x$ and $y$ be the neighbor of $v_1$ and $v_{2k}$, respectively in $G-X$ other than $u_1$ and $u_{2k}$. If there exists a $(0,0,117)$-coloring $\varphi$ of $G-X$ such that $\varphi(u_i)=c$ for every $i$ with $1 \leq i \leq 2k$, then $\varphi(x)=\varphi(y)$. By Lemma \[lem:unbal3:ext-coloring2\], we may assume $\varphi(x) \neq \varphi(u_1)=c$. Define $\varphi(v_1)=\{a,b\}\setminus\{\varphi(x)\}$ and $\varphi(v_i)=\{a,b\}\setminus\{\varphi(v_{i-1})\}$ for every $2 \leq i \leq 2k$. Since $\lvert X \rvert$ is even, $\varphi(v_1) \neq \varphi(v_{2k})$. That is, $\varphi(v_{2k})=\varphi(x)$. As this must not extend $\varphi$ to be a $(0,0,117)$-coloring of $G$, $\varphi(y)=\varphi(v_{2k})$. Therefore $\varphi(y)=\varphi(x)$. A face $f$ is [annoying]{} if exactly one vertex incident with $f$ is a $120^+$-vertex and all other vertices incident with $f$ are $3$-vertices. We say that two faces are [adjacent]{} if they share at least one edge. \[lem:unbal3:annoying\] If an annoying $5$-face $f$ is adjacent to only annoying $3$-faces and annoying $5$-faces, then $f$ is adjacent to at most two $3$-faces. Let $f=wx'xyy'$ where $w$ is the $120^+$-vertex on $f$ and $x',x,y,y'$ are all $3$-vertices. For $e\in\{wx', x'x, xy, yy', y'w\}$, let $f_e$ be the face incident with $e$ other than $f$. Suppose to the contrary that $f$ is adjacent to three annoying $3$-faces. Since $G$ has no $4$-cycles, two $3$-faces cannot share an edge, and two faces incident with the same $3$-vertex must share an edge. Since $x', x, y, y'$ are all $3$-vertices, this implies that $f_{xy}, f_{wx'}, f_{y'w}$ must be the annoying $3$-faces adjacent to $f$. Let $f_{xy}=xyz$ so that $z$ is a $120^+$-vertex and the common neighbor of $x$ and $y$. Also, let $f_{wx'}=wx'x$ and $f_{y'w}=wy'y_1$ so that $x_1$ and $y_1$ is the common neighbor of $w, x'$ and $w,y'$, respectively, which must be a $3$-vertex. Note that $z, x_1, y_1$ must be all distinct since otherwise that would imply the existence of a $4$-cycle. Let $z_1$ be the common neighbor of $z$ and $x_1$. Since $f_{xx'}$ is an annoying $5$-face, $z_1$ must be a $3$-vertex. Also, $z_1 \not \in \{z, x, y, x', y', x_1, y_1, w\}$ since there are no $4$-cycles. Let $z_2$ be the common neighbor of $z$ and $y_1$. Similarly, $z_2$ is a $3$-vertex and $z_2\not\in\{z, x, y, x', y', x_1, y_1, w\}$. Note that $z_1 \neq z_2$, since $z$ has degree at least $120$, $f_{xx'}$ and $f_{yy'}$ are 5-faces and $xyz$ is a 3-face. Note that the subgraph of $G$ induced on $\{x_1,x',x,y,y',y_1\}$ is a path. See Figure \[fig:unbal3:annoying\] for an illustration. Suppose that $z_1z_2$ is not an edge of $G$. Set $H=(G-\{x, y, x', y'\})\cup x_1y_1$. Note that $H$ is still a plane graph with no $4$-cycles, since $z_1z_2$ is not an edge. Since $H$ is a graph with fewer vertices than $G$, there is a $(0, 0, 117)$-coloring $\varphi$ of $H$. Note that $\varphi$ is a $(0,0,117)$-coloring of $G-\{x,y,x',y'\}$. If $\varphi(z) \neq c$, then let $$\begin{aligned} \varphi(x)&=c& \varphi(x')&\in\{a, b, c\}\setminus\{\varphi(x_1), \varphi(w)\}\\ \varphi(y')&\in\{a, b, c\}\setminus\{\varphi(w), \varphi(y_1)\}& \varphi(y)&\in\{a, b, c\}\setminus\{\varphi(z), \varphi(y')\}\end{aligned}$$ to extend $\varphi$ to all of $G$. Hence $\varphi(z)=c$. Since $w$ is a $120^+$-vertex, by Lemma \[lem:unbal3:ext-coloring2\], $\varphi(w)=c$ and $\{\varphi(x_1),\varphi(y_1)\} \subseteq \{a,b\}$. Since $\varphi(z)=\varphi(w)=c$, Lemma \[lem:unbal3:ext-coloring3\] implies that $\varphi(x_1)=\varphi(y_1)$. However, $\{\varphi(x_1),\varphi(y_1)\} \subseteq \{a,b\}$ and $x_1y_1$ is an edge of $H$, so $\varphi(x_1) \neq \varphi(y_1)$, a contradiction. Therefore, $z_1z_2$ is an edge of $G$. Since $G-\{x, y, x', y', x_1, y_1\}$ is a graph with fewer vertices than $G$, there exists a $(0, 0, 117)$-coloring $\varphi$ for $G-\{x, y, x', y', x_1, y_1\}$. If $\varphi(z) \neq c$, then let $$\begin{aligned} \varphi(x)&=c& \varphi(x_1)&\in\{a, b, c\}\setminus\{\varphi(z_1), \varphi(w)\}\\ \varphi(x')&\in\{a, b, c\}\setminus\{\varphi(x_1), \varphi(w)\}& \varphi(y_1)&\in\{a, b, c\}\setminus\{\varphi(z_2), \varphi(w)\}\\ \varphi(y')&\in\{a, b, c\}\setminus\{\varphi(w), \varphi(y_1)\}& \varphi(y)&\in\{a, b, c\}\setminus\{\varphi(z), \varphi(y')\}\end{aligned}$$ to extend $\varphi$ to all of $G$, which is a contradiction. Hence $\varphi(z)=c$. Since $w$ is a $120^+$-vertex, by Lemma \[lem:unbal3:ext-coloring2\], $\varphi(w)=c$ and $\{\varphi(z_1),\varphi(z_2)\} \subseteq \{a,b\}$. Since $\varphi(z)=\varphi(w)=c$, Lemma \[lem:unbal3:ext-coloring3\] implies that $\varphi(z_1)=\varphi(z_2)$. However, $\{\varphi(z_1),\varphi(z_2)\} \subseteq \{a,b\}$ and $z_1z_2$ is an edge of $G-\{x, y, x', y', x_1, y_1\}$, so $\varphi(z_1) \neq \varphi(z_2)$, a contradiction. ![Figure for Lemma \[lem:unbal3:annoying\][]{data-label="fig:unbal3:annoying"}](fig-unbal3-annoying.pdf) Discharging ----------- We now define the initial charge at each vertex and each face. For every $z\in V(G)\cup F(G)$, let $\mu(z)=d(z)-4$. The total initial charge is negative since $$\begin{aligned} \sum_{z\in V(G)\cup F(G)} \mu(z) =\sum_{v\in V(G)} (d(v)-4)+\sum_{f\in F(G)} (d(f)-4) =-4\|V(G)\|+4\|E(G)\|-4\|F(G)\| =-8 <0.\end{aligned}$$ The last equality holds by Euler’s formula. Here are the discharging rules: 1. Each $5^+$-face $f$ sends charge $\frac{k_{f,v}}{r}(d(f)-4)$ to each incident 3-vertex $v$, where $r=\sum k_{f,u}$ and the sum is over all 3-vertices $u$ incident with $f$. 2. Let $v$ be a $120^+$-vertex. 1. $v$ sends charge $2\over 3$ to each neighbor. 2. $v$ sends charge $3\over 5$ to each incident $3$-face. 3. Each vertex $v$ where $d(v)\in\{4, \ldots, 119\}$ sends charge $1\over 3$ to each incident $3$-face. 4. Each $3$-vertex that is not incident with a $3$-face sends charge $1\over 15$ to each adjacent $3$-vertex. The discharging rule (R1) shows how a face distributes its initial charge, (R2) shows how a $120^+$-vertex sends charge, (R3) shows how a vertex with degree in $\{4, \ldots, 119\}$ sends charge, and (R4) shows how a $3$-vertex that is not incident with a $3$-face sends charge to an adjacent $3$-vertex. See Figure \[fig:unbal3-rules\] for an illustration of the discharging rules. ![Discharging rules[]{data-label="fig:unbal3-rules"}](fig-unbal3-rules.pdf) The rest of this section will prove that the sum of the final charge $\mu^(z)$ is nonnegative for $z\in V(G)\cup F(G)$. Note that every $5^+$-face has nonnegative final charge since it only distributes its initial charge, which is positive. There are no $4$-faces since there are no $4$-cycles, and each edge is incident with at most one $3$-face since there are no $4$-cycles. We will first show that each $4^+$-vertex has nonnegative final charge. Then, instead of counting $3$-vertices and $3$-faces separately, we will compute the final charge of $3$-faces and $3$-vertices together. Every $4^+$-vertex $v$ has nonnegative final charge. Note that $v$ is incident with at most $\lfloor{d(v)\over 2}\rfloor$ $3$-faces since there are no $4$-cycles. If $v$ is a $119^-$-vertex, then by Lemma \[lem:unbal3:vx-degree\], $v$ has a neighbor $u$ that is a $120^+$-vertex. By (R2A), $u$ sends charge $2\over 3$ to $v$, and by (R3), $v$ sends charge at most ${1\over 3}\cdot\lfloor{d(v)\over2}\rfloor$ to its incident $3$-faces. Thus, $\mu^(v)\geq d(v)-4+{2\over 3}-{1\over 3}\cdot\lfloor{d(v)\over2}\rfloor\geq 0$ when $d(v)\geq 4$. Now assume $v$ is a $120^+$-vertex. Then $v$ sends charge at most $2d(v)\over 3$ to its neighbors by (R2A) and $v$ sends charge at most ${3\over 5}\cdot{\lfloor{d(v)\over 2}\rfloor}$ to its incident $3$-faces by (R2B). Thus, $\mu^(v)\geq d(v)-4-{2d(v)\over 3}-{3\over 5}\cdot{\lfloor{d(v)\over 2}\rfloor}\geq 0$ when $d(v)\geq 120$. Note that a $6^+$-face and $5^+$-face sends charge at least $1\over 3$ and at least $1\over 5$, respectively, to each incident $3$-vertex. In particular, a $5$-face that is incident with at least one and at least two $4^+$-vertices sends charge at least $1\over 4$ and at least $1\over 3$, respectively, to each incident $3$-vertex. \[claim:unbal3:3x\] Each $3$-vertex $v$ that is not incident with a $3$-face has positive final charge. By Lemma \[lem:unbal3:vx-degree\], $v$ has a $120^+$-vertex $u$ as a neighbor. The faces incident with $v$ sends charge at least $3 \cdot \frac{1}{5}$ to $v$ by (R1) and $u$ sends charge $2\over 3$ to $v$ by (R2). Also $v$ loses charge $1\over 15$ at most twice by (R4). Thus, $\mu^(v)\geq -1+{3\over 5}+{2\over 3}-{2\over 15}>0$. \[claim:unbal3:33\] If $f$ is a $3$-face that is incident with three $3$-vertices $x, y, z$, then the sum of the final charge of $f, x, y, z$ is nonnegative. Let $x', y', z'$ be the neighbor of $x, y, z$, respectively, that is not on $f$. Since there are no $4$-cycles, $x',y',z'$ are pairwise distinct. By Lemma \[lem:unbal3:vx-degree\], $x', y', z'$ are all $120^+$-vertices. Since $x,y,z$ are 3-vertices, $xx',yy',zz'$ are not contained in $3$-faces. Therefore each face that is adjacent to $f$ is incident with at least two $120^+$-vertices. Thus, each of $x, y, z$ receives charge at least $2\over 3$ from the incident faces by (R1). Now $x', y', z'$ sends charge ${2\over 3}$ to $x, y, z$, respectively, by (R2A). Thus, $\mu^(f)+\mu^(x)+\mu^(y)+\mu^(z)\geq -4+3\cdot{2\over 3}+3\cdot{2\over 3}=0$. \[claim:unbal3:32\] If $f$ is a $3$-face $xyz$ that is incident with exactly two $3$-vertices $x$ and $y$, then the sum of the final charge of $f, x, y$ is nonnegative. Let $x'$ and $y'$ be the neighbor of $x$ and $y$, respectively, that is not on $f$. Note that $x'$ and $y'$ are distinct, and $xx'$ and $yy'$ are not contained in any $3$-faces since $x$ and $y$ are 3-vertices and $G$ has no $4$-cycles. Assume $z$ is not a $120^+$-vertex. This implies that $x'$ and $y'$ are both $120^+$-vertices by Lemma \[lem:unbal3:vx-degree\]. Therefore each face that is adjacent to $f$ is incident with at least two $4^+$-vertices. Thus, each of $x$ and $y$ receives charge at least $2\over 3$ from the incident faces by (R1). Now $x'$ and $y'$ sends charge ${2\over 3}$ to $x$ and $y$, respectively, by (R2A). Also $z$ sends charge $1\over 3$ to $f$ by (R3). Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+2\cdot{2\over 3}+2\cdot{2\over 3}+{1\over 3}=0$. Assume $z$ is a $120^+$-vertex. This implies that $x, y, f$ receives charge $2\over 3$, $2\over 3$, at least $3\over 5$, from $z$ by (R2A), (R2A), (R2B), respectively; note that the sum of these charge is ${29\over 15}$. Let $f_{xy}, f_{zx}, f_{zy}$ be the face incident with $xy, zx, zy$, respectively, that is not $f$. It is possible that $f_{xy},f_{zy}$, and $f_{zx}$ are not pairwise distinct. Assume that one of $x',y'$ is a $4^+$-vertex. Without loss of generality, we may assume that $x'$ is a $4^+$-vertex. By (R1), $f_{zx}$ and $f_{xy}$ gives charge at least ${1\over 3}$ and at least ${1\over 2}$ to $x$ and $x, y$, respectively. Also, $f_{zy}$ gives charge ${1\over 4}$ to $y$ by (R1). Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+{29\over 15}+{1\over 3}+{1\over 2}+{1\over 4}>0$. So we may assume both $x'$ and $y'$ are $3$-vertices. By Lemma \[lem:unbal3:vx-degree\], $x'$ and $y'$ must have a neighbor $x''$ and $y''$, respectively, that is a $120^+$-vertex. Note that $x''=y''$ is possible. If none of $x''$ and $y''$ is incident with $f_{xy}$, then $f_{xy}$ sends charge at least ${2\over 5}$ to $x$ and $y$ by (R1), and each $f_{zx}$ and $f_{zy}$ sends charge at least ${1\over 3}$ to $x$ and $y$, respectively, by (R1). Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+{29\over 15}+{2\over 5}+2\cdot{1\over 3}=0$. If exactly one of $x''$ and $y''$ is incident with $f_{xy}$, then without loss of generality, we may assume $x''$ is incident with $f_{zx}$ and $y''$ is incident with $f_{xy}$. Now, by (R1), $f_{xy}$ sends charge at least $1\over 4$ to each of $x$ and $y$, and $f_{zx}$ and $f_{zy}$ sends charge at least $1\over 3$ and at least $1\over 4$ to $x$ and $y$, respectively. Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+{29\over 15}+{1\over 3}+3\cdot{1\over 4}>0$. Assume both $x''$ and $y''$ are incident with $f_{xy}$. If $x''\neq y''$, then $d(f_{xy}) \geq 6$ and $f_{xy}$ sends charge at least ${1\over 3}$ to each of $x$ and $y$ by (R1), and $f_{zx}$ and $f_{zy}$ sends charge at least ${1\over 4}$ to $x$ and $y$, respectively, by (R1). Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+{29\over 15}+{2\over 3}+2\cdot{1\over 4}>0$. Now consider the case when $x''=y''$, so $f_{xy}$ sends charge $1\over 4$ to each of $x$ and $y$ by (R1). If one of $f_{zy}$ and $f_{zx}$ is either a $6^+$-face or a $5$-face that is not annoying, then it sends charge at least $1\over 3$ to $y$ or $x$ by (R1) and the other face still sends charge to $x$ or $y$ at least $1\over 4$ by (R1). Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+{29\over 15}+{1\over 3}+3\cdot{1\over 4}>0$. So assume each of $f_{zx}$ and $f_{zy}$ is an annoying $5$-face, which sends charge $1\over 4$ by (R1). In particular, $x'$ and $y'$ have degree 3. Therefore, $x$ and $y$ receive a total of charge $1$ by the surrounding faces. If $f'$ is a 3-face incident with $x'$, then it is incident with $x',x''$, and a vertex on $f_{zx}$ other than $z$. Since $f_{zx}$ is an annoying 5-face, $f'$ is an annoying 3-face. So every 3-face incident with $x'$ is annoying. Similarly, every 3-face incident with $y'$ is annoying. Since $xyy'x''x'$ is an annoying 5-face and $xyz$ is an annoying 3-face, either one of $x',y'$ is not incident with any 3-face, or some 3-face incident with both $x',y'$, by Lemma \[lem:unbal3:annoying\]. The later implies that $x'$ is adjacent to $y'$, which is a contradiction since $x'y'yx$ is now a $4$-cycle. Hence one of $x',y'$ is not incident with any 3-face, and that vertex sends charge $1\over 15$ to either $x$ or $y$ by (R4). Thus, $\mu^(f)+\mu^(x)+\mu^(y)\geq -3+{29\over 15}+1+{1\over 15}= 0$. \[claim:unbal3:31\] If $f$ is a $3$-face $xyz$ that is incident with exactly one $3$-vertex $x$, then the sum of the final charge of $f$ and $x$ is nonnegative. By Lemma \[lem:unbal3:vx-degree\], $x$ has a neighbor $x'$ that is a $120^+$-vertex. Assume $x'\not\in\{y, z\}$. The sum of charge received from the faces incident with $x$ is at least $2 \cdot \frac{1}{3}$ by (R1). Also, $x'$ sends charge $2\over 3$ to $x$ by (R2A). Each of $y$ and $z$ sends charge at least $1\over 3$ to $f$ by either (R2B) or (R3). Thus, $\mu^(f)+\mu^(x)\geq -2+4\cdot{1\over 3}+{2\over 3}= 0$. So we may assume $x' \in \{y,z\}$. Without loss of generality, assume $x'=y$. The sum of charge received from the faces incident with $x$ is at least $2 \cdot \frac{1}{4}$ by (R1). Now $x'$ sends charge $2\over 3$ and $3\over 5$ to $x$ and $f$ by (R2A) and (R2B), respectively. Also, $z$ sends charge at least $1\over 3$ to $f$ by either (R2B) or (R3). Thus, $\mu^(f)+\mu^(x)\geq -2+2\cdot{1\over 4}+{2\over 3}+{3\over 5}+{1\over 3}>0$. \[claim:unbal3:30\] If $f$ is a $3$-face $xyz$ that is incident with no $3$-vertices, then the final charge of $f$ is nonnegative. Since each of $x, y, z$ is a $4^+$-vertex, each of $x, y, z$ sends charge at least $1\over 3$ to $f$ by either (R2B) or (R3). Thus, $\mu^(f)\geq -1+3\cdot{1\over 3}=0$. Since no $3$-vertex is contained in two different $3$-faces, the sum of the final charge on all $3$-faces and all $3$-vertices is nonnegative by Claims \[claim:unbal3:3x\], \[claim:unbal3:33\], \[claim:unbal3:32\], \[claim:unbal3:31\], and \[claim:unbal3:30\]. Acknowledgment {#acknowledgment .unnumbered} ============== We thank the referees for carefully reading the manuscript and helpful suggestions. [10]{} K. Appel and W. Haken. Every planar map is four colorable. [I]{}. [D]{}ischarging. , 21(3):429–490, 1977. K. Appel, W. Haken, and J. Koch. Every planar map is four colorable. [II]{}. [R]{}educibility. , 21(3):491–567, 1977. O. V. Borodin, A. O. Ivanova, M. Montassier, P. Ochem, and A. Raspaud. 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[^1]: Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2015R1C1A1A02036398). This work was supported by Hankuk University of Foreign Studies Research Fund. Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do, Republic of Korea `ilkyoo@hufs.ac.kr` [^2]: Partially supported by the National Science Foundation under Grant No. DMS-1664593. Department of Mathematics, Princeton University, Princeton, NJ, USA. `chliu@math.princeton.edu` [^3]: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2017R1A2B4005020). Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea. School of Mathematics, KIAS, Seoul, Republic of Korea. `sangil@kaist.edu`
--- author: - \| L.A. Bokut[^1]\ [ School of Mathematical Sciences, South China Normal University]{}\ [Guangzhou 510631, P.R. China]{}\ [Sobolev Institute of Mathematics, Russian Academy of Sciences]{}\ [Siberian Branch, Novosibirsk 630090, Russia]{}\ [bokut@math.nsc.ru]{}\ \ Yuqun Chen[^2] and Qiuhui Mo\ [ School of Mathematical Sciences, South China Normal University]{}\ [Guangzhou 510631, P.R. China]{}\ [yqchen@scnu.edu.cn]{}\ [scnuhuashimomo@126.com]{} title: 'Gröbner-Shirshov bases and embeddings of algebras[^3] ' --- Abstract: In this paper, by using Gröbner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. We show that in the following classes, each countably generated algebra over a countable field $k$ can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. Also we prove that any countably generated module over a free associative algebra $k\langle X \rangle$ can be embedded into a cyclic $k\langle X \rangle$-module, where $\|X\|>1$. We give another proofs of the well known theorems: each countably generated group (resp. associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (resp. associative algebra, semigroup, Lie algebra). Key words: Gröbner-Shirshov basis, group, associative algebra, Lie algebra, associative differential algebra, associative $\Omega$-algebra, module. [AMS]{} Mathematics Subject Classification(2000): 20E34, 16Sxx, 17B05, 08Cxx, 16D10, 16S15, 13P10 Introduction ============ G. Higman, B.H. Neumann and H. Neuman[@HNN49] proved that any countable group is embeddable into a 2-generated group. It means that the basic rank of variety of groups is equal to two. In contrast, for example, the basic ranks of varieties of alternative and Malcev algebras are equal to infinity (I.P. Shestakov [@She]): there is no such $n$, that a countably generated alternative (Malcev) algebra can be embeddable into $n$-generated alternative (Malcev) algebra. Even more, for any $n\geq1$, there exists an alternative (Malcev) algebra generated by $n+1$ elements which can not be embedded into an $n$-generated alternative (Malcev) algebra (V.T. Filippov [@Fi1981; @Fi1984]). For Jordan algebras, it is known that the basic rank is bigger than 2, since any 2-generated Jordan algebra is special (A.I. Shirshov [@Sh56]), but there exist (even finitely dimensional) non-special Jordan algebras (A.A. Albert [@Al]). A.I. Malcev [@Ma] proved that any countably generated associative algebra is embeddable into a 2-generated associative algebra. A.I. Shirshov [@Sh58] proved the same result for Lie algebras and T. Evans [@Ev] proved the same result for semigroup. The first example of finitely generated infinite simple group was constructed by G. Higman [@Hi]. Later P. Hall [@Ha] proved that any group is embeddable into a simple group which is generated by 3 prescribed subgroups with some cardinality conditions. In particular, any countably generated group is embeddable into a simple 3-generated group. B. Neumann proved that any non-associative algebra is embeddable into a non-associative division algebra such that any equation $ax=b$, $xa=b$, $a\neq 0$ has a solution in the latter. Any division algebra is simple. P.M. Cohn [@Co] proved that any associative ring without zero divisors is embeddable into a simple associative ring without zero divisors such that any equation $ax - xa=b$, $a\neq 0$, has a solution in the latter. L.A. Skornyakov [@Sk] proved that any non-associative algebra without zero divisors is embeddable into a non-associative division algebra without zero divisors. I.S. Ivanov [@Iv65; @Iv67] prove the same result for $\Omega$-algebras (see also A.G. Kurosh [@Ku2]). P.M. Cohn [@Co] proved that any Lie algebra is embeddable into a division Lie algebra. E.G. Shutov [@Shu] and L.A. Bokut [@Bo63] proved that any semigroup is embeddable into a simple semigroup, and L.A. Bokut [@Bo76] proved that any associative algebra is embeddable into a simple associative algebra such that any equation $xay=b$, $a\neq 0$ is solvable in the latter. L.A. Bokut [@Bo62; @Bo62L] proved that any Lie (resp. non-associative, commutative, anti-commutative) algebra $A$ is embeddable into an algebraically closed (in particular simple) Lie (resp. non-associative, commutative, anti-commutative) algebra $B$ such that any equation $f(x_1,...,x_n)=0$ with coefficient in $B$ has a solution in $A$ (an equation over $B$ is an element of a free product of $B$ with a corresponding free algebra $k(X)$). L.A. Bokut [@Bo76; @Bo78; @BoK] proved that any associative (Lie) algebra is embeddable into a simple associative (algebraically closed Lie) algebra which is a sum of 4 prescribed (Lie) subalgebras with some cardinality conditions. In particular any countable associative (Lie) algebra is embeddable into a simple finitely generated associative (Lie) algebra. A.P. Goryushkin [@Go] proved that any countable group is embeddable into a simple 2-generated group. In this paper, by using Gröbner-Shirshov bases and some ideas from [@Bo76; @Bo78], we prove that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. We show that in the following classes, each countably generated algebra over a countable field $k$ can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. Also we prove that any countably generated module over a free associative algebra $k\langle X \rangle$ can be embedded into a cyclic $k\langle X \rangle$-module, where $\|X\|>1$. We give another proofs of Higman-Neumann-Neumann’s and Shirshov’s results mentioned above. We systematically use Gröbner-Shirshov bases theory for associative algebras, Lie algebras, associative $\Omega$-algebras, associative differential algebras, modules, see [@Sh62; @BoCQ; @ChCL; @ChCZ]. Preliminaries ============= We first cite some concepts and results from the literature [@Bo72; @Bo76; @Sh62] which are related to Gröbner-Shirshov bases for associative algebras. Let $X$ be a set and $k$ a field. Throughout this paper, we denote by $k\langle X\rangle$ the free associative algebra over $k$ generated by $X$, by $X^$ the free monoid generated by $X$ and by $N$ the set of natural numbers. A well ordering $<$ on $X^$ is called monomial if for $u, v\in X^$, we have $$u < v \Rightarrow w\|_u < w\|_v \ for \ all \ w\in X^,$$ where $w\|_u=w\|_{x_i\mapsto u}, \ w\|_v=w\|_{x_i\mapsto v}$ and $x_{i}$’s are the same individuality of the letter $x_{i}\in X$ in $w$. A standard example of monomial ordering on $X^$ is the deg-lex ordering which first compare two words by degree and then by comparing them lexicographically, where $X$ is a well ordered set. Let $X^$ be a set with a monomial ordering $<$. Then, for any polynomial $f\in k\langle X\rangle$, $f$ has the leading word $\overline{f}$. We call $f$ monic if the coefficient of $\overline{f}$ is 1. By $deg(f)$ we denote the degree of $\overline{f}$. Let $f,\ g\in k\langle X\rangle$ be two monic polynomials and $w\in X^$. If $w=\overline{f}b=a\overline{g}$ for some $a,b\in X^$ such that $deg(\overline{f})+deg(\overline{g})>deg(w)$, then $(f,g)_w=fb-ag$ is called the intersection composition of $f,g$ relative to $w$. If $w=\overline{f}=a\overline{g}b$ for some $a, b\in X^$, then $(f,g)_w=f-agb$ is called the inclusion composition of $f,g$ relative to $w$. In $(f,g)_w$, $w$ is called the ambiguity of the composition. Let $S\subset k\langle X\rangle$ be a monic set. A composition $(f,g)_w$ is called trivial modulo $(S,w)$, denoted by $$(f,g)_w\equiv0 \ \ \ mod(S,w)$$ if $(f,g)_w=\sum\alpha_ia_is_ib_i,$ where every $\alpha_i\in k, \ s_i\in S,\ a_i,b_i\in X^$, and $a_i\overline{s_i}b_i<w$. Recall that $S$ is a Gröbner-Shirshov basis in $k\langle X\rangle$ if any composition of polynomials from $S$ is trivial modulo $S$. The following lemma was first proved by Shirshov [@Sh62] for free Lie algebras (with deg-lex ordering) (see also Bokut [@Bo72]). Bokut [@Bo76] specialized the approach of Shirshov to associative algebras (see also Bergman [@Be]). For commutative polynomials, this lemma is known as Buchberger’s Theorem (see [@Bu65; @Bu70]). \[l1\] [(Composition-Diamond lemma for associative algebras)]{} Let $k$ be a field, $A=k \langle X\|S\rangle=k\langle X\rangle/Id(S)$ and $<$ a monomial ordering on $X^$, where $Id(S)$ is the ideal of $k \langle X\rangle$ generated by $S$. Then the following statements are equivalent: 1. $S$ is a Gröbner-Shirshov basis in $k\langle X\rangle$. 2. $f\in Id(S)\Rightarrow \bar{f}=a\bar{s}b$ for some $s\in S$ and $a,b\in X^$. 3. $Irr(S) = \{ u \in X^* \| u \neq a\bar{s}b,s\in S,a ,b \in X^\}$ is a $k$-basis of the algebra $A=k\langle X \| S \rangle=k\langle X\rangle/Id(S)$. If a subset $S$ of $k\langle X \rangle$ is not a Gröbner-Shirshov basis then one can add all nontrivial compositions of polynomials of $S$ to $S$. Continue this process repeatedly, we finally obtain a Gröbner-Shirshov basis $S^{comp}$ that contains $S$. Such a process is called Shirshov algorithm. Let $A=sgp\langle X\|S\rangle$ be a semigroup presentation. Then $S$ is also a subset of $k\langle X \rangle$ and we can find Gröbner-Shirshov basis $S^{comp}$. We also call $S^{comp}$ a Gröbner-Shirshov basis of $A$. $Irr(S^{comp})=\{u\in X^\|u\neq a\overline{f}b,\ a ,b \in X^,\ f\in S^{comp}\}$ is a $k$-basis of $k\langle X\|S\rangle$ which is also the set of all normal words of $A$. The following lemma is well known and can be easily proved. \[l2.2\] Let $k$ be a field, $S\subset k\langle X\rangle$. Then for any $f\in k\langle X\rangle$, $f$ can be expressed as $f=\sum_{u_i\in Irr(S),u_i\leq\bar f}\alpha_iu_i + \sum_{a_j\overline{s_j}b_j\leq\overline{f}}\beta_ja_js_jb_j$, where $\alpha_i, \beta_j\in k, a_j, b_j\in X^, s_j\in S$. The analogous lemma is valid for the free Lie algebra $Lie(X)$ (see, for example, [@BoC1]). \[l2.3\] Let $k$ be a field, $S\subset Lie(X)$. Then for any $f\in Lie(X)$, $f$ can be expressed as $f=\sum_{[u_i]\in Irr(S),[u_i]\leq\bar f}\alpha_i[u_i] + \sum_{a_j\overline{s_j}b_j\leq\overline{f}}\beta_j[a_js_jb_j]$, where $\alpha_i, \beta_j\in k, a_j, b_j\in X^, s_j\in S$, and $Irr(S) = \{[u]\|[u] \mbox{ is a non-associative Lyndon-Shirshov word on}\ X, \ u \neq a\bar{s}b,s\in S,a ,b \in X^\}$. Associative algebras, Groups and Semigroups =========================================== In this section we give another proofs for the following theorems mentioned in the introduction: every countably generated group (resp. associative algebra, semigroup) can be embedded into a two-generated group (resp. associative algebra, semigroup). Even more, we prove the following theorems: (i) Every countably generated associative algebra over a countable field $k$ can be embedded into a simple two-generated associative algebra. (ii) Every countably generated semigroup can be embedded into a (0-)simple two-generated semigroup. In this section, all the algebras we mention contain units. In 1949, G. Higman, B.H. Neumann, and H. Neumann [@HNN49] prove that every countable group can be embedded into a two-generated group. Now we give another proof for this theorem. (G. Higman, B.H. Neumann and H. Neumann) Every countable group can be embedded into a two-generated group. Proof We may assume that the group $G=\{g_{0}=1,g_{1},g_{2},g_{3},\dots \}$. Let $$H=gp\langle G\setminus\{g_0\},a,b,t\|g_{j}g_{k}=[g_{j},g_{k}],\ at=tb, b^{-i}ab^{i}t=tg_{i} a^{-i}ba^{i},\ i,j,k\in N \rangle.$$ G. Higman, B.H. Neumann and H. Neumann [@HNN49] (see also [@LySc]) proved that $G$ can be embedded into $H$. Now, we use the Composition-Diamond lemma, i.e., Lemma \[l1\] to reprove this theorem. Clearly, $H$ can also be expressed as $$H=gp\langle G\setminus\{g_0\},a,b,t\| S \rangle,$$ where $S$ consists of the following polynomials ($\varepsilon=\pm 1,i,j,k\in N$): $$\begin{aligned} 1.&&g_{j}g_{k}=[g_{j},g_{k}]\\ 2.&&a^\varepsilon t=tb^{\varepsilon}\\ 3.&&b^{\varepsilon} t^{-1}=t^{-1}a^{\varepsilon}\\ 4.&&ab^{i}t=b^{i}tg_{i}a^{-i}ba^{i}\\ 5.&&a^{-1}b^{i}t=b^{i}t(g_{i}a^{-i}ba^{i})^{-1}\\ 6.&&ba^{i}t^{-1}=a^{i}g_{i}^{-1}t^{-1}b^{-i}ab^{i}\\ 7.&&b^{-1}a^{i}g_{i}^{-1}t^{-1}=a^{i}t^{-1}b^{-i}a^{-1}b^{i}\\ 8.&&a^{\varepsilon} a^{-\varepsilon}=b^{\varepsilon} b^{-\varepsilon}=t^{\varepsilon} t^{-\varepsilon}=1\end{aligned}$$ We order $\{g_{i},a^{\pm 1}, b^{\pm 1}\}^{} $ by deg-lex ordering with $ g_{i}<a<a^{-1}<b<b^{-1}. $ Denote by $X=\{g_{i},a^{\pm 1}, b^{\pm 1}, t^{\pm 1}\}$. For any $u \in X^{}$, $u$ can be uniquely expressed without brackets as $$u=u_{0}t^{\varepsilon_{1}}u_{1}t^{\varepsilon_{2}}u_{2}\cdots t^{\varepsilon_{n}}u_{n},$$ where $u_{i}\in \{g_{i},a^{\pm 1}, b^{\pm 1}\}^{} , n\geq 0, \varepsilon_{i}=\pm 1$. Denote by $$wt(u)=(n,u_{0},t^{\varepsilon_{1}},u_{1},t^{\varepsilon_{2}},u_{2},\dots, t^{\varepsilon_{n}},u_{n}).$$ Then, we order $X^$ as follows: for any $u,v\in X^$ $$u>v\Leftrightarrow wt(u)>wt(v)\ \ \ \ \ \ \ \ lexicographically,$$ where $t>t^{-1}$. With this ordering, we can check that $S$ is a Gröbner-Shirshov basis in the free associative algebra $k\langle X\rangle$. By Lemma \[l1\], the group $G$ can be embedded into $H$ which is generated by $\{a,b\}$. $\blacksquare$ A.I. Malcev [@Ma] proved that any countably generated associative algebra is embeddable into a two-generated associative algebra, and T. Evans [@Ev] proved that every countably generated semigroup can be embedded into a two-generated semigroup. Now, by applying Lemma \[l1\], we give another proofs of this two embedding theorems. \[t4.2\] (A.I. Malcev) Every countably generated associative algebra can be embedded into a two-generated associative algebra. Proof* We may assume that $ A= k \langle X \|S\rangle $ is an associative algebra generated by $X$ with relations $S$, where $X=\{x_{i},i=1,2,\dots\}$. By Shirshov algorithm, we can assume that $S$ is a Gröbner-Shirshov basis in the free associative algebra $k\langle X\rangle$ with deg-lex ordering on $X^$. Let $$H=k\langle X, a,b \| S, aab^{i}ab=x_{i}, \ i=1,2,\dots\rangle.$$ We can check that $$\{S, aab^{i}ab=x_{i}, \ i=1,2,\dots\}$$ is a Gröbner-Shirshov basis in $ k\langle X,a,b \rangle$ with deg-lex ordering on $(X\cup\{a,b\})^$ where $a>b>x,\ x\in X$ since there are no new compositions. By Lemma \[l1\], $A$ can be embedded into $H$ which is generated by $\{a,b\}$. $\blacksquare$ By the proof of Theorem \[t4.2\], we have immediately the following corollary. (T. Evans) Every countably generated semigroup can be embedded into a two-generated semigroup. \[t0.0\] Every countably generated associative algebra over a countable field $k$ can be embedded into a simple two-generated associative algebra. Proof Let $A$ be a countably generated associative algebra over a countable field $k$. We may assume that $A$ has a countable $k$-basis $\{1\}\cup X_0$, where $X_0=\{x_{i}\| i=1,2,\ldots\}$ and 1 is the unit of $A$. Then $A$ can be expressed as $A= k\langle X_0 \|x_{i}x_{j}=\{x_{i},x_{j}\}, i,j\in N \rangle$, where $\{x_{i},x_{j}\}$ is a linear combination of $x_t, \ x_t\in X_0$. Let $A_0=k\langle X_0\rangle$, $A_0^+=A_0\backslash\{0\}$ and fix the bijection $$(A_0^+,A_0^+)\longleftrightarrow\{(x_m^{(1)},y_m^{(1)}), m\in N\}.$$ Let $X_1=X_0\cup\{x_m^{(1)},y_m^{(1)},a,b\|m\in N\}$, $A_1=k\langle X_1\rangle$, $A_1^+=A_1\backslash\{0\}$ and fix the bijection $$(A_1^+,A_1^+)\longleftrightarrow\{(x_m^{(2)},y_m^{(2)}), m\in N\}.$$ $$\vdots$$ Let $X_{n+1}=X_n\cup\{x_m^{(n+1)},y_m^{(n+1)}\|m\in N\}$, $n\geq1$, $A_{n+1}=k\langle X_{n+1}\rangle$, $A_{n+1}^+=A_{n+1}\backslash\{0\}$ and fix the bijection $$(A_{n+1}^+,A_{n+1}^+)\longleftrightarrow\{(x_m^{(n+2)},y_m^{(n+2)}), m\in N\}.$$ $$\vdots$$ Consider the chain of the free associative algebras $$A_0\subset A_1\subset A_2\subset\ldots\subset A_n\subset\ldots.$$ Let $X=\bigcup_ {n=0}^ {\infty}X_n$. Then $k\langle X\rangle=\bigcup_ {n=0}^ {\infty}A_n$. Now, define the desired algebra $\mathcal{A}$. Take the set $X$ as the set of the generators for this algebra and take the following relations as one part of the relations for this algebra $$\label{1} x_{i}x_{j}=\{x_{i},x_{j}\}, \ i,j\in N$$ $$\label{2} aa(ab)^nb^{2m+1}ab=x_m^{(n)},\ m,n\in N$$ $$\label{3} aa(ab)^nb^{2m}ab=y_m^{(n)}, \ m,n\in N$$ $$\label{4} aabbab=x_1$$ Before we introduce the another part of the relations on $\mathcal{A}$, let us define canonical words of the algebra $A_n$, $n\geq0$. A word in $X_0$ without subwords that are the leading terms of (\[1\]) is called a canonical word of $A_0$. A word in $X_1$ without subwords that are the leading terms of (\[1\]), (\[2\]), (\[3\]), (\[4\]) and without subwords of the form $$(x_m^{(1)})^{deg(g^{(0)})+1}\overline{f^{(0)}}y_m^{(1)},$$ where $(x_m^{(1)},y_m^{(1)})\longleftrightarrow(f^{(0)},g^{(0)})\in(A_0^+,A_0^+)$ such that $f^{(0)},g^{(0)}$ are non-zero linear combination of canonical words of $A_0$, is called a canonical word of $A_1$. Suppose that we have defined canonical word of $A_{k}$, $k<n$. A word in $X_n$ without subwords that are the leading terms of (\[1\]), (\[2\]), (\[3\]), (\[4\]) and without subwords of the form $$(x_m^{(k+1)})^{deg(g^{(k)})+1}\overline{f^{(k)}}y_m^{(k+1)},$$ where $(x_m^{(k+1)},y_m^{(k+1)})\longleftrightarrow(f^{(k)},g^{(k)})\in(A_{k}^+,A_{k}^+)$ such that $f^{(k)},g^{(k)}$ are non-zero linear combination of canonical words of $A_{k}$, is called a canonical word of $A_{n}$. Then the another part of the relations on $\mathcal{A}$ are the following: $$\label{5} (x_m^{(n)})^{deg(g^{(n-1)})+1}f^{(n-1)}y_m^{(n)}-g^{(n-1)}=0,\ m,n\in N$$ where $(x_m^{(n)},y_m^{(n)})\longleftrightarrow(f^{(n-1)},g^{(n-1)})\in(A_{n-1}^+,A_{n-1}^+)$ such that $f^{(n-1)},g^{(n-1)}$ are non-zero linear combination of canonical words of $A_{n-1}$. By Lemma \[l2.2\], we have that in $\mathcal{A}$ every element can be expressed as linear combination of canonical words. Denote by $S$ the set constituted by the relations (\[1\])-(\[5\]). We can see that $S$ is a Gröbner-Shirshov basis in $k \langle X\rangle$ with deg-lex ordering on $X^$ since in $S$ there are no compositions except for the ambiguity $x_{i}x_{j}x_{k}$ which is a trivial case. By Lemma \[l1\], $A$ can be embedded into $\mathcal{A}$. By (\[5\]), $\mathcal{A}$ is a simple algebra. By (\[2\])-(\[5\]), $\mathcal{A}$ is generated by $\{a,b\}$. $\blacksquare$ A semigroup $S$ without zero is called simple if it has no proper ideals. A semigroup $S$ with zero is called 0-simple if $\{0\}$ and $S$ are its only ideals, and $S^2\neq\{0\}$. ([@Ho]) A semigroup $S$ with 0 is 0-simple if and only if $SaS=S$ for every $a\neq0$ in $S$. A semigroup $S$ without 0 is simple if and only if $SaS=S$ for every $a$ in $S$. The following theorem follows from the proof of Theorem \[t0.0\]. \[t3.0\] Every countably generated semigroup can be embedded into a simple two-generated semigroup. Remark:* Let ${S}$ be a simple semigroup. Then the semigroup ${S}^0$ with 0 attached is a 0-simple semigroup. Therefore, by Theorem \[t3.0\], each countably generated semigroup can be embedded into a 0-simple two-generated semigroup. Lie algebras ============ In this section, we give another proof of the Shirshov’s theorem: every countably generated Lie algebra can be embedded into a two-generated Lie algebra. And even more, we show that every countably generated Lie algebra over a countable field $k$ can be embedded into a simple two-generated Lie algebra. We start with the Lyndon-Shirshov associative words. Let $X=\{x_i\|i\in I\}$ be a well-ordered set with $x_i>x_p$ if $i>p$ for any $i,p\in I$. We order $X^$ by the lexicographical ordering. ([@Ly; @Sh58], see [@BoC1; @Uf]) Let $u\in X^\ast$ and $u\neq 1$. Then $u$ is called an $ALSW$ (associative Lyndon-Shirshov word) if $$(\forall v,w\in X^\ast, v,w\neq 1) \ u=vw\Rightarrow vw>wv.$$ ([@Chen; @Sh58], see [@BoC1; @Uf]) A non-associative word $(u)$ in $X$ is called a $NLSW$ (non-associative Lyndon-Shirshov word) if 1. $u$ is an $ALSW$, 2. if $(u)=((v)(w))$, then both $(v)$ and $(w)$ are $NLSW$’s, 3. in (ii) if $(v)=((v_1)(v_2))$, then $v_2\leq w$ in $X^\ast$. ([@Chen; @Sh58], see [@BoC1; @Uf]) Let $u$ be an $ALSW$. Then there exists a unique bracketing way such that $(u)$ is a $NLSW$. Let $X^{\ast\ast}$ be the set of all non-associative words $(u)$ in $X$. If $(u)$ is a $NLSW$, then we denote it by $[u]$. ([@Chen; @Sh58], see [@BoC1; @Uf]) $NLSW$’s forms a linear basis of $Lie(X)$, the free Lie algebra generated by $X$. Composition-Diamond lemma for free Lie algebras (with deg-lex ordering) is given in [@Sh62] (see also [@BoC1]). By applying this lemma, we give the following theorem. (A.I. Shirshov) Every countably generated Lie algebra can be embedded into a two-generated Lie algebra. Proof* We may assume that $$L=Lie( X\|S)$$ is a Lie algebra generated by $X$ with relations $S$, where $X=\{x_{i},i=1,2,\dots\}$. By Shirshov algorithm, we can assume that $S$ is a Gröbner-Shirshov basis in the free Lie algebra $Lie(X)$ on deg-lex ordering. Let $$H=Lie( X, a,b \| S, [aab^{i}ab]=x_{i}, \ i=1,2,\dots).$$ We can check that $$\{ S, [aab^{i}ab]=x_{i}, \ i=1,2,\dots\}$$ is a Gröbner-Shirshov basis of $Lie( X,a,b )$ on deg-lex ordering with $a>b>x_i$ since there are no new compositions. By the Composition-Diamond lemma for Lie algebras, $L$ can be embedded into $H$ which is generated by $\{a,b\}$. $\blacksquare$ Every countably generated Lie algebra over a countable field $k$ can be embedded into a simple two-generated Lie algebra. Proof Let $L$ be a countably generated Lie algebra over a countable field $k$. We may assume that $L$ has a countable $k$-basis $X_0=\{x_{i}\| i=1,2,\ldots\}$. Then $L$ can be expressed as $L= Lie(X_0 \|[x_{i}x_{j}]=\{x_{i},x_{j}\}, i,j\in N)$. Let $L_0=Lie(X_0)$, $L_0^+=L_0\backslash\{0\}$ and fix the bijection $$(L_0^+,L_0^+)\longleftrightarrow\{(x_m^{(1)},y_m^{(1)}), m\in N\}.$$ Let $X_1=X_0\cup\{x_m^{(1)},y_m^{(1)},a,b\|m\in N\}$, $L_1=Lie( X_1)$, $L_1^+=L_1\backslash\{0\}$ and fix the bijection $$(L_1^+,L_1^+)\longleftrightarrow\{(x_m^{(2)},y_m^{(2)}), m\in N\}.$$ $$\vdots$$ Let $X_{n+1}=X_n\cup\{x_m^{(n+1)},y_m^{(n+1)}\|m\in N\}$, $n\geq1$, $L_{n+1}=Lie(X_{n+1})$, $L_{n+1}^+=L_{n+1}\backslash\{0\}$ and fix the bijection $$(L_{n+1}^+,L_{n+1}^+)\longleftrightarrow\{(x_m^{(n+2)},y_m^{(n+2)}), m\in N\}.$$ $$\vdots$$ Consider the chain of the free Lie algebras $$L_0\subset L_1\subset L_2\subset\ldots\subset L_n\subset\ldots.$$ Let $X=\bigcup_ {n=0}^ {\infty}X_n$. Then $Lie(X)=\bigcup_ {n=0}^ {\infty}L_n$. Now, define the desired Lie algebra $\mathcal{L}$. Take the set $X$ as the set of the generators for this algebra and take the following relations as one part of the relations for this algebra $$\label{11} [x_{i}x_{j}]=\{x_{i},x_{j}\}, \ i,j\in N$$ $$\label{12} [aa(ab)^nb^{2m+1}ab]=x_m^{(n)},\ m,n\in N$$ $$\label{13} [aa(ab)^nb^{2m}ab]=y_m^{(n)}, \ m,n\in N$$ $$\label{14} [aabbab]=x_1$$ Before we introduce the another part of the relations on $\mathcal{L}$, let us define canonical words of the Lie algebra $L_n$, $n\geq0$. A NLSW $[u]$ in $X_0$ where $u$ without subwords that are the leading terms of (\[11\]) is called a canonical word of $L_0$. A NLSW $[u]$ in $X_1$ where $u$ without subwords that are the leading terms of (\[11\]), (\[12\]), (\[13\]), (\[14\]) and without subwords of the form $$x_m^{(1)}\overline{f^{(0)}}x_m^{(1)}(y_m^{(1)})^{deg(g^{(0)})+1},$$ where $(x_m^{(1)},y_m^{(1)})\longleftrightarrow(f^{(0)},g^{(0)})\in(L_0^+,L_0^+)$ such that $f^{(0)},g^{(0)}$ are non-zero linear combination of canonical words of $L_0$, is called a canonical word of $L_1$. Suppose that we have defined canonical word of $L_{k}$, $k<n$. A NLSW $[u]$ in $X_n$ where $u$ without subwords that are the leading terms of (\[11\]), (\[12\]), (\[13\]), (\[14\]) and without subwords of the form $$x_m^{(k+1)}\overline{f^{(k)}}x_m^{(k+1)}(y_m^{(k+1)})^{deg(g^{(k)})+1},$$ where $(x_m^{(k+1)},y_m^{(k+1)})\longleftrightarrow(f^{(k)},g^{(k)})\in(L_k^+,L_k^+)$ such that $f^{(k)},g^{(k)}$ are non-zero linear combination of canonical words of $L_k$, is called a canonical word of $L_n$. Then the another part of the relations on $\mathcal{L}$ are the following: $$\label{15} (x_m^{(n)}f^{(n-1)})[x_m^{(n)}(y_m^{(n)})^{deg(g^{(n-1)})+1}]-g^{(n-1)}=0,\ \ m, n\in N$$ where $(x_m^{(n)},y_m^{(n)})\longleftrightarrow(f^{(n-1)},g^{(n-1)})\in(L_{n-1}^+,L_{n-1}^+)$ such that $f^{(n-1)},g^{(n-1)}$ are non-zero linear combination of canonical words of $L_{n-1}$. By Lemma \[l2.3\], we have in $\mathcal{L}$ every element can be expressed as linear combination of canonical words. Denote by $S$ the set constituted by the relations (\[11\])-(\[15\]). Define $\ldots>x_q^{(2)}>x_m^{(1)}>a>b>x_i>y_n^{(1)}>y_p^{(2)}>\ldots$. We can see that in $S$ there are no compositions unless for the ambiguity $x_{i}x_{j}x_{k}$. But this case is trivial. Hence $S$ is a Gröbner-Shirshov basis in $Lie(X)$ on deg-lex ordering which implies that $L$ can be embedded into $\mathcal{L}$. By (\[12\])-(\[15\]), $\mathcal{L}$ is a simple Lie algebra generated by $\{a,b\}$. $\blacksquare$ Associative differential algebras ================================= Composition-Diamond lemma for associative differential algebras with unit is established in a recent paper [@ChCL]. By applying this lemma in this section, we show that: (i). Every countably generated associative differential algebra can be embedded into a two-generated associative differential algebra. (ii). Any associative differential algebra can be embedded into a simple associative differential algebra. (iii). Every countably generated associative differential algebra with countable set $\mathcal{D}$ of differential operations over a countable field $k$ can be embedded into a simple two-generated associative differential algebra. Let $\mathcal{A}$ be an associative algebra over a field $k$ with unit. Let $\mathcal{D}$ be a set of multiple linear operations on $\mathcal{A}$. Then $\mathcal{A}$ is called an associative differential algebra with differential operations $\mathcal{D}$ or $\mathcal{D}$-algebra, for short, if for any $D\in \mathcal{D}, \ a, b \in \mathcal{A}$, $$D(ab)=D(a)b+aD(b).$$ Let $\mathcal{D}=\{D_j\|j\in J\}$. For any $m=0,1,\cdots$ and $\bar{j}=(j_1,\cdots,j_m)\in J^m$, denote by $D^{\bar{j}}=D_{j_1}D_{j_2}\cdots D_{j_{m}}$ and $D^{\omega}(X)=\{D^{\bar{j}}(x)\|x\in X, \ \bar{j}\in J^m, \ m\geq 0\}$, where $D^{0}(x)=x$. Let $T=(D^{\omega}(X))^$ be the free monoid generated by $D^{\omega}(X)$. For any $u=D^{\overline{i_1}}(x_1)D^{\overline{i_2}}(x_2)\cdots D^{\overline{i_n}}(x_n)$ $\in T$, the length of $u$, denoted by $\|u\|$, is defined to be $n$. In particular, $\|1\|=0$. Let $k\langle X;\mathcal{D} \rangle=kT$ be the $k$-algebra spanned by $T$. For any $D_j\in \mathcal{D}$, we define the linear map $ D_j: \ k\langle X;\mathcal{D} \rangle \rightarrow k\langle X;\mathcal{D} \rangle$ by induction on $\|u\|$ for $u\in T$: 1. $D_j(1)=0$. 2. Suppose that $u=D^{\bar{i}}(x)=D_{i_1}D_{i_2}\cdots D_{i_{m}}(x)$. Then $D_j(u)=D_jD_{i_1}D_{i_2}\cdots D_{i_{m}}(x)$. 3. Suppose that $u=D^{\bar{i}}(x)\cdot v, \ v\in T$. Then $D_j(u)=(D_jD^{\bar{i}}(x))\cdot v+D^{\bar{i}}(x)\cdot D_j(v)$. Then, $k\langle X;\mathcal{D} \rangle$ is a free associative differential algebra generated by $X$ with differential operators $\mathcal{D}$ (see [@ChCL]). Let $\mathcal{D}=\{D_j\|j\in J\}$, $X$ and $J$ well ordered sets, $D^{\bar{i}}(x)=D_{i_1}D_{i_2}\cdots D_{i_{m}}(x)\in D^{\omega}(X)$. Denote by $$wt(D^{\bar{i}}(x))=(x; m, i_1, i_2,\cdots, i_{m}).$$ Then, we order $D^{\omega}(X)$ as follows: $$D^{\bar{i}}(x)< D^{\bar{j}}(y)\Longleftrightarrow wt(D^{\bar{i}}(x))< wt(D^{\bar{j}}(y)) \ \mbox{ lexicographically}.$$ It is easy to check this ordering is a well ordering on $D^{\omega}(X)$. Now, we order $T=(D^{\omega}(X))^$ by deg-lex ordering. We will use this ordering in this section. For convenience, for any $u\in T$, we denote $\overline{D^{\bar{j}}(u)}$ by ${d^{\bar{j}}(u)}$. Every countably generated associative differential algebra can be embedded into a two-generated associative differential algebra. Proof Suppose that $\mathcal{A}=k\langle X;\mathcal{D}\|S \rangle$ is an associative differential algebra generated by $X$ with relations $S$, where $X=\{x_{i},i=1,2,\dots\}$. By Shirshov algorithm, we can assume that with the deg-lex ordering on $(D^{\omega}(X))^$ defined as above, $S$ is a Gröbner-Shirshov basis of the free associative differential algebra $k\langle X;\mathcal{D}\rangle$ in the sense of the paper [@ChCL]. Let $\mathcal{B}=k\langle X,a,b; \mathcal{D}\| S, aab^{i}ab=x_{i} \rangle$. We have that with the deg-lex ordering on $(D^{\omega}(X,a,b))^$, $\{S, aab^{i}ab=x_{i}, \ i=1,2,\dots\}$ is a Gröbner-Shirshov basis in the free associative differential algebra $k\langle X,a,b;\mathcal{D}\rangle$ since there are no new compositions. By the Composition-Diamond lemma in [@ChCL], $\mathcal{A}$ can be embedded into $\mathcal{B}$ which is generated by $\{a,b\}$. $\blacksquare$ Every associative differential algebra can be embedded into a simple associative differential algebra. Proof Let ${A}$ be an associative differential algebra over a field $k$ with $k$-basis $\{1\}\cup X$, where $X=\{x_i\mid i\in I\}$ and $I$ is a well ordered set. It is clear that $S_0=\{x_{i}x_{j}=\{x_{i},x_{j}\}, \ D(x_i)=\{D(x_i)\}, i,j\in I,\ D\in \cal D\}$ where $\{D(x_i)\}$ is a linear combination of $x_j, j\in I$, is a Gröbner-Shirshov basis in the free associative differential algebra $k\langle X;\mathcal{D}\rangle$ with the deg-lex ordering on $(D^{\omega}(X))^$, and ${A}$ can be expressed as $${A}=k\langle X;\mathcal{D}\|x_ix_j=\{x_i,x_j\}, D(x_i)=\{D(x_i)\}, i,j\in I,\ D\in \cal D\rangle.$$ Let us totally order the set of monic elements of ${A}$. Denote by $T$ the set of indices for the resulting totally ordered set. Consider the totally ordered set $T^2=\{(\theta,\sigma)\|\theta,\sigma\in T\}$ and assign $(\theta,\sigma)<(\theta',\sigma')$ if either $\theta<\theta'$ or $\theta=\theta'$ and $\sigma<\sigma'$. Then $T^2$ is also totally ordered set. For each ordered pair of elements $f_\theta, f_\sigma\in{A}, \ \ \theta, \sigma\in T$, introduce the letters $x_{\theta\sigma},y_{\theta\sigma}$. Let ${A}_{1}$ be the associative differential algebra given by the generators $$X_1=\{x_i,y_{\theta\sigma},x_{\varrho\tau}\| i\in I,\ \theta, \sigma, \varrho, \tau\in T\}$$ and the defining relations $$S=\{x_ix_j=\{x_i,x_j\}, D(x_i)=\{D(x_i)\}, x_{\theta\sigma}f_\theta y_{\theta\sigma}=f_\sigma\mid i, j\in I, \ D\in {\cal D},\ ({\theta,\sigma})\in T^2\}.$$ We can have that with the deg-lex ordering on $(D^{\omega}(X_1))^$, $S$ is a Gröbner-Shirshov basis in the free associative differential algebra $k\langle X_1;\mathcal{D}\rangle$ in the sense of the paper [@ChCL] since there are no new compositions. Thus, by the Composition-Diamond lemma in [@ChCL], ${A}$ can be embedded into ${A}_1$. The relations $x_{\theta\sigma}f_\theta y_{\theta\sigma}=f_\sigma$ of ${A}_1$ provide that in ${A}_1$ every monic element $f_\theta$ of the subalgebra ${A}$ generates an ideal containing algebra $A$. Mimicking the construction of the associative differential algebra ${A}_1$ from the ${A}$, produce the associative differential algebra ${A}_2$ from ${A}_1$ and so on. As a result, we acquire an ascending chain of associative differential algebras $${A}={A}_0\subset{A}_1\subset{A}_2\subset\cdots$$ such that every monic element $f\in A_k$ generates an ideal in ${A}_{k+1}$ containing $A_k$. Therefore, in the associative differential algebra $${\cal A}=\bigcup_ {k=0}^ {\infty}{A}_k,$$ every nonzero element generates the same ideal. Thus, ${\cal A}$ is a simple associative differential algebra. $\blacksquare$ Every countably generated associative differential algebra with countable set $\mathcal{D}$ of differential operations over a countable field $k$ can be embedded into a simple two-generated associative differential algebra. Proof Let $A$ be a countably generated associative differential algebra with countable set $\mathcal{D}$ of differential operations over a countable field $k$. We may assume that $A$ has a countable $k$-basis $\{1\}\cup X_0$, where $X_0=\{x_{i}\| i=1,2,\ldots\}$. Then $A$ can be expressed as $$A=k\langle X_0;\mathcal{D}\|x_ix_j=\{x_i,x_j\}, D(x_i)=\{D(x_i)\}, i,j\in N,\ D\in \cal D\rangle.$$ Let $A_0=k\langle X_0;\mathcal{D}\rangle$, $A_0^+=A_0\backslash\{0\}$ and fix the bijection $$(A_0^+,A_0^+)\longleftrightarrow\{(x_m^{(1)},y_m^{(1)}), m\in N\}.$$ Let $X_1=X_0\cup\{x_m^{(1)},y_m^{(1)},a,b\|m\in N\}$, $A_1=k\langle X_1;\mathcal{D}\rangle$, $A_1^+=A_1\backslash\{0\}$ and fix the bijection $$(A_1^+,A_1^+)\longleftrightarrow\{(x_m^{(2)},y_m^{(2)}), m\in N\}.$$ $$\vdots$$ Let $X_{n+1}=X_n\cup\{x_m^{(n+1)},y_m^{(n+1)}\|m\in N\}$, $n\geq1$, $A_{n+1}=k\langle X_{n+1};\mathcal{D}\rangle$, $A_{n+1}^+=A_{n+1}\backslash\{0\}$ and fix the bijection $$(A_{n+1}^+,A_{n+1}^+)\longleftrightarrow\{(x_m^{(n+2)},y_m^{(n+2)}), m\in N\}.$$ $$\vdots$$ Consider the chain of the free associative differential algebras $$A_0\subset A_1\subset A_2\subset\ldots\subset A_n\subset\ldots.$$ Let $X=\bigcup_{n=0}^{\infty}X_n$. Then $k\langle X;\mathcal{D}\rangle=\bigcup_{n=0}^{\infty}A_n$. Now, define the desired associative differential algebra $\mathcal{A}$. Take the set $X$ as the set of the generators for this algebra and take the following relations as one part of the relations for this algebra $$\label{16} x_{i}x_{j}=\{x_{i},x_{j}\}, \ D(x_i)=\{D(x_i)\}, \ i,j\in N,\ D\in \cal D$$ $$\label{17} aa(ab)^nb^{2m+1}ab=x_m^{(n)},\ m,n\in N$$ $$\label{18} aa(ab)^nb^{2m}ab=y_m^{(n)}, \ m,n\in N$$ $$\label{19} aabbab=x_1$$ Before we introduce the another part of the relations on $\mathcal{A}$, let us define canonical words of the algebras $A_n$, $n\geq0$. An element in $(D^{\omega}(X_0))^$ without subwords of the form ${d^{\bar{i}}(u)}$ where $u$ is the leading terms of (\[16\]), is called a canonical word of $A_0$. An element in $(D^{\omega}(X_1))^$ without subwords of the form ${d^{\bar{i}}(u)}$ where $u$ is the leading terms of (\[16\]), (\[17\]), (\[18\]), (\[19\]) and $$(x_m^{(1)})^{\|\overline{g^{(0)}}\|+1}\overline{f^{(0)}}y_m^{(1)},$$ where $(x_m^{(1)},y_m^{(1)})\longleftrightarrow(f^{(0)},g^{(0)})\in(A_0^+,A_0^+)$ such that $f^{(0)},g^{(0)}$ are non-zero linear combination of canonical words of $A_0$, is called a canonical word of $A_1$. Suppose that we have defined canonical word of $A_{k}$, $k<n$. An element in $(D^{\omega}(X_n))^$ without subwords of the form ${d^{\bar{i}}(u)}$ where $u$ is the leading terms of (\[16\]), (\[17\]), (\[18\]), (\[19\]) and $$(x_m^{(k+1)})^{\|\overline{g^{(k)}}\|+1}\overline{f^{(k)}}y_m^{(k+1)},$$ where $(x_m^{(k+1)},y_m^{(k+1)})\longleftrightarrow(f^{(k)},g^{(k)})\in(A_k^+,A_k^+)$ such that $f^{(k)},g^{(k)}$ are non-zero linear combination of canonical words of $A_k$, is called a canonical word of $A_n$. Then the another part of the relations on $\mathcal{A}$ are the following: $$\label{20} (x_m^{(n)})^{\|\overline{g^{(n-1)}}\|+1}f^{(n-1)}y_m^{(n)}-g^{(n-1)}=0, \ \ m, n\in N$$ where $(x_m^{(n)},y_m^{(n)})\longleftrightarrow(f^{(n-1)},g^{(n-1)})\in(A_{n-1}^+,A_{n-1}^+)$ such that $f^{(n-1)},g^{(n-1)}$ are non-zero linear combination of canonical words of $A_{n-1}$. We can get that in $\mathcal{A}$ every element can be expressed as linear combination of canonical words. Denote by $S$ the set constituted by the relations (\[16\])-(\[20\]). We can have that with the deg-lex ordering on $(D^{\omega}(X))^$ defined as above, $S$ is a Gröbner-Shirshov basis in $k \langle X;\mathcal{D}\rangle$ since in $S$ there are no compositions except for the ambiguity $x_{i}x_{j}x_{k}$ which is a trivial case. This implies that $A$ can be embedded into $\mathcal{A}$. By (\[17\])-(\[20\]), $\mathcal{A}$ is a simple associative differential algebra generated by $\{a,b\}$.$\blacksquare$ Associative algebras with multiple operations ============================================= Composition-Diamond lemma for associative algebra with multiple operations $\Omega$ (associative $\Omega$-algebra, for short) is established in a recent paper [@BoCQ]. By applying this lemma, we show in this section that: (i). Every countably generated associative $\Omega$-algebra can be embedded into a two-generated associative $\Omega$-algebra. (ii). Any associative $\Omega$-algebra can be embedded into a simple associative $\Omega$-algebras. (iii). Each countably generated associative $\Omega$-algebra with countable multiple operations $\Omega$ over a countable field $k$ can be embedded into a simple two-generated associative $\Omega$-algebra. The concept of multi-operations algebras ($\Omega$-algebras) was first introduced by A.G. Kurosh in [@Ku; @Ku2]. Let $k$ be a field. An associative algebra with multiple linear operations is an associative $k$-algebra $A$ with a set $\Omega$ of multi-linear operations. Let $X$ be a set and $$\Omega=\bigcup_{n=1}^{\infty}\Omega_{n},$$ where $\Omega_{n}$ is the set of $n$-ary operations, for example, ary $(\delta)=n$ if $\delta\in \Omega_n$. Denote by $S(X)$ the free semigroup without identity generated by $X$. For any non-empty set $Y$ (not necessarily a subset of $S(X)$), let $$\Omega(Y)=\bigcup\limits_{n=1}^{\infty}\{\delta(x_1,x_2,\cdots,x_n)\|\delta\in \Omega_n, x_i\in Y, \ i=1,2,\cdots,n\}.$$ Define $$\begin{aligned} \mathfrak{S}_{0}&=&S(X),\\ \mathfrak{S}_{1}&=&S(X\cup \Omega(\mathfrak{S}_{0})), \\ \vdots\ \ & &\ \ \ \ \vdots\\ \mathfrak{S}_{n}&=&S(X\cup \Omega(\mathfrak{S}_{n-1})),\\ \vdots\ \ & &\ \ \ \ \vdots\end{aligned}$$ Then we have $$\mathfrak{S}_{0}\subset\mathfrak{S}_{1}\subset\cdots \subset\mathfrak{S}_{n}\subset\cdots.$$ Let $$\mathfrak{S}(X)=\bigcup_{n\geq0}\mathfrak{S}_{n}.$$ Then, we can see that $\mathfrak{S}(X)$ is a semigroup such that $ \Omega(\mathfrak{S}(X))\subseteq \mathfrak{S}(X). $ For any $u\in \mathfrak{S}(X)$, $dep(u)=\mbox{min}\{n\|u\in\mathfrak{S}_{n} \}$ is called the depth of $u$. Let $k\langle X; \Omega\rangle$ be the $k$-algebra spanned by $\mathfrak{S}(X)$. Then, the element in $\mathfrak{S}(X)$ (resp. $k\langle X; \Omega\rangle$) is called a $\Omega$-word (resp. $\Omega$-polynomial). Extend linearly each map $\delta\in\Omega_n$, $$\delta:\mathfrak{S}(X)^n\rightarrow \mathfrak{S}(X), \ (x_1,x_2,\cdots,x_n)\mapsto \delta(x_1,x_2,\cdots,x_n)$$ to $k\langle X; \Omega\rangle$. Then, $k\langle X; \Omega\rangle$ is a free associative algebra with multiple linear operators $\Omega$ on the set $X$ (see [@BoCQ]). Let $X$ and $\Omega$ be well ordered sets. We order $X^$ by the deg-lex ordering. For any $u\in \mathfrak{S}(X)$, $u$ can be uniquely expressed without brackets as $$u=u_0\delta_{i_{_{1}}}\overrightarrow{x_{i_1}}u_1\cdots \delta_{i_{_{t}}}\overrightarrow{x_{i_{t}}}u_t,$$ where each $u_i\in X^,\delta_{i_{_{k}}}\in \Omega_{i_{_{k}}}, \ \overrightarrow{x_{i_k}}=( x_{k_{1}},x_{k_{2}},\cdots, x_{k_{i_{_{k}}}} )\in \mathfrak{S}(X)^{i_k}$. Denote by $$wt(u)=(t,\delta_{i_{_{1}}},\overrightarrow{x_{i_{1}}}, \cdots,\delta_{i_{_{t}}},\overrightarrow{x_{i_t}}, u_0, u_1, \cdots, u_t ).$$ Then, we order $\mathfrak{S}(X)$ as follows: for any $u,v\in \mathfrak{S}(X)$, $$\label{o1} u>v\Longleftrightarrow wt(u)>wt(v)\ \mbox{ lexicographically}$$ by induction on $dep(u)+dep(v)$. It is clear that the ordering (\[o1\]) is a monomial ordering on $\mathfrak{S}(X)$ (see [@BoCQ]). Denote by $deg_{\Omega}(u)$ the number of $\delta$ in $u$ where $\delta\in\Omega$, for example, if $u=x_1\delta_1(x_2)\delta_3(x_2,x_1,\delta_1(x_3))$, then $deg_{\Omega}(u)=3$. Every countably generated associative $\Omega$-algebra can be embedded into a two-generated associative $\Omega$-algebra. Proof Suppose that $A= k\langle X ; \Omega \|S\rangle$ is an associative $\Omega$-algebra generated by $X$ with relations $S$, where $X=\{x_{i},i=1,2,\dots\}$. By Shirshov algorithm, we can assume that $S$ is a Gröbner-Shirshov basis of the free associative $\Omega$-algebra $k\langle X ; \Omega\rangle$ in the sense of the paper [@BoCQ] with the ordering (\[o1\]). Let $H=k\langle X, a,b ;\Omega \| S, aab^{i}ab=x_{i}, \ i=1,2,\dots\rangle$. We can check that $\{S, aab^{i}ab=x_{i}, \ i=1,2,\dots\}$ is a Gröbner-Shirshov basis in the free associative $\Omega$-algebra $k\langle X,a,b ; \Omega \rangle$ since there are no new compositions. By the Composition-Diamond lemma in [@BoCQ], $A$ can be embedded into $H$ which is generated by $\{a,b\}$. $\blacksquare$ Every associative $\Omega$-algebra can be embedded into a simple associative $\Omega$-algebra. Proof Let $A$ be an associative $\Omega$-algebra over a field $k$ with $k$-basis $X=\{x_i\mid i\in I\}$ where $I$ is a well ordered set. Denote by $$\begin{aligned} S&=&\{x_ix_j=\{x_i,x_j\}, \delta_n(x_{k_1},\ldots,x_{k_n})= \{\delta_n(x_{k_1},\ldots,x_{k_n})\}\|\\ && \ \ \ \ \ \ \ \ \ i,j,k_1,\ldots,k_n\in I,\ \delta_n\in\Omega_n,n\in N\},\end{aligned}$$ where $\{\delta_n(x_{k_1},\ldots,x_{k_n})\}$ is a linear combination of $x_i, i\in I$. Then in the sense of the paper [@BoCQ], $S$ is a Gröbner-Shirshov basis in the free associative $\Omega$-algebra $k\langle X;\Omega \rangle$ with the ordering (\[o1\]). Therefore $A$ can be expressed as $$\begin{aligned} A=k\langle X;\Omega \|S\rangle.\end{aligned}$$ Let us totally order the set of monic elements of $A$. Denote by $T$ the set of indices for the resulting totally ordered set. Consider the totally ordered set $T^2=\{(\theta,\sigma)\}$ and assign $(\theta,\sigma)<(\theta',\sigma')$ if either $\theta<\theta'$ or $\theta=\theta'$ and $\sigma<\sigma'$. Then $T^2$ is also totally ordered set. For each ordered pair of elements $f_\theta, f_\sigma\in\ A, \ \theta, \sigma\in T$, introduce the letters $x_{\theta\sigma},y_{\theta\sigma}$. Let $A_{1}$ be the associative $\Omega$-algebra given by the generators $$X_1=\{x_i,y_{\theta\sigma},x_{\varrho\tau}\mid i\in I, \ \theta, \sigma, \varrho, \tau\in T\}$$ and the defining relations $S_1$ where $S_1$ is the union of $S$ and $$\begin{aligned} x_{\theta\sigma}f_\theta y_{\theta\sigma}=f_\sigma,\ ({\theta,\sigma})\in T^2.\end{aligned}$$ We can have that in $S_1$ there are no compositions unless for the ambiguity $x_{i}x_{j}x_{k}$. But this case is trivial. Hence $S_1$ is a Gröbner-Shirshov basis of the free associative $\Omega$-algebra $k\langle X_1; \Omega \rangle$ in the sense of the paper [@BoCQ] with the ordering (\[o1\]). Thus, by the Composition-Diamond lemma in [@BoCQ], $A$ can be embedded into $A_1$. The relations $x_{\theta\sigma}f_\theta y_{\theta\sigma}=f_\sigma$ of $A_1$ provide that in ${A}_1$ every monic element $f_\theta$ of the subalgebra ${A}$ generates an ideal containing algebra $A$. Mimicking the construction of the associative $\Omega$-algebra $A_1$ from the $A$, produce the associative $\Omega$-algebra $A_2$ from $A_1$ and so on. As a result, we acquire an ascending chain of associative $\Omega$-algebras $ A=A_0\subset A_1\subset A_2\subset\cdots$ such that every nonzero element generates the same ideal. Let $ {\cal A}=\bigcup_ {k=0}^ {\infty}A_k. $ Then ${\cal A}$ is a simple associative $\Omega$-algebra. $\blacksquare$ Every countably generated associative $\Omega$-algebra with countable multiple operations $\Omega$ over a countable field $k$ can be embedded into a simple two-generated associative $\Omega$-algebra. Proof Let $A$ be a countably generated associative $\Omega$-algebra with countable multiple operations $\Omega$ over a countable field $k$. We may assume that $A$ has a countable $k$-basis $X_0=\{x_{i}\| i=1,2,\ldots\}$. Denote by $$\begin{aligned} S&=&\{x_ix_j=\{x_i,x_j\}, \delta_n(x_{k_1},\ldots,x_{k_n})= \{\delta_n(x_{k_1},\ldots,x_{k_n})\}\|\\ && \ \ \ \ \ \ \ \ \ i,j,k_1,\ldots,k_n\in N,\ \delta_n\in\Omega_n,n\in N\}.\end{aligned}$$ Then $A$ can be expressed as $ A=k\langle X_0;\Omega \|S\rangle. $ Let $A_0=k\langle X_0;\Omega\rangle$, $A_0^+=A_0\backslash\{0\}$ and fix the bijection $$(A_0^+,A_0^+)\longleftrightarrow\{(x_m^{(1)},y_m^{(1)}), m\in N\}.$$ Let $X_1=X_0\cup\{x_m^{(1)},y_m^{(1)},a,b\|m\in N\}$, $A_1=k\langle X_1;\Omega\rangle$, $A_1^+=A_1\backslash\{0\}$ and fix the bijection $$(A_1^+,A_1^+)\longleftrightarrow\{(x_m^{(2)},y_m^{(2)}), m\in N\}.$$ $$\vdots$$ Let $X_{n+1}=X_n\cup\{x_m^{(n+1)},y_m^{(n+1)}\|m\in N\}$, $n\geq1$, $A_{n+1}=k\langle X_{n+1};\Omega\rangle$, $A_{n+1}^+=A_{n+1}\backslash\{0\}$ and fix the bijection $$(A_{n+1}^+,A_{n+1}^+)\longleftrightarrow\{(x_m^{(n+2)},y_m^{(n+2)}), m\in N\}.$$ $$\vdots$$ Consider the chain of the free associative $\Omega$-algebras $$A_0\subset A_1\subset A_2\subset\ldots\subset A_n\subset\ldots.$$ Let $X=\bigcup_{n=0}^{\infty}X_n$. Then $k\langle X;\Omega\rangle=\bigcup_ {n=0}^ {\infty}A_n$. Now, define the desired algebra $\mathcal{A}$. Take the set $X$ as the set of the generators for this algebra and take the union of $S$ and the following relations as one part of the relations for this algebra $$\label{a2} aa(ab)^nb^{2m+1}ab=x_m^{(n)},\ m,n\in N$$ $$\label{a3} aa(ab)^nb^{2m}ab=y_m^{(n)}, \ m,n\in N$$ $$\label{a4} aabbab=x_1$$ Before we introduce the another part of the relations on $\mathcal{A}$, let us define canonical words of the algebras $A_n$, $n\geq0$. A $\Omega$-word in $X_0$ without subwords that are the leading terms of $s\ (s\in S)$ is called a canonical word of $A_0$. A $\Omega$-word in $X_1$ without subwords that are the leading terms of $s\ (s\in S\cup\{(\ref{a2}), (\ref{a3}), (\ref{a4})\})$ and without subwords of the form $$(\delta_1(x_m^{(1)}))^{deg_{\Omega}(\overline{g^{(0)}})+1}\overline{f^{(0)}}y_m^{(1)},$$ where $(x_m^{(1)},y_m^{(1)})\longleftrightarrow(f^{(0)},g^{(0)})\in(A_0^+,A_0^+)$ such that $f^{(0)},g^{(0)}$ are non-zero linear combination of canonical words of $A_0$, is called a canonical word of $A_1$. Suppose that we have defined canonical word of $A_{k}$, $k<n$. A $\Omega$-word in $X_n$ without subwords that are the leading terms of $s\ (s\in S\cup\{(\ref{a2}), (\ref{a3}), (\ref{a4})\})$ and without subwords of the form $$(\delta_1(x_m^{(k+1)}))^{deg_{\Omega}(\overline{g^{(k)}})+1}\overline{f^{(k)}}y_m^{(k+1)},$$ where $(x_m^{(k+1)},y_m^{(k+1)})\longleftrightarrow(f^{(k)},g^{(k)})\in(A_k^+,A_k^+)$ such that $f^{(k)},g^{(k)}$ are non-zero linear combination of canonical words of $A_k$, is called a canonical word of $A_n$. Then the another part of the relations on $\mathcal{A}$ are the following: $$\label{a5} (\delta_1(x_m^{(n)}))^{deg_{\Omega}(\overline{g^{(n-1)}})+1}f^{(n-1)}y_m^{(n)}-g^{(n-1)}=0, \ \ m, n\in N$$ where $(x_m^{(n)},y_m^{(n)})\longleftrightarrow(f^{(n-1)},g^{(n-1)})\in(A_{n-1}^+,A_{n-1}^+)$ such that $f^{(n-1)},g^{(n-1)}$ are non-zero linear combination of canonical words of $A_{n-1}$. We can see that in $\mathcal{A}$ every element can be expressed as linear combination of canonical words. Denote by $S_1=S\cup\{(\ref{a2}), (\ref{a3}), (\ref{a4}),(\ref{a5})\}$. We can have that with the ordering (\[o1\]), $S_1$ is a Gröbner-Shirshov basis in $k \langle X;\Omega\rangle$ in the sense of the paper [@BoCQ] since in $S_1$ there are no compositions except for the ambiguity $x_{i}x_{j}x_{k}$ which is a trivial case. This implies that $A$ can be embedded into $\mathcal{A}$. By (\[a2\])-(\[a5\]), $\mathcal{A}$ is a simple associative $\Omega$-algebra generated by $\{a,b\}$. $\blacksquare$ Associative $\lambda$-differential algebras =========================================== In this section, by applying the Composition-Diamond lemma for associative $\Omega$-algebras in [@BoCQ], we show that: (i). Each countably generated associative $\lambda$-differential algebra can be embedded into a two-generated associative $\lambda$-differential algebra. (ii). Each associative $\lambda$-differential algebra can be embedded into a simple associative $\lambda$-differential algebra. (iii). Each countably generated associative $\lambda$-differential algebra over a countable field $k$ can be embedded into a simple two-generated associative $\lambda$-differential algebra. Let $k$ be a commutative ring with unit and $\lambda\in k$. An associative $\lambda$-differential algebra over $k$ ([@GuK08]) is an associative $k$-algebra $R$ together with a $k$-linear operator $D:R\rightarrow R$ such that $$D(xy)=D(x)y+xD(y)+\lambda D(x)D(y),\ \forall x, y \in R.$$ Any associative $\lambda$-differential algebra is also an associative algebra with one operator $\Omega=\{D\}$. In this section, we will use the notations given in the Section 6. Let $X$ be well ordered and $k\langle X;D\rangle$ the free associative algebra with one operator $\Omega=\{D\}$ defined in the Section 6. For any $u\in \mathfrak{S}(X)$, $u$ has a unique expression $$u=u_1u_2\cdots u_n,$$ where each $u_i\in X\cup D(\mathfrak{S}(X))$. Denote by $deg_{_{X}}(u)$ the number of $x\in X$ in $u$, for example, if $u=D(x_1x_2)D(D(x_1))x_3\in \mathfrak{S}(X)$, then $deg_{_{X}}(u)=4$. Let $$wt(u)=(deg_{_{X}}(u), u_1,u_2,\cdots, u_n).$$ Now, we order $\mathfrak{S}(X)$ as follows: for any $u,v\in \mathfrak{S}(X)$, $$\label{o2} u>v\Longleftrightarrow wt(u)>wt(v)\ \mbox{ lexicographically}$$ where for each $t, \ u_t>v_t$ if one of the following holds: \(a) $u_t, v_t\in X$ and $u_t>v_t$; \(b) $u_t=D(u_{t}^{'}), v_t\in X$; \(c) $u_t=D(u_{t}^{'}),v_t=D(v_{t}^{'})$ and $u_{t}^{'}>v_{t}^{'}$. Then the ordering (\[o2\]) is a monomial ordering on $\mathfrak{S}(X)$ (see [@BoCQ]). ([@BoCQ], Theorem 5.1)\[t3.5\] With the ordering (\[o2\]) on $\mathfrak{S}(X)$, $$S_0=\{D(uv)-D(u)v-uD(v)-\lambda D(u)D(v) \|\ u,v \in \mathfrak{S}(X)\}$$ is a Gröbner-Shirshov basis in the free $\Omega$-algebra $k\langle X;D\rangle$ where $\Omega=\{D\}$. \[l4.4\] Let $A$ be an associative $\lambda$-differential algebra with $k$-basis $X=\{x_{i}\| i\in I\}$. Then $A$ has a representation $A= k\langle X ; D \| S \rangle$, where $S=\{x_{i}x_{j}=\{x_{i},x_{j}\},D(x_{i})=\{D(x_{i})\}, D(x_{i}x_{j})=D(x_{i})x_{j}+x_{i}D(x_{j})+\lambda D(x_{i})D(x_{j})\mid i, j\in I\}$. Moreover, if $I$ is a well ordered set, then with the ordering (\[o2\]) on $\mathfrak{S}(X)$, $S$ is a Gröbner-Shirshov basis in the free $\Omega$-algebra $k\langle X; D\rangle$ in the sense of [@BoCQ]. Proof Clearly, $k\langle X ; D \| S \rangle$ is an associative $\lambda$-differential algebra. By the Composition-Diamond lemma in [@BoCQ], it suffices to check that with the ordering (\[o2\]) on $\mathfrak{S}(X)$, $S$ is a Gröbner-Shirshov basis in $k\langle X; D\rangle$ in the sense of [@BoCQ]. The ambiguities $w$ of all possible compositions of $\Omega$-polynomials in $S$ are: 1. $x_ix_jx_k,\ i,j,k\in I,$ 2. $D(x_ix_j),\ i,j\in I$. We will check that each composition in $S$ is trivial $mod(S,w)$. For 1), the result is trivial. For 2), let $f=D(x_{i}x_{j})-D(x_{i})x_{j}-x_{i}D(x_{j})-\lambda D(x_{i})D(x_{j})$, $g=x_{i}x_{j}-\{x_{i},x_{j}\},\ i,j\in I$. Then $w=D(x_{i}x_{j})$ and $$\begin{aligned} (f,g)_{w}&=&-D(x_{i})x_{j}-x_{i}D(x_{j})-\lambda D(x_{i})D(x_{j})+D(\{x_{i},x_{j}\})\\ &\equiv&\{D(\{x_{i},x_{j}\})\}-\{\{D(x_{i})\},x_{j}\}-\{x_{i},\{D(x_{j})\}\}-\lambda \{\{D(x_{i})\},\{D(x_{j})\}\}\\ &\equiv&0 \ \ mod(S,w).\end{aligned}$$ This shows that $S$ is a Gröbner-Shirshov basis in the free $\Omega$-algebra $k\langle X; D\rangle$. $\blacksquare$ Now we get the embedding theorems for associative $\lambda$-differential algebras. \[t3.6\] Every countably generated associative $\lambda$-differential algebra over a field can be embedded into a two-generated associative $\lambda$-differential algebra. Proof Let $A$ be a countably generated associative $\lambda$-differential algebra over a field $k$. We may assume that $A$ has a countable $k$-basis $X=\{x_{i}\| i=1,2,\ldots\}$. By Lemma \[l4.4\], $A= k\langle X ; D \| S \rangle$, where $S=\{x_{i}x_{j}=\{x_{i},x_{j}\},D(x_{i})=\{D(x_{i})\}, D(x_{i}x_{j})=D(x_{i})x_{j}+x_{i}D(x_{j})+\lambda D(x_{i})D(x_{j})\mid i, j\in N\}$. Let $H=k\langle X, a,b; D \| S_{1} \rangle$ where $$\begin{aligned} &&S_{1}=\{x_{i}x_{j}=\{x_{i},x_{j}\},D(x_{i})=\{D(x_{i})\}, D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\\ &&\ \ \ \ \ \ \ \ aab^{i}ab=x_{i}\| u, v \in \mathfrak{S}(X,a,b), i, j\in N\}.\end{aligned}$$ We want to prove that $S_{1}$ is also a Gröbner-Shirshov basis in the free $\Omega$-algebra $k\langle X,a,b;D\rangle$ with the ordering (\[o2\]). Now, let us check all the possible compositions in $S_{1}$. The ambiguities $w$ of all possible compositions of $\Omega$-polynomials in $S_{1}$ are:\ $$\begin{array}{lllll} 1)\ \ x_{i}x_{j}x_{k}& 2)\ D(u\|_{x_{i}x_{j}}v)& 3)\ D(uv\|_{x_{i}x_{j}})& 4)\ D(u\|_{D(x_{i})}v)& 5)\ D(uv\|_{D(x_{i})})\\ 6)\ D(uv\|_{D(u_{1}v_{1})})& 7) \ D(u\|_{D(u_{1}v_{1})}v)& 8)\ D(u\|_{aab^{i}ab}v)& 9)\ D(uv\|_{aab^{i}ab}) \end{array}$$ where $u,v,u_{1},v_{1}\in \mathfrak{S}(X,a,b),x_{i},x_{j},x_{k}\in X $. We have to check that all these compositions are trivial $mod(S_1,w)$. In fact, by Lemma \[t3.5\] and since $S$ is a Gröbner-Shirshov basis in $k\langle X ;D\rangle$, we need only to check $2)-5),8),9)$. Here, for example, we just check $3),4),8)$. Others are similarly proved. For 3), let $f=D(uv\|_{x_{i}x_{j}})-D(u)v\|_{x_{i}x_{j}}-uD(v\|_{x_{i}x_{j}})-\lambda D(u)D(v\|_{x_{i}x_{j}}), \ \ g=x_{i}x_{j}-\{x_{i},x_{j}\}, \ \ u, v\in \mathfrak{S}(X,a,b), x_{i},x_{j}\in X $. Then $w=D(uv\|_{x_{i}x_{j}})$ and $$\begin{aligned} (f,g)_{w}&=&- D(u)v\|_{x_{i}x_{j}} - uD(v\|_{x_{i}x_{j}})- \lambda D(u)D(v\|_{x_{i}x_{j}})+D(uv\|_{\{x_{i},x_{j}\}})\\ &\equiv&- D(u)v\|_{\{x_{i},x_{j}\}} - uD(v\|_{\{x_{i},x_{j}\}})- \lambda D(u)D(v\|_{\{x_{i},x_{j}\}})+D(uv\|_{\{x_{i},x_{j}\}})\\ &\equiv&0.\end{aligned}$$ For 4), let $f=D(u\|_{D(x_{i})}v)-D(u\|_{D(x_{i})})v-u\|_{D(x_{i})}D(v)- \lambda D(u\|_{D(x_{i})})D(v), \ \ g=D(x_{i})-\{D(x_{i})\},\ \ u,v,D(x_{i})\in \mathfrak{S}(X,a,b),x_{i}\in X $. Then $w=D(u\|_{D(x_{i})}v)$ and $$\begin{aligned} (f,g)_{w}&=&- D( u\|_{D(x_{i})})v - u\|_{D(x_{i})}D(v)- \lambda D(u\|_{D(x_{i})})D(v)+D(u\|_{\{D(x_{i})\}}v)\\ &\equiv&- D( u\|_{\{D(x_{i})\}})v - u\|_{\{D(x_{i})\}}D(v)- \lambda D(u\|_{\{D(x_{i})\}})D(v)+D(u\|_{\{D(x_{i})\}}v)\\ &\equiv&0.\end{aligned}$$ For 8), let $f=D(u\|_{aab^{i}ab}v) - D( u\|_{aab^{i}ab})v - u\|_{aab^{i}ab}D(v)-\lambda D(u\|_{aab^{i}ab})D(v) , \ g=aab^{i}ab-x_{i},\ u, v \in \mathfrak{S}(X,a,b),\ x_{i}\in X$. Then $w=D(u\|_{aab^{i}ab}v)$ and $$\begin{aligned} (f,g)_{w}&=&- D( u\|_{aab^{i}ab})v - u\|_{aab^{i}ab}D(v)- \lambda D(u\|_{aab^{i}ab})D(v)+D(u\|_{x_{i}}v)\\ &\equiv&D(u\|_{x_{i}}v)- D( u\|_{x_{i}})v - u\|_{x_{i}}D(v)- \lambda D(u\|_{x_{i}})D(v)\\ &\equiv&0.\end{aligned}$$ So $S_{1}$ is a Gröbner-Shirshov basis in $k\langle X,a,b;D\rangle$. By the Composition-Diamond lemma in [@BoCQ], $A$ can be embedded into $H$ which is generated by $\{a,b\}$. $\blacksquare$ \[t3.7\] Every associative $\lambda$-differential algebra over a field can be embedded into a simple associative $\lambda$-differential algebra. Proof Let $A$ be an associative $\lambda$-differential algebra over a field $k$ with basis $X=\{x_i\mid i\in I\}$ where $I$ is a well ordered set. Then by Lemma \[l4.4\], $A$ can be expressed as $A= k\langle X ; D \| S \rangle$ where $S=\{x_{i}x_{j}=\{x_{i},x_{j}\},D(x_{i})=\{D(x_{i})\}, D(x_{i}x_{j})=D(x_{i})x_{j}+x_{i}D(x_{j})+\lambda D(x_{i})D(x_{j})\mid i, j\in I\}$ and $S$ is a Gröbner-Shirshov basis in $k \langle X ;D\rangle$ with the ordering (\[o2\]). Let us totally order the set of monic elements of ${A}$. Denote by $T$ the set of indices for the resulting totally ordered set. Consider the totally ordered set $T^2=\{(\theta,\sigma)\}$ and assign $(\theta,\sigma)<(\theta',\sigma')$ if either $\theta<\theta'$ or $\theta=\theta'$ and $\sigma<\sigma'$. Then $T^2$ is also totally ordered set. For each ordered pair of elements $f_\theta, f_\sigma\in\ A,\ \theta, \sigma\in T$, introduce the letters $x_{\theta\sigma},y_{\theta\sigma}$. Let $A_{1}$ be the associative $\lambda$-differential algebra given by the generators $$X_1=\{x_i, y_{\theta\sigma}, x_{\varrho\tau} \| i\in I, \ \theta, \sigma, \varrho, \tau\in T\}$$ and the defining relations $$x_ix_j=\{x_i,x_j\}, \ \ i,j\in I,$$ $$D(x_{i})=\{D(x_{i})\}, \ \ i\in I,$$ $$D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\ \ u,v \in \mathfrak{S}(X_1),$$ $$x_{\theta\sigma}f_\theta y_{\theta\sigma}=f_\sigma, \ \ ({\theta,\sigma})\in T^2.$$ We want to prove that these relations is also a Gröbner-Shirshov basis in $k \langle X_1 ;D\rangle$ with the same ordering (\[o2\]). Now, let us check all the possible compositions. The ambiguities $w$ of all possible compositions of $\Omega$-polynomials are:\ $$\begin{array}{lllll} 1) \ x_{i}x_{j}x_{k}& 2)\ D(u\|_{x_{i}x_{j}}v)& 3)\ D(uv\|_{x_{i}x_{j}})& 4)\ D(u\|_{D(x_{i})}v)& 5)\ D(uv\|_{D(x_{i})})\\ 6)\ D(uv\|_{D(u_{1}v_{1})})& 7)\ D(u\|_{D(u_{1}v_{1})}v)& 8)\ D(u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}}v)& 9)\ D(uv\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}}) \end{array}$$ where $u,v,u_{1},v_{1}\in \mathfrak{S}(X_1), x_{i}, x_{j}, x_{k}\in X, ({\theta,\sigma})\in T^2$. In fact, by Lemma \[t3.5\] and since $S$ is a Gröbner-Shirshov basis in $K\langle X ;D\rangle$, we just need to check $2)-5),8),9)$. Here, for example, we just check $8)$. Others are similarly proved. Let $f=D(u\|_{x_{\theta\sigma}\overline{f_\theta }y_{\theta\sigma}}v) - D( u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}})v - u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}}D(v)-\lambda D(u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}})D(v), \ g=x_{\theta\sigma}f_\theta y_{\theta\sigma}-f_\sigma=x_{\theta\sigma}\overline{f_\theta }y_{\theta\sigma}+x_{\theta\sigma}f_\theta'y_{\theta\sigma}-f_\sigma$, where $f_\theta=\overline{f_\theta }+f_\theta'$, $u, v \in \mathfrak{S}(X_1), ({\theta,\sigma})\in T^2$. Then $w=D(u\|_{x_{\theta\sigma}\overline{f_\theta }y_{\theta\sigma}}v)$ and $$\begin{aligned} (f,g)_{w}&=&- D( u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}})v - u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}}D(v)-\lambda D(u\|_{x_{\theta\sigma}\overline{f_\theta} y_{\theta\sigma}})D(v)+ D(u\|_{(-x_{\theta\sigma}f_\theta'y_{\theta\sigma}+f_\sigma)}v)\\ &\equiv&D(u\|_{(-x_{\theta\sigma}f_\theta'y_{\theta\sigma}+f_\sigma)}v)- D( u\|_{(-x_{\theta\sigma}f_\theta'y_{\theta\sigma}+f_\sigma)})v - u\|_{(-x_{\theta\sigma}f_\theta'y_{\theta\sigma}+f_\sigma)}D(v)\\ &&- \lambda D(u\|_{(-x_{\theta\sigma}f_\theta'y_{\theta\sigma}+f_\sigma)})D(v)\\ &\equiv&0.\end{aligned}$$ Thus, by the Composition-Diamond lemma in [@BoCQ], $A$ can be embedded into $A_1$. The relations $x_{\theta\sigma}f_\theta y_{\theta\sigma}=f_\sigma$ of $A_1$ provide that in ${A}_1$ every monic element $f_\theta$ of the subalgebra ${A}$ generates an ideal containing algebra $A$. Mimicking the construction of the associative $\lambda$-differential algebra $A_1$ from the $A$, produce the associative $\lambda$-differential algebra $A_2$ from $A_1$ and so on. As a result, we acquire an ascending chain of associative $\lambda$-differential algebras $ A=A_0\subset A_1\subset A_2\subset\dots$ such that every nonzero element generates the same ideal. Let $ {\cal A}=\bigcup_ {k=0}^ {\infty}A_k. $ Then ${\cal A}$ is a simple associative $\lambda$-differential algebra. $\blacksquare$ Every countably generated associative $\lambda$-differential algebra over a countable field $k$ can be embedded into a simple two-generated associative $\lambda$-differential algebra. Proof Let $A$ be a countably generated associative $\lambda$-differential algebra over a countable field $k$. We may assume that $A$ has a countable $k$-basis $X_0=\{x_{i}\| i=1,2,\ldots\}$ and it can be expressed as, by Lemma \[l4.4\], $A= k\langle X_0 ; D \| S_0 \rangle$ where $S_0=\{x_{i}x_{j}=\{x_{i},x_{j}\},D(x_{i})=\{D(x_{i})\}, D(x_{i}x_{j})=D(x_{i})x_{j}+x_{i}D(x_{j})+\lambda D(x_{i})D(x_{j})\mid i, j\in N\}$ and $S_0$ is a Gröbner-Shirshov basis in $k \langle X_0 ;D\rangle$ with the ordering (\[o2\]). Let $A_0=k\langle X_0;D \rangle$, $A_0^+=A_0\backslash\{0\}$ and fix the bijection $$(A_0^+,A_0^+)\longleftrightarrow\{(x_m^{(1)},y_m^{(1)}), m\in N\}.$$ Let $X_1=X_0\cup\{x_m^{(1)},y_m^{(1)},a,b\|m\in N\}$, $A_1=k\langle X_1;D\rangle$, $A_1^+=A_1\backslash\{0\}$ and fix the bijection $$(A_1^+,A_1^+)\longleftrightarrow\{(x_m^{(2)},y_m^{(2)}), m\in N\}.$$ $$\vdots$$ Let $X_{n+1}=X_n\cup\{x_m^{(n+1)},y_m^{(n+1)}\|m\in N\}$, $n\geq1$, $A_{n+1}=k\langle X_{n+1};D \rangle$, $A_{n+1}^+=A_{n+1}\backslash\{0\}$ and fix the bijection $$(A_{n+1}^+,A_{n+1}^+)\longleftrightarrow\{(x_m^{(n+2)},y_m^{(n+2)}), m\in N\}.$$ $$\vdots$$ Consider the chain of the free $\Omega$-algebras $$A_0\subset A_1\subset A_2\subset\ldots\subset A_n\subset\ldots.$$ Let $X=\bigcup_ {n=0}^ {\infty}X_n$. Then $k\langle X;D\rangle=\bigcup_ {n=0}^ {\infty}A_n$. Now, define the desired algebra $\mathcal{A}$. Take the set $X$ as the set of the generators for this algebra and take the following relations as one part of the relations for this algebra $$\label{b11} x_{i}x_{j}=\{x_{i},x_{j}\}, D(x_{i})=\{D(x_{i})\},\ i,j\in N$$ $$\label{b12} D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\ u,v \in \mathfrak{S}(X)$$ $$\label{b2} aa(ab)^nb^{2m+1}ab=x_m^{(n)},\ m,n\in N$$ $$\label{b3} aa(ab)^nb^{2m}ab=y_m^{(n)}, \ m,n\in N$$ $$\label{b4} aabbab=x_1$$ Before we introduce the another part of the relations on $\mathcal{A}$, let us define canonical words of the algebras $A_n$, $n\geq0$. A $\Omega$-word in $X_0$ without subwords that are the leading terms of (\[b11\]) and (\[b12\]) is called a canonical word of $A_0$. A $\Omega$-word in $X_1$ without subwords that are the leading terms of (\[b11\]), (\[b12\]), (\[b2\]), (\[b3\]), (\[b4\]) and without subwords of the form $$(x_m^{(1)})^{deg_X(\overline{g^{(0)}})}\overline{f^{(0)}}y_m^{(1)},$$ where $(x_m^{(1)},y_m^{(1)})\longleftrightarrow(f^{(0)},g^{(0)})\in(A_0^+,A_0^+)$ such that $f^{(0)},g^{(0)}$ are non-zero linear combination of canonical words of $A_0$, is called a canonical word of $A_1$. Suppose that we have defined canonical word of $A_{k}$, $k<n$. A $\Omega$-word in $X_n$ without subwords that are the leading terms of (\[b11\]), (\[b12\]), (\[b2\]), (\[b3\]), (\[b4\]) and without subwords of the form $$(x_m^{(k+1)})^{deg_X(\overline{g^{(k)}})}\overline{f^{(k)}}y_m^{(k+1)},$$ where $(x_m^{(k+1)},y_m^{(k+1)})\longleftrightarrow(f^{(k)},g^{(k)})\in(A_k^+,A_k^+)$ such that $f^{(k)},g^{(k)}$ are non-zero linear combination of canonical words of $A_k$, is called a canonical word of $A_n$. Then the another part of the relations on $\mathcal{A}$ are the following: $$\label{b5} (x_m^{(n)})^{deg_X(\overline{g^{(n-1)}})}f^{(n-1)}y_m^{(n)}-g^{(n-1)}=0,\ \ m, n\in N$$ where $(x_m^{(n)},y_m^{(n)})\longleftrightarrow(f^{(n-1)},g^{(n-1)})\in(A_{n-1}^+,A_{n-1}^+)$ such that $f^{(n-1)},g^{(n-1)}$ are non-zero linear combination of canonical words of $A_{n-1}$. We can get that in $\mathcal{A}$ every element can be expressed as linear combination of canonical words. Denote by $S$ the set constituted by the relations (\[b11\])-(\[b5\]). We want to prove that $S$ is also a Gröbner-Shirshov basis in the free $\Omega$-algebra $k\langle X;D\rangle$ with the ordering (\[o2\]). The ambiguities $w$ of all possible compositions of $\Omega$-polynomials in $S$ are:\ $$\begin{array}{lll} 1)\ \ x_{i}x_{j}x_{k}& 2)\ D(u\|_{x_{i}x_{j}}v)& 3)\ D(uv\|_{x_{i}x_{j}})\\ 4)\ D(u\|_{D(x_{i})}v)& 5)\ D(uv\|_{D(x_{i})})& 6)\ D(uv\|_{D(u_{1}v_{1})})\\ 7) \ D(u\|_{D(u_{1}v_{1})}v)& 8)\ D(u\|_{aa(ab)^nb^{2m+1}ab}v)& 9)\ D(uv\|_{aa(ab)^nb^{2m+1}ab}) \\10)\ D(u\|_{aa(ab)^nb^{2m}ab}v)& 11)\ D(uv\|_{aa(ab)^nb^{2m}ab}) &12)\ D(u\|_{aabbab}v)\\ 13)\ D(uv\|_{aabbab})& 14)\ D(u\|_{(x_m^{(n)})^{deg_X(\overline{g^{(n-1)}})}\overline{f^{(n-1)}}y_m^{(n)}}v)& 15)\ D(uv\|_{(x_m^{(n)})^{deg_X(\overline{g^{(n-1)}})}\overline{f^{(n-1)}}y_m^{(n)}}) \end{array}$$ where $u,v,u_{1},v_{1}\in \mathfrak{S}(X_1), x_{i}, x_{j}, x_{k}\in X$. The proof of all possible compositions to be trivial $mod(S,w)$ is similar to that of Theorems \[t3.6\], \[t3.7\]. Here we omit the details. So $S$ is a Gröbner-Shirshov basis in $k\langle X;D\rangle$ with the ordering (\[o2\]), which implies that $A$ can be embedded into $\mathcal{A}$. By (\[b2\])-(\[b5\]), $\mathcal{A}$ is a simple associative $\lambda$-differential algebra generated by $\{a,b\}$. $\blacksquare$ Modules ======== In this section, by applying the Composition-Diamond lemma for modules (see [@ChCZ; @Ch]), we show that every countably generated $k\langle X \rangle$-module can be embedded into a cyclic $k\langle X \rangle$-module, where $\|X\|>1$. Let $X,Y$ be well ordered sets and $mod_{k\langle X\rangle} \langle Y\rangle$ a free left $k\langle X\rangle$-module with the basis $Y$. Suppose that $<$ is the deg-lex ordering on $X^$. Let $X^Y=\{uy\|u\in X^, \ y\in Y\}$. We define an ordering $\prec$ on $X^Y$ as follows: for any $w_1=u_1y_i,w_2=u_2y_j\in X^Y$, $$\label{03} w_1\prec w_2\Leftrightarrow u_1<u_2 \ \ \mbox{ or } u_1=u_2, \ y_i<y_j$$ It is clear that the ordering $\prec$ is left compatible in the sense of $$w\prec w'\Rightarrow aw \prec aw' \ \mbox{ for any } a\in X^.$$ \[t8.8\] Let $X$ be a set with $\|X\|>1$. Then every countably generated $k\langle X \rangle$-module can be embedded into a cyclic $k\langle X \rangle$-module. Proof We may assume that $M=Mod_{k\langle X \rangle}\langle Y\| T \rangle$ where $Y=\{y_{i},i=1,2,\dots\}$. By Shirshov algorithm, we may assume that $T$ is a Gröbner-Shirshov basis in the free module $Mod_{k\langle X \rangle}\langle Y\rangle$ in the sense of the paper [@ChCZ] with the ordering (\[03\]) on $X^Y$. Assume that $ a, b\in X,\ a\neq b$. Consider the ${k\langle X \rangle}$-module $$_{{k\langle X \rangle}}M'=Mod_{k\langle X \rangle} \langle Y, y \| T,\ ab^{i}y-y_{i},y_{i}\in Y, i=1,2,\dots \rangle.$$ We can check that $\{T,\ ab^{i}y-y_{i},i=1,2,\dots\}$ is also a Gröbner-Shirshov basis in the free module $Mod_{k\langle X \rangle} \langle Y, y \rangle $ with the same ordering (\[03\]) on $X^(Y\cup\{y\})$ since there are no new compositions. By the Composition-Diamond lemma in [@ChCZ], $M$ can be embedded into $_{{k\langle X \rangle}}M'$ which is a cyclic $k\langle X \rangle$-module generated by $y$. $\blacksquare$ Remark In Theorem \[t8.8\], the condition $\|X\|>1$ is essential. For example, let $_{k[x]}M=\oplus_{i\in I}k[x]y_i$ be a free $k[x]$-module with $k[x]$-basis $Y=\{y_i\|i\in I\}$, where $\|I\|>1$. Then $_{k[x]}M$ can not be embedded into a cyclic $k[x]$-module. Indeed, suppose that $_{k[x]}M$ can be embedded into a cyclic $k[x]$-module $k[x]y$. Let $y_1,y_2\in Y$ with $y_1\neq y_2$. 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Shestakov, A problem of Shirshov, [Algebra Logic]{}, [16]{}, 153-166 (1978). A.I. Shirshov, On free Lie rings, [Mat. Sb.]{}, [45]{}, 13-21 (1958). A.I. Shirshov, On special J-rings, [Mat. Sbornik]{}, [38]{}, 149-166 (1956). A.I. Shirshov, Some algorithmic problem for Lie algebras, [Sibirsk. Mat. Z.]{}, [3]{}, 292-296 (1962) (in Russian); English translation in [SIGSAM Bull.]{}, [33]{}(2), 3-6 (1999). L.A. Skornyakov, Non-associative free T-sums of fields, [Mat. Sb.]{}, [44]{}, 297-312 (1958). E.G. Shutov, Embeddings of semigroups in simple and complete semigroups, [Mat. Sb.]{}, [62]{}(4), 496-511 (1963). V.A. Ufnarovski, combinatorial and Asymptotic Methods in Algebra, [Encyclopaedia Mat. Sci.]{}, [57]{}, 1-196 (1995). [^1]: Supported by RFBR 01-09-00157, LSS–344.2008.1 and SB RAS Integration grant No. 2009.97 (Russia). [^2]: Corresponding author. [^3]: Supported by the NNSF of China (No.10771077) and the NSF of Guangdong Province (No.06025062).
--- abstract: 'The scarcity of large parallel corpora is an important obstacle for neural machine translation. A common solution is to exploit the knowledge of language models () trained on abundant monolingual data. In this work, we propose a novel approach to incorporate a as prior in a neural translation model (). Specifically, we add a regularization term, which pushes the output distributions of the to be probable under the prior, while avoiding wrong predictions when the “disagrees” with the . This objective relates to knowledge distillation, where the can be viewed as teaching the about the target language. The proposed approach does not compromise decoding speed, because the is used only at training time, unlike previous work that requires it during inference. We present an analysis on the effects that different methods have on the distributions of the . Results on two low-resource machine translation datasets show clear improvements even with limited monolingual data.' author: - 'Christos Baziotis, Barry Haddow' - \| Alexandra Birch\ Institute for Language, Cognition and Computation\ School of Informatics, University of Edinburgh\ 10 Crichton Street, Edinburgh EH8 9AB\ `c.baziotis@sms.ed.ac.uk`,\ `bhaddow@inf.ed.ac.uk`, `a.birch@ed.ac.uk` bibliography: - 'refs.bib' title: 'Language Model Prior for Low-Resource Neural Machine Translation' ---
--- abstract: 'Using molecular dynamics computer simulations we investigate the dynamics of supercooled silica in the frequency range 0.5-20 THz and the wave-vector range 0.13-1.1Å$^{-1}$. We find that for small wave-vectors the dispersion relations are in very good agreement with the ones found in experiments and that the frequency at which the boson-peak is observed shows a maximum at around 0.39Å$^{-1}$.' address: 'Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, D-55099 Mainz, Germany' author: - 'Jürgen Horbach, Walter Kob and Kurt Binder' title: 'The Dynamics of Supercooled Silica: Acoustic modes and Boson peak' --- Introduction ============ In the last few years a significant effort was undertaken to understand the nature of the so-called boson-peak, a prominent dynamical feature at around 1 THz which is observed in strong glassformers [@boson_peak; @taraskin97ab; @wischnewski97; @dellanna97]. Various theoretical approaches have been proposed to explain this peak, such as localized vibrational modes or scattering of acoustic modes, but so far no clear picture has emerged yet. Recently also computer simulations have been used in order to gain insight into the mechanism that gives rise to this peak, but due to the high cooling rates with which the samples were prepared (on the order of $10^{12}$K/s) and small system sizes (20-40Å) the results of these investigations were not able to give a final answer either [@taraskin97ab; @dellanna97]. In the present work we present the results of a large scale computer simulation of supercooled silica. At the temperature investigated we are still able to fully equilibrate the sample, thus avoiding the problem of the high cooling rates, and by using a large system we can minimize the possibility of finite size effects [@horbach96]. Thus, by making a large computational effort, we are able to study the dynamics of this strong glassformer in a frequency and wave-vector range which is not accessible to real experiments and can therefore investigate the properties of the boson-peak in greater detail than was possible so far. Details of the Simulations ========================== The silica model we use for our simulation is the one proposed by van Beest [et al.]{} [@beest90] and has been shown to give a good description of the static properties of silica glass [@vollmayr96] as well as the dynamical properties of the supercooled melt, such as the activation energy of the diffusion constant [@horbach97]. In this model the potential between ions $i$ and $j$ is given by $$\phi(r_{ij})=\frac{q_i q_j e^2}{r_{ij}}+A_{ij}e^{-B_{ij}r_{ij}}- \frac{C_{ij}}{r_{ij}^6}\quad . \label{eq1}$$ The values of the parameters $q_i$, $A_{ij}$, $B_{ij}$ and $C_{ij}$ can be found in the original publication [@beest90]. The non-Coulombic part of the potential was truncated and shifted at 5.5Å. The simulations were done at constant volume and the density of the system was fixed to 2.3 g/cm$^3$. The system size was 8016 ions, giving a size of the box of (48.37Å)$^3$, and the equations of motion were integrated over $4\cdot 10^6$ times steps of 1.6fs, thus over a time span of 6.4ns. This time is sufficiently long to fully equilibrate the system at 2900K, the temperature considered in this study [@horbach97]. More details on the simulations can be found in Ref. [@horbach98]. Results ======= In the present work we study the dynamics of the system by means of $J_L(q,\nu)$ and $J_T(q,\nu)$, the longitudinal and transverse current-current correlation functions for wave-vector $q$ at frequency $\nu$ [@boon80]. These are defined as the longitudinal and transverse part of the current-current correlation function, i.e. $$J_{\alpha}(q,\nu)=N^{-1}\int_{-\infty}^{\infty} dt \exp(i2\pi\nu t) \sum_{kl} \langle {\bf u}_k(t) \cdot {\bf u}_l(0) \exp(i{\bf q} \cdot [{\bf r}_k(t)-{\bf r}_l(0)]) \rangle \quad,$$ where ${\bf u}_k(t)$ is equal to ${\bf q}\cdot \dot{\bf{r}}_k(t)/q$ for $\alpha=L$ and equal to ${\bf q}\times \dot{{\bf r}}_k(t)/q$ for $\alpha=T$. Note that $J_L(q,\nu)$ is also equal to $\nu^2 S(q,\nu)/q^2$, where $S(q,\nu)$ is the dynamical structure factor as measured in scattering experiments. In the following we will focus on the silicon-silicon correlation only, but we have found that the oxygen-oxygen correlation function behave very similarly. In Fig. 1 we show the frequency dependence of $J_L(q,\nu)$ for wave-vectors between 0.13Å$^{-1}$, the smallest wave-vector compatible with our box, and 0.8Å$^{-1}$, a wave-vector which is still significantly smaller than the location of the first sharp diffraction peak in $S(q)$, which is around 1.6Å$^{-1}$. From the figure we recognize that this correlation function has a peak at a frequency $\nu_L(q)$ which increases with increasing $q$ and which correspond to the longitudinal acoustic modes. A similar picture is obtained for $J_T(q,\nu)$ [@horbach98]. The fact that also this correlation function shows an acoustic mode shows that even at this relatively high temperatures the system is able to sustain transverse acoustic modes and thus is visco-elastic. The boson-peak is seen best in the dynamic structure factor $S(q,\nu)$, which is shown in Fig. 2 for small values of $q$. From this figure we see that $\nu_{BP}(q)$, the location of the boson-peak, is $q$-dependent in that it moves from small frequencies for small values of $q$ (dashed lines) to a maximum frequency for $q\approx 0.39$Å$^{-1}$ (bold dotted line) and then back to small frequencies for large values of $q$ (solid lines) (see also inset of Fig. 3). Also included is the location of the boson-peak as determined from neutron scattering experiments at 1673K which is around 1.5 THz [@wischnewski97]. We will comment more on this work below. From the wave-vector dependence of $\nu_L$ and $\nu_T$ we obtain the dispersion relation which is presented in Fig. 3. We see that, as expected, for small wave-vectors $\nu_L$ depends linearly on $q$. Also included in the figure is a line with slope $c_L$=6370m/s, the experimental value of the longitudinal sound velocity of silica at around 1600K[@wischnewski97]. We see that the data points for $\nu_L$ for small $q$ are very close to this line and thus we conclude that the sound velocity of this system is very close to the one of real silica, thus giving further support for the validity of the model potential. For the transverse acoustic modes the agreement between the experiment and the simulation data is a bit inferior, but still good. We also note that for wave-vectors larger than 1.4Å$^{-1}$, a bit less than the location of the first sharp diffraction peak, $\nu_L$ and $\nu_T$ do not increase anymore. The reason for this is likely the fact that at this $q$ value the system has a quasi-Brillouin zone [@taraskin97ab]. Also included in the figure is $\nu_{BP}$, the location of the boson-peak. We find that for large wave-vectors $\nu_{BP}$ is around 1.8 THz, a value that is a bit larger than the experimental value of 1.5 THz reported by Wischnewski [et al.]{} at 1673K [@wischnewski97]. However, these authors also found that $\nu_{BP}$ increases with increasing temperature and a rough extrapolation of their data for $\nu_{BP}$ to T=2900K shows that a value of 1.8 THz is quite reasonable, thus giving further support for the validity of our model. In the inset of Fig. 3 we show the dispersion curves at small values of $q$. Interestingly we observe that $\nu_{BP}(q)$ shows a maximum at around $q\approx 0.39$Å$^{-1}$. This maximum might be due to the fact that the mechanism leading to the boson-peak is particularly effective at this wave-vector or that there are [two]{} mechanisms giving rise to the boson-peak, one dominating at small wave-vectors and the second one dominating at larger wave-vectors and that in the vicinity of $q\approx 0.39$Å$^{-1}$ the sum of the contribution of the two mechanisms is largest. Finally we mention that at small wave-vectors the curve for $\nu_{BP}(q)$ seems to join smoothly the one for $\nu_L$. Within the accuracy of our data it is not clear, whether the boson-peak and the longitudinal acoustic mode become identical or whether the boson-peak ceases to exist for wave-vectors smaller than approximately 0.2Å$^{-1}$ and thus we cannot use this feature to exclude one of the possible mechanisms proposed for the boson-peak. To summarize we can say that our simulation allows to investigate the dynamics of supercooled silica in a wave-vector and frequency range which is not accessible to real experiments. We find that the dispersion relations for the longitudinal and transverse acoustic modes agree very well with the experimental values and that $\nu_{BP}$ shows a maximum at around 0.39Å$^{-1}$. This $q$ corresponds to a length scale on the order of 15Å. Thus we have evidence that the mechanism leading to the boson-peak is very effective on the length scale of several tetrahedra. Acknowledgements: We thank U. Buchenau, G. Ruocco and F. Sciortino for valuable discussions and the DFG, through SFB 262, and the BMBF, through grant 03N8008C, for financial support. U. Buchenau, M. Prager, N. Nücker, A. J. Dianoux, N. Ahmad and W. A. Phillips, Phys. Rev. B. [34]{}, 5665 (1986); P. Benassi, M. Krisch, C. Masciovecchio, V. Mazzacurati, G. Monaco, G. Ruocco, F. Sette, and R. Verbeni, Phys. Rev. Lett. [77]{}, 3835 (1996); M. Foret, E. Courtens, R. Vacher, and J.-B. Suck, Phys. Rev. Lett. [77]{}, 3831 (1996). S. N. Taraskin and S. R. Elliott, Europhys. Lett. [39]{}, 37 (1997); S. N. Taraskin and S. R. Elliott (preprint 1997). A. Wischnewski, U. Buchenau, A. J. Dianoux, W. A. Kamitakahara, and J. L. Zarestky (preprint 1997). R. Dell’Anna, G. Ruocco, M. Sampoli, and G. Viliani, (preprint 1997). J. Horbach, W. Kob, K. Binder and C.A. Angell, Phys. Rev. E. [54]{}, R5897 (1996). B. W. H. van Beest, G. J. Kramer and R. A. van Santen, Phys. Rev. Lett. [64]{}, 1955 (1990). K. Vollmayr, W. Kob and K. Binder, Phys. Rev. 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--- abstract: 'We study the problem of a particle hopping on the Bethe lattice in the presence of a Coulomb potential. We obtain an exact solution to the particle’s Green’s function along with the full energy spectrum. In addition, we present a mapping of a generalized radial potential problem defined on the Bethe lattice to an infinite number of one dimensional problems that are easily accessible numerically. The latter method is particularly useful when the problem admits no analytical solution.' author: - Olga Petrova - Roderich Moessner bibliography: - 'bethebib.bib' title: 'The Coulomb potential $V(r)=1/r$ and other radial problems on the Bethe lattice' --- Introduction ============ In many body physics, exactly solvable models are few and far between. In their absence, a common strategy to pursue is the use of mean field approximations [@Weiss1907], where the interactions between a finite number of the system’s constituents and the rest are modeled through an effective field approximating the effects of the latter on the former. One of such approaches bears the name of its inventor, Hans Bethe [@Bethe1935], and turns out to be exact on cycle-free graphs [@Kurata1953]. This coined the term “Bethe lattice”, which, despite its seeming unphysicalness, has since been successfully used to describe a plethora of physical phenomena, including excitations in antiferromagnets [@Brinkman], Anderson localization [@0022-3719-6-10-009], percolation [@percolation], and hopping of ions in ice [@Chen] to name a few. In particular, the Coulomb potential problem on the Bethe lattice, first appearing in [@Gallinar] and addressed in this work, recently surfaced in a study of quantum spin ice by the authors [@diluted]. In this paper we treat the problem of a single particle hopping on the Bethe lattice in the presence of a radial potential: $$H=T+V(n) \label{eq:Hgen}$$ where $T$ is $-t$ times the adjacency matrix, and the potential $V(n)$ is a function of the Bethe lattice generation $n$ only – i.e., a radial potential. The two central results of this work are: (a) a mapping of the general radial $V(n)$ problem to a family of one dimensional chains, and (b) the exact solution for the attractive Coulomb potential, $V(n)=\frac{C}{n}$ where $C<0$. The latter comes in the form of a closed form expression for the lattice Green’s function. With that, one can obtain the energy levels of the model, and the local density of states as a function of $n$. The mapping of the Bethe lattice model (\[eq:Hgen\]) to a family of $1d$ problems, is particularly useful when the exact solution to the problem cannot be obtained. Normally, numerical treatments of Bethe lattice models are of limited use owing to the large fraction (no less than half the total) of vertices at the edges. Mapping to $1d$ chains lets us circumvent this obstacle. The paper is structured as follows. In Section \[sec:tightbinding\] we discuss some features common to all models where a single particle is hopping on the Bethe lattice, or a finite Cayley tree, in the presence of a radial potential: the symmetries of the Hamiltonian (\[eq:Hgen\]) (\[sec:symmetries\]), the mapping of different symmetry sectors to a family of $1d$ problems (\[sec:1Dmapping\]), and the continued fraction technique to solving Eq. (\[eq:Hgen\]) perturbatively (\[sec:greens\]). In the technical heart of the paper, Section \[sec:coulomb\], we present the exact solution to the Coulomb potential problem on the Bethe lattice, obtained by carrying out the aforementioned perturbative calculation to infinite order. We summarize our findings and discuss possible applications in Section \[sec:conclusion\]. Tight-binding radial Hamiltonians on the Bethe lattice {#sec:tightbinding} ====================================================== In this Section we discuss some general properties of radial tight-binding models (\[eq:Hgen\]) defined on the Bethe lattice, a rooted infinite cycle-free graph with coordination number $z$ and connectivity $K$, defined as $K=z-1$. The root vertex is labeled by 1. To simplify further discussion, we reduce the coordination number at 1 to $z-1$, as shown in Fig. \[fig:lattice\](a), such that every vertex of the Bethe lattice generation $n$ is connected to $K$ vertices at generation $(n+1)$. Generations $n=1$ and $n=2$ correspond to the root and its nearest neighbors respectively. Due to the absence of closed cycles in the Bethe lattice, there is a unique path connecting any two vertices. The distance from the root to a given node is therefore given by the node’s generation $n-1$. Much of the discussion presented here applies to finite Cayley trees as well: the symmetry analysis (\[sec:symmetries\]) carries over directly, whereas the $1d$ chains and the continued fractions that one gets from the mapping (\[sec:1Dmapping\]) and the expansion of the Green’s function in powers of $t$ (\[sec:greens\]) respectively, are finite, rather than infinite, for the Cayley tree problems. One particularly simple result that follows is that a Cayley tree with $M$ generations has $K^{M-2}$ degenerate edge modes with energy $V(M)$. Symmetries of the model {#sec:symmetries} ----------------------- Here and in Section \[sec:1Dmapping\], we treat the $z=3$ case for concreteness unless stated otherwise. We refer to two vertices at generation $(r+1)$ as siblings if they are connected to the same parent vertex at generation $r$. Exchanging left and right siblings at a given generation leaves the Hamiltonian (\[eq:Hgen\]) invariant [@Chen]. These operations commute, which allows us to associate a separate quantum number, denoting the parity under such exchanges, with each generation $n>1$. Each such symmetry sector can be mapped onto a $1d$ half-line, whose origin is offset by the number equal to the highest Bethe lattice generation with an odd quantum number for a given sector. To understand the physical meaning of the mapping to follow, consider three vertices depicted in Fig. \[fig:lattice\](b): parent site 1 at generation $r$, and siblings 2 and 3 at generation $r+1$ that are connected to it. Let us construct two states: a symmetric and an antisymmetric combination of the particle being at sites 2 and 3, $\|\Psi_S\rangle$ and $\|\Psi_A\rangle$ respectively. When the hopping term $T$ in Hamiltonian (\[eq:Hgen\]) acts on $\|\Psi_S\rangle$, the contributions from sites 2 and 3 add up enabling the particle to hop back to the parent site 1 with amplitude $2t$. By contrast, $T$ acting on $\|\Psi_A\rangle$ results in back-hopping canceling out. From the extension of this argument to the entire generation of vertices rather than a single sibling pair, it follows that a particle starting out at a state that is odd at the $k^\mathrm{th}$ generation cannot hop to generations $r<k$. Therefore, the particle’s wavefunction has zero amplitude at all generations $r<k$: in particular, only the all even states have a nonzero amplitude at the origin. This property allows us to make a statement about the degeneracies of the energy levels: if the sector’s highest generation with an odd quantum number is $n$, then each of the lower generations $r>2$ contributes a factor of 2 to the total degeneracy of that sector’s energy levels. ![Left: the first four generations of the Bethe lattice with $z=3$. The origin ($r=1$) has coordination number $z-1$. Right: a three site fragment of the Bethe lattice.[]{data-label="fig:lattice"}](betheL){width="0.9\linewidth"} Mapping to infinite half-lines {#sec:1Dmapping} ------------------------------ We now describe how different Hilbert space sectors, labeled by the even and odd quantum numbers discussed above, can be mapped onto one-dimensional chains. The connection between the spectra of a Hamiltonian defined on a half-line and a related problem on the Bethe lattice was known previously [@ASW]. The technique we present can be used to calculate the full spectrum of the Bethe lattice problem from $1d$ models that have the advantage of being trivial to simulate numerically. Moreover, one can also use it to obtain the eigenstates of the original problem via exact diagonalization, or their probabilities by the analytic method described in Section \[sec:greens\]. The mapping can be understood in the following way. We start in the $\|i\rangle$ basis, where $i$ identifies individual nodes of the Bethe lattice. Having labeled the vertices in Fig. \[fig:lattice\](a) from left to right at each row, starting with $1$ for the root, we end up with a Hamiltonian matrix of the following form: $$H = \begin{bmatrix} V(1) & -t & -t & 0 & ... \\ -t & V(2) & 0 & -t & ...\\ -t & 0 & V(2) & 0 & ... \\ 0 & -t & 0 & V(3) & ... \\ ... & ... & ... & ... & ... \end{bmatrix}. \label{eq:mappingH1}$$ The matrix above can be brought to a block diagonal form via a unitary transformation into the basis of states defined by their even/odd quantum numbers: $$H = \begin{bmatrix} V(1) & -\sqrt{2}t &0 & 0 &0&0& ... \\ -\sqrt{2}t & V(2) & -\sqrt{2}t & 0&0 &0& ...\\ 0 & -\sqrt{2}t & V(3) & -\sqrt{2}t &0&0& ... \\ ...&...&...&...&...&...&... \\ 0 & 0 & 0 & 0 &V(2)&-\sqrt{2}t&... \\ 0 & 0 & 0 & 0 &-\sqrt{2}t&V(3)&... \\ ... & ... & ... & ... &...&...&... \end{bmatrix}$$ where the upper left block corresponds to the all-even sector of the Hilbert space, and the next one to the states which are odd at the second generation and thus have zero amplitude at site 1. The problem of a single particle hopping on the Bethe lattice in the presence of a radial potential $V(n)$ has thus been decomposed into a series of one-dimensional problems with hopping strength multiplied by a factor of $\sqrt{2}$ compared to the original, such that in each $k^\mathrm{th}$ infinite half-line $V(n)$ is offset by $k$. The degeneracies discussed in the previous Section are reflected in the fact that the block diagonal form of the Hamiltonian (\[eq:mappingH1\]) contains multiple identical $1d$ problems starting with the third generation, and their number is equal to the degeneracy of the respective sector. The energy levels of the Bethe lattice problem with hopping amplitude $-t$ and radial potential $V(n)$ have a trivial relation to those of the related $1d$ chains: namely, Bethe lattice energies from the sector with highest odd generation at $k$ are given by the energies of the $1d$ chain with hopping $-\sqrt{2}t$ and potential $V(n+k-1)$. In that sector, the probability amplitude for finding the particle in a state with energy $E_n$ at Bethe lattice generation $r$ is given by the $(r-k+1)^\mathrm{th}$ component of the eigenvector with eigenvalue $E_n$ of the corresponding $1d$ problem. Our next step depends on whether the form of $V(n)$ in Eq. (\[eq:Hgen\]) allows for an exact solution. In the case that it does not, exact diagonalization will readily provide us with both the energy levels and the wavefunctions for each one-dimensional chain, which can then be used to derive the corresponding observables for the original Bethe lattice problem. The advantage of employing the mapping technique lies in the fact that for a one dimensional chain, boundary effects quickly become negligible as we increase the system size. On the other hand, Cayley trees are notoriously difficult to deal with numerically due to the large fraction of nodes at the systems’ edges. On the analytic front, there are various ways to approach one-dimensional hopping problems on the lattice. The one that we discuss in the next Section can be applied to the original Bethe lattice problem directly, in addition to the special $z=2$ case of one-dimensional chains. When the technique that we are about to discuss results in an exact solution for the lattice Green’s function, there is no benefit in first decomposing the Bethe lattice problem to a series of $1d$ ones, so the generalized coordination number $z$ and connectivity $z=K-1$ will be used in the rest of the paper. Before we turn to the discussion of the Green’s function calculation, we note that both the symmetry analysis and the mapping to $1d$ have a straightforward generalization to Bethe lattices with larger coordination numbers $z>3$ [@Chen]. In this case, each generation’s quantum number takes one of $K$ values. Going back to our three site example in Section \[sec:symmetries\], consider connecting site 1 in the Bethe lattice fragment in Fig. \[fig:lattice\](b) to $K$, rather than 2, descendants, labeled from 2 to $K+1$. The argument remains almost identical, except for there now being $K-1$ states that do not give rise to back-hopping: $\|\Psi_A^j\rangle=(\|2\rangle+e^{1\times(i2\pi j/K)}\|3\rangle+...+e^{(K-1)\times(i2\pi j/K)}\|K+1\rangle)/\sqrt{K}$, where $j$ is an index going from $1$ to $K-1$. The continued fraction method for calculating the lattice Green’s function {#sec:greens} -------------------------------------------------------------------------- The cycle-free nature of the Bethe lattice makes it well-suited for recursive approaches [@0022-3719-6-10-009; @Brinkman]. In order to calculate the Green’s function we separate the Hamiltonian (\[eq:Hgen\]) into two parts $$H_0 = V(n) \qquad \mathrm{and} \qquad H'=T$$ and generate a perturbation series in $t$ using the Dyson equation $$\mathcal{G}(\omega)=\mathcal{G}_0(\omega)+\mathcal{G}_0(\omega)\Sigma(\omega)\mathcal{G}(\omega) \label{eq:Dyson}$$ where $\mathcal{G}_0(\omega)$ is the unperturbed Green’s function $$\mathcal{G}_0(\omega)=\cfrac{1}{\omega-H_0},$$ and $\Sigma(\omega)$ is the self-energy of the particle. If we consider a diagonal element $G_i(\omega)$ of the Green’s function operator, its self-energy is given by a sum of terms associated with all paths on the Bethe lattice going away from node $i$ and back to it. For each hop one such term gains a factor of $-t$ and is divided by $\omega-H_0$ evaluated at the “arrival” node (excluding the starting vertex $i$). The lack of closed cycles on the lattice makes all paths that we need to count self-retracing, allowing for a straightforward way to sum them up. We can write $G_i(\omega)$ as a power series expansion in $t$, or, equivalently, as a continued fraction. For instance, for the root vertex $1$ the diagonal element of the lattice Green’s function is given by $$G_1(\omega)=\cfrac{1}{\omega-V(1)-\cfrac{Kt^2}{\omega-V(2)-\cfrac{Kt^2}{\omega-V(3)-...}}}. \label{eq:G1}$$ The self energies for other nodes of the Bethe lattice involve, in addition to “forward” hops to higher generations, paths that go through the root. Their Green’s functions $G_{i\in n}(\omega)$ (where $i$ is a node at the $n$th generation) can be defined in terms of a finite number of $G_{k}^{F}(\omega)$, infinite continued fractions involving only the forward hops starting from a node at the $k^\mathrm{th}$ generation: $$G_{i\in n}(\omega)=\cfrac{1}{\omega-V(n)-Kt^2G^F_{n+1}(\omega)-\cfrac{t^2}{\left[G_{j\in n-1}(\omega)\right]^{-1}+t^2G^F_n(\omega)}} \label{eq:Gfull}$$ where $$G_{k}^{F}(\omega)=\cfrac{1}{\omega-V(k)-\cfrac{Kt^2}{\omega-V(k+1)-...}}.%\cfrac{Kt^2}{\omega-V(k+2)-...}}}. \label{eq:Gfor}$$ Provided we can obtain a closed form expression for Eq. (\[eq:Gfor\]), the diagonal elements $G_i(\omega)$ of the full Green’s function give us the system’s energy levels (poles of $G_i(\omega)$ occur at the energies $\omega$ whose corresponding eigenstates are visible at the lattice node $i$) and the local densities of states (the local DOS at node $i$ is proportional to Im$\left[G_i(\omega)\right]$). The probability of the bound state with an energy $\omega_n$ at node $i$ is given by the residue of $G_i(\omega)$ evaluated at $\omega_n$. A continuous imaginary part of $G_i(\omega)$ points to the existence of a continuum energy band. Constant potential on a sublattice ---------------------------------- Given the particle’s Green’s function, we have the full knowledge of its energy spectrum, but not of its eigenstates. Although a direct way to obtain the exact eigenstates for a general radial $V(n)$ is often unavailable, there are specific cases when a large fraction of them can be deduced. One such case is that of a potential that takes a constant value on at least one of the two sublattices (corresponding to alternating generations of the Bethe lattice), e.g., $V_A$ on sublattice A. The bipartite nature of the lattice allows for a straightforward way of constructing eigenstates with the energy $V_{A}$: amplitudes on the A nodes are assigned in such a way that hopping to the B nodes cancels out. For instance, for a non-normalized eigenstate corresponding to eigenvalue $V_A$, start with amplitude $+1$ at the root (taking it to belong to sublattice A), and assign zeros on all even (B) generations and $\left(-1/K\right)^{\frac{n-1}{2}}$ on all odd (A) ones. Alternatively, we can choose a sibling set at any sublattice A generation $k$, assign amplitudes to the siblings such that hopping back to the parent node cancels out, $$\sum_{i=1}^Ks_i=0,$$ and construct the rest of the state by assigning amplitude $s_i\times\left(-1/K\right)^{\frac{n-k}{2}}$ to all the A nodes in the $i^\mathrm{th}$ sibling branch, and zeros elsewhere. Such bipartite eigenstates with energy $V_A$ constitute fraction $1/2K$ of the total number of eigenstates. The Coulomb potential problem {#sec:coulomb} ============================= We consider a single particle hopping on the Bethe lattice in the presence of an attractive Coulomb potential: $$H=T+C/n \label{eq:Hcoul}$$ where $T$ is the hopping matrix whose non zero elements are equal to $-t$, $n$ is the generation of the Bethe lattice ($n=1$ at the root), and $C<0$. The exact solution {#sec:exact} ------------------ What makes the Coulomb problem exactly solvable is the fact that the infinite continued fractions of the form (\[eq:Gfor\]) with $V(n)=C/n$ can be written in closed form, with the use of special functions [@Ramanujan]: $$G^{F}_k(\omega)=\cfrac{2k/\omega}{\sqrt{1+x^2}+1} \cfrac{1}{k-\cfrac{C/\omega}{\sqrt{1+x^2}}} \cfrac{F^2_1\left(1-\cfrac{C/\omega}{\sqrt{1+x^2}},k+1,k+1-\cfrac{C/\omega}{\sqrt{1+x^2}},\cfrac{1-\sqrt{1+x^2}}{1+\sqrt{1+x^2}}\right) }{F^2_1\left(1-\cfrac{C/\omega}{\sqrt{1+x^2}},k,k-\cfrac{C/\omega}{\sqrt{1+x^2}},\cfrac{1-\sqrt{1+x^2}}{1+\sqrt{1+x^2}}\right)} \label{eq:Gexact}$$ where $x^2=-\cfrac{4Kt^2}{\omega^2}$ and $F^2_1(a,b,c,z)$ is Gauss hypergeometric function. We can thus obtain the diagonal elements of the Green’s function $\mathcal{G}(\omega)$ for our problem exactly, and use them to calculate the energy levels and the local densities of states, which we do next. The all-even bound states {#sec:alleven} ------------------------- We already know from Section \[sec:symmetries\] that only the all-even states are allowed at the root. Therefore, all the energy levels of the all-even sector are given by the poles of $G_1(\omega)=G^{F}_1(\omega)$. Obtained from Eq. (\[eq:Gexact\]), it is proportional to a ratio of two hypergeomeric functions. In the $k=1$ case, the hypergeometric function in the denominator has no zeros. The ground state energy corresponds to the pole that $G_1(\omega)$ has when $1-\cfrac{C/\omega}{\sqrt{1+x^2}}$ in the denominator is equal to zero, whereas the excited energy levels are given by the poles of the hypergeometric function in the numerator: $$G_1(\omega)\propto F^2_1\left(1-\cfrac{C/\omega}{\sqrt{1+x^2}},2,2-\cfrac{C/\omega}{\sqrt{1+x^2}},\cfrac{1-\sqrt{1+x^2}}{1+\sqrt{1+x^2}}\right).$$ To find them, we make use of the hypergeometric series representation of $F^2_1(a,b,c,z)$: $$F^2_1(a,b,c,z)=\sum^{\infty}_{j=0}\cfrac{(a)_j(b)_j}{(c)_j}\cfrac{z^2}{j!}$$ where $(x)_j$ is the Pochhammer symbol $(x)_j=x(x+1)...(x+j-1)$. Evidently, $F^2_1(a,b,c,z)$ has a pole whenever $c+j=0$, where $j$ is an integer from $0$ to $\infty$. Together with the pole when $1-\cfrac{C/\omega}{\sqrt{1+x^2}}=0$, this leads to the following energy levels: $$\omega_n=-\sqrt{\cfrac{C^2}{n^2}+4Kt^2}, \qquad n=1,2,3,... \label{eq:levels}$$ In the continuum limit $n\to\infty$, the energy levels (\[eq:levels\]) reduce to the familiar $1/n^2$ dependence of the Bohr formula. It is interesting to note that the same result is obtained by solving the continuum Hydrogen atom problem with a $C/r$ potential in the limit of infinitely many dimensions [@bohrmodel]. The bound state energy levels of all other sectors can be obtained numerically as poles of the appropriate $G^F_k(\omega)$, given in Eq. (\[eq:Gexact\]). Scattering states ----------------- A closer inspection of the Green’s function (\[eq:Gexact\]) reveals an imaginary part in the $\|\omega\|<2\sqrt{K}t$ range, corresponding to the continuum energy band, which follows the accumulation of bound states below $-2\sqrt{K}t$. Interestingly, the free particle problem [@Brinkman] has the same continuum spectrum, although its distribution of states within the band is different. It is thus instructive to first consider the case where $C=0$. For the time being, we restore the coordination number at the root to $z$, which makes the symmetric equal-amplitude superposition of all vertices an eigenstate with energy $-zt$. The lattice Green’s function is given by the expression: $$\mathcal{G}_\mathrm{free}(\omega)=\cfrac{1}{\omega-\cfrac{2zt^2}{\omega+\sqrt{\omega^2-4Kt^2}}}. \label{eq:freeG}$$ which indeed indicates a continuous spectrum between $-2\sqrt{K}t$ and $2\sqrt{K}t$. The gap from the $-zt$ uniform state to the lower edge of the band, $-2\sqrt{K}t$, can be explained by the peculiar dimensionality of the Bethe lattice. The imaginary part of Eq. (\[eq:freeG\]), which is proportional to the density of states $N(\omega)$, has a singularity at the edge, whereas the expected form is $N(\omega)\propto C(\omega-\epsilon)^{d/2}$ where $\epsilon$ is the edge of the band and $d$ is the dimension of the system. Given the Bethe lattice is “infinitely dimensional”, $d\to\infty$, the band tails disappear, and the system acquires a gap. Introduction of closed loops in a way that results in a finite $d$ therefore leads to appearance of band tails stretching below $\epsilon=-2\sqrt{K}t$ towards the uniform state at $-zt$. Summary and Discussion {#sec:conclusion} ====================== We have obtained an exact solution for the spectrum of the Coulomb potential problem on the $(K+1)$-coordinated Bethe lattice. The energy levels of different symmetry sectors are given by the poles of the forward hopping Green’s functions (\[eq:Gexact\]). The sectors correspond to a choice of even or one of the $(K-1)$ odd quantum numbers for each generation of the Bethe lattice $n>1$. If the highest generation with an odd quantum number is $k$, then all of the states in the corresponding sectors have zero amplitude at nodes belonging to generations $n<k$, and the sectors’ energy levels, obtained from the poles of $G_k^F(\omega)$, are $K^{k-2}$-fold degenerate. Each such sector can be mapped onto a hopping problem on an infinite half-line, where $V(n)$ is offset by $k$ and the hopping amplitude is multiplied by $\sqrt{K}$. This mapping exists for any problem whose Hamiltonian has the form (\[eq:Hgen\]), and is particularly useful when, unlike in the Coulomb potential case, an exact solution cannot be obtained. Models which involve approximating various physical structures as well as mathematical constructions by cycle-free graphs are plentiful. A single particle hopping on such graph can capture complex many body physics. For instance, the motion of a free particle on the Bethe lattice has been used to study antiferromagnets [@Brinkman] and water ice [@Chen], whereas a particle hopping in the presence of random and radial potentials was used to model Anderson localization [@0022-3719-6-10-009] and spin ice [@diluted] respectively. Apart from the few known exactly solvable cases, problems defined on Cayley trees are difficult to treat. While the exact diagonalizaion of a single particle problem may seem manageable using numerical methods, the large number of nodes at the boundary poses an immense difficulty. One may circumvent the issue by introducing closed cycles at the ends and/or focusing on the properties associated with the interior of the tree. We demonstrate that there is an alternative point of view for radial potentials. The mapping of each sector onto a one-dimensional problem allows for numerical treatment with tractable boundary effects stemming from the single edge node. In addition to the broadly applicable mapping above, the exact solution derived in the present work may prove to be of use across various fields of physics, the inverse $r$ potentials being among the most common. In fact, the motivation behind this study came from considering the problem of emergent magnetic monopoles arising in a particular class of frustrated magnets – spin ice – where the Bethe lattice turns out to be a good approximation for the state space of the problem, and the Coulomb potential defined on it served to represent interactions between pairs of monopoles [@diluted]. Acknowledgments =============== The authors thank S. L. Sondhi for useful discussions and collaboration on a related project [@diluted], and Yen Ting Lin and Nikos Bagis for pointing us to Ramanujan’s work on continued fractions. The authors acknowledge support of the Helmholtz Virtual Institute New States of Matter and their Excitations, the Alexander von Humboldt Foundation (OP), and the German Science Foundation (DFG) via SFB 1143.
--- abstract: 'We report Molecular Dynamics (MD) simulations of a generic hydrophobic nanopore connecting two reservoirs which are initially at different Na${\bf{^+}}$ concentrations, as in a biological cell. The nanopore is impermeable to water under equilibrium conditions, but the strong electric field caused by the ionic concentration gradient drives water molecules in. The density and structure of water in the pore are highly field dependent. In a typical simulation run, we observe a succession of cation passages through the pore, characterized by approximately bulk mobility. These ion passages reduce the electric field, until the pore empties of water and closes to further ion transport, thus providing a possible mechanism for biological ion channel gating.' author: - 'J. Dzubiella' - 'R. J. Allen' - 'J.-P. Hansen' title: 'Electric field-controlled water permeation coupled to ion transport through a nanopore' --- Water and ion permeation of nanopores is a key issue for biological membrane-spanning ion channels and aquaporins, as well as for materials like zeolites, gels and carbon nanotubes. Recent simulations report intermittent filling of hydrophobic nanopores by water under equilibrium conditions [@hummer:nature; @beckstein:jpc; @allen:prl]. However, imbalances in ion concentrations in the inside and outside of cell membranes create strong electric fields [@hille]. Experiments on water near interfaces show that strong fields can induce considerable electrostriction of water [@toney:nature]. The nonequilibrium behavior of confined water and ions in strong fields should therefore be very important for ion permeation and ion channel function. In particular, biological ion channels “open” and “close” to ion transport in response to changes in the electric field across the membrane. This behavior, know as [voltage gating]{}, is crucial to their function, but its mechanism is so far not well understood [@hille]. Related approaches [@kuyucak:2001] to ion transport include the application of a uniform external electric field [@crozier:prl; @chung:2002], more specific models of particular proteins [@berneche:nature; @tieleman:2001; @roux:1994; @lopez:2002], Brownian Dynamics [@chung:2002] and continuum theories [@nonner:1998]. Here we present the results of MD simulations in which a strong electric field across the pore is explicitly created by an ionic charge imbalance, as in a cell. We follow the relaxation of this nonequilibrium system to equilibrium. Our generic model ion channel consists of a cylindrical hydrophobic pore of length $L_{\rm p}= 16$Å and radius $R_{\rm p}=5$ to $7$Å, through a membrane slab which separates two reservoirs containing water and Na$^+$ and Cl$^-$ ions, as shown in Fig. 1. One reservoir has initial concentrations $c_{\rm {Na^{+}}}\approx 0.9$M (12 cations) and $c_{\rm Cl^{-}}\approx 0.6$M (8 anions), while for the second reservoir, $c_{\rm Na^{+}}\approx 0.3$M (4 cations) and $c_{\rm Cl^{-}}\approx 0.6$M (8 anions). This imbalance of charge generates an average electric field of $0.37$V/Å across the membrane. These ion concentrations and electric field are typically five times larger than under ”normal“ physiological conditions, but they could be achieved in the course of a rare, large fluctuation at the pore entrance. The enhanced ion concentrations in the initial state were chosen to improve the signal-to-noise ratio in the simulation and to allow the detection of novel transport mechanisms. The chosen pore dimensions are comparable to those of the selectivity filter of a K$^{+}$ channel [@zhou:nature; @berneche:nature]. The simulation cell contains two pores in sequence along the z-axis, one of which is shown in Fig.1. The reservoir to the right of this pore thus forms the left reservoir for the other pore. Due to periodic boundary conditions, the right reservoir of the latter is also the left reservoir of the first. In this arrangement, the simplest to allow the use of full three-dimensional periodic boundaries, the flows of ions in response to the concentration gradient will be anti-parallel in the two channels. The relaxation towards equilibrium, where the two reservoirs are individually electroneutral, will thus involve an indirect coupling between the two pores. The water molecules are modelled by the SPC/E potential [@berendsen:jpc] which consists of an O atom, carrying an electric charge $q=-0.8476$e, and two H atoms with $q=0.4238$e. The O-atoms on different water molecules interact via a Lennard-Jones (LJ) potential with parameters $\epsilon=0.6502$kJmol$^{-1}$ and $\sigma=3.169$Å . The model is rigid, with OH bond length $1$Å and HOH angle $109.5^\circ$. The Na$^+$ ions have LJ parameters $\epsilon=0.3592$kJmol$^{-1}$, $\sigma=2.73$Å and $q=+$e and for the Cl$^-$, $\epsilon=0.1686$kJmol$^{-1}$, $\sigma=4.86$Å and $q=-$e [@spohr:1999]. Ions and water O-atoms interact with the confining pore and membrane surfaces by a potential of the LJ form $V=\epsilon'[(\sigma'/r)^{12}-(\sigma'/r)^{6}]$, where $r$ is the orthogonal distance from the nearest confining surface. The potential parameters are $\epsilon'=1.0211$kJmol$^{-1}~$ and $\sigma'=0.83$Å . If $R_{\rm p}$ is the geometric radius of the cylindrical pore, one may conveniently define an effective radius $R$ by the radial distance from the cylinder axis at which the interaction energy of a water O-atom with the confining surface is $k_{B}T$, leading at room temperature to $R\approx R_{\rm p}-2$Å ; similarly the effective length of the pore is $L\approx L_{\rm p}+4$Å . Ion-water, ion-ion, water-surface and ion-surface cross terms are defined using the usual Lorentz-Berthelot combining rules. Polarizability of the membrane [@allen:jpcm] and of the water molecules and ions is neglected. The total simulation cell including both of the channels is of dimensions $l_{x}=l_{y}=23.5\pm 0.3$Å and $l_{z}=112.9\pm1.5$Å and contains 1374 water molecules, 16 Na$^+$ and 16 Cl$^-$ ions. Molecular Dynamics simulations were carried out with the DLPOLY2 package [@dlpoly], using the Verlet algorithm [@frenkelsmit; @allen] with a timestep of 2fs. The pressure was maintained at $P=1$bar and the temperature at $T=300$K using a Berendsen barostat and thermostat [@berendsen:jcp]. Electrostatic interactions were calculated using the particle-mesh Ewald method [@essmann:jcp]. Water permeation of the pore is strongly affected by the electric field. The effective channel radius chosen for most of the simulations ($R=3$Å) is such that under equilibrium conditions (i.e. with equal numbers of anions and cations on both sides and hence no electric field), the channel is empty of water and ions [@hille]. However, the ionic charge imbalance across the the membrane causes the pore to fill spontaneously with water. The electric field throughout the system was monitored by measuring the electrostatic force on phantom test particles on a three dimensional grid [@crozier:prl]. Fig. 2a shows the average local electric field around one pore before the first ion moves through a channel. It is nearly zero in the reservoirs. Inside the pore the field is very strong ($\sim 0.37$V/Å) and has a small inward radial component. The profile of the $z$-component of the field $E_{z}$ is shown in Fig. 2b and is constant inside the pore. During the course of the simulation, a number of Na$^+$ ions move through the pore. Each of these events changes the reservoir charge imbalance and reduces the electric field in the pore. This has a dramatic effect on the behavior of the water, as shown in Fig. 3 in which the number $n_{\rm H_{2}O}$ of water molecules inside one pore is plotted as a function of time for a typical simulation run. Initially, the water in the pore undergoes strong electrostriction, comparable to experimental observations [@toney:nature], with an average density $$\label{mth1} \rho\approx n_{\rm H_{2}O}/(\pi R^{2}L),$$ twice as large as that of bulk water in equilibrium. If we assume bulk density of water $\rho_{0}$ inside a channel of radius $R$ and length $L$ we expect an average number $n_{\rm bulk}$ of molecules inside the channel with $$\label{mth2} n_{\rm bulk}=\rho_{0}(\pi R^{2}L).$$ $n_{\rm bulk}$ is indicated with an arrow on the right-hand side in Fig. 3. At each ion crossing, the number $n_{\rm H_2O}$ of water molecules inside the pore drops but is still larger than $n_{\rm bulk}$. In this particular simulation run shown in Fig.3 three cations went through this pore. After the third ion crosses ($t\approx 2.25$ns), the electric field is no longer strong enough to sustain channel filling and the pore spontaneously empties of water, thereby becoming impermeable to further ion transport. However the other pore in the simulation cell remains filled and the final ion crossing eventually occurs through this pore which after that also empties of water. Finally equilibrium is restored to the system and both channels are empty of water. On repeating the simulation 5 times we observe that the closing of one pore after the third ion passed through it occurs in all runs. The structure of water in the filled pore is strongly affected by the field, as shown in the inset of Fig. 3. Before the first ion crossing, water forms clear layers near the pore wall and along the z-axis. The central layer disappears after the first ion crossing and the outer layer becomes less well-defined. After a further ion crossing, water is more-or-less evenly distributed. Ion transport through the pore is found to occur essentially at constant velocity. Fig. 1 shows snapshots from a typical simulation run just before (a) and while (b) a sodium ion passes through the channel. Within the reservoirs the anions and cations diffuse among the water molecules. When a cation in the Na$^{+}$-rich reservoir comes close to the channel entrance, it experiences the strong axial field shown in Fig. 2 and is dragged into the channel. Analysis of $15$ simulation runs, with the same initial charge imbalance but different initial configurations, shows that once the first ion enters the channel, it moves with a constant velocity which is approximately the same in all runs, and then reverts to diffusive motion at the other end of the channel. Fig. 4 shows typical cation positions along the z-axis, as a function of time, for the first, second and third ion crossings, as shown in Fig. 3. The second ion also traverses the channel at constant, somewhat reduced velocity, although it appears to pause for approximately 10ps at the channel entrance, perhaps in order to shed its bulk-like solvation shell. We observe that this “pausing time” is rather widely distributed between simulation runs. The cation mobility $\mu_+$, defined by $\vec v = \mu_{+} e \vec E$, can be calculated from the slopes of the trajectories in Fig. 4, together with the measured electric fields, as in Fig.2. The resulting values are $\mu_{+}\approx 4.5\times 10^{11}{\rm\;s\,kg^{-1}}$ for the first ion, $\mu_{+}\approx 3.8\times 10^{11}{\rm\;s\,kg^{-1}}$ for the second ion, and $\mu_{+}\approx 2.4\times 10^{11}{\rm\;s\,kg^{-1}}$ for the third ion. These values are close to the value of $\mu_{+}\approx2.3\times 10^{11}{\rm\;s\,kg^{-1}}$ obtained from the self-diffusion constant in the reservoir $D_+$, using Einstein’s relation $\mu_{+}=D_{+}/k_{B}T$, but seem to increase with the magnitude of the electric field inside the pore. This enhancement of the mobility correlates with the change of structure of the water inside the channel, shown in the inset of Fig. 3. The tetrahedral hydrogen bond network which water forms under equilibrium conditions is disrupted inside the pore under high electric fields. Simulations of wider ($R=5$Å) pores and with different lengths give qualitatively the same results. The critical electric field for water permeation is, however, sensitive to the pore radius and length. This suggests that voltage-dependent gating in ion channels, if it were to occur by changes in water permeation of a hydrophobic section of the pore [@allen:prl; @hille], might be strongly dependent on channel geometry. The key finding which emerges from our simulations is the strong correlation between water and ion behavior under non-equilibrium conditions. Ionic charge imbalance across the membrane induces water permeation of the hydrophobic pore and thus makes it permeable to ions. This suggests that voltage gating of ion channels may be linked to the coupling between water and ion permeation in pores far from equilibrium. The structure and density of water in the pore is dramatically affected by the strong electric field. The passage of a cation through the channel causes an abrupt jump of the electric field, and an ensuing jump in the number of water molecules inside the pore. Ion passage through the pore occurs at constant velocity and with a mobility coefficient similar to that of the bulk solution at equilibrium. The authors are grateful to Jane Clarke and Michele Vendruscolo for a careful reading of the manuscript. This work was supported in part by the EPSRC. R. J. A. is grateful to Unilever for a Case award. [10]{} bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}url \#1[`#1`]{}urlprefixbibinfo\#1\#2[\#2]{}eprint\#1[\#1]{} , , , **, (). , , , , (). , , , , (). , (, ). , **, (). , , , , (). , , , , , , (). , , , , (). , , (). , , , , , (). , , (). , , , , , , (). , , (). , , , , , (). , , , , (). , , (). , , , , (). (), . , (, ). , (, ). , , , , , , (). , , , , , **, ().
--- abstract: 'A homogeneous Gibbons–Hawking ansatz is described, leading to $4$-dimensional hyperkähler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkähler metrics that are, in a suitable sense, complete. Our construction also gives rise to incomplete metrics on $3$-dimensional contact manifolds that induce complete Carnot–Carathéodory distances.' address: - 'Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany' - 'Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain.' author: - Hansjörg Geiges - Jesús Gonzalo Pérez title: 'A homogeneous Gibbons–Hawking ansatz and Blaschke products' --- [^1] [^2] Introduction ============ The aim of this paper is to present an intriguing construction of hyperkähler structures. In [@gego08 Theorem 10] we exhibited a global rigidity of $4$-dimensional hyperkähler metrics: if such a metric $g$ admits a homothetic vector field with a compact transversal, then $g$ is flat. We proved this by an argument involving the integral formula for the signature of a compact $4$-manifold, applied to a quotient of a neighbourhood of the transversal. That line of reasoning obviously suggests the question whether the compactness hypothesis can be weakened to a completeness condition. In the present paper we define the natural notion of completeness for a Riemannian metric with a homothety (slice-completeness) and give a construction leading to non-flat, slice-complete hyperkähler structures. In Section \[section:GH\] we discuss a homogeneous Gibbons–Hawking ansatz, which forms the basis for our construction. In Section \[section:canonical\] we derive explicit formulæ for various metrics that arise in this construction. The main part of the construction of our examples is contained in Section \[section:example\]. In view of a related [incompleteness]{} result from [@gego08], see Theorem \[thm:incomplete\] below, it was to be expected (and is confirmed here) that the construction of such slice-complete examples would be quite delicate. So it is all the more surprising that our construction actually yields an infinite-dimensional family of isometry classes of such structures (Section \[section:many\]). In Section \[section:non-flat\] it is shown that our construction does indeed give rise to [non-flat]{} hyperkähler metrics. In Section \[section:CC\] we relate our construction to the theory of taut contact spheres developed in [@gego08]; this relation originally motivated the search for the hyperkähler metrics described here. In that context we describe another surprising phenomenon, namely, examples of incomplete Riemannian metrics giving rise to complete Carnot–Carathéodory distances. A homogeneous Gibbons–Hawking ansatz {#section:GH} ==================================== The Gibbons–Hawking ansatz [@giha78] allows one to construct hyperkähler metrics with an $S^1$-invariance. We want to study metrics arising from this ansatz, subject to an additional homogeneity property amounting to the existence of a homothetic vector field. A vector field $Y$ is [homothetic]{} for the Riemannian metric $g$ if it satisfies $L_Yg=g$. The [canonical slice]{} corresponding to such a vector field is the subset defined by the equation $g(Y,Y)=1$. The canonical slice is a hypersurface transverse to $Y$ and, if the flow of $Y$ is complete, it intersects each orbit exactly once. For a cone metric $g=e^{2s}(ds^2+\overline{g})$ and $Y=\partial_s$, the canonical slice is the hypersurface $\{ s=0\}$ orthogonal to $Y$. Most homothetic fields, however, are not orthogonal to any hypersurface; in such situations our definition still gives a natural choice of transversal. A Riemannian metric on a product $M\times{{\mathbb R}}$ with translation along the ${{\mathbb R}}$-factor as homotheties is necessarily incomplete in the ${{\mathbb R}}$-direction: proper paths of the form $\{ p\}\times (-\infty ,s_0]\subset M\times{{\mathbb R}}$ have finite length in such a metric. Therefore, the best one can aim for is completeness in the transverse directions. A Riemannian metric on a product $M\times{{\mathbb R}}$ with translation along the ${{\mathbb R}}$-factor as homotheties is [slice-complete]{} if the canonical slice is complete in the induced metric. For the construction of our examples, we shall be working on a $4$-manifold $W$ of the form $W=\Sigma \times{{\mathbb R}}_t\times S^1_\theta$ with $\Sigma$ an open surface; the subscripts denote the respective coordinates. We look for hyperkähler structures $(g,\Omega_1,\Omega_2,\Omega_3)$ with the following properties: - The flow of $\partial_\theta$ preserves the metric $g$, and $\partial_{\theta}$ is a Hamiltonian vector field for each of the symplectic forms $\Omega_i$. - The vector field $\partial_t$ satisfies $L_{\partial_t}\Omega_i=\Omega_i$, $i=1,2,3$, hence also $L_{\partial_t}g=g$. Notice that this is stronger than just being homothetic. The partial differential equations for a hyperkähler structure linearise under condition (i) to the $3$-dimensional Laplace equation. Under the additional condition (ii), one can reduce these equations further to the Cauchy–Riemann equations in real dimension 2. We next expand on these two claims. Given a $3$-manifold $M$, any hyperkähler structure on the product $M\times S^1_\theta$ satisfying condition (i) can be described by the Gibbons–Hawking ansatz. In this ansatz, one selects Hamiltonian functions $x_1,x_2,x_3$ such that $dx_i=\partial_\theta{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i$. Then there exist a unique $1$-form $\eta$ and a unique positive function $V$ giving the following expressions for the symplectic forms $(\Omega_1,\Omega_2,\Omega_3)$, where $(i,j,k)$ runs over the cyclic permutations of $(1,2,3)$: $$\label{eqn:GH-forms} \Omega_i=(d\theta +\eta )\wedge dx_i+V\, dx_j\wedge dx_k,$$ and the following one for the hyperkähler metric: $$\label{eqn:GH-metric} g= V^{-1}\cdot (d\theta +\eta )^2+V\cdot (dx_1^2+dx_2^2+dx_3^2).$$ Here the forms $dx_1,dx_2,dx_3$ are a basis for the annihilator of $\partial_\theta$, and so $(x_1,x_2,x_3)$ are (at least locally) coordinates for the orbit space of $\partial_\theta$. The function $V$ satisfies $\partial_\theta V\equiv 0$ and is thus locally a function of only $(x_1,x_2,x_3)$. The $1$-form $\eta$ annihilates $\partial_\theta$ and is invariant under its flow, so it is locally pulled back from the orbit space of $\partial_\theta$. All this means that $\eta$ and $V$ are locally objects on $(x_1,x_2,x_3)$-space, and we shall treat them as such for the purpose of local calculations. The projection onto the orbit space of $\partial_\theta$ is a Riemannian submersion from the metric $(1/V)\, g=g(\partial_\theta ,\partial_\theta )\, g$ to the Euclidean metric $dx_1^2+dx_2^2+dx_3^2$. The condition for (\[eqn:GH-forms\]) to define a triple of closed $2$-forms is then $$\label{eqn:curl} d\eta +dV=0,$$ where the Hodge star operator is in terms of that Euclidean metric and the orientation defined by $(dx_1,dx_2,dx_3)$. For the (local) existence of $\eta$ it is necessary and sufficient that $V$ be harmonic with respect to the Euclidean metric. We call $V$ the [Gibbons–Hawking potential]{}. The systematic construction of a hyperkähler structure on $M\times S^1_\theta$ proceeds as follows. Start with a local diffeomorphism $x=(x_1,x_2,x_3)\co M\to {{\mathbb R}}^3$, and consider the metric $x^g_{{{\mathbb R}}^3}$, where $g_{{{\mathbb R}}^3}$ is the standard Euclidean metric on ${{\mathbb R}}^3$. Use $x$ also to pull the standard orientation of ${{\mathbb R}}^3$ back to $M$. Let now $\eta$ and $V$ be a $1$-form and a function, respectively, defined on $M$ and satisfying (\[eqn:curl\]) with respect to the metric $x^g_{{{\mathbb R}}^3}$ and the pulled-back orientation. By lifting $x,\eta ,V$ to $M\times S^1_\theta$ in the obvious way, and inserting them into the defining equations (\[eqn:GH-forms\]) and (\[eqn:GH-metric\]), we obtain a hyperkähler structure on $M\times S^1_\theta$ invariant under the flow of $\partial_\theta$. One may regard $(M,x^g_{{{\mathbb R}}^3})$ as a [non-schlicht]{} domain in ${{\mathbb R}}^3$, and $\eta$ and $V$ as multiple-valued objects in ${{\mathbb R}}^3$. On $M$, however, they are perfectly well defined, and so equation (\[eqn:curl\]) can be read as an identity on $M$. We now restrict our attention to $3$-manifolds $M$ of the form $M=\Sigma \times{{\mathbb R}}_t$ with $\Sigma$ an open surface, and impose both conditions (i) and (ii). The following definition is useful for describing the special features of this case. A tensorial object [o]{} (on $W=M\times S^1_{\theta}$ or on $M$) is called [homogeneous of degree $k$]{} if $L_{\partial_t}{\bf o}=k\cdot{\bf o}$. Condition (ii) requires that $g$ and the symplectic forms be homogeneous of degree $1$. Since $\partial_\theta$ is invariant under the flow of $\partial_t$, the potential $V=1/g(\partial_\theta ,\partial_\theta )$ must be homogeneous of degree $-1$. Condition (ii) provides a convenient choice for the Hamiltonian functions $x_i$, because one easily checks that $x_i:=\Omega_i (\partial_\theta ,\partial_t)$ satisfies $dx_i=\partial_\theta{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i$ in this case. This choice has the virtue that the $x_i$ are homogeneous of degree $1$, i.e.that $\partial_tx_i=x_i$. We call $x:=(x_1,x_2,x_3)\co\Sigma \times{{\mathbb R}}_t\to{{\mathbb R}}^3\setminus\{ 0\}$ the [momentum map]{}. Then the equations $\partial_tx_i=x_i$ say that $\partial_t$ is $x$-related to the position vector field on ${{\mathbb R}}^3$. The uniqueness of $\eta$ for given Hamiltonian functions $x_i$ implies that in the present situation both $\eta$ and the function $\eta (\partial_t)$ are homogeneous of degree $0$. Consider now the function $\rho :=\|x\|\co\Sigma \times {{\mathbb R}}_t\to{{\mathbb R}}^+$. The $1$-form $\frac{d\rho}{\rho}$ is homogeneous of degree $0$ and satisfies $\frac{d\rho}{\rho}\, (\partial_t)\equiv 1$. Hence $$\eta = \eta (\partial_t)\, \frac{d\rho}{\rho}+\xi ,$$ where the $1$-form $\xi$ is homogeneous of degree $0$ (i.e.$\partial_t$-invariant), $\partial_{\theta}$-invariant, and it annihilates both $\partial_t$ and $\partial_{\theta}$. So $\xi$ is the pull-back of a $1$-form on $\Sigma$, for which we continue to write $\xi$. The map $x/\|x\|\co\Sigma\times{{\mathbb R}}_t\to S^2$ is independent of $t$ and thus the pull-back of a unique map $\Phi\co\Sigma\to S^2$. The latter is a local diffeomorphism. We endow the unit sphere $S^2$ with the metric induced from the standard metric on ${{\mathbb R}}^3$, and with the orientation as boundary of the $3$-ball. This determines on $S^2$ a holomorphic structure and a standard volume form $\mbox{\rm Vol}_{S^2}$. We endow $\Sigma $ with the holomorphic structure $J$ lifted from $S^2$ by $\Phi$, i.e.the structure that turns $\Phi\co\Sigma\to S^2$ into a local biholomorphism. \[thm:GH\] The function $V$ and the $1$-form $\eta = \eta (\partial_t)\, \frac{d\rho}{\rho}+\xi$ satisfy (\[eqn:curl\]) on $\Sigma\times{{\mathbb R}}_t$ if and only if the following two conditions are satisfied: - $d\xi = \rho V\,\Phi^\mbox{\rm Vol}_{S^2}$, - $\varphi :=\eta (\partial_t)+{{\bf i}}\rho V$ is the pullback to $\Sigma\times{{\mathbb R}}_t$ of a holomorphic function on $(\Sigma ,J)$. Let $\tilde{u}+{{\bf i}}\tilde{v}$ be a local holomorphic coordinate on $S^2$, and let $u+{{\bf i}}v$ be its pullback under $\Phi$. Then $u,v$ are homogeneous of degree $0$ and $(\rho ,u,v)$ are local coordinates on $\Sigma\times{{\mathbb R}}_t$ giving the same orientation as $(x_1,x_2,x_3)$. Since $V$ is homogeneous of degree $-1$, we have $V_\rho =-\rho^{-1}V$, therefore $$dV=-\rho^{-1}V\, d\rho +V_u\, du +V_v\, dv.$$ On the other hand, $$d\rho =\rho^2\,\Phi^\mbox{\rm Vol}_{S^2},\quad du=-d\rho\wedge dv,\quad dv=d\rho\wedge du,$$ and so $$dV= -\rho V\,\Phi^\mbox{\rm Vol}_{S^2} - V_u\, d\rho\wedge dv +V_v\, d\rho\wedge du.$$ Since $\xi$ is homogeneous of degree $0$ and annihilates $\partial_t$, we have $\xi =\xi_1(u,v)\, du+\xi_2(u,v)\, dv$ and $$d\eta = (\xi_{2u}-\xi_{1v})\, du\wedge dv- \eta (\partial_t)_v\,\frac{d\rho}{\rho}\wedge dv -\eta (\partial_t)_u\,\frac{d\rho}{\rho}\wedge du.$$ Then (\[eqn:curl\]) is seen to be equivalent to the system $$\begin{aligned} d\xi & = & \rho V\,\Phi^\mbox{\rm Vol}_{S^2}, \\ \eta (\partial_t)_u & = & \;\;\;\rho V_v \; = \;\;\;\: (\rho V)_v, \\ \eta (\partial_t)_v & = & -\rho V_u \; = \; -(\rho V)_u.\end{aligned}$$ The last two equations are the Cauchy–Riemann equations for $\varphi$. In order to describe the systematic construction of hyperkähler structures satisfying (i) and (ii), it is convenient to fix the complex structure $J$ on $\Sigma$ in advance. We then use [holomorphic data]{} of the following kind to construct the desired structures: - a local biholomorphism $\Phi\co (\Sigma ,J)\to S^2$, - a holomorphic function $\varphi\co (\Sigma ,J)\to{{\mathbb H}}$ with values in the upper half-plane ${{\mathbb H}}$. Being homogeneous of degree $1$, the function $\rho$ must be of the form $\rho=e^t\rho_0$ with $\rho_0$ the pullback of a positive function on $\Sigma$. Given a choice of $\Phi , \varphi ,\rho_0$, the construction is as follows. The momentum map is given by $x=\rho\,\Phi$. We take an antiderivative $\xi$ for $(\mbox{\rm Im}\,\varphi )\,\Phi^\mbox{\rm Vol}_{S^2}$ on $\Sigma$. Set $\eta =(\mbox{\rm Re}\,\varphi )\, \frac{d\rho}{\rho} +\xi\;$ and $\; V=(\mbox{\rm Im}\,\varphi)\,\rho^{-1}$, both pulled back to $W$. The hyperkähler structure is given by (\[eqn:GH-forms\]) and (\[eqn:GH-metric\]) with these values for $x,\eta ,V$. Given any pair of functions $h_1,h_2\co\Sigma\to{{\mathbb R}}$, the diffeomorphism of $W$ given by $$\bigl( p,t,\theta \bigr)\longmapsto \bigl( p,t+h_1(p),\theta +h_2(p)\bigr)$$ preserves $\partial_t$ and $\partial_{\theta}$ and pulls the hyperkähler structure with data $(\Phi ,\varphi ,\rho_0 ,\xi)$ back to the one corresponding to $(\Phi ,\varphi ,e^{h_1}\rho_0 ,\xi +dh_2)$. This implies that if $\Sigma$ is simply-connected, then the triple $$\bigl(\mbox{\rm hyperk\"ahler structure},\partial_\theta , \partial_t\bigr)$$ is determined up to isomorphism by the holomorphic data $(\Phi ,\varphi )$ alone, thus allowing us to make any choice for $\rho_0$ and $\xi$. For general $\Sigma$, the choice of $\xi$ matters, but the function $\rho_0$ can always be chosen freely. We shall presently establish a convenient choice for $\rho_0$. Canonical metrics {#section:canonical} ================= We continue to consider structures satisfying (i) and (ii). Recall that the canonical slice is the hypersurface $$S=\{p\in W\co g(\partial_t,\partial_t )_p=1\}.$$ The vector field $\partial_{\theta}$ is a Killing field for $g$ and commutes with $\partial_t$. So $\partial_{\theta}$ is tangent to $S$, and the restriction of $\partial_{\theta}$ to $S$ is a Killing field for the $3$-dimensional metric $g_3$ induced on $S$ by the hyperkähler metric. That metric $g_3$, in turn, induces the quotient metric on $S/(\mbox{\rm flow of}\;\partial_\theta )$ which makes the projection a Riemannian submersion. We now introduce the holomorphic function $\psi =-1/\varphi$, which still takes values in the upper half-plane ${{\mathbb H}}$. The next lemma shows that the choice $\rho_0= \mbox{\rm Im}\,\psi$ is especially convenient, because then $\{ t=0\}$ is the canonical slice. \[lem:slice\] The canonical slice is the product $G\times S^1_\theta$, where $G$ is the surface in $\Sigma\times{{\mathbb R}}_t$ described equivalently by any of the following equations: $$\begin{array}{ccl} V & = & \|\varphi \|^2 ,\\ \rho & = & \mbox{\rm Im}\,\psi ,\\ t & = & \log \mbox{\rm Im}\,\psi -\log\rho_0. \end{array}$$ Formula (\[eqn:GH-metric\]) gives $$g(\partial_t,\partial_t)=V^{-1}\cdot(\eta (\partial_t))^2 +V\cdot (x_1^2+x_2^2+x_3^2).$$ By Theorem \[thm:GH\], this can be written as $$g(\partial_t,\partial_t)=V^{-1}\cdot (\hbox{\rm Re}\,\varphi )^2+V\cdot\rho^2 .$$ So the canonical slice is given by the equation $V^{-1}\cdot (\hbox{\rm Re}\,\varphi)^2 +V\cdot\rho^2=1$, which again by Theorem \[thm:GH\] transforms to $$V=(\hbox{\rm Re}\,\varphi )^2+(\hbox{\rm Im}\,\varphi )^2 =\|\varphi \|^2 .$$ Using $\rho =(\rho V)/V$ and Theorem \[thm:GH\], this is seen to be equivalent to $$\rho =\frac{\rho V} {\|\varphi \|^2} =\frac{\hbox{\rm Im}\,\varphi}{\|\varphi \|^2} = \hbox{\rm Im}\,\psi .$$ The third description of the canonical slice then follows from $\rho =e^t\rho_0$. The surface $G$ is the graph of a function $\Sigma\to{{\mathbb R}}_t$. Hence the map $(p,t,\theta )\mapsto p$ induces a diffeomorphism $\sigma\co S/(\mbox{\rm flow of}\;\,\partial_\theta )\to\Sigma$. The diffeomorphism $\sigma$ sends the quotient metric to the metric $$\label{eqn:g-sigma} g_\Sigma :=\frac{1}{\|\psi \|^2}\,\left[ (d\,\hbox{\rm Im}\,\psi)^2 +(\hbox{\rm Im}\,\psi )^2\,\Phi^g_{S^2} \right]$$ on $\Sigma$, with $g_{S^2}$ denoting the standard metric on $S^2$. It follows from (\[eqn:GH-metric\]) that the orthogonal complement to $\partial_\theta$ is described by the equation $d\theta +\eta =0$, both on $W$ and on $S$. The restriction of the hyperkähler metric to this complement is then the same as the restriction of the quadratic form $q:=V\cdot (dx_1^2+dx_2^2+dx_3^2)$. This $q$ is $\partial_{\theta}$-invariant and has $\partial_\theta$ as an isotropic direction, therefore its restriction to the canonical slice $S$ is the pullback of the quotient metric under the quotient projection. The standard metric of ${{\mathbb R}}^3$ equals $dr^2 +r^2\, g_{S^2}$ in spherical coordinates. This and the equations $\rho =\mbox{\rm Im}\,\psi$ and $V=\|\varphi \|^2$ from Lemma \[lem:slice\] imply that the restriction of $q$ to $S$ is given by the right-hand side of (\[eqn:g-sigma\]), with $\Phi\co\Sigma\to S^2$ replaced by $x/\rho\co S\to S^2$. It is then clear that $\sigma$ pulls $g_\Sigma$ given by (\[eqn:g-sigma\]) back to the quotient metric. A slice-complete example {#section:example} ======================== The orbits of the Killing field $\partial_\theta$ in the canonical slice are circles (of variable length $2\pi V^{-1/2}=2\pi\, \|\psi \|$). So it is clear that the induced metric $g_3$ is complete if and only if the quotient metric is complete; the latter in turn is isometric to $g_\Sigma$. In the sequel the Riemann surface $(\Sigma ,J)$ will be the unit disc $\D\subset{{\mathbb C}}$. \[thm:example\] There are holomorphic data $(\Phi ,\psi )$ on $\D$ such that formula (\[eqn:g-sigma\]) defines a complete metric $g_{\D}$ on $\D$. Thus, the pair $(\Phi ,\psi )$ gives rise to a slice-complete hyperkähler metric on $\D\times{{\mathbb R}}_t\times S^1_{\theta}$. The construction of the pair $(\Phi ,\psi )$ will take up the rest of this section. Saying that $g_{\D}$ is complete means that any proper path $\gamma\co [0, +\infty )\rightarrow{\D}$ has infinite length in this metric. Intuitively, for the metric $\Phi^* g_{S^2}$ to be complete the map $\Phi\co \D\to S^2$ would have to wrap $\D$ around $S^2$ so as to push the boundary of $\D$ infinitely far away. Such a map, however, would have to be a covering. The $2$-sphere being simply connected, this is impossible. Still, [most]{} of that boundary can be pushed infinitely far away. This is achieved as follows. Equip $\D$ with the Poincaré metric and consider the conformal covering projection $$\Phi \co {\D}\longrightarrow S^2\setminus\{ p_1,p_2,p_3\}$$ of the sphere with three punctures by the unit disc. Let us recall the construction of such a covering map. Take a hyperbolic triangle with its three vertices on $\partial{\D}$ (a so-called [ideal]{} hyperbolic triangle), tessellate $\D$ by this triangle and the infinitely many images under successive reflection on the sides (Figure \[figure:tess\]), then consider the quotient ${\D}/\Gamma$ where $\Gamma$ is the group consisting of the hyperbolic translations within the group generated by those reflections. This quotient space ${\D}/\Gamma$ is the result of gluing corresponding sides of two copies of the original triangle, hence a sphere with three punctures. As a consequence of the uniformisation theorem, there is a biholomorphism $${{\mathbb D}}/\Gamma\longrightarrow S^2\setminus\{ p_1,p_2,p_3\} ;$$ see [@thur97], in particular pp. 117 and 147–149. ![The hyperbolic tessellation of $\D$ corresponding to $\Phi$.[]{data-label="figure:tess"}](tess) The map $\Phi$ is the composition $$\D\longrightarrow{{\mathbb D}}/\Gamma\longrightarrow S^2\setminus\{ p_1,p_2,p_3\} ,$$ and it has rank $2$ everywhere. There is a distinguished sequence $(z_n)$ of points on the unit circle $\partial{\D}$, namely the vertices of all triangles in the tessellation. For $j=1,2,3$ choose a small closed metric ball $B_j$, centred at the puncture $p_j$ in $S^2$, such that the balls $2B_1,2B_2,2B_3$ of double radius have disjoint closures. Their preimages under $\Phi$ are shown in Figure \[figure:horo\]. For each $n$ let $D_n$ be the connected component of $\Phi^{-1}(B_1\cup B_2\cup B_3)$ having $z_n$ as limit point, and let $2D_n$ be the same for $\Phi^{-1}(2B_1\cup 2B_2\cup 2B_3)$. Then both $(D_n)$ and $(2D_n)$ form a sequence of pairwise disjoint horodisc-like regions. ![The hyperbolic tessellation and the horodisc-like regions.[]{data-label="figure:horo"}](horo) \[lem:longpath\] Let $\gamma \co[0,+\infty )\rightarrow\D$ be any proper path. Either $\gamma$ has infinite length in the metric $\Phi^* g_{S^2}$, or it has an end $\gamma ([t_0,+\infty ))$ contained in some region $2D_{n_0}$. If $\gamma$ is proper in $\D$ and contained in $2D_{n_0}$, we must of course have $\gamma (t)\to z_{n_0}$ as $t\to +\infty$. So the lemma says that a path of finite length in the metric $\Phi^g_{S^2}$ can only escape the disc through one of the points $z_n$. Intuitively, the covering map $\Phi$ wraps around the punctured sphere so as to push the boundary of the disc infinitely far away, with the exception of the countable set $\{ z_n\co n\in\N\}$. Suppose first that the path visits $\cup_{n=1}^{\infty} D_n$ only finitely often. By deleting an initial segment, we may then assume that the trace of $\gamma$ is disjoint from all the $D_n$. Each triangle $T_k$ of the tessellation intersects exactly three of the $D_n$. Let $T'_k$ be the compact region obtained by removing from $T_k$ the interior of those three intersections. With the metric $\Phi^g_{S^2}$ the $T'_k$ are pairwise congruent spherical regions in the shape of a hexagon, with three sides coming from the boundaries $\Phi^{-1}(\partial B_j)$, $j=1,2,3$ (call these the odd sides), and three sides coming from the sides of $T_k$ (call these the even sides). The path $\gamma$ is proper and contained in the union $\cup_{k=1}^{\infty}T'_k$, therefore it must visit infinitely many different regions $T'_k$, and among those there must be infinitely many $T'_k$ where $\gamma$ enters on one (necessarily even) side and exists on another (also even) side. If $c_1>0$ is the minimum spherical distance between even sides, which is the same for all $T'_k$, then the length of $\gamma$ in the metric $\Phi^* g_{S^2}$ is bounded from below by the series $c_1+c_1+\cdots$ and therefore infinite. Suppose now that $\gamma$ visits the disjoint union $\cup_{n=1}^{\infty} D_n$ infinitely often and there are at least two regions $D_{n_1}$ and $D_{n_2}$ which the path visits infinitely many times. Then the length of $\gamma$ in the metric $\Phi^* g_{S^2}$ is bounded below by the series $c_2+c_2+\cdots$, where $c_2$ is the minimum spherical distance between the balls $B_1,B_2,B_3$, and hence infinite. There remains the case when $\gamma$ visits but a single region $D_{n_0}$ infinitely often. If it has an end $\gamma ([t_0,+\infty ))$ contained in $2D_{n_0}$, we are done. Otherwise $\gamma$ takes infinitely many journeys from $\partial D_{n_0}$ to $\partial (2D_{n_0})$ and the length of $\gamma$ in the metric $\Phi^* g_{S^2}$ is bounded below by the series $c_3+c_3+\cdots$, where $c_3$ is the radius of the metric ball $B_j=\Phi (D_{n_0})$, and so again this length is infinite. A path in $2B_j\setminus\{ p_j\}$ may well have finite spherical length and limit $p_j$. Thus $2D_{n_0}$ contains paths with $z_{n_0}$ as limit and finite length in the metric $\Phi^g_{S^2}$. To ensure that $g_\D$ be a complete metric, we need to choose the function $\psi$ so that the integral of $\|d\,\hbox{\rm Im}\,\psi \|/\|\psi \|$ along such paths is infinite. Such a $\psi$ can be found with the help of so-called Blaschke products. For the reader’s convenience we first review this notion. Given $a\in{\D}$ with $a\neq 0$, let $F_a(z)$ be the orientation-preserving isometry for the Poincaré metric that exchanges $0$ with $a$. The line segment joining 0 to $a$ is a Poincaré geodesic, and $F_a$ is the $180^{\hbox{\scriptsize o}}$ Poincaré rotation about the Poincaré midpoint of that segment (Figure \[figure:rotation\]). It is given by $\displaystyle{F_a(z)=\frac{a-z}{1-\overline{a}\, z}}$. $0$ \[br\] at 207 222 $a$ \[br\] at 352 294 ![Poincaré rotation.[]{data-label="figure:rotation"}](rotation "fig:") The [Blaschke factor]{} $B_a(z)$ is the function $$B_a(z)=\frac{\overline{a}}{\|a\|}\,\frac{a-z}{1-\overline{a}\, z}\, ,$$ i.e. the result of multiplying the isometry $F_a$ by a unitary constant so that $B_a(0)$ is a positive real number. The purpose of this normalisation is to get a simple convergence condition for [Blaschke products]{}, which are finite or infinite products of the form $$B(z)=z^m\cdot\prod_k B_{a_k}(z).$$ It is well known [@haym64 Section 6.7] that under the condition $\sum_k (1-\|a_k\|)<\infty$ this product converges to a holomorphic function $B\co\D\rightarrow\D$. Let $(z_n)$ be a sequence of pairwise distinct points on $\partial{\D}$. Let $D\subset {\D}$ be a convex open domain with its boundary interior to $\D$ except for the point $1\in\partial{\D}$. Then there exists a Blaschke product $B\co\D\rightarrow\D$ such that for each $n$ we have a finite number $\lambda_n$ with $$\frac{B(z)-1}{z-z_n}\longrightarrow \lambda_n \;\;\hbox{as}\;\; z \longrightarrow z_n,\;\; z\in z_n D:= \{\, z_n\zeta\co\zeta\in D\,\}.\qed$$ We are now going to apply this theorem to the sequence $(z_n)$ of vertices in our hyperbolic tessellation. In the Euclidean metric on $\D$, the size of the regions $2D_n$ is bounded above by the size of the three of those horodisc-like regions at the vertices of the hyperbolic triangle containing the origin $0\in\D$. Thus we can choose a circular disc $D\subset{\D}$ whose boundary is interior to $\D$ except for 1, and such that for each $n$ the rotated image $z_n D$ contains $2D_n$. By the theorem above we have a Blaschke product $B\co\D\to\D$ such that for all $n$ $$B(z)\longrightarrow 1 \;\;\hbox{as}\;\; z\longrightarrow z_n,\;\; z\in 2D_n.$$ The holomorphic function $1-B$ maps $\D$ into the open disc of radius $1$ and centre $1$. Moreover, for each $n$ this function has limit $0$ as $z$ approaches $z_n$ inside $2D_n$. The open disc of radius $1$ and centre $1$ is contained in the right half-plane $\{ z\co\mbox{\rm Re}\, z>0\}$, on which there is a well-defined holomorphic square root with values in the quadrant $Q_1:=\{ z\co\hbox{arg}\, z\in (-\pi/4,\pi /4)\}$. So there is a holomorphic function $\sqrt{1-B}\co\D\rightarrow Q_1$ mapping $\D$ into the region bounded by a half lemniscate symmetric about the positive real axis. This function likewise has limit $0$ as $z$ approaches $z_n$ inside $2D_n$. Finally, we set $\psi={{\bf i}}\cdot\sqrt{1-B}$. This holomorphic function maps $\D$ into the quadrant $$Q_2=\{ z\co \hbox{arg}\, z\in (\pi/4,3\pi /4)\} = \{ u+{{\bf i}}\, v\co \|u\|<\|v\|\}\subset{{\mathbb H}}.$$ In fact, $\psi ({\D})$ is contained in the region bounded by a half lemniscate symmetric about the positive imaginary axis (Figure \[figure:map0\]). Moreover, $\psi (z)$ converges to $0$ as $z$ approaches $z_n$ inside $2D_n$. $\psi$ \[b\] at 270 121 $z_n$ \[tl\] at 161 15 $2D_n$ \[b\] at 148 67 $Q_2$ at 563 164 \[b\] at 468 153 ![The map $\psi$.[]{data-label="figure:map0"}](map0 "fig:") This finishes the construction of the pair $(\Phi ,\psi )$. We are now going to verify that the metric $g_{\D}$ defined by this pair as in equation (\[eqn:g-sigma\]) is indeed complete. This will conclude the proof of Theorem \[thm:example\]. Since $\psi$ satisfies $\|\hbox{Re}\,\psi \|<\|\hbox{Im}\,\psi \|$, we have $\|\hbox{Im}\,\psi \|/\|\psi \|> 1/\sqrt{2}$. By Lemma \[lem:longpath\], a proper path in $\D$ is infinitely long in the metric $(\hbox{Im}\,\psi /\|\psi \|)^2\cdot\Phi^ g_{S^2}$, hence also in the metric $g_{\D}$, unless it has an end contained in some region $2D_{n_0}$. But if a proper path $\gamma\co [t_0,+\infty )\rightarrow\D$ is contained in $2D_{n_0}$, then we must have $$\gamma (t)\longrightarrow z_{n_0}\;\;\hbox{and}\;\;\psi(\gamma (t)) \longrightarrow 0 \;\;\hbox{as}\;\; t\longrightarrow +\infty .$$ Writing $\psi(\gamma (t))=u(t)+{{\bf i}}\, v(t)$, we have $$\int_{\gamma}\frac{\|d\,\hbox{Im}\,\psi \|}{\|\psi \|}> \int_{t_0}^{+\infty}\frac{\|v'(t)\|}{\sqrt{2}\, v(t)}\, dt = \frac{1}{\sqrt{2}}\,\bigl(\hbox{total variation of}\;\log v(t)\bigr) .$$ Since $\log v(t)\to\log 0= -\infty$ as $t\to +\infty$, we deduce that the integral of $\|d\,\hbox{Im}\,\psi \|/\|\psi \|$ along $\gamma$ is infinite, and so is the length of $\gamma$ in the metric $g_{\D}$. An infinite-dimensional family of examples {#section:many} ========================================== The preceding construction goes through if we replace $\psi$ by $\psi_\mu =\mu\circ\psi$, where $\mu$ is any holomorphic function whose domain contains $\{ 0\}\cup\psi ({{\mathbb D}})$ and such that $\mu (0)=0$ and $\mu (\psi ({{\mathbb D}}))\subset Q_2$, see Figure \[figure:map1\]. With $\Phi$ the covering map described in Section \[section:example\], we thus obtain an infinite-dimensional family of holomorphic data $(\Phi ,\psi_{\mu})$ that yield slice-complete hyperkähler structures in $\D\times{{\mathbb R}}_t\times S^1_\theta$. $\psi_{\mu}$ \[b\] at 270 121 \[b\] at 487 166 ![The map $\psi_{\mu}$.[]{data-label="figure:map1"}](map1 "fig:") We now want to show that the family of pairs $(\Phi ,\psi_{\mu})$ gives rise to an infinite-dimensional family of slice-complete hyperkähler metrics. Consider triples $({\mathcal H},X,Y)$ made of a hyperkähler structure ${\mathcal H}=(g,J_1,J_2,J_3)$, a vector field $X$ preserving the entire structure, and a vector field $Y$ such that $L_Y\Omega_i=\Omega_i$ and $[X,Y]\equiv 0$. If two such triples are isomorphic, then the image $x(S)$ of the canonical slice of $Y$ under the momentum map is the same for both. For a triple in our family, the image $x(S)$ is the radial graph in ${{\mathbb R}}^3$ of the multi-valued function $(\mbox{\rm Im}\, \psi_\mu )\circ\Phi^{-1}\co S^2\rightarrow {{\mathbb R}}^+$. As $\mu$ varies, these graphs form an infinite-dimensional family of immersed surfaces in ${{\mathbb R}}^3$, so we have an infinite-dimensional family of isomorphism classes of triples. Then the metrics also constitute an infinite-dimensional family of isometry classes, because for a given hyperkähler metric $g$ the number of degrees of freedom for $J_1,J_2,J_3,X,Y$ is bounded [a priori]{} by a finite constant: the complex structures $J_1,J_2,J_3$ are parallel with respect to the Riemannian connection; the Lie algebra of Killing vector fields is finite-dimensional [@kono63 Thm. VI.3.3]; any two homothetic vector fields differ by a Killing field. Non-flatness {#section:non-flat} ============ The following proposition implies, as we shall see below, that the hyperkähler metrics in our examples are non-flat. This proposition may be regarded as a converse of [@gego08 Thm. 10]. \[prop:non-flat\] If a metric with a homothetic vector field is slice-complete and flat, then the canonical slice is compact. Consider a tubular neighbourhood $U$ of the canonical slice $S$. Endow the universal cover $\widetilde{U}$ with the lifted flat metric. The lift of the homothetic vector field to $\widetilde{U}$ is a vector field $\widetilde{Y}$ homothetic for the lifted metric, and its canonical slice is the inverse image $\widetilde{S}$ of $S\subset U$ under the covering map $\widetilde{U}\to U$. This $\widetilde{S}$ is connected, because its tubular neighbourhood $\widetilde{U}$ is connected. We conclude that $\widetilde{S}$ is a universal cover of $S$ under the restricted projection $\widetilde{S}\to S$, which is a local isometry for the induced metrics. Since $S$ is complete with the induced metric, so is $\widetilde{S}$. There is a local isometry $F\co \widetilde{U}\to\E^n$ into Euclidean space. If $U'\subset\widetilde{U}$ is an open domain on which $F$ is injective, then on the image $F(U')$ we have the vector field $F_\widetilde{Y}$ that is homothetic for the Euclidean metric of $\E^n$ restricted to $F(U')$. Then there is a vector field $Y_0$, defined on all of $\E^n$ and homothetic for the Euclidean metric, that coincides with $F_\widetilde{Y}$ on $F(U')$. On $\widetilde{U}$ we have the homothetic vector fields $\widetilde{Y}$ and $F^Y_0$, and they have to be identical because they coincide on $U'$. This means that the local isometry $F$ sends $\widetilde{Y}$ to $Y_0$, hence it maps the canonical slice $\widetilde{S}$ of $\widetilde{Y}$ to the canonical slice $S_0$ of $Y_0$. Since $\widetilde{S}$ is complete, the restriction $F\co\widetilde{S}\to S_0$ must be a covering. But homothetic vector fields in $\E^n$ are linear, and $S_0$ is thus an ellipsoid. This forces $\widetilde{S}$ to be diffeomorphic with $S^{n-1}$ and in particular compact. [A fortiori]{}, the slice $S$ must be compact. Hyperkähler metrics are Ricci flat [@bess87 p. 284], and thus in particular Einstein metrics. Regularity results of DeTurck–Kazdan [@deka81], cf. [@bess87 Sections 5.E–F], say that Einstein metrics (in dimension at least $3$) are real analytic in harmonic coordinates (the reason being that the equations for an Einstein metric form a quasi-linear elliptic system in such coordinates). Thus, if such a metric is flat in some domain, then it must be flat everywhere. With Proposition \[prop:non-flat\] we conclude that the hyperkähler metrics described in Sections \[section:example\] and \[section:many\] are non-flat on every open set, because the canonical slice of each of these metrics is complete but non-compact. According to [@gego08 Prop. 34], $4$-dimensional hyperkähler metrics that are cone metrics are necessarily flat, so the metrics constructed above are not cone metrics in any open set, i.e. none of their homothetic vector fields are hypersurface orthogonal, not even locally. A Carnot–Carathéodory phenomenon {#section:CC} ================================ In this section we show how the examples constructed in Sections \[section:example\] and \[section:many\] lead in a natural way to incomplete Riemannian metrics that nonetheless induce complete Carnot–Carathéodory distances. We begin by explaining these concepts. A $1$-form $\alpha$ on a $3$-manifold $M$ is a [contact form]{} if $d(e^t\alpha )$ is a symplectic form on $M\times{{\mathbb R}}_t$. This is equivalent to $\alpha\wedge d\alpha$ being a volume form on $M$. The pair $(M,\alpha )$ is then called a [contact manifold]{}. It is well known [@geig08 Section 3.3] that any two points on a contact manifold $(M,\alpha )$ can be joined by a [Legendre path]{}, i.e. a path $\gamma (s)$ such that $\alpha (\gamma'(s))\equiv 0$. Given a Riemannian metric $g$ on $M$, the induced [Carnot–Carathéodory distance]{} is, for each pair of points $p_1,p_2\in M$, the infimum of the lengths of Legendre paths joining $p_1$ to $p_2$, see [@mont02] or [@grom96]. Notice that this is bounded below by the Riemannian distance, hence it is certainly a complete distance if $g$ happens to be complete. A triple of contact forms $(\alpha_1,\alpha_2,\alpha_3)$ on $M$ is called a [contact sphere]{} if for each ${\bf c}=(c_1,c_2,c_3)\in{{\mathbb R}}^3\setminus \{ 0\}$ the linear combination $\alpha_{\bf c}=c_1\alpha_1+c_2\alpha_2+c_3\alpha_3$ is a contact form. The contact sphere is called [taut]{} if all volume forms $\alpha_{\bf c}\wedge d\alpha_{\bf c}$ with $\|{\bf c}\|=1$ are equal, see [@gego95]. Multiplying the three $1$-forms of a taut contact sphere by the same non-vanishing function $w$ gives a new taut contact sphere, thanks to the identity $w\alpha\wedge d(w\alpha )=w^2\,\alpha\wedge d\alpha$. We now recall from [@gego08] a correspondence between hyperkähler structures on $M\times{{\mathbb R}}_t$, satisfying condition (ii) of Section \[section:GH\], and taut contact spheres on $M$. Given the hyperkähler structure, consider the symplectic forms $\Omega_{\bf c}=c_1\,\Omega_1+c_2\,\Omega_2+c_3\,\Omega_3$ for ${\bf c}\in S^2$. We know that the volume form $\Omega_{\bf c}^2$ is the same for all unitary $\bf c$. On the other hand, the equation $L_{\partial_t}\Omega_{\bf c}=\Omega_{\bf c}$ is equivalent to $\Omega_{\bf c}=d(\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_{\bf c})$, and this implies $$\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }(\Omega_{\bf c}^2)=2\, (\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_{\bf c})\wedge d(\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_{\bf c}) .$$ Since the $3$-form $\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }(\Omega_{\bf c}^2)$ does not depend on $\bf c$, the family $\bigl( \partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_{\bf c}\bigr)_{{\bf c}\in S^2}$ induces a taut contact sphere on any transversal for $\partial_t$. If $(\alpha_1,\alpha_2,\alpha_3)$ is the contact sphere induced on the transversal $\{ t=0\}$, the conditions $L_{\partial_t}(\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i)=\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i$ and $(\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i)(\partial_t)=0$ lead to the expressions $$\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i =e^t\alpha_i,\quad i=1,2,3,$$ where the $\alpha_i$ have been pulled back to $M\times{{\mathbb R}}_t$. Then also $\Omega_i=d(e^t\alpha_i)$. Conversely, if we are given a taut contact sphere $(\alpha_1,\alpha_2,\alpha_3)$ on $M$, then — subject only to the sign condition $(\alpha_1\wedge d\alpha_1)/(\alpha_1\wedge\alpha_2\wedge\alpha_3)>0$, which will be satisfied after a suitable permutation of the $\alpha_i$ — the symplectic forms $\Omega_i=d(e^t\alpha_i)$ determine a hyperkähler structure on $M\times{{\mathbb R}}_t$ that obviously satisfies condition (ii) of Section \[section:GH\]. These two processes, passing from a hyperkähler structure to a taut contact sphere and vice versa, are inverses of each other. Now we can define the natural metric mentioned at the beginning of this section. For a hyperkähler structure satisfying condition (ii) of Section \[section:GH\], we write $(\omega_1,\omega_2,\omega_3)$ for the taut contact sphere induced by the $1$-forms $\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i$ on the canonical slice and call $g_s:=\omega_1^2+\omega_2^2+\omega_3^2$ the [short metric]{} on the canonical slice. Here is how the short metric $g_s$ and the canonical metric $g_3$ defined in Section \[section:canonical\] are related. Any taut contact sphere $(\alpha_1,\alpha_2,\alpha_3)$ on a $3$-manifold satisfies the following structure equations, where $(i,j,k)$ ranges over the cyclic permutations of $(1,2,3)$, with $\beta_0$ a unique $1$-form and $\Lambda_0$ a unique function: $$\label{eqn:structure} d\alpha_i=\beta_0\wedge\alpha_i+\Lambda_0\,\alpha_j\wedge\alpha_k.$$ If we have a hyperkähler structure satisfying (ii) on $M\times{{\mathbb R}}_t$, and $(\alpha_1,\alpha_2,\alpha_3)$ is the contact sphere induced by the $1$-forms $\partial_t{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }\Omega_i$ on the transversal $\{ t=0\}$, then the function $\Lambda_0$ from the structure equations is positive and we have the formulæ$$\Omega_i=d(e^t\alpha_i)=e^t\bigl( \Lambda_0^{-1/2}(dt+\beta_0)\wedge \Lambda_0^{1/2}\alpha_i+ \Lambda_0^{1/2}\alpha_j\wedge \Lambda_0^{1/2} \alpha_k\bigl) .$$ It follows that the hyperkähler metric is given by $$\label{eqn:metric} g=e^t\bigl( \Lambda_0^{-1}\cdot (dt+\beta_0 )^2+\Lambda_0\cdot (\alpha_1^2+\alpha_2^2+\alpha_3^2)\bigr) .$$ The canonical slice is thus given by the equation $t=\log\Lambda_0$. The taut contact sphere $(\omega_1,\omega_2,\omega_3)$ induced on the canonical slice $S$ is given by $\omega_i=e^t\alpha_i\|_{TS}$. With $\beta:=(dt+\beta_0)\|_{TS}$ we have $$g_3:=g\|_{TS}=\beta^2+\omega_1^2+\omega_2^2+\omega_3^2.$$ Clearly, $g_s$ is shorter than $g_3$. The short metric is incomplete in all the examples described in Sections \[section:example\] and \[section:many\], because of the following result. \[thm:incomplete\] If a hyperkähler structure on $\Sigma\times{{\mathbb R}}_t\times S^1_\theta$ satisfies conditions (i) and (ii) of Section \[section:GH\] and the canonical slice is non-compact, then the short metric is incomplete. From the definitions of $\omega_i$ we have $d\omega_i=d(e^t\alpha_i)\|_{TS}$. A straightforward computation yields the following structure equations for the taut contact sphere $(\omega_1,\omega_2,\omega_3)$ on the canonical slice: $$\label{eqn:structure-omega} d\omega_i=\beta\wedge\omega_i+\omega_j\wedge\omega_k.$$ These structure equations can be used to give an alternative definition of the $1$-form $\beta$. The next proposition is the last ingredient we need in order to display the announced Carnot–Carathéodory phenomenon. \[prop:CC\] Given a Riemann surface $(\Sigma ,J)$ and holomorphic data $(\Phi ,\varphi )$, consider the corresponding hyperkähler structure on $W=\Sigma\times{{\mathbb R}}_t\times S^1_\theta$, as explained in Section \[section:GH\]. Let $(\omega_1,\omega_2,\omega_3)$ be the taut contact sphere induced on the canonical slice $S$, and let $\beta$ be the $1$-form on $S$ defined by the structure equations (\[eqn:structure-omega\]). If $\varphi$ is non-constant, then $\beta$ vanishes only along a discrete set of orbits of the $S^1$-action on $S$, and it is a contact form in the rest of $S$, defining the opposite orientation to that defined by the contact forms $\omega_i$. Therefore, any two points on $S$ can be joined by a path tangent to $\ker\beta$. The proof of this proposition is given in the appendix. We can use the paths tangent to $\ker\beta$ to define Carnot–Carathéodory distances on $S$. The one induced by $g_3$ is a complete distance, because $g_3$ is a complete metric. Now the relation $g_3=g_s+\beta^2$ tells us that the [incomplete]{} metric $g_s$ coincides with $g_3$ on those paths and thus induces exactly the same Carnot–Carathéodory distance as $g_3$. So we have an incomplete metric $g_s$ inducing a complete Carnot–Carathéodory distance. The geometric interpretation of this fact is that the proper paths tangent to $\ker\beta$ are so wrinkled that they always have infinite length in the incomplete metric $g_s$. This phenomenon is interesting because such paths can be arbitrarily $C^0$-close to any given path — so $g_s$ is, in some sense, very close to being complete. Appendix {#appendix .unnumbered} ======== Here we prove Proposition \[prop:CC\]. We first derive an explicit formula for $\beta$, from which all the properties claimed in Proposition \[prop:CC\] can then be deduced. In analogy to Section \[section:GH\], we define the momentum map $x=(x_1,x_2,x_3)\co W\to{{\mathbb R}}^3\,$ by $$x_i=\Omega_i (\partial_\theta ,\partial_t)=-e^t\alpha_i(\partial_\theta ) ,$$ and we define $\rho\co S\to{{\mathbb R}}$ by $\rho =\|x\|$. Observe that $x$ is $\partial_{\theta}$-invariant, which allowed us in Section \[section:GH\] to view it as a function on $\Sigma\times{{\mathbb R}}_t$. Below, however, we want to consider the $x_i$ as functions on $S$, where they equal $-\omega_i(\partial_{\theta})$. On $TS$ we then have the identity $dx_i=\partial_{\theta}{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }d\omega_i$. We also regard the functions $\varphi$ and $\psi =-1/\varphi$ as functions on the canonical slice $S$, by first pulling them back to $\Sigma\times{{\mathbb R}}_t\times S^1_\theta$ and then restricting them to $S$. \[lem:psi\] On $S$ we have the identity $\psi =-\beta (\partial_\theta )+{{\bf i}}\rho$. Lemma \[lem:slice\] states that on $S$ the imaginary part of $\psi$ is $\rho$. In order to determine the real part, we use the following alternative expressions for the hyperkähler metric $g$ on $W$: $$\label{eqn:g} V^{-1}\cdot (d\theta +\eta )^2+V\cdot (dx_1^2+dx_2^2+dx_3^2) = e^t\bigl( \Lambda_0^{-1}\cdot (dt+\beta_0)^2+\Lambda_0\cdot (\alpha_1^2+\alpha_2^2+\alpha_3^2)\bigr) .$$ Computing $g(\partial_\theta ,\partial_t)$ in the two possible ways, we get $V^{-1}\eta (\partial_t)=e^t\Lambda_0^{-1}\beta_0 (\partial_\theta )$. On the canonical slice we have $e^t=\Lambda_0$ and, by Lemma \[lem:slice\], $V=\|\varphi \|^2$. Moreover, $\eta (\partial_t)=\mathrm{Re}\,\varphi$ by the definition of $\varphi$ in Theorem \[thm:GH\]. So on $S$ we have $$\beta (\partial_\theta )=\beta_0(\partial_{\theta})= \frac{\mathrm{Re}\,\varphi}{\|\varphi \|^2}=-\mathrm{Re}\,\psi ,$$ as claimed. From (\[eqn:g\]) we find the following alternative expressions for the metric $g_3$ induced on $S$: $$\label{eqn:g3} \|\varphi\|^{-2}\cdot (d\theta +\eta )^2+\|\varphi \|^2 \cdot (dx_1^2+dx_2^2+dx_3^2) = \beta^2+\omega_1^2+\omega_2^2+\omega_3^2.$$ Introduce the auxiliary $1$-form $\gamma :=\sum_{i=1}^3x_i\omega_i$ on $S$. By taking the interior product with $\partial_\theta$ in (\[eqn:g3\]) we find, with Lemma \[lem:psi\], $$\label{eqn:1} \|\varphi \|^{-2}\cdot (d\theta +\eta )= \beta (\partial_\theta )\beta -\gamma = -(\mathrm{Re}\,\psi ) \beta -\gamma.$$ With the structure equations (\[eqn:structure-omega\]) for $d\omega_i$, we obtain from $dx_i=\partial_\theta{\,\rule{2.3mm}{.2mm}\rule{.2mm}{2.3mm}\; }d\omega_i$ the equations $$dx_i=\beta (\partial_\theta )\omega_i+x_i\beta -x_j\omega_k+x_k\omega_j .$$ This yields $$\label{eqn:2} \rho\, d\rho =\sum_ {i=1}^3x_i\, dx_i= \rho^2\beta +\beta (\partial_\theta)\gamma = \rho^2\beta -(\mathrm{Re}\,\psi )\gamma .$$ Formulæ (\[eqn:1\]) and (\[eqn:2\]) constitute a linear system for $\beta$ and $\gamma$. We solve it for $\beta$, observing that from Lemma \[lem:psi\] we have $\beta (\partial_{\theta})^2+\rho^2=\|\psi \|^2$, to obtain $$\begin{aligned} \beta & = & \|\psi \|^{-2}\cdot\bigl( \rho\, d\rho -(\mathrm{Re}\,\psi )\|\varphi \|^{-2}\cdot (d\theta +\eta )\bigr) \\ & = & \|\psi \|^{-2}\rho\, d\rho - (\mathrm{Re}\,\psi ) (d\theta +\eta ).\end{aligned}$$ Using the expression $\eta =(\mathrm{Re}\,\varphi)\frac{d\rho}{\rho}+\xi =-\|\psi \|^{-2}(\mathrm{Re}\,\psi )\frac{d\rho}{\rho}+\xi$ from Theorem \[thm:GH\], as well as Lemma \[lem:psi\] and the identities in Lemma \[lem:slice\], we transform the last equality as follows: $$\begin{aligned} \beta & = & \|\psi \|^{-2}\cdot \bigl(\rho^2 +(\mathrm{Re}\,\psi )^2\bigr) \frac{d\rho}{\rho} -(\mathrm{Re}\,\psi )(d\theta +\xi )\\ & = & \frac{d\rho}{\rho} - (\mathrm{Re}\,\psi )(d\theta +\xi )\\ & = & d\log\mathrm{Im}\,\psi - (\mathrm{Re}\,\psi )(d\theta +\xi ).\end{aligned}$$ This is the desired explicit expression for $\beta$. We next want to determine the subset of $S$ where $\beta$ vanishes. Write the canonical slice as $S=G\times S^1_{\theta}$ as in Lemma \[lem:slice\], and points in $G$ as $(p,t(p))$ with $p\in\Sigma$. We then see that $\beta$ vanishes precisely along the circles $\{ (p,t(p))\}\times S^1_\theta$ for those points $p\in\Sigma$ where $d\,\mathrm{Im}\,\psi$ and $\mathrm{Re}\,\psi$ vanish simultaneously. It is easy to verify that for a non-constant holomorphic function $\psi$ such points form a discrete subset of $\Sigma$. Therefore $\beta$ is non-zero outside a discrete set of circles, all of which are orbits of the $S^1$-action on the canonical slice. Finally, we want to prove that $\beta$ is a contact form outside the described vanishing set. From $d(d\omega_i)=0$ and the structure equations (\[eqn:structure-omega\]) one gets $$d\beta\wedge\omega_i+\beta\wedge\omega_j\wedge\omega_k=0.$$ Write $\beta =b_1\omega_1+b_2\omega_2+b_3\omega_3$. Then $$d\beta = - b_1\,\omega_2\wedge\omega_3-b_2\,\omega_3\wedge\omega_1-b_3 \,\omega_1\wedge\omega_2,$$ and so $$\beta\wedge d\beta =-(b_1^2+b_2^2+b_3^2)\,\omega_1\wedge\omega_2\wedge\omega_3 =-(b_1^2+b_2^2+b_3^2)\,\omega_i\wedge d\omega_i.$$ So $\beta$ is indeed a contact form where it is non-zero, and we also observe that the induced orientation is opposite to the one defined by the $\omega_i$. Dragan Vukotić helped us with many questions and discussions about analytic functions on the unit disc, setting us on the right track several times. Francisco Martín Serrano and Antonio Martínez López gave us valuable information on the same subject. Maria José Martín Gómez provided us with the fundamental reference [@bcp83]. Ernesto Girondo made the computer generated hyperbolic tessellation. It is a pleasure to thank all of them. [99]{} C. L. Belna, F. W. Carroll and G. Piranian, Strong Fatou-1-points of Blaschke products, [Trans. Amer. Math. Soc.]{} [280]{} (1983), 695–702. A. L. Besse, [Einstein Manifolds]{}, Ergeb. Math. Grenzgeb. (3) [10]{}, Springer-Verlag, Berlin (1987). D. M. DeTurck and J. L. Kazdan, Some regularity theorems in Riemannian geometry, [Ann. Sci. École Norm. Sup. (4)]{} [14]{} (1981), 249–260. H. Geiges, [An Introduction to Contact Topology]{}, Cambridge Stud. Adv. Math. [109]{}, Cambridge University Press (2008). H. Geiges and J. Gonzalo Pérez, Contact geometry and complex surfaces, [Invent. Math.]{} [121]{} (1995), 147–209. H. Geiges and J. Gonzalo Pérez, Contact spheres and hyperkähler geometry, [Comm. Math. Phys.]{}, to appear. G. W. Gibbons and S. W. Hawking, Gravitational multi-instantons, [Phys. Lett. B]{} [78]{} (1978), 430–432. M. Gromov, Carnot–Carathéodory spaces seen from within, in: [Sub-Riemannian Geometry]{}, Progr. Math. [144]{}, Birkhäuser, Basel (1996), pp. 79–323. W. K. Hayman, [Meromorphic Functions]{}, Oxford Math. Monogr., Clarendon Press, Oxford (1964). S. Kobayashi and K. Nomizu, [Foundations of Differential Geometry]{}, Vol. 1, Interscience Publishers, New York (1963). R. Montgomery, [A Tour of Subriemannian Geometries, their Geodesics and Applications]{}, Math. Surveys Monogr. [91]{}, American Mathematical Society, Providence (2002). W. P. Thurston, [Three-dimensional Geometry and Topology, Vol. 1]{}, edited by Silvio Levy, Princeton Math. Ser. [35*]{}, Princeton University Press (1997). [^1]: H. G. is partially supported by DFG grant GE 1245/1-2 within the framework of the Schwerpunktprogramm “Globale Differentialgeometrie”. [^2]: J. G. is partially supported by grants MTM2004-04794 and MTM2007-61982 from MEC Spain.
--- abstract: 'We use magnetic flux-tubes to stabilize zero-energy modes in a lattice realization of a 2-dimensional superconductor from class D of classification table of topological condensed matter systems. The zero modes are exchanged by slowly displacing the flux-tubes and an application of the adiabatic theorem demonstrates the geometric nature of the resulting unitary time-evolution operators. Furthermore, an explicit numerical evaluation reveals that the evolutions are in fact topological, hence supplying a representation of the braid group, which turns out to be non-abelian. This physical representation is further formalized using single-strand planar diagrams. Lastly, we discuss how these predictions can be implemented with and observed in classical meta-materials and how the standard Majorana representation of the braid group can be generated by measuring derived physical observables.' address: - 'School of Physics, Nankai University, Tianjin 300071, China' - 'Department of Physics, Yeshiva University, New York, NY 10016, USA' author: - 'Yifei Liu, Yingkai Liu' - Emil Prodan title: 'Braiding Flux-Tubes in Topological Quantum and Classical Lattice Models from Class-D' --- Introduction ============ In strongly correlated condensed matter systems, fractional and non-abelian statistics of topological defects, called anyons, have been predicted and simulated in a large number of models [@BarabanPRL2009; @ProdanPRB2009; @KapitPLR2012; @WuPRL2014]. These properties have been recognized for their potential in quantum computing, because controlled braiding of anyons induce unitary transformations of their Hilbert sub-space, which can be concatenated into quantum gates [@BonesteelPRL2005]. Since the braids are topological in nature, these unitary transformations do not depend on the geometric details of the braids, hence they can be physically reproduced again and again with high fidelity. Furthermore, the anyons do not couple directly with the fermionic degrees of freedom, hence they are immune to the environment fluctuations and this, together with the topological nature of the braids, supply an error protection mechanism at the hardware level [@KitaevBook; @FreedmanCMP2002; @KitaevAOP2003; @WangBook; @PachosBook]. Evidence of non-abelian statistics in correlated electron systems is experimentally sought, indirectly, from anyon interferometry [@ChamonPRB1997; @BondersonAPhys2008; @WillettRPP2013], topological spectroscopies [@ChangRMP2003; @CooperPRL2015; @MorampudiPRL2017; @PapicRPX2018]. However, demonstrating controlled concatenated braidings of the anyons, [e.g.]{} of Majorana fermions via the T-junction procedure [@AliceaNatPhys2011], is quite far in the future. In a recent work [@BarlasArxiv2019], one of the authors pointed out that topological point defects can be stabilized and braided in classical meta-materials. This offers new straightforward experimental venues where controlled braiding and direct observation of non-abelian statistics can be achieved. Let us point out that braiding of four Majorana-like modes in an experimental photonic setup was recently achieved in [@XuSciAdv2018]. Following a similar strategy as in [@BarlasArxiv2019], we here present a 2-dimensional lattice model where large numbers of topological point defects can be stabilized by magnetic flux-tubes, to form a highly quasi-degenerate resonant level in the middle of a topological spectral gap. These are somewhat similar to the Aharonov-Casher zero modes [@AharonovPRA1979], whose braiding characteristics have been analyzed in [@KennethAP2014]. There are, however, several important differences. Our model is on a discrete lattice rather than on the continuum plane and we insisted on this feature because such models can be more straightforwardly implement experimentally with meta-materials (see section \[Sec:ClassicalD\]). Also, in our case, the spectral stabilization of the defects modes in the middle of the topological gap is due to a particle-hole symmetry of the model, which is achieved only at flux values of $\pi$ in the natural units. Since the models are formulated on a lattice, there is a fundamental difficulty related to displacing the flux-tubes, because the Peierls factors behave discontinuosly when links are crossed. Nevertheless, by properly augmenting the hopping amplitudes, we succeeded on adiabatically and smoothly exchanging the flux-tubes and on ultimately obtaining an adiabatic representation of the braid group. This representation is numerically computed following the procedure from [@ProdanPRB2009] and the calculations confirm that the representation is non-abelian. The physical representation of the braid group is further analyzed in terms of single-strand planar diagrams for arbitrary number of flux tubes. As demonstrated in [@BarlasPRB2018], any quantum lattice model can be simulated with classical waves supported by passive meta-materials, in a manner that respects all the symmetries, in particular, the anti-unitary particle-hole symmetry. Using the algorithm discovered in [@BarlasPRB2018], we supply concrete tight-binding dynamical matrices that can be implemented with networks of coupled mechanical resonators, such as springs and balls [@ProdanNatComm2017] or magnetically coupled spinners [@ApigoPRM2018]. As we shall see, implementing the braid operations in a laboratory can be achieved by slowly changing the coupling strengths between the resonators and no transfers of mass or re-configurations of the system is required. The representations of the braid group generated by such adiabatic deformations are fundamentally different from the one generated by braiding Majorana fermions in a superconductor. The latter, however, is an induced representation which simply lifts the braid group representations generated at BdG effective level to the many particle realm. Guided by this fact, we introduce certain derived physical observables that reside in the Clifford algebra associated to the classical zero-mode space and demonstrate that the derived representation coincides with the one generated by brading the Majorana fermions. The Bulk Model ============== In this section we introduce the translational invariant bulk model defined on a 2-dimensional lattice and we discuss its spectral and topological characteristics as well as its particle-hole symmetry. The Bulk Hamiltonian and its Spectral Properties ------------------------------------------------ We will work with the minimal 2-dimensional model from the D-class of classification table of topological condensed matter [@SRFL2008; @QiPRB2008; @Kit2009; @RSFL2010], which relates to BdG effective Hamiltonians describing the fermionic excitations in a $p+\imath p$ super-conductor. The model is formulated on a square lattice and it has two degrees of freedom per each node. Hence, the Hilbert space is ${{\mathbb C}}^2 \otimes \ell^2({{\mathbb Z}}^2)$. If $S_j$’s are shift operators on the lattice: $$S_j \|\bm m \rangle =\| m + \bm e_j \rangle, \quad S_j^\dagger \|\bm m \rangle = \|\bm m - \bm e_j \rangle, \quad j=1,2,$$ where $\bm e_j$ are the generators of ${{\mathbb Z}}^2$, then the Hamiltonian of our model can be written in the following compact form: $$\label{Eq:BulkH} H=\sigma_3\otimes\bigg [ M \, I+\tfrac{1}{2} \sum_{j=1,2} (S_j +S_j^\ast)\bigg ] + \tfrac{1}{2 \imath} \sum_{j=1,2} \sigma_j \otimes (S_j -S_j^\ast).$$ Throughout, the $\sigma$’s will denote Pauli’s matrices. As one can see, we fixed the energy and space units such that the Hamiltonian displays only one parameter, the mass $M$. The energy dispersion spectrum can be computed explicitly: $$E_{\bm k} = \sqrt{ \Big(M+\sum_{j=1,2}\cos(k_j)\Big)^2+\sum_{j=1,2} \sin(k_j)^2 }, \quad \bm k \in [-\pi,\pi]^2,$$ and plots of the dispersion bands are supplied in Fig. \[Fig:EnergyKxKy\]. The data reveals bulk gap closings at $M=0$ and $\pm 2$ and, as we shall see, these are all topological phase transitions. ![Band structure of the Hamiltonian for selected values of $M$ parameter. Band touchings are observed at $M= \pm 2$ and $M=0$.[]{data-label="Fig:EnergyKxKy"}](EnergyKxKy.pdf){width="\textwidth"} Topological characteristics --------------------------- The Berry curvature of the lower band can be also computed explicitly: $$\begin{aligned} F(\bm k) =\frac{\cos k_1+\cos k_2+M \cos k_1 \cos k_2}{\big ((M+\cos k_1+\cos k_2)^2+\sin k_1 ^2+\sin k_2^2\big)^{3/2}} \end{aligned}$$ and plots of it are supplied in Fig. \[Fig:BerryCurvature\]. The bulk invariant, which labels the topological phases of the model, is the Chern number of the lower band: $$\begin{aligned} \int_{{{\mathbb T}}^2}{\rm d} \bm k \ F(\bm k)= \begin{cases} 1&\text{for}\ 0<M<2\\ -1&\text{for}\ -2<M<0\\ 0&\text{for}\ 2<\|M\|. \end{cases} \end{aligned}$$ There are two topological phases with a common boundary at $M=0$ and these topological phases are surrounded by a trivial phase which resides in the region $\|M\|>2$ of the parameter space. The hallmark of the topological phases is the emergence of chiral edge states whenever a bulk sample is halved [@ProdanBook1]. In this work, however, we will be interested in point defects rather than edges. Stabilization of zero-dimensional topological modes inside the bulk gap is enabled by an additional and fundamental property of the model, namely, its particle-hole symmetry: $$\label{Eq:PHSymmetry} \Theta_{\rm PH} H \Theta_{\rm PH}^{-1} = -H, \quad \Theta_{\rm PH} = (\sigma_1 \otimes I) \mathcal K, \quad \Theta_{\rm PH}^2 = 1,$$ where ${{\mathcal K}}$ is the ordinary complex conjugation. This symmetry forces the energy spectrum to be symmetric relative to $E=0$ mark. ![Plots of the Berry curvature as function of quasi-momentum, for selected values of $M$ parameter.[]{data-label="Fig:BerryCurvature"}](BerryCurvature.pdf){width="\textwidth"} Magnetic Flux Insertion ======================= In this section, we insert infinitely thin magnetic flux-tubes through the plane of the lattice and study the spectral properties of the resulting Hamiltonian. As a result of the topological character of the Hamiltonian and of its particle-hole symmetry, a quasi-degenerate energy level develops at zero energy when the flux values are properly adjusted. The invariant space corresponding to this quasi-degenerate level will supply the representation space for the braid group. Peierls substitution -------------------- For a generic lattice Hamiltonian on the Hilbert space ${{\mathbb C}}^2 \otimes \ell^2({{\mathbb Z}}^2)$: $$\begin{aligned} H=\sum_{\bm n,\bm n'} h_{\bm n',\bm n} \otimes \|\bm n ' \rangle \langle \bm n \|, \end{aligned}$$ a flux-tube insertion at position $\bm x$ is taken into account by the Peierls substitution [@Peierls], which transforms the Hamiltonian into: $$\begin{aligned} H(\bm x)=\sum_{\bm n,\bm n'} \exp \Big (\imath \int_{\gamma_{\bm n \bm n'}}\vec{A}\cdot{\rm d} \vec{\ell} \, \Big) \, h_{\bm n',\bm n} \otimes \|\bm n'\rangle \langle\bm n \|, \end{aligned}$$ where $\vec{A}$ is the corresponding magnetic vector potential. A geometric representation of the situation is supplied in Fig. \[Fig:FluxInsertion\]. For a flux-tube of infinitesimal radius and carrying a magnetic flux $\Phi$, the vector potential takes the form $\vec A = \frac{\Phi}{2 \pi r} \hat e_\varphi$ in the standard polar coordinates centered at $\bm x$. As such, the Peierls phase has an explicit expression: $$\theta_{\bm n\bm n'}(\bm x) = \int_{\gamma_{\bm n \bm n'}}\vec{A}\cdot {\rm d}{\vec{\ell}}= \frac{\Phi}{2\pi} \Delta \varphi_{\bm n\bm n'},$$ with the $\Delta \varphi$ angles as specified in Fig. \[Fig:FluxInsertion\]. \[Re:Peierls\] Several important observations are in place: - The angles $\Delta \varphi_{\bm n \bm n'}$ are measured from $\bm n$ towards $\bm n'$, hence they come with definite sign, which is positive if the rotation of $\bm n$ towards $\bm n'$ is in the trigonometric sense, and negative otherwise. - Up to an additive factor of $\pm 2\pi$, $\Delta \varphi_{\bm n \bm n'}=\varphi_{\bm n'}-\varphi_{\bm n}$, where the latter are the angular coordinates of $\bm n$ and $\bm n'$, respectively, in a coordinate system centered at $\bm x$. This statement can be written more precisely as: $$\label{Eq:Mod2PI} \Delta \varphi_{\bm n \bm n'}=\big (\varphi_{\bm n'}-\varphi_{\bm n}\big ) {\rm mod}\, 2\pi.$$ - All angles defined above depend on the position of the flux-tube and, when needed, we will specify this dependency explicitly. In particular, $\Delta \varphi_{\bm n \bm n'}(\bm x)$ jumps by $\pm 2\pi$ whenever $\bm x$ crosses the segment joining $\bm n$ and $\bm n'$. This will have important consequences for the braiding of the flux-tubes. ![Geometry associated with the flux-tube insertion.[]{data-label="Fig:FluxInsertion"}](GAngleDifference.pdf){width="70.00000%"} With the above data, our specific model with $N$ flux-tubes inserted at positions $\bm x_1,\ldots,\bm x_N$ becomes: $$\begin{aligned} \label{Eq:Hamiltonian} & H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N)= M \sigma_3 \otimes I \\ \nonumber & \quad +\tfrac{1}{2} \sum_{j=1,2}\sum_{\bm n \in \mathbb Z^2} \bigg [ e^{\imath \theta_{\bm n, \bm n+\bm e_j}} (\sigma_3 - \imath \sigma_j)\otimes \|\bm n +\bm e_j\rangle \langle \bm n\|+ h.c. \bigg] ,\end{aligned}$$ with $\theta_{\bm n \bm n'}=\sum_{k=1}^N \theta_{\bm n \bm n'}(\bm x_k)$. The mass parameter $M$ will be fixed in the middle of the topological phase from now on, namely, $M=1$. Spectral flow with the flux --------------------------- We present first numerical results for two flux-tubes positioned at fixed locations $\bm x_1$ and $\bm x_2$. Since our computations are performed with periodic boundary conditions, we are constrained to consider $\Phi_1=-\Phi_2=\Phi$, for a total of zero-flux through the entire lattice. The spectrum of the Hamiltonian as function of $\Phi$ is reported in Fig. \[Fig:SpecVsFlux\], together with samples of eigen-functions profiles. As it is well known [@AvronCMP1994], the flux-tubes trap a number of electron states equal to the Chern number of the bulk model and, as the value of the flux increases/decreases, the eigen-energy of the trapped states flows from the upper/lower band towards the lower/upper band, setting in motion a spectral flow. This is precisely what can be observed in Fig. \[Fig:SpecVsFlux\]. Let us point out that the pair of chiral bands seen there correspond to the two flux-tubes and, since the tubes are spatially separated, there is practically no interference between the two and they can be analyzed one at a time. In particular, the pairs of eigen-functions shown in Figs. \[Fig:SpecVsFlux\] are practically the topological modes trapped by the individual flux-tubes. ![(a) Spectrum of $H(\bm x_1\|\Phi,\bm x_2\|-\Phi)$ as function of $\Phi$. (b-c) The spatial profile of the eigen-vectors associated with the eigenvalues marked by the red dots in panel (a). Spatial localization of the eigen-vectors becomes sharper and sharper as the eigenvalues approach the mid-gap level.[]{data-label="Fig:SpecVsFlux"}](SpectrumVSFlux.pdf){width="70.00000%"} The spectral flow repeats itself with a period of $\Phi=2\pi$ in Fig. \[Fig:SpecVsFlux\]. To understand this feature in the most general context, let us go back to the general case of $N$ tubes and consider an increase by $2 \pi$ of one flux $\Phi_j$. We define the unitary transformation: $$\|\bm n \rangle \rightarrow U_k(\bm x_j) \|\bm n \rangle =e^{\imath k \varphi_{\bm n}(\bm x_j)} \, \|\bm n \rangle, \quad \bm n \in {{\mathbb Z}}^2, \quad k \in \mathbb Z,$$ with the action on the hopping operators: $$\label{Eq:UFactor} U_k(\bm x_j)\, \|\bm n' \rangle \langle \bm n \| \, U_k^{-1}(\bm x_j)= e^{\imath k (\varphi_{\bm n'}(\bm x_j)-\varphi_{\bm n}(\bm x_j))} \, \|\bm n' \rangle \langle \bm n \|.$$ When the flux $\Phi_j$ is increased by $2\pi$, the Peierls phase factor is modified by the multiplicative term $e^{\imath \Delta \varphi_{\bm n\bm n'}(\bm x_j)}$ and, from , we see that this factor is the same as the one in for $k=1$. Therefore: $$H(\Phi_j+2\pi) = U_1(\bm x_j)H(\Phi_j)U_1^{-1}(\bm x_j),$$ and this explains why an increase of any of the fluxes by multiples of $2\pi$ leaves the energy spectrum unchanged. The space of zero-modes ----------------------- The second interesting feature in Fig. \[Fig:SpecVsFlux\] is the intersection of the spectral flows, which occurs at zero energy and at $\Phi = (2p +1)\pi$, $p \in \mathbb Z$. This phenomenon is related to the particle-hole symmetry of the original model. Indeed, let us consider the general case of $N$ flux tubes with $\Phi_j = (2 p_j +1)\pi$. Then a conjugation by $\Theta_{\rm PH}$ transforms the Peierls phase factors from $e^{\imath (p_j +\frac{1}{2}) \Delta \varphi_{\bm n \bm n'}(\bm x_j)}$ to $e^{-\imath (p_j +\frac{1}{2}) \Delta \varphi_{\bm n \bm n'}(\bm x_j)}$. These two terms differ by a multiplicative factor of $e^{-\imath (2p_j +1) \Delta \varphi_{\bm n \bm n'}(\bm x_j)}$ and, by recalling , this can be easily assessed to be a purely gauge conjugation. More precisely: $$\begin{aligned} \Theta_{\rm PH} \, H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N ) \, \Theta_{\rm PH}^{-1} = - U H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N) U^{-1},\end{aligned}$$ with: $$\begin{aligned} \label{Eq:PHGauge} U= U_{2p_1+1}(\bm x_1) \ldots U_{2p_N+1}(\bm x_N).\end{aligned}$$ It is convenient to re-define the PH-symmetry operation as: $$\Theta_{\rm PH} = U^{-1} (\sigma_1 \otimes I) \mathcal K, \quad \Theta_{\rm PH}^2 = I,$$ such that: $$\begin{aligned} \Theta_{\rm PH} \, H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N ) \, \Theta_{\rm PH}^{-1} = - H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N).\end{aligned}$$ It now becomes clear that the energy spectra must be mirror symmetric relative to $E=0$, whenever the fluxes $\Phi_j$ are odd multiples of $\pi$. This implies that, if the bulk Chern number is odd and the flux-tubes are well separated such that they can be treated separately, then each of their chiral spectral flows must cross the level $E=0$ at $\Phi=(2p+1)\pi$. This is illustrated by an explicit numerical simulation in Fig. \[Fig:OddVsEvenChern\]. The conclusion is that, if $\Phi_j$’s are fixed at $\pm \pi$, the flux-tubes are well separated and our model is in a topological phase, then $H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N )$ displays a $N$-fold quasi-degenerate energy level centered at zero. The spectrum corresponding to this quasi-degenerate level will be denoted by $\Sigma_0$ and eigen-vectors associated with $\Sigma_0$ will be referred as zero-modes. The linear $N$-dimensional space spanned by them will be referred as the zero-modes space and will be denoted by ${{\mathcal H}}_0(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N )$, with the understanding that $\Phi_j$’s are fixed at $\pm \pi$. Note that this sub-space of ${{\mathbb C}}^2 \otimes \ell^2({{\mathbb Z}}^2)$ changes with the positions of the flux-tubes. ![Spectral flow under a single flux-tube insertion for three values of the Chern number. The models with higher Chern numbers were obtained by stacking appropriate numbers of minimal models and by turning on a soft coupling between the stacked copies. The cases with odd Chern numbers display exact zero-modes while, in the case of even Chern number, the mid-gap modes are unstable and split from $E=0$ mark.[]{data-label="Fig:OddVsEvenChern"}](OddVsEvenChern.pdf){width="90.00000%"} Adiabatic Displacement of the Flux-Tubes ======================================== In this section, we investigate the time evolution of the quantum states induced by a slow displacement of the flux-tubes. In its current form, the Hamiltonian evolves discontinuously whenever a flux-tube crosses an edge of the lattice. Our main goal here is to describe a solution which resolves this issue and, consequently, it allows us to achieve the adiabatic regime where the time evolution of the zero-modes becomes purely geometric in nature. Resolving the discontinuity of the Hamiltonian {#SubS:ResDisc} ---------------------------------------------- Let us recall the discussion from Remark \[Re:Peierls\], where we learned that $\Delta \varphi_{\bm n \bm n'}$ jumps by an additive factor of $\pm 2 \pi$ whenever a flux-tube crosses the segment joining $\bm n$ and $\bm n'$. Because the flux-tubes are fixed at odd multiples of $\pi$, this implies that the Peierls phase factors in the Hamiltonian \[Eq:Hamiltonian\] change sign. The only way to correct for this undesirable effect, is to smoothly turn off the coefficient of the Hamiltonian that is affected by this phenomenon. This can be done by modifying the Hamiltonian to the following expression: $$\begin{aligned} \label{Eq:NewHamiltonian} & \qquad \qquad H(\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N)= M \sigma_3 \otimes I \\ \nonumber & \quad +\tfrac{1}{2} \sum_{j=1,2}\sum_{\bm n \in \mathbb Z^2} \bigg [ g(\bm n,\bm n+\bm e_j)e^{\imath \theta_{\bm n, \bm n+\bm e_j}} (\sigma_3 - \imath \sigma_j)\otimes \|\bm n +\bm e_j\rangle \langle \bm n\|+ h.c. \bigg] ,\end{aligned}$$ where $$g(\bm n,\bm n') =\prod_{k=1}^N \Big (1-\exp\Big(-(\|0.5(\bm n+\bm n')-\bm x_k\|/\gamma)^2\Big)\Big ).$$ This particular choice was made to ensure that, whenever the position $\bm x_k$ of any of the flux-tubes traverses a link $(\bm n,\bm n +\bm e_j)$ through the middle of it, the hopping term corresponding to this link, and only to this link, is smoothly turned off by the factor $g$ inserted in the Hamiltonian. Hence, from now on, we work exclusively with the Hamiltonian and the flux-tubes will be displaced such that they always cross the links of the lattice through the middle. The parameter $\gamma$ will be fixed at $1$ throughout. ![Evolution of the spectrum of (a) Hamiltonian ($\gamma=1$), and (b) Hamiltonian , when a (+)-flux tube is circled around a (-)-flux tube.[]{data-label="Fig:SmoothEvolution"}](PotentialSpectrum.pdf){width="90.00000%"} In Fig. \[Fig:SmoothEvolution\], we report the evolution of the energy spectrum computed with the Hamiltonian \[Eq:NewHamiltonian\], as a (+) flux-tube is circled around a (-) flux-tube. It demonstrates that the quasi-zero energy spectrum remains isolated from the rest of the spectrum, which is the pre-requisite for achieving the adiabatic regime. Further tests and comparisons between the Hamiltonians and are illustrated in Fig. \[Fig:UTest\], after more background is developed. As we shall see, it is practically impossible to achieve the adiabatic regime with the Hamiltonian , due to the sudden discontinuities discussed above, but it is possible with the Hamiltonian . The time evolution operators ---------------------------- In the case when the positions of the flux-tubes change in time, a quantum state $\Psi_0$ prepared at time $t_0$ evolves as: $$\Psi(t)=U(t,t_0)\Psi_0,$$ where $U(t,t_0)$ is the unitary time evolution operator supplied by the unique solution of the equation: $$\label{Eq:TimeEvolution1} \imath \partial_t U(t,t_0) = H\big (\bm x_1(t)\|\Phi_1,\ldots,\bm x_N(t)\|\Phi_N \big ) U(t,t_0), \quad U(t_0,t_0)=I.$$ The following standard property of the family of evolution operators will play a role later on: $$\label{Eq:GroupProperty} U(t'',t')U(t',t) = U(t'',t), \quad \forall \, t,t',t'' \in {{\mathbb R}}.$$ Adiabatic time-evolution and geometric monodromies -------------------------------------------------- The time evolution considered above can be thought of as induced by the motion of a point $x$ in the multi-dimensional space ${{\mathbb R}}^{2N}$. Consider now a path $\zeta : [0,1]\rightarrow {{\mathbb R}}^{2N}$ in this space and a point that slides along this path with speed $v=1/T$, resulting in the time-dependent Hamiltonian: $$H_t = H\big (\bm x_1(v t)\|\Phi_1,\ldots,\bm x_N(vt)\|\Phi_N \big ), \quad t \in [0,T].$$ Let: $$P_x = \chi_{[-\epsilon,\epsilon]}\Big (H\big (\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N \big ) \Big )$$ be the spectral projector onto the zero-modes space, when the flux-tubes are arranged in an arbitrary instantaneous configuration $x \in {{\mathbb R}}^{2N}$. Above, $\epsilon$ is a small positive constant accounting for the spreading of $\Sigma_0$ when the flux-tubes are at finite separations from one each other and $\chi$ denotes the indicator function. This operator projects the large Hilbert space ${{\mathcal H}}={{\mathbb C}}^2 \otimes \ell^2({{\mathbb Z}}^2)$ onto the zero modes space: $$P_x {{\mathcal H}}= {{\mathcal H}}_0\big (\bm x_1\|\Phi_1,\ldots,\bm x_N\|\Phi_N \big ).$$ Then the adiabatic theorem [@Kato1950] assures us that, under certain regularity conditions related to the smoothness of the time-dependence of $H_t$: $$\label{Eq:AdiabaticTh} U(t,t_0) \,P_{x_0} = W_\zeta (x_t)\, P_{x_0} + o(v),$$ where $W_\zeta(x):P_{x_0}{{\mathcal H}}\rightarrow P_x {{\mathcal H}}$ is the monodromy along $\zeta$, [i.e.]{} the unique unitary solution of the equation: $$\imath \bm \nabla_x W(x) = \imath [\bm \nabla_x P_x,P_x] \, W(x), \quad W(x_0)=P_{x_0},$$ when integrated along the path $\zeta$. Numerically, the monodromy can be conveniently computed as [@ProdanPRB2009]: $$\label{Eq:PracticalUa} W_\zeta(x) = \lim_{K \rightarrow \infty} P_{x_K}\ldots P_{x_0},$$ where $\{ x_0,\ldots,x_K \}$ is a discretization of the segment of path $\zeta$ joining $x_0$ and $x$. In practice, $K$ will have to be set to a finite value which will be referred to as the discretization parameter. ![The unitarity test when a (+)-flux tube is circled around a (-)-flux tube, for (a,b,c) Hamiltonian ($\gamma=1$), and (d,e,f) Hamiltonian . The monodromies were computed with using different discretizations of the world-lines, namely $K=1000$, $5000$ and $10000$.[]{data-label="Fig:UTest"}](MonodromiesTest.pdf){width="80.00000%"} Note that is an adaptation of the general adiabatic theorem, as formulated in [@Kato1950], to our particular context. It takes that form precisely because the spectrum $\Sigma_0$ remains pinned at zero for all flux-tube configurations, which is a remarkable consequence of the topological character of the model and its PH-symmetry. In these conditions, says that the unitary adiabatic time evolution of the zero modes is purely geometric in nature. Indeed, let $T$ be the time it takes the point $x$ to slide along $\zeta$. Then: $$U(T+t_0,t_0)P_{x_0} = W_\zeta(x_f) P_{x_0} + o(1/T),$$ and, according to the above, the time variable is completely absent in the definition of $W_\zeta$, which in fact is entirely determined by the path $\zeta$. When the curve $\zeta$ closes into itself, the unitary time evolution operator maps the initial zero modes space into itself, hence supplying a unitary operator $W_\zeta : P_{x_0} {{\mathcal H}}\rightarrow P_{x_0} {{\mathcal H}}$. Our last comment here is that, if one moves the flux-tubes and uses the original Hamiltonian , then one will find out that the resulting time-dependent Hamiltonian $H_t$ does not satisfy the regularity conditions required by the adiabatic theorem. It is at this point where the solution supplied in section \[SubS:ResDisc\] is to be appreciated. To convince the reader about the striking difference between the two cases, we show in Fig. \[Fig:UTest\] a test on the unitarity of $W_\zeta(x)$: $$\label{Eq:UTest1} W_\zeta (x)^\dagger W_\zeta (x) = I_{P_{x_0}{{\mathcal H}}},$$ as one flux-tube is circled around another, as in Fig. \[Fig:SmoothEvolution\]. Instead of using directly, we use the quantitative test: $$\label{Eq:UTest2} {\rm Tr}\big ( W_\zeta (x)^\dagger W_\zeta (x) \big ) = {\rm dim} \, P_{x_0} {{\mathcal H}}.$$ As one can see in Fig. \[Fig:UTest\], when increasing the discretization parameter, the test becomes more and more accurate when the braiding is performed with Hamiltonian , but this is not at all the case when the Hamiltonian is used instead. The Braid Group ${{\mathcal B}}_N$ and its Physical Representations =================================================================== ![Schematic of a physical process which permutes the flux-tubes $i$ and $j$. The lines represent the world-lines of the two flux-tubes which encode their positions at any instantaneous moment of time.[]{data-label="Fig:BraidingSchematic"}](BraidingSchematic.pdf){width="0.5\linewidth"} In this section we consider physical processes as the ones illustrated in Fig. \[Fig:BraidingSchematic\](a), which result in a permutation of the flux-tubes. The unitary operators induced by such actions are not fully determined by the final permutations of the flux-tubes, but they depend on the world-lines, [e.g.]{} on whether these world-lines enclose or not other flux-tubes. The goal of this section is to connect the adiabatic unitary operators resulting from such actions and the braid group [@BraidBook]. The braid group --------------- ![ []{data-label="Fig:PictorialBraidGroup"}](PictorialBraidGroup.pdf){width="0.8\linewidth"} The braid group ${{\mathcal B}}_N$ consists of all finite braid operations that can be applied to $N$ strands. Two braids can be concatenated to obtain a new braid and this simple rule defines the group composition. Pictorially, a braid can be represented as $N$ strands entering a box and being acted on by a set of braid moves that are defined up to a topological equivalence or, equivalently, to the so called Reidemeister moves [@KauffmanBook1]. The $N$ strands exit the box at the same locations they entered. The composition of two braids reduces to stacking the boxes, erasing the middle segment and compressing the resulting rectangle until becomes the box. The unit of the group consists of $N$ parallel strands. All these are illustrated in Fig. \[Fig:PictorialBraidGroup\]. ![Sequences of adiabatic cycles which exchange flux tubes. If the resulting unitary transformations depend solely on the topology of the world lines, then $U_{j-1,j}U_{j,j+1}U_{j-1,j}$ coincides with $U_{j,j+1}U_{j-1,j}U_{j,j+1}$ because the (a) and (b) world lines are topologically equivalent. Since these are the defining relations of the braid group [@BraidBook], the adiabatic exchanges supply a unitary representation of the abstract braid group. []{data-label="Fig:AdiabaticBraidGroup"}](BraidGroup.pdf){width="0.7\linewidth"} While the braid group ${{\mathcal B}}_N$ has an infinite number of elements, it has a finite number of generators. These are $\beta_{j,j+1}$, $j=1,\ldots,N-1$, where $\beta_{j,j+1}$ weaves the strands $j$ and $j+1$. An illustration of such generator and its inverse can be found in Fig. \[Fig:PictorialBraidGroup\]. In terms of these generators, the braid group can be defined as the group generated by the $\beta_{j,j+1}$’s together with the relations: $$\begin{aligned} \label{Eq:BRelations} & \beta_{j,j+1} \beta_{j+1,j+2} \beta_{j,j+1} = \beta_{j+1,j+2} \beta_{j.j+1} \beta_{j+1,j+2}, \quad j=1,\ldots,N-2, \\ & \beta_{j,j+1} \beta_{k,k+1} = \beta_{k,k+1} \beta_{j,j+1}, \quad j,k=1,\ldots,N-1, \quad \|j-k\| \geq 2.\end{aligned}$$ This is known as the Artin presentation of the braid group [@ArtinAM1947]. An element of the group is the equivalence class w.r.t. of a word like: $$\beta_{i_1,i_1+1}^{n_1} \beta_{i_2,i_2+1}^{n_2} \ldots \beta_{i_k,i_k+1}^{n_k},$$ where the indices $i_\alpha$ are drawn from $\{1,\ldots,N-1\}$ and the powers $n_\alpha$ can be any integer number. Note that the size of a word can grow indefinitely. ![Evolution of the spectrum during (a) $(-\pi, -\pi)$, (b) $(+\pi, +\pi)$ and (c) $(+\pi, -\pi)$ fusions. The tubes are separated at $t=0$ and overlap at $t=1$ and for $(-\pi, -\pi)$ and $(+\pi, +\pi)$ fusion, two pairs of $(+\pi, -\pi)$ tubes were created. The zero modes seen after fusion correspond to the un-fused flux-tubes.[]{data-label="Fig:FusionSpectrum"}](FusionSpectrum.pdf){width="0.8\linewidth"} Adiabatic representation of the braid group ------------------------------------------- We consider now the physical processes illustrated in Fig. \[Fig:AdiabaticBraidGroup\]. They consists of adiabatic displacements of (+)-flux-tubes which result in permutations of the tubes. Note that at each marked times, $t_0$, …, $t_3$, the system is in the same exact configuration. Let $H_t$ and $U(t,t')$ denote the time-dependent Hamiltonian and the unitary time evolution induced by the processes in Fig. \[Fig:AdiabaticBraidGroup\](a), and $H'_t$ and $U'(t,t')$ be the same objects for the processes in Fig. \[Fig:AdiabaticBraidGroup\](b). Clearly: $$U(t_1,t_0) = U(t_3,t_2) = U'(t_2,t_1)$$ and these identical unitary operators will be denoted as $U_{j-1,j}$. Similarly: $$U(t_2,t_1) = U'(t_1,t_0) = U'(t_3,t_2),$$ and these identical unitary operators will be denoted as $U_{j,j+1}$. Since after each of these time-intervals the system returns in the original configuration $x_0$, both $U_{j-1,j}$ and $U_{j,j+1}$ act on $P_{x_0}{{\mathcal H}}$. Furthermore, it follows straight from that: $$U(t_3,t_0)=U(t_3,t_2)U(t_2,t_1)U(t_1,t_0) = U_{j-1,j}U_{j,j+1} U_{j-1,j},$$ and similarly: $$U'(t_3,t_0)=U'(t_3,t_2)U'(t_2,t_1)U'(t_1,t_0) = U_{j,j+1}U_{j-1,j} U_{j,j+1}.$$ In general, there is no reason to believe that $U(t_3,t_0)=U'(t_3,t_0)$ because, after all, the physical processes in panels (a) and (b) of Fig. \[Fig:AdiabaticBraidGroup\] are distinct. However, recall that the adiabatic evolutions are geometric in nature and, since the zero-modes are concentrated near the flux-tubes, the world lines can be deformed without any physical consequences as long as the flux-tubes are kept far apart from each other. If that is the case, then the world lines in panels (a) and (b) of Fig. \[Fig:AdiabaticBraidGroup\] can be deformed into each other without affecting the final outcomes. The conclusion is that: $$\label{Eq:Relation} U_{j-1,j}U_{j,j+1} U_{j-1,j} = U_{j,j+1}U_{j-1,j} U_{j,j+1},$$ and these are exactly the defining relations of the braid group. The conclusion is that the adiabatic evolutions supply a representation of the braid group on $P_{x_0}{{\mathcal H}}$, which we call the adiabatic representation of the braid group. The reason why one can expect a non-trivial representation of the braid group is because the parameter space of the flux-tubes is punctured. Indeed, in Fig. \[Fig:FusionSpectrum\] we illustrate the behavior of the spectrum as pairs of flux tubes are fused. As one can see, the degeneracy is lifted and, as such, the adiabatic theorem, in the form we presented, does not apply anymore because dynamical terms must be also taken into account. As shown in [@ProdanPRB2009], a Berry curvature can be associated to the adiabatic deformations and this curvature is usually concentrated near such singularities, leading in general to non-trivial monodromies. Braiding the Zero Modes {#Sec:Braiding0s} ======================= ![Plots of $\sum_{\alpha=1,2} \langle \bm n,\alpha\|P_{x_t}\|\bm n,\alpha \rangle$ defined in as function of lattice position $\bm n$, at three instances along the adiabatic cycle shown by the arrows. The cycle was discretized in 1000 steps. The value at a point $\bm n$ of the $25 \times 25$ lattice is proportional with the size of the disk centered at that position.[]{data-label="Fig:2U12Braiding"}](2U12Braiding.pdf){width="0.8\linewidth"} In this section we compute explicitly the braid matrices and confirm that they supply an adiabatic representations of the braid group. In all instances, the flux-tubes are nucleated in pairs of $\Phi=\pm \pi$ but the (-) flux-tubes will be fixed and only the (+) flux-tubes will be braided. ![Plots of the eigen-vectors of $U_{12}$ with the latter computed with the process shown in Fig. \[Fig:2U12Braiding\]. The panels show the amplitude $\|\psi(\bm n)\|^2$ of the vectors as function of position $\bm n$ on the lattice. The corresponding eigen-values are specified in each panel.[]{data-label="Fig:2U12Vectors"}](2U12Vectors.pdf){width="\linewidth"} Two (+) flux-tubes (${{\mathcal B}}_2$) {#Sec:AdiabaticB2} --------------------------------------- The exchange route of the (+) flux-tubes as well as the positions of the fixed (-) flux tubes are shown in Fig. \[Fig:2U12Braiding\]. The dimension of the zero modes space is 4. The spatial profiles of the zero-modes $\varphi_j$ trapped by the four flux-tubes are conveniently captured by the diagonal part of $P_x$, the projection onto the zero modes space: $$\label{Eq:ZProj} \sum_{\alpha=1,2} \langle \bm n,\alpha\|P_x\|\bm n,\alpha \rangle = \sum_{j=1}^4 \|\varphi_j(\bm n,\alpha)\|^2,$$ which is what is plotted in Fig. \[Fig:2U12Braiding\]. As one can see, the zero-modes remain relatively well separated during the exchange operation. Furthermore, since the flux-tubes that are exchanged carry the same flux, the Hamiltonian returns to its initial value, hence the adiabatic cycle is closed. The monodromy $U_{12}$ was computed with Eq. \[Eq:PracticalUa\] and, for a discretization of $K=5 \times 10^4$, its spectrum was found to be: $$\label{Eq:2U12Spec} {\rm Spec}(U_{12}) = \big \{-\exp\big(\tfrac{\imath \pi}{4}\big),\exp\big ( \tfrac{\imath \pi}{4}\big ),1,1 \big \} \pm 1\%.$$ The corresponding eigen-vectors are reported in Fig. \[Fig:2U12Vectors\]. As expected, the trivial eigenvalues 1 correspond to the zero modes trapped by the (-) flux-tubes, which do not participate in the braiding. The eigen-vectors corresponding to the eige-nvalues $\pm\exp\big(\tfrac{\imath \pi}{4}\big)$ are shared between the (+) flux tubes and they are mapped into each other by the particle-hole symmetry operation . The above numerical results indicate that when a flux tube is circled around another, the resulting adiabatic monodromy is $\pm \imath I$. We have verified this statement, directly, for the process described in Fig. \[Fig:UTest\]. Hence, the flux-tube must circle four times for the monodromy to return to identity. The conclusion is that the adiabatic representation supply a cyclic representation of ${{\mathcal B}}_2$ of period 8: $U_{12}^8 = I$. ![The initial configuration of the flux-tubes together with the world-lines for the exchange matrices marked in each panel. The initial configuration is represented by the plot of $\sum_{\alpha=1,2} \langle \bm n,\alpha\|P_{x_0} \| \bm n,\alpha\rangle$ as function of $\bm n$ on a $30\times 30$ lattice.[]{data-label="Fig:3UBraidings"}](3UBraidings.pdf){width="0.6\linewidth"} Three (+) flux-tubes (${{\mathcal B}}_3$) {#Sec:AdiabaticB3} ----------------------------------------- The initial configurations and the exchange routes of flux-tubes are shown in Fig. \[Fig:3UBraidings\] and the monodromies $U_{12}$ and $U_{23}$ were again computed with Eq. \[Eq:PracticalUa\]. The eigenvalues of both monodromies were numerically found to be: $$\label{Eq:SpecU12U23} {\rm Spec}(U_{12}) = {\rm Spec}(U_{23}) = \big \{-\exp(\tfrac{\imath\pi}{4}),\exp(\tfrac{\imath\pi}{4}),1,1,1,1 \big \} \pm 1\%,$$ when the adiabatic cycle was discretized in a $K=5 \times 10^4$ number of steps. The corresponding eigen-vectors are shown in Fig. \[Fig:3U12Vectors\] for $U_{12}$ and in Fig. \[Fig:3U23Vectors\] for $U_{23}$. At this point we computed the physical representation of the generators of the braid group and we can verify directly the fundamental relation . Numerically, when the cycles were discretized in $K=5 \times 10^4$ steps, we obtained: $$U_{12}U_{23}U_{12} - U_{23}U_{12}U_{23} = 0\pm 0.01,$$ which confirms that the adiabatic braids supply a representation of the braid group. The spectra of these unitary transformations was found to be: $$\label{Eq:SpecU123} {\rm Spec}(U_{12}U_{23}U_{12}) =\{-\imath,\ \imath,\ \imath,\ 1,\ 1,\ 1\} \pm 1\%,$$ and similarly for ${\rm Spec}(U_{23}U_{12}U_{23})$. ![The eigen-vectors of $U_{12}$ for the eigen-values specified for each panel. []{data-label="Fig:3U12Vectors"}](3U12Eigenvectors.pdf){width="0.8\linewidth"} Since three of the flux-tubes are mere spectators to the braidings, we expect that three of the eigen-vectors corresponding to trivial eigenvalues 1 of $U_{12}$ and $U_{23}$ to be pinned at these (-) flux-tubes and to be irrelevant for the braiding process. This is not clear from Figs. \[Fig:3U12Vectors\] and \[Fig:3U23Vectors\] and this is because the computer out-putted arbitrary linear combinations of the four eige-vectors corresponding to the degenerate eigenvalue 1. However, to confirm our supposition, we have verified that the two eigen-vectors of $U_{12}$ and the two eige-vectors of $U_{23}$, corresponding to the non-trivial eigen-values, span a 3-dimensional linear space, [i.e.]{} that there is one and only one linear dependency between them. Furthermore, this 3-dimensional linear space coincides with the 3-dimensional linear space spanned by the three eigen-vectors of $U_{23}U_{12}U_{23}$ corresponding to the three non-trivial eigenvalues. The important conclusion is that the adiabatic representation of the braid group revealed by our calculations is an irreducible representation of dimension 3. ![Eigen-vectors of $U_{23}$ for the eigen-values specified for each panel.[]{data-label="Fig:3U23Vectors"}](3U23Eigenvectors.pdf){width="0.8\linewidth"} The Resulting Irreducible Representation of ${{\mathcal B}}_N$ ============================================================== The numerical data supplied in the previous section enable us to identify the irreducible representation of the braid group for a generic number of flux tubes. This representation is described first in a matrix form and then in a diagramatic form. The matrix presentation {#Sec:MatrixPres} ----------------------- Consider the following matrices $N\times N$ complex matrices: $$\label{Eq:Qs} Q_{j,j+1} = \begin{pmatrix} \ddots & \vdots & \vdots & \vdots & \vdots & \iddots \\ \ldots & \gamma & 0 & 0 & 0 & \ldots \\ \ldots & 0 & 0 & \alpha & 0 & \ldots \\ \ldots & 0 & \beta & 0 & 0 & \ldots \\ \ldots & 0 & 0 & 0 & \gamma & \ldots \\ \iddots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix},$$ where the $2 \times 2$ square matrix containing the coefficients $\alpha$ and $\beta$ sits at rows $j$ and $j+1$, as well as at columns $j$ and $j+1$. The $\gamma$’s fill in the diagonal all the way to the upper-left and lower-right corners. We can let the index $j$ run from $1$ to $N-1$, hence there are $N-1$ such $Q$ matrices. The adjoint matrices are: $$Q_{j,j+1}^\ast = \begin{pmatrix} \ddots & \vdots & \vdots & \vdots & \vdots & \iddots \\ \ldots & \gamma^\ast & 0 & 0 & 0 & \ldots \\ \ldots & 0 & 0 & \beta^\ast & 0 & \ldots \\ \ldots & 0 & \alpha^\ast & 0 & 0 & \ldots \\ \ldots & 0 & 0 & 0 & \gamma^\ast & \ldots \\ \iddots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix},$$ and a direct computation gives: $$Q_{j,j+1} Q_{j,j+1}^\ast = \begin{pmatrix} \ddots & \vdots & \vdots & \vdots & \vdots & \iddots \\ \ldots & \|\gamma\|^2 & 0 & 0 & 0 & \ldots \\ \ldots & 0 & \|\alpha\|^2 & 0 & 0 & \ldots \\ \ldots & 0 & 0 & \|\beta\|^2 & 0 & \ldots \\ \ldots & 0 & 0 & 0 & \|\gamma\|^2 & \ldots \\ \iddots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix},$$ as well as: $$Q_{j,j+1}^\ast Q_{j,j+1} = \begin{pmatrix} \ddots & \vdots & \vdots & \vdots & \vdots & \iddots \\ \ldots & \|\gamma\|^2 & 0 & 0 & 0 & \ldots \\ \ldots & 0 & \|\beta\|^2 & 0 & 0 & \ldots \\ \ldots & 0 & 0 & \|\alpha\|^2 & 0 & \ldots \\ \ldots & 0 & 0 & 0 & \|\gamma\|^2 & \ldots \\ \iddots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}.$$ As such, the matrices are unitary if we force the complex coefficients $\alpha$, $\beta$ and $\gamma$ to lie on the unit circle. Furthermore: $$Q_{j,j+1} Q_{j+1,j+2} Q_{j,j+1} = \begin{pmatrix} \ddots & \vdots & \vdots & \vdots & \vdots & \iddots \\ \ldots & \gamma^3 & 0 & 0 & 0 & \ldots \\ \ldots & 0 & 0 & 0 & \gamma \alpha^2 & \ldots \\ \ldots & 0 & 0 & \gamma \alpha \beta & 0 & \ldots \\ \ldots & 0 & \gamma \beta^2 & 0 & 0 & \ldots \\ \ldots & 0 & 0 & 0 & \gamma^3 & \ldots \\ \iddots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix},$$ where the $3\times 3$ off-diagonal matrix sits at rows $j$, $j+1$, $j+2$ and columns $j$, $j+1$, $j+2$, and a direct computation will show that: $$Q_{j,j+1} Q_{j+1,j+2} Q_{j,j+1} = Q_{j+1,j+2} Q_{j,j+1} Q_{j+1,j+2},$$ as well as: $$Q_{j,j+1} Q_{j',j'+1} = Q_{j',j'+1} Q_{j,j+1},$$ if $\|j-j'\|\geq 2$. Comparing with , we conclude that the $Q$-matrices supply an $N \times N$ representation of the braid group ${{\mathcal B}}_N$. Connection with the adiabatic physical representation {#Sec:PhysicalRep} ----------------------------------------------------- Specializing for $N=3$, we have: $$\label{Eq:QN3} Q_{12} = \begin{pmatrix} 0 & \alpha & 0 \\ \beta & 0 & 0 \\ 0 & 0 & \gamma \\ \end{pmatrix}, \quad Q_{23} = \begin{pmatrix} \gamma & 0 & 0 \\ 0 & 0 & \alpha \\ 0 & \beta & 0 \\ \end{pmatrix}.$$ Guided by the numerical results from section \[Sec:Braiding0s\], we make the choice: $$\label{Eq:AlphaBeta} \alpha = e^{-\frac{\imath \pi}{4}}, \quad \beta = - e^{-\frac{\imath \pi}{4}}, \quad \gamma=1.$$ and find: $${\rm Spec}(Q_{12}) = {\rm Spec}(Q_{12}) = \big \{ - e^{\frac{\imath \pi}{4}}, e^{\frac{\imath \pi}{4}}, 1 \big \}$$ as well as: $${\rm Spec}(Q_{12}Q_{23}Q_{12}) = {\rm Spec}(Q_{23}Q_{12}Q_{23})=\big \{ - \imath, \imath, \imath \big \},$$ in full agreement with and . A diagramatic presentation -------------------------- A single-strand $N$-diagram is a box with $N$-marked points at the bottom edge and $N$-marked points on the top edge, and a smooth line connecting one marked point at the base with one marked point at the top. The smooth line exists and dives normally to the edges. A few examples for $N=3$ are supplied below: $$\label{Eq:B3Diags} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} , \ \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) -- (0.3,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} , \ \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \end{tikzpicture} \end{matrix}$$ For a generic $N$, there are $N^2$ such single-strand diagrams. Two single-strand diagrams can be composed in the following way. One stacks the diagrams on top of each other, respecting the order, and, if the reunion of the paths results in a smooth curve then that is the result of the composition. If not, then the composition is zero. Below are some example: $$\label{Eq:CExample1} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.9,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \ \cdot \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \ = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \draw[color=red] (0,0.7) rectangle (1.2,1.4); \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \filldraw (0.3,1.4) circle [radius=1pt]; \filldraw (0.6,1.4) circle [radius=1pt]; \filldraw (0.9,1.4) circle [radius=1pt]; \draw (0.9,0.7) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,1.4); \end{tikzpicture} \end{matrix} \begin{matrix} \ = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \, , \end{matrix}$$ and: $$\label{Eq:CExample1} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \ \cdot \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.9,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \ = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.9,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \draw[color=red] (0,0.7) rectangle (1.2,1.4); \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \filldraw (0.3,1.4) circle [radius=1pt]; \filldraw (0.6,1.4) circle [radius=1pt]; \filldraw (0.9,1.4) circle [radius=1pt]; \draw (0.6,0.7) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,1.4); \end{tikzpicture} \end{matrix} \begin{matrix} \ = 0 . \end{matrix}$$ As opposed to the link diagrams, not every single-strand diagram has an inverse. In fact there is no single-strand diagram that can play the role of unity, hence the set of these diagrams together with the composition form only a semi-group. This semi-group is not very useful by himself and we expand to an algebra $T^{(1)}_N$, whose elements are formal series: $$\label{Eq:TLElement} a = \sum_{j} a_j \, \boxed{D_j} \, ,$$ where $\boxed{D_j}$’s are single strand $N$-diagrams and the coefficients $a_j$’s are simple complex numbers. The addition and multiplication of the algebra are: $$\Big ( \sum_{j} a_j \, \boxed{D_j} \Big ) + \Big (\sum_{j} a'_j \, \boxed{D_j} \Big ) = \sum_{j} (a_j+a'_j) \, \boxed{D_j}$$ and: $$\Big ( \sum_{j} a_j \, \boxed{D_j} \Big ) \cdot \Big (\sum_{i} a'_i \, \boxed{D_j} \Big ) = \sum_{j,i} a_j \, a'_i \ \boxed{\begin{matrix}D_j \\ D_i \end{matrix}} \, ,$$ respectively. The algebra $T^{(1)}_N$ does have a unit. For $N=3$, for example, the unit is supplied by: $$\begin{matrix} 1 = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \ + \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \end{tikzpicture} \end{matrix} \begin{matrix} \ + \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.9,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \end{tikzpicture} \end{matrix}$$ We can endow the algebra $T^{(1)}_N$ with a $\ast$-operation: $$\Big ( \sum_{j} a_j \, \boxed{D_j} \Big )^\ast = \sum_{j} a_j^\ast \, \boxed{D_j}^{\, \ast},$$ where $\boxed{D_j}^{\, \ast}$ for a single-strand diagram means reflection relative to the middle horizontal axis of the box. Furthermore, $T^{(1)}_N$ accepts a positive and faithful trace, [i.e.]{} an additive map ${{\mathcal T}}: T^{(1)}_N \rightarrow {{\mathbb C}}$ such that ${{\mathcal T}}(gag^{-1}) = {{\mathcal T}}(a)$ for any $a$ from $T^{(1)}_N$ and $g$ from $GL(T_N)$ and ${{\mathcal T}}(a^\ast a) \geq 0$ for all $a \in T^{(1)}_N$, with equality only when $a=0$. Being a linear map, we only need to specify how ${{\mathcal T}}$ acts on the individual diagrams. The rule is that, when applied on a single-strand diagram, the trace gives $1$ if the strand is straight and $0$ otherwise. For example: $$\begin{matrix} {{\mathcal T}}\end{matrix} \begin{pmatrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \end{tikzpicture} \end{pmatrix} \begin{matrix} \ = 1, \quad \end{matrix} \begin{matrix} {{\mathcal T}}\end{matrix} \begin{pmatrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \end{tikzpicture} \end{pmatrix} \begin{matrix} \ = 0. \end{matrix}$$ Note that the trace is normalized such that ${{\mathcal T}}(1) = N$. As for any algebra with a unit, there exists the group $GL(T_N)$ of all invertible elements from $T_N$. Below we supply an explicit group homomorphism $$\rho: {{\mathcal B}}_N \rightarrow GL(T_N),$$ via formal replacements of links by the linear combination: $$\begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (0.9,0); \draw[color=red] (0,-0.7) -- (0.9,-0.7); \braid[number of strands=2,width=0.3cm,height=0.7cm,border height=0cm] a_1; \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,-0.7) circle [radius=1pt]; \filldraw (0.6,-0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ \rightarrow \ \end{matrix}\ \begin{matrix} \alpha \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (0.9,0); \draw[color=red] (0,0.7) -- (0.9,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \, \beta \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (0.9,0); \draw[color=red] (0,0.7) -- (0.9,0.7); \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ and: $$\begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (0.9,0); \draw[color=red] (0,-0.7) -- (0.9,-0.7); \braid[number of strands=2,width=0.3cm,height=0.7cm,border height=0cm] a_1^{-1}; \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,-0.7) circle [radius=1pt]; \filldraw (0.6,-0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ \rightarrow \ \end{matrix} \begin{matrix} \alpha^\ast \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (0.9,0); \draw[color=red] (0,0.7) -- (0.9,0.7); \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \, \beta^\ast \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (0.9,0); \draw[color=red] (0,0.7) -- (0.9,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ Furthermore, after all the links are resolved by applying the above rule, the result will be a linear combination of planar diagrams with strands that do not cross each other. These diagrams are further decomposed into linear combinations of single-strand diagrams. For example: $$\begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (1.2,0); \draw[color=red] (0,0.7) -- (1.2,0.7); \draw (0.3,0) -- (0.3,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ \rightarrow \ \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (1.2,0); \draw[color=red] (0,0.7) -- (1.2,0.7); \draw (0.3,0) -- (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \ \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) -- (1.2,0); \draw[color=red] (0,0.7) -- (1.2,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ It is a simple exercise to verify that: $$\tau_{j,j+1}=\rho(\beta_{j,j+1}), \quad \tau^\ast_{j,j+1} = \rho(\beta^{-1}_{j,j+1}), \quad j=1,\ldots,N-1,$$ satisfy the relations . For example, the generators of ${{\mathcal B}}_3$ become: $$\label{Eq:T12} \begin{matrix} \tau_{12} = \ \end{matrix} \begin{matrix} \alpha \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \, \beta \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.3,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.9,0) -- (0.9,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ and: $$\label{Eq:T23} \begin{matrix} \tau_{23} = \ \end{matrix}\ \begin{matrix} \alpha \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.9,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \, \beta \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.3,0) -- (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ and, after applying the rules of calculus, we obtain: $$\label{Eq:T12T23T12} \begin{matrix} \tau_{12} \tau_{23} \tau_{12} = \ \end{matrix} \begin{matrix} \alpha^2 \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.9,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \, \alpha\beta \, \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0) -- (0.6,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ + \beta^2 \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.9,0.7) .. controls +(0,-0.2) and +(0,0.2) .. (0.3,0); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ and same for $\tau_{23} \tau_{12} \tau_{23}$. We now construct the GNS-representation [@DixBook] of $T^{(1)}_N$ induced by ${{\mathcal T}}$. For this, one considers the linear space ${{\mathcal V}}_N$ that coincides with the algebra $T^{(1)}_N$ when the latter is considered only with the additive structure. To distinguish between $T^{(1)}_N$ and ${{\mathcal V}}_N$, we will write the elements of the latter as: $$\label{Eq:Vector} \|\psi\rangle = \sum_j a_j \left \| \boxed{D_j} \right \rangle, \quad a_j \in {{\mathbb C}},$$ Where the sum is over all available single-strand diagrams. Alternatively, ${{\mathcal V}}_N$ can be thought as the ${{\mathbb C}}$-linear span of all single-strand diagrams. As such, ${\rm dim}({{\mathcal V}}_N) = N^2$. Furthermore, ${{\mathcal V}}_N$ can be equipped with a non-degenerate scalar product, which is the unique sesqui-linear map acting on the single strand diagrams as: $$\left \langle \boxed{D}, \boxed{D'} \right \rangle = {{\mathcal T}}\left ( \boxed{D}^\ast \cdot \boxed{D'} \right ).$$ Then $\big ({{\mathcal V}}_N, \, \langle , \rangle \big )$ becomes a finite-dimensional Hilbert space. The regular representation $\eta$ of $T^{(1)}_N$ on ${{\mathcal V}}_N$ is supplied by the natural action $\eta(a)\|\psi\rangle = \|a\psi \rangle$. Explicitly, if $\psi$ is as in , then: $$\label{Eq:TnAction} \eta \Big ( \sum_i a'_i \, \boxed{D_i}\Big )\|\psi \rangle = \sum_{i,j} a'_i a_j \left \|\boxed{\begin{matrix} D_i \\ D_j \end{matrix}} \right \rangle,$$ where on the right we have the composition of single-strand diagrams. Let us pause and appreciate that, through the composition: $$\label{Eq:RegularRep} \eta \circ \rho : {{\mathcal B}}_N \rightarrow {\rm GL}({{\mathcal V}}_N) \simeq {\rm GL}({{\mathbb C}}^{N^2})$$ we constructed a finite-dimensional representation of the braid group. Furthermore, it is immediate to check that the representation is unitary, that is, the map in lands in ${\rm U}({{\mathbb C}}^{N^2})$ rather than the whole ${\rm GL}({{\mathbb C}}^{N^2})$. Let us point out that, if one tries the same procedure directly on the link diagrams, one will find that the resulting linear space is infinitely dimensional, because there are an infinite number of inequivalent link diagrams. The representation is not irreducible but the irreducible representations can be easily generated once the left-ideal structure of $T^{(1)}_N$ is mapped out, which is our next task. The left-ideals enter the discussion in the following way. If $J \subset T^{(1)}_N$ is a left-ideal, that is, $a b \in J$ for all $a\in T^{(1)}_N$ and $b \in J$, then the linear sub-space ${{\mathcal V}}_J \subset {{\mathcal V}}_N$ spanned by all $\|\psi\rangle $’s from $J$ is invariant to the regular action of $T^{(1)}_N$: $$\eta(a)\|\psi \rangle = \|a \psi \rangle \in {{\mathcal V}}_J.$$ Since the action of $T^{(1)}_N$ gets trapped by these sub-spaces, we can study them one at a time, instead of dealing with the large ${{\mathcal V}}_N$. One can quickly convince himself that the left-ideals are all direct sums of the normal left-ideals generated by one single-strand diagram. Hence an irreducible invariant sub-space of ${{\mathcal V}}_N$ consists of the ${{\mathbb C}}$-linear span of single-strand diagrams originating from the same marked point at the base. There are $N$ such diagrams, hence the dimension of these invariant sub-spaces are $N$. For $N=3$, one of such invariant sub-space is generated by the following diagrams: $$\begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.3,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ , \quad \end{matrix}\ \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.6,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ , \quad \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0) .. controls +(0,0.2) and +(0,-0.2) .. (0.9,0.7); \filldraw (0.3,0) circle [radius=1pt]; \filldraw (0.6,0) circle [radius=1pt]; \filldraw (0.9,0) circle [radius=1pt]; \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ If one changes the base marked point, one obtains $N$ distinct such invariant sub-spaces and their direct sum is a linear space of dimension $N \times N = {\rm dim}({{\mathcal V}}_N)$, hence the whole ${{\mathcal V}}_N$. Furthermore, the these invariant sub-spaces are canonically isomorphic, hence it is enough to study one of them. In fact, the position of the base point is irrelevant and we can think of this essentially unique invariant sub-space as the ${{\mathbb C}}$-linear span of the diagrams: $$\begin{matrix} \varphi_1 = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.3,0.3) -- (0.3,0.7); \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ , \quad \varphi_2 = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.6,0.3) -- (0.6,0.7); \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix} \begin{matrix} \ , \quad \varphi_3 = \ \end{matrix} \begin{matrix} \begin{tikzpicture} \draw[color=red] (0,0) rectangle (1.2,0.7); \draw (0.9,0.3) -- (0.9,0.7); \filldraw (0.3,0.7) circle [radius=1pt]; \filldraw (0.6,0.7) circle [radius=1pt]; \filldraw (0.9,0.7) circle [radius=1pt]; \end{tikzpicture} \end{matrix}$$ shown here for $N=3$. It is now easy to verify that the matrices supplied by the matrix elements: $$\langle \varphi_n \| \tau_{j,j+1} \| \varphi_m \rangle, \quad j=1,\ldots,N_1, \quad m,n=1,\ldots,N,$$ coincide with the $Q_{j,j+1}$ matrices introduced in section \[Sec:MatrixPres\]. For the case $N=3$, using $\tau_{j,j+1}$ from and , these matrices have been computed explicitly and confirmed to be identical to . Implementation with Classical Meta-Materials ============================================ In this section we briefly recall the algorithmic procedure from [@BarlasPRB2018] of transforming quantum lattice models into dynamical matrices governing the dynamics of small oscillations of coupled mechanical resonators. We also demonstrate here how to recover the standard Majorana representation of the braid group using derived physical observables. Classical Dynamical Matrices {#Sec:ClassicalD} ---------------------------- The representation of the braid group generated by the adiabatic displacements of the flux-tubes supplies an example of a non-abelian statistics that can be implemented with and observed in a classical meta-material. This is important because it represent a new and more straightforward route towards the long-sought demonstration of a controlled non-abelian braiding of arbitrary number of anyons in a physical system. Below, we briefly elaborate the main steps of this implementation. The key is the map $\rho$ supplied in [@BarlasPRB2018], running from linear operators over ${{\mathbb C}}^2 \otimes \ell^2({{\mathbb Z}}^2)$ to linear operators over ${{\mathbb C}}^4 \otimes \ell^2({{\mathbb Z}}^2)$. If we write the Hamiltonian as: $$H = \sum_{\bm n,\bm n'} h_{\bm n',\bm n} \otimes \|\bm n'\rangle \langle \bm n \|,$$ with $h_{\bm n',\bm n}$ being the $2 \times 2$ hopping matrices appearing in , then we generated a classical dynamical matrices $D$ governing the small oscillations of coupled mechanical resonators via: $$\label{Eq:D} D = \omega_0^2 I + \rho(H) = \omega_0^2 I + \sum_{\bm n,\bm n'} \begin{pmatrix}{\rm Re}[h_{\bm n',\bm n}] & {\rm Im}[h_{\bm n',\bm n}] \\ - {\rm Im}[h_{\bm n',\bm n}] & {\rm Re}[h_{\bm n',\bm n}] \\ \end{pmatrix} \otimes \|\bm n'\rangle \langle \bm n \|,$$ where the first term is chosen such that the spectrum of $D$ is contained by the positive real axis. As one can see, by doubling the degrees of freedom per lattice site, all entries of the dynamical matrix were made real and such dynamical matrices can be implemented in a laboratory with passive meta-materials, for example, using the platform of magnetically coupled spinners [@ApigoPRM2018]. This is elaborated at length in [@BarlasPRB2018]. As pointed out in [@BarlasPRB2018], except for a rigid shift by $\omega_0^2$, the resonant energy spectrum of $D$ is identical to that of $H$ but its degeneracy is doubled. Hence, the zero modes of $H$ double and become mid-gap normal modes for $D$ that oscillate with pulsation $\omega_0$. Furthermore, $D$ has an intrinsic and un-avoidable $U(1)$ symmetry given by the conjugation with ${\small \begin{pmatrix} 0 & I_2 \\ -I_2 & 0 \end{pmatrix}} \otimes I$. The spectrum of this symmetry operator consists of just two points $\pm \imath$, hence the Hilbert space of $D$ decouples into two dynamically invariant symmetry-sectors: $${{\mathbb C}}^4 \otimes \ell^2({{\mathbb Z}}^2) = \Pi_- \big ({{\mathbb C}}^4 \otimes \ell^2({{\mathbb Z}}^2) \big ) \oplus \Pi_+ \big ( {{\mathbb C}}^4 \otimes \ell^2({{\mathbb Z}}^2) \big )$$ where $\Pi_\pm$ are the spectral projectors onto the $\pm \imath$ eigenvalues. As shown in [@BarlasPRB2018], $\Pi_+ D \Pi_+$ defined over $\Pi_+ \big ( {{\mathbb C}}^4 \otimes \ell^2({{\mathbb Z}}^2)\big )$ is unitarily equivalent to the original Hamiltonian $H$. In particular, it inherits the PH-symmetry, which in the classical setting is referenced from the mid-gap point: $$\widetilde \Theta_{\rm PH} \, \big ( \Pi_+ (D-\omega_0^2) \Pi_+ \big ) \, \widetilde \Theta_{\rm PH}^{-1} = - \Pi_+ (D-\omega_0^2) \Pi_+ ,$$ with $\widetilde \Theta_{\rm PH}$ as defined in [@BarlasPRB2018]: $$\widetilde \Theta_{\rm PH} = J \, \rho\big (U^{-1}(\sigma_1 \otimes I) \big ) \, {{\mathcal K}}, \quad J = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix} \otimes I.$$ The implication is that the whole braiding program is reproduced by the classical system defined by $D$. Derived physical observables and representations ------------------------------------------------ Even in the classical regime, there are interesting derived physical observables, whose dynamics supply representations of the braid group that derive from the one found in the previous sections, yet they are fundamentally different. In particular, we show below that, by using such derived physical observables, we can reproduce the standard Majorana representation of the braid group. We assume that all initial loads of the lattice occur in the symmetry sector $\Pi_+$. An explicit and practical way to achieve such task has been elaborated at length in [@BarlasPRB2018]. Since $\Pi_+ (D-\omega_0^2) \Pi_+$ is unitarily equivalent to $H$ for all configurations of the flux-tubes, we will work with $H$ in the following, primarily because we already fixed the notation. Let $C\ell\big({{\mathcal H}}_0,\langle,\rangle\big )$ be the complex Clifford algebra associated to the space ${{\mathcal H}}_0$ of zero modes with $N$ flux-tubes arranged in a particular configuration. Then, for any $\psi \in {{\mathcal H}}_0$, we have an element $\Gamma(\psi)$ in $C\ell\big({{\mathcal H}}_0,\langle,\rangle\big )$, such that: $$\Gamma(\psi)^\dagger \Gamma(\psi') + \Gamma(\psi')\Gamma(\psi)^\dagger = 2 \langle \psi,\psi' \rangle, \quad \Gamma(\Theta_{\rm PH}\psi) = \Gamma(\psi)^\dagger.$$ We can make things more explicitly by considering the basis $\{\varphi_j\}_{j=\overline{1,N}}$ of ${{\mathcal H}}_0$ consisting of particle-hole symmetric modes localized at each of the flux tubes. Note that this is precisely the basis that supply the matrix form of the braid transformations. Let: $$\Gamma_j = \Gamma(\varphi_j), \quad \Gamma_j^\dagger = \Gamma_j, \quad j=1,\ldots N,$$ which satisfy the canonical relations: $$\Gamma_i \Gamma_j + \Gamma_j \Gamma_i = 2 \delta_{ij}.$$ Note that the $\Gamma$’s can be canonically identified with matrices. Now, each of the resonant modes $\varphi_j$ can be loaded with arbitrary amplitude and phase, leading to an oscillatory state: $$\psi(t) = e^{\imath \omega_0 t} \psi = e^{\imath \omega_0 t} \sum_{j=1}^N \alpha_j \varphi_j,$$ where the coefficients $\alpha_j$ are complex amplitudes. Then: $$\label{Eq:GammaPsi} \Gamma(\psi) = \sum_{j=1}^N \alpha_j \Gamma_j.$$ The $\Gamma(\psi)$’s can play the role of our derived physical observables. Indeed, nothing stops us in feeding the measurement of the small oscillations state $\psi$ of the mechanical system into the matrix $\Gamma(\psi)$. Furthermore, note that the oscillatory state of the system can be fully recovered from : $$\alpha_j = \tfrac{1}{\rm dim} {\rm Tr}\big ( \Gamma(\psi) \Gamma_j \big ), \quad j = 1,\ldots,N,$$ where ${\rm dim}$ is the dimension of $\Gamma$’s. In fact, we can observe and manipulate products of the form $\Gamma(\psi_1) \ldots \Gamma(\psi_k)$, by creating $k$ identical copies of the system and by loading these copies, coherently, into the oscillatory states $\psi_1$, …, $\psi_k$. We now compute the induced braid operations on these derived physical observables, via the defining relation: $$\label{Eq:InduceActions} \Gamma(U_{j,j+1} \psi) = {{\mathbb U}}_{j,j+1} \, \Gamma(\psi)\, {{\mathbb U}}_{j,j+1}^\dagger, \quad \forall \ \psi \in {{\mathcal H}}_0, \quad i=1,\dots,N-1,$$ where ${{\mathbb U}}$’s are sought inside the Clifford algebra, more precisely in the sub-algebra generated by $\Gamma_j$ and $\Gamma_{j+1}$. For simplicity, we will ignore the abelian phase factor $e^{-\imath \frac{\pi}{4}}$ in and we will assume $\alpha=1$ and $\beta=-1$. Then the solutions to are supplied by the following expressions: $${{\mathbb U}}_{j,j+1} = \tfrac{1}{\sqrt{2}}(1 - \Gamma_j \Gamma_{j+1}), \quad {{\mathbb U}}_{j,j+1} {{\mathbb U}}_{j,j+1}^\dagger = {{\mathbb U}}_{j,j+1}^\dagger {{\mathbb U}}_{j,j+1} = I.$$ Indeed, one can verify directly that: $$\begin{aligned} {{\mathbb U}}_{j,j+1} \Gamma(\varphi_j) {{\mathbb U}}_{j,j+1}^\dagger={{\mathbb U}}_{j,j+1} \Gamma_j {{\mathbb U}}_{j,j+1}^\dagger = \Gamma_{j+1} = \Gamma(\alpha \varphi_{j+1}) = \Gamma(U_{j,j+1}\varphi_j),\\ {{\mathbb U}}_{j,j+1} \Gamma_(\varphi_{j+1}) {{\mathbb U}}_{j,j+1}^\dagger={{\mathbb U}}_{j,j+1} \Gamma_{j+1} {{\mathbb U}}_{j,j+1}^\dagger = - \Gamma_{j} =\Gamma(\beta \varphi_j)= \Gamma(U_{j,j+1}\varphi_{j+1}).\end{aligned}$$ and for $k$ different from either $j$ and $j+1$: $$\nonumber {{\mathbb U}}_{j,j+1} \Gamma(\varphi_k) {{\mathbb U}}_{j,j+1}^\dagger={{\mathbb U}}_{j,j+1} \Gamma_k {{\mathbb U}}_{j,j+1}^\dagger = \Gamma_k = \Gamma(\alpha \varphi_{k}) = \Gamma(U_{j,j+1}\varphi_k).$$ By linearity, follows. We now can see explicitly that the derived representation $\beta_{j,j+1} \rightarrow {{\mathbb U}}_{j,j+1}$ coincides with the $SU(2)_2$ representation supplied by the braiding of Majorana fermions [@AliceaNatPhys2011]. Conclusions and Outlook ======================= To observe in a laboratory the phenomena described in the previous section, we need to supply first the experimentalists with the $h$-coefficients in and their variation during the adiabatic cycles. The experimentalists need to engineer couplings between four layers of square lattices of mechanical resonators, and to slowly vary in time these couplings. With the platform of magnetically coupled spinners introduced in [@ApigoPRM2018], layering has been already achieved and variable couplings can be implemented by replacing the permanent magnets by electro-magnets whose strength can be programmed at will. While this were still in the planning, the exciting work [@ChenArxiv2019] appeared, showing an experimental demonstration and characterization of mechanical Majorana-like modes. In fact, our community is rapidly learning that classical topological meta-materials can be used quite effectively for information processing (see [e.g.]{} the recent work by Fruchart et al [@FruchartArxiv2019]). For us, it will be extremely important to figure out how to generate and observe the derived physical observables at the hardware level. If this can be indeed achieved, then the quantum algorithms generated for the Majorana fermions program can be simulated and tested with classical hardware. An interesting aspect which was revealed to us during this work was the duality between the representations of the braid group generated with first and second quantizations, that is, between $U_{i,i+1}$ and ${{\mathbb U}}_{i,i+1}$. As is well known [@WangBook; @KauffmanBook2], the latter can be generated with $N$-strands planar diagrams and our work shows that there is a relation between the representations generated with single-strand and $N$-strand planar diagrams. The passage from one to another can be achieved by simply passing to the Clifford algebra over the representation space of the former. We feel that it is important to understand this mechanism at the level of diagrams, because there are many other derived physical observables, some which naturally connect with $k$-strand diagrams, $1<k<N$. It is not excluded that dualities also exist for these cases. An even more intriguing question for us is what happens when, instead of Clifford algebra, we use generic parafermionic algebras to generate the derived physical observables. This will certainly be among our future investigations. 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--- abstract: 'We report unphysical irregularities and discontinuities in some key experimentally-measurable quantities computed within the GW approximation of many-body perturbation theory applied to molecular systems. In particular, we show that the solution obtained with partially self-consistent GW schemes depends on the algorithm one uses to solve self-consistently the quasi-particle (QP) equation. The main observation of the present study is that each branch of the self-energy is associated with a distinct QP solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy as a function of the internuclear distance. Moreover, we clearly observe “ripple” effects, i.e., a discontinuity in one of the QP energies induces (smaller) discontinuities in the other QP energies. Going from one branch to another implies a transfer of weight between two solutions of the QP equation. The case of occupied, virtual and frontier orbitals are separately discussed on distinct diatomics. In particular, we show that multisolution behavior in frontier orbitals is more likely if the HOMO-LUMO gap is small.' author: - Mickaël Véril - Pina Romaniello - 'J. A. Berger' - 'Pierre-François Loos' title: Unphysical Discontinuities in GW Methods --- Background ========== Many-body perturbation theory methods based on the one-body Green function $G$ are fascinating as they are able to transform an unsolvable many-electron problem into a set of non-linear one-electron equations, thanks to the introduction of an effective potential $\Sigma$, the self-energy. Electron correlation is explicitly incorporated via a sequence of self-consistent steps connected by Hedin’s equations. [@Hedin_1965] In particular, Hedin’s approach uses a dynamically screened Coulomb interaction $W$ instead of the standard bare Coulomb interaction. Important experimental properties such as ionization potentials, electron affinities as well as spectral functions, which are related to direct and inverse photo-emission, can be obtained directly from the one-body Green function. [@Onida_2002] A particularly successful and practical approximation to Hedin’s equations is the so-called GW approximation [@Onida_2002; @Aryasetiawan_1998; @Reining_2017] which bypasses the calculation of the most complicated part of Hedin’s equations, the vertex function. [@Hedin_1965] Although (perturbative) [[G$_0$W$_0$]{}]{} is probably the simplest and most widely used GW variant, [@Hybertsen_1985a; @vanSetten_2013; @Bruneval_2012; @Bruneval_2013; @vanSetten_2015; @vanSetten_2018] its starting point dependence has motivated the development of partially [@Hybertsen_1986; @Shishkin_2007; @Blase_2011; @Faber_2011; @Faleev_2004; @vanSchilfgaarde_2006; @Kotani_2007; @Ke_2011; @Kaplan_2016] and fully [@Stan_2006; @Stan_2009; @Rostgaard_2010; @Caruso_2012; @Caruso_2013; @Caruso_2013a; @Caruso_2013b; @Koval_2014; @Wilhelm_2018] self-consistent versions in order to reduce or remove this undesirable feature. Here, we will focus our attention on partially self-consistent schemes as they have demonstrated comparable accuracy and are computationally lighter than the fully self-consistent version. [@Caruso_2016] Moreover, they are routinely employed for solid-state and molecular calculations and are available in various computational packages. [@Blase_2011; @Blase_2018; @Bruneval_2016; @vanSetten_2013; @Kaplan_2015; @Kaplan_2016; @Krause_2017; @Caruso_2016; @Maggio_2017] Recently, an ever-increasing number of successful applications of partially self-consistent GW methods have sprung in the physics and chemistry literature for molecular systems, [@Ke_2011; @Bruneval_2012; @Bruneval_2013; @Bruneval_2015; @Bruneval_2016; @Bruneval_2016a; @Koval_2014; @Hung_2016; @Blase_2018; @Boulanger_2014; @Jacquemin_2017; @Li_2017; @Hung_2016; @Hung_2017; @vanSetten_2015; @vanSetten_2018] as well as extensive and elaborate benchmark sets. [@vanSetten_2015; @Maggio_2017; @vanSetten_2018; @Richard_2016; @Gallandi_2016; @Knight_2016; @Dolgounitcheva_2016; @Bruneval_2015; @Jacquemin_2015] There exist two main types of partially self-consistent GW methods: i) “eigenvalue-only quasiparticle” GW ([[evGW]{}]{}), [@Hybertsen_1986; @Shishkin_2007; @Blase_2011; @Faber_2011] where the quasiparticle (QP) energies are updated at each iteration, and ii) “quasiparticle self-consistent” GW ([[qsGW]{}]{}), [@Faleev_2004; @vanSchilfgaarde_2006; @Kotani_2007; @Ke_2011; @Kaplan_2016] where one updates both the QP energies and the corresponding orbitals. Note that a starting point dependence remains in [[evGW]{}]{} as the orbitals are not self-consistently optimized in this case. In a recent article, [@Loos_2018] while studying a model two-electron system, [@Seidl_2007; @Loos_2009a; @Loos_2009c; @Loos_2010e; @Loos_2011b; @Gill_2012] we have observed that, within partially self-consistent GW (such as [[evGW]{}]{} and [[qsGW]{}]{}), one can observe, in the weakly correlated regime, (unphysical) discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, HOMO-LUMO gap, total and correlation energies, as well as vertical excitation energies). In the present manuscript, we provide further evidences and explanations of this undesirable feature in real molecular systems. For sake of simplicity, the present study is based on simple closed-shell diatomics (, and ). However, the same phenomenon can be observed in many other molecular systems, such as , , , , , etc. Although we mainly focus on [[G$_0$W$_0$]{}]{} and [[evGW]{}]{}, similar observations can be made in the case of [[qsGW]{}]{} and second-order Green function (GF2) methods. [@SzaboBook; @Casida_1989; @Casida_1991; @Stefanucci_2013; @Ortiz_2013; @Phillips_2014; @Phillips_2015; @Rusakov_2014; @Rusakov_2016; @Hirata_2015; @Hirata_2017; @Loos_2018] Unless otherwise stated, all calculations have been performed with our locally-developed GW software, which closely follows the MOLGW implementation. [@Bruneval_2016] ![image](fig1a){width="\linewidth"} ![image](fig1b){width="\linewidth"} ![ \[fig:H2-zoom\] HF orbital energies (dotted lines) and QP energies as functions of the internuclear distance ${R_{\ce{H2}}}$ for the LUMO+1 and LUMO+2 orbitals of at the [[G$_0$W$_0$]{}]{}@HF/6-31G (solid lines) and [[evGW]{}]{}@HF/6-31G (dashed lines) levels. For convenience, the intermediate (center) branch is presented in lighter green for the LUMO+2. ](fig2){width="0.6\linewidth"} Theory ====== Here, we provide brief details about the main equations and quantities behind [[G$_0$W$_0$]{}]{} and [[evGW]{}]{} considering a (restricted) Hartree-Fock (HF) starting point. [@SzaboBook] More details can be found, for example, in Refs. . For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy $$\label{eq:SigC} {\Sigma^\text{c}_{p}}(\omega) = {\Sigma^\text{p}_{p}}(\omega) + {\Sigma^\text{h}_{p}}(\omega),$$ $$\begin{aligned} \label{eq:SigCh} {\Sigma^\text{h}_{p}}(\omega) & = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi\|x]^2}{\omega - {\epsilon_{i}} + {\Omega_{x}} - i \eta}, \\ \label{eq:SigCp} {\Sigma^\text{p}_{p}}(\omega) & = 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa\|x]^2}{\omega - {\epsilon_{a}} - {\Omega_{x}} + i \eta},\end{aligned}$$ where $\eta$ is a positive infinitesimal. The screened two-electron integrals $$[pq\|x] = \sum_{ia} (pq\|ia) ({\boldsymbol{X}}+{\boldsymbol{Y}})_{ia}^{x}$$ are obtained via the contraction of the bare two-electron integrals [@Gill_1994] $(pq\|rs)$ and the transition densities $({\boldsymbol{X}}+{\boldsymbol{Y}})_{ia}^{x}$ originating from a random phase approximation (RPA) calculation [@Casida_1995; @Dreuw_2005] $$\label{eq:LR} \begin{pmatrix} {\boldsymbol{A}}& {\boldsymbol{B}}\\ {\boldsymbol{B}}& {\boldsymbol{A}}\\ \end{pmatrix} \begin{pmatrix} {\boldsymbol{X}}\\ {\boldsymbol{Y}}\\ \end{pmatrix} = {\boldsymbol{\Omega}}\begin{pmatrix} \boldsymbol{1} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{-1} \\ \end{pmatrix} \begin{pmatrix} {\boldsymbol{X}}\\ {\boldsymbol{Y}}\\ \end{pmatrix},$$ with $$\begin{aligned} \label{eq:RPA} A_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia\|jb), & B_{ia,jb} & = 2 (ia\|bj),\end{aligned}$$ and $\delta_{pq}$ is the Kronecker delta. [@NISTbook] Equation also provides the neutral excitation energies ${\Omega_{x}}$. In practice, there exist two ways of determining the [[G$_0$W$_0$]{}]{} QP energies. [@Hybertsen_1985a; @vanSetten_2013] In its “graphical” version, they are provided by one of the many solutions of the (non-linear) QP equation $$\label{eq:QP-G0W0} \omega = {\epsilon^\text{HF}_{p}} + \Re[{\Sigma^\text{c}_{p}}(\omega)].$$ In this case, special care has to be taken in order to select the “right” solution, known as the QP solution. In particular, it is usually worth calculating its renormalization weight (or factor), ${Z_{p}}({\epsilon^\text{HF}_{p}})$, where $$\label{eq:Z} {Z_{p}}(\omega) = \qty[ 1 - \pdv{\Re[{\Sigma^\text{c}_{p}}(\omega)]}{\omega} ]^{-1}.$$ Because of sum rules, [@Martin_1959; @Baym_1961; @Baym_1962; @vonBarth_1996] the other solutions, known as satellites, share the remaining weight. In a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$. Within the linearized version of [[G$_0$W$_0$]{}]{}, one assumes that $$\label{eq:SigC-lin} {\Sigma^\text{c}_{p}}(\omega) \approx {\Sigma^\text{c}_{p}}({\epsilon^\text{HF}_{p}}) + (\omega - {\epsilon^\text{HF}_{p}}) \left. \pdv{{\Sigma^\text{c}_{p}}(\omega)}{\omega} \right\|_{\omega = {\epsilon^\text{HF}_{p}}},$$ that is, the self-energy behaves linearly in the vicinity of $\omega = {\epsilon^\text{HF}_{p}}$. Substituting into yields $$\label{eq:QP-G0W0-lin} {\epsilon^\text{{G$_0$W$_0$}}_{p}} = {\epsilon^\text{HF}_{p}} + {Z_{p}}({\epsilon^\text{HF}_{p}}) \Re[{\Sigma^\text{c}_{p}}({\epsilon^\text{HF}_{p}})].$$ Unless otherwise stated, in the remaining of this paper, the [[G$_0$W$_0$]{}]{} QP energies are determined via the linearized method. In the case of [[evGW]{}]{}, the QP energy, ${\epsilon^\text{{GW}}_{p}}$, are obtained via Eq. , which has to be solved self-consistently due to the QP energy dependence of the self-energy \[see Eq. \]. [@Hybertsen_1986; @Shishkin_2007; @Blase_2011; @Faber_2011] At least in the weakly correlated regime where a clear QP solution exists, we believe that, within [[evGW]{}]{}, the self-consistent algorithm should select the solution of the QP equation with the largest renormalization weight ${Z_{p}}({\epsilon^\text{{GW}}_{p}})$. In order to avoid convergence issues, we have used the DIIS convergence accelerator technique proposed by Pulay. [@Pulay_1980; @Pulay_1982] Moreover, throughout this paper, we have set $\eta = 0$. ![ \[fig:H2-QPvsOm\] ${\Sigma^\text{c}_{{\text{HOMO}}}}(\omega)$ and ${\Sigma^\text{c}_{{\text{LUMO}}+2}}(\omega)$ (in eV) as functions of the frequency $\omega$ obtained at the [[evGW]{}]{}@HF/6-31G level for at ${R_{\ce{H2}}}= 1.0$ bohr. The solutions of the QP equation are given by the intersection of the and blue curves.](fig3){width="0.6\linewidth"} Results ======= Virtual orbitals ---------------- As a first example, we consider the hydrogen molecule in a relatively small gaussian basis set (6-31G) in order to be able to study easily the entire orbital energy spectrum. Although the number of irregularities/discontinuities as well as their locations may vary with the basis set, the conclusions we are going to draw here are general. Figure \[fig:H2\] reports three key quantities as functions of the internuclear distance ${R_{\ce{H2}}}$ for various orbitals at the [[G$_0$W$_0$]{}]{} and the self-consistent [[evGW]{}]{} levels: i) the QP energies \[${\epsilon^\text{{G$_0$W$_0$}}_{p}}$ or ${\epsilon^\text{{GW}}_{p}}$\], ii) the correlation part of the self-energy \[${\Sigma^\text{c}_{p}}({\epsilon^\text{HF}_{p}})$ or ${\Sigma^\text{c}_{p}}({\epsilon^\text{{GW}}_{p}})$\], and iii) the renormalization factor/weight \[${Z_{p}}({\epsilon^\text{HF}_{p}})$ or ${Z_{p}}({\epsilon^\text{{GW}}_{p}})$\]. ### [[G$_0$W$_0$]{}]{} Let us first consider the results of the [[G$_0$W$_0$]{}]{} calculations reported in the top row of Fig. \[fig:H2\]. Looking at the curves of ${\epsilon^\text{{G$_0$W$_0$}}_{p}}$ as a function of ${R_{\ce{H2}}}$ (top left graph of Fig. \[fig:H2\]), one notices obvious irregularities in the LUMO+2 around ${R_{\ce{H2}}}= 1.0$ bohr and in the LUMO+1 around ${R_{\ce{H2}}}= 2.1$ bohr. For information, the experimental equilibrium geometry of is around ${R_{\ce{H2}}}= 1.4$ bohr. [@HerzbergBook] These irregularities are unphysical, and occur in correspondence with a series of poles in ${\Sigma^\text{c}_{{\text{LUMO}}+1}}$ and ${\Sigma^\text{c}_{{\text{LUMO}}+2}}$ (see top center graph of Fig. \[fig:H2\]). For example, one can notice two poles in ${\Sigma^\text{c}_{{\text{LUMO}}+2}}$ just before and after ${R_{\ce{H2}}}= 1.0$ bohr, giving birth to three branches. The origin of the irregularities in ${\epsilon_{{\text{LUMO}}+1}}$ and ${\epsilon_{{\text{LUMO}}+2}}$ can, therefore, be traced back to the wrong assumption that ${\Sigma^\text{c}_{{\text{LUMO}}+1}}(\omega)$ and ${\Sigma^\text{c}_{{\text{LUMO}}+2}}(\omega)$ are linear functions of $\omega$ in the vicinity of, respectively, $\omega = {\epsilon^\text{HF}_{{\text{LUMO}}+1}}$ and $\omega = {\epsilon^\text{HF}_{{\text{LUMO}}+2}}$ \[see Eq. \]. However, despite the divergencies in the self-energy, the QP energies ${\epsilon^\text{{G$_0$W$_0$}}_{{\text{LUMO}}+1}}$ and ${\epsilon^\text{{G$_0$W$_0$}}_{{\text{LUMO}}+2}}$ remain finite thanks to a rapid decrease of the renormalization factor at the ${R_{\ce{H2}}}$ values for which the self-energy diverges \[see Eq. and top right graph of Fig. \[fig:H2\]\]. For example, note that ${Z_{{\text{LUMO}}+2}}$ reaches exactly zero at the pole locations. A very similar scenario unfolds for the LUMO+1, except that a single pole is present in ${\Sigma^\text{c}_{{\text{LUMO}}+1}}$. Let us analyze this point further. Since the self-energy behaves as ${\Sigma^\text{c}_{p}} \sim \delta^{-1}$ (with $\delta \to 0$) in the vicinity of a singularity, one can easily show that ${Z_{p}} \sim (1+\delta^{-2})^{-1} \sim \delta^{2}$, which yields ${\epsilon^\text{{G$_0$W$_0$}}_{p}} \sim {\epsilon^\text{HF}_{p}} +\delta$. In plain words, ${\epsilon^\text{{G$_0$W$_0$}}_{p}}$ remains finite near the poles of the self-energy thanks to the linearization of the QP equation \[see Eq. \]. It also evidences that, at the pole locations (i.e. $\delta = 0$), we have ${\epsilon^\text{{G$_0$W$_0$}}_{p}} = {\epsilon^\text{HF}_{p}}$, i.e., by construction the QP energy is forced to remain equal to the zeroth-order energy. This is nicely illustrated in Fig. \[fig:H2-zoom\], where we have plotted the HF orbital energies (dotted lines) as well as the [[G$_0$W$_0$]{}]{} QP energies (solid lines) around the two “problematic” internuclear distances. The behavior of ${\epsilon^\text{{G$_0$W$_0$}}_{{\text{LUMO}}+1}}$ (solid orange line on the right panel of Fig. \[fig:H2-zoom\]) is particularly instructive and shows that the [[G$_0$W$_0$]{}]{} QP energies can have an erratic behavior near the poles of the self-energy. It is interesting to investigate further the origin of these poles. As evidenced by Eq. , for a calculation involving $2n$ electrons and $N$ basis functions, the self-energy has exactly $n N (N-n)$ poles originating from the combination of the $N$ poles of the Green function $G$ (at frequencies ${\epsilon_{p}}$) and the $n(N-n)$ poles of the screened Coulomb interaction $W$ (at the RPA singlet excitations ${\Omega_{x}}$). For example, at ${R_{\ce{H2}}}= 2.11$ bohr, the combination of ${\epsilon^\text{HF}_{{\text{LUMO}}}} = 3.83$ eV and the HOMO-LUMO-dominated first neutral excitation energy ${\Omega_{1}} = 22.24$ eV are equal to the LUMO+1 energy ${\epsilon^\text{{G$_0$W$_0$}}_{{\text{LUMO}}+1}} = 26.07$ eV. Around ${R_{\ce{H2}}}=1.0$ bohr, the two poles of ${\Sigma^\text{c}_{{\text{LUMO}}+1}}$ are due to the following accidental equalities: ${\epsilon^\text{{G$_0$W$_0$}}_{{\text{LUMO}}+1}} = {\epsilon^\text{HF}_{{\text{LUMO}}}} + {\Omega_{2}}$, and ${\epsilon^\text{{G$_0$W$_0$}}_{{\text{LUMO}}+1}} = {\epsilon^\text{HF}_{{\text{LUMO}}+1}} + {\Omega_{1}}$. Because the number of poles in $G$ and $W$ are both proportional to $N$, these spurious poles in the self-energy become more and more frequent for larger gaussian basis sets. For virtual orbitals, the higher in energy the orbital is, the earlier the singularities seem to appear. Finally, the irregularities in the [[G$_0$W$_0$]{}]{} QP energies as a function of ${R_{\ce{H2}}}$ can also be understood as follows. Since within [[G$_0$W$_0$]{}]{} only one pole of $G$ is calculated, i.e., the QP energy, all the satellite poles are discarded. Mixing between QP and satellites poles, which is important when they are close to each other, hence, is not considered. This situation can be compared to the lack of mixing between single and double excitations in adiabatic time-dependent density-functional theory and the Bethe-Salpeter equation [@Maitra_2004; @Cave_2004; @Romaniello_2009_JCP; @Sangalli_2011] (see also Refs. ). ### [[evGW]{}]{} Within partially self-consistent schemes, the presence of poles in the self-energy at a frequency similar to a QP energy has more dramatic consequences. The results for at the [[evGW]{}]{}@HF/6-31G level are reported in the bottom row of Fig. \[fig:H2\]. Around ${R_{\ce{H2}}}= 1.0$ bohr, we observe that, for the LUMO+2, one can fall onto three distinct solutions depending on the algorithm one relies on to solve self-consistently the QP equation (see bottom left graph of Fig. \[fig:H2\]). In order to obtain each of the three possible solutions in the vicinity of ${R_{\ce{H2}}}= 1.0$ bohr, we have run various sets of calculations using different starting values for the QP energies and sizes of the DIIS space. In particular, we clearly see that each of these solutions yield a distinct energy separated by several electron volts (see zoom in Fig. \[fig:H2-zoom\]), and each of them is associated with a well-defined branch of the self-energy, as shown by the center graph in the bottom row of Fig. \[fig:H2\]. For convenience, the intermediate (center) branch is presented in lighter green in Figs. \[fig:H2\] and \[fig:H2-zoom\], while the left and right branches are depicted in darker green. Interestingly, the [[evGW]{}]{} iterations are able to “push” the QP solution away from the poles of the self-energy, which explains why the renormalization factor is never exactly equal to zero (see bottom right graph of Fig. \[fig:H2\]). However, one cannot go smoothly from one branch to another, and each switch between solutions implies a significant energetic discontinuity. Moreover, we observe “ripple” effects in other virtual orbitals: a discontinuity in one of the QP energies induces (smaller) discontinuities in the others. This is a direct consequence of the global energy dependence of the self-energy \[see Eq. \], and is evidenced on the left graph in the bottom row of Fig. \[fig:H2\] around ${R_{\ce{H2}}}= 2.1$ bohr. The main observation of the present study is that each branch of the self-energy is associated with a distinct QP solution. We clearly see that, when one goes from one branch to another, there is a transfer of weight between the QP and one of the satellites, which becomes the QP on the new branch. [@Loos_2018] As opposed to the strongly correlated regime where the QP picture breaks down, i.e., there is no clear QP, here there is alway a clear QP except at the vicinity of the poles where the weight transfer occurs. As for [[G$_0$W$_0$]{}]{}, this sudden transfer is caused by the artificial removal of the satellite poles. However, in the [[evGW]{}]{} results the problem is amplified by the self-consistency. We expect that keeping the full frequency dependence of the self-energy would solve this problem. It is also important to mention that the self-consistent algorithm is fairly robust as it rarely selects a solution with a renormalization weight lower than $0.5$, as shown by the center graph in the bottom row of Fig. \[fig:H2\]. In other words, when the renormalization factor of the QP solution becomes too small, the self-consistent algorithm switches naturally to a different solution. From a technical point of view, around the poles of the self-energy, it is particularly challenging to converge self-consistent calculations, and we heavily relied on DIIS to avoid such difficulties. We note that an alternative ad hoc approach to stabilize such self-consistent calculations is to increase the value of the positive infinitesimal $\eta$. Figure \[fig:H2-QPvsOm\] shows the correlation part of the self-energy for the HOMO and LUMO+2 orbitals as a function of $\omega$ ( curves) obtained at the self-consistent [[evGW]{}]{}@HF/6-31G level for with ${R_{\ce{H2}}}= 1.0$ bohr. The solutions of the QP equation are given by the intersections of the and blue curves. On the one hand, in the case of the HOMO, we have an unambiguous QP solution (at $\omega \approx -20$ eV) which is well separated from the other solutions. In this case, one can anticipate a large value of the renormalization factor ${Z_{{\text{HOMO}}}}$ as the self-energy is flat around the intersection of the two curves. On the other hand, for the LUMO+2, we see three solutions of the QP equation very close in energy from each other around $\omega = 50$ eV. In this particular case, there is no well-defined QP peak as each solution has a fairly small weight. Therefore, one may anticipate multiple solution issues when a solution of the QP equation is close to a pole of the self-energy. Finally, we note that the multiple solutions discussed here are those of the QP equation, i.e., multiple QP poles associated to a single Green function. This is different from the multiple solutions discussed in Refs. , in which it is shown that, in general, the nonlinear Dyson equation admits multiple Green functions, which can be physical but also unphysical. ![image](fig4){width="\linewidth"} Occupied orbitals ----------------- So far, we have seen that multiple solutions seem to only appear for virtual orbitals (LUMO excluded). However, we will show here that it can also happen in occupied orbitals. We take as an example the fluorine molecule () in a minimal basis set (STO-3G), and perform [[evGW]{}]{}@HF calculations within the frozen-core approximation, that is, we do not update the orbital energies associated with the core orbitals. Figure \[fig:F2\] shows the behavior (as a function of the distance between the two fluorine atoms ${R_{\ce{F2}}}$) of the same quantities as in Fig. \[fig:H2\] but for some of the occupied orbitals of (HOMO-6, HOMO-5 and HOMO-4). Similarly to the case of discussed in the previous section, we see discontinuities in the QP energies around ${R_{\ce{F2}}}= 2.3$ bohr (for the HOMO-6) and ${R_{\ce{F2}}}= 2.7$ bohr (for the HOMO-5). For information, the experimental equilibrium geometry of is ${R_{\ce{F2}}}= 2.668$ bohr, which evidences that the second discontinuity is extremely close to the experimental geometry. Let us mention here that we have not found any discontinuity in the HOMO orbital. The case of the frontier orbitals will be discussed below. For , here again, we clearly observe ripple effects on other occupied orbitals. Similarly to virtual orbitals, we have found that the lower in energy the orbital is, the earlier the singularities seem to appear. ![ \[fig:BeO\] ${\Sigma^\text{c}_{{\text{HOMO}}}}(\omega)$ and ${\Sigma^\text{c}_{{\text{LUMO}}}}(\omega)$ (in eV) as functions of the frequency $\omega$ obtained at the [[G$_0$W$_0$]{}]{}@PBE/cc-pVDZ level for at its experimental geometry. [@HerzbergBook] The solutions of the QP equations are given by the intersection of the and blue curves.](fig5){width="0.6\linewidth"} Frontier orbitals ----------------- Before concluding, we would like to know, whether or not, this multisolution behavior can potentially appear in frontier orbitals. This is an important point to discuss as these orbitals are directly related to the ionization potential and the electron affinity, hence to the gap. Let us take the HOMO orbital as an example. A similar rationale holds for the LUMO orbital. According to the expression of the hole and particle parts of the self-energy given in Eqs. and respectively, ${\Sigma^\text{c}_{{\text{HOMO}}}}(\omega)$ has poles at $\omega = {\epsilon_{i}} - {\Omega_{x}}$ and $\omega = {\epsilon_{a}} + {\Omega_{x}}$ with ${\Omega_{x}} > 0$. Evaluating the self-energy at $\omega = {\epsilon_{{\text{HOMO}}}}$ would yield ${\epsilon_{{\text{HOMO}}}} - {\epsilon_{i}} = - {\Omega_{x}}$ and ${\epsilon_{{\text{HOMO}}}} - {\epsilon_{a}} = + {\Omega_{x}}$, which is in clear contradiction with the assumption that ${\Omega_{x}} > 0$. Therefore, the self-energy is never singular at $\omega={\epsilon_\text{HOMO}}$ and $\omega={\epsilon_\text{LUMO}}$ and the linearized [[G$_0$W$_0$]{}]{} equations can be solved without any problem for the frontier orbitals. This is true for any $G_0$, that is, it does not depend on the starting point. As can be seen from Eqs. and , the two poles of the self-energy closest to the Fermi level are located at $\omega = {\epsilon_\text{HOMO}}- {\Omega_{1}}$ and $\omega = {\epsilon_\text{LUMO}}+ {\Omega_{1}}$. As a consequence, there is a region equal to ${\epsilon_\text{HOMO}}- {\epsilon_\text{LUMO}}+ 2{\Omega_{1}}$ around the Fermi level in which the self-energy does not have poles. Because ${\Omega_{1}} \approx {\epsilon_\text{HOMO}}- {\epsilon_\text{LUMO}}= {E_\text{gap}}$, this region is approximately equal to $3{E_\text{gap}}$. For “graphical” [[G$_0$W$_0$]{}]{}, the solution might lie outside this range, even for the frontier orbitals. This can happen when ${E_\text{gap}}$ is much smaller than the true GW gap. In particular, this could occur for graphical [[G$_0$W$_0$]{}]{} on top of a Kohn-Sham starting point, which is known to yield gaps that are (much) smaller than GW gaps. Within graphical [[G$_0$W$_0$]{}]{}, multiple solution issues for the HOMO have been reported by van Setten and coworkers [@vanSetten_2015; @Maggio_2017] in several systems (, , and ). In their calculations, they employed PBE orbital energies [@Perdew_1996] as starting point, and this type of functionals is well known to drastically underestimate ${E_\text{gap}}$. [@ParrBook] As an example, we have computed, within the frozen-core approximation, ${\Sigma^\text{c}_{{\text{HOMO}}}}(\omega)$ and ${\Sigma^\text{c}_{{\text{LUMO}}}}(\omega)$ as functions of $\omega$ at the [[G$_0$W$_0$]{}]{}@PBE/cc-pVDZ level for beryllium monoxide () at its experimental geometry (i.e. ${R_{\ce{BeO}}}= 2.515$ bohr). [@HerzbergBook] These calculations have been performed with MOLGW. [@Bruneval_2016] The results are gathered in Fig. \[fig:BeO\], where one clearly sees that multiple solutions appear for both the HOMO and LUMO orbitals. Note that performing the same set of calculations with a HF starting point yields a perfectly unambiguous single QP solution. For this system, PBE is a particularly bad starting point for a GW calculation with a HOMO-LUMO gap equal to $1.35$ eV. Using the same basis set, HF yields a gap of $8.96$ eV, while [[G$_0$W$_0$]{}]{}@HF and [[G$_0$W$_0$]{}]{}@PBE yields $7.54$ and $5.60$ eV. The same observations can be made for the other systems reported as problematic by van Setten and coworkers. [@vanSetten_2015; @Maggio_2017] As a general rule, it is known that HF is usually a better starting point for GW in small molecular systems. [@Blase_2011; @Bruneval_2013; @Loos_2018; @Langre_2018] For larger systems, hybrid functionals [@Becke_1993] might be the ideal compromise, thanks to the increase of the HOMO-LUMO gap via the addition of (exact) HF exchange. [@Bruneval_2013; @Boulanger_2014; @Bruneval_2015; @Jacquemin_2015; @Gui_2018] Concluding remarks ================== The GW approximation of many-body perturbation theory has been highly successful at predicting the electronic properties of solids and molecules. [@Onida_2002; @Aryasetiawan_1998; @Reining_2017] However, it is also known to be inadequate to model strongly correlated systems. [@Romaniello_2009; @Romaniello_2012; @DiSabatino_2015; @DiSabatino_2016; @Tarantino_2017] Here, we have found severe shortcomings of two widely-used variants of GW in the weakly correlated regime. We have evidenced that one can hit multiple solution issues within [[G$_0$W$_0$]{}]{} and [[evGW]{}]{} due to the location of the QP solution near poles of the self-energy. Within linearized [[G$_0$W$_0$]{}]{}, this implies irregularities in key experimentally-measurable quantities of simple diatomics, while, at the partially self-consistent [[evGW]{}]{} level, discontinues arise. Because the RPA correlation energy [@Casida_1995; @Dahlen_2006; @Furche_2008; @Bruneval_2016] and the Bethe-Salpeter excitation energies [@Strinati_1988; @Leng_2016; @Blase_2018] directly dependent on the QP energies, these types of discontinuities are also present in these quantities, hence in the energy surfaces of ground and excited states. Illustrative examples can be found in our previous study. [@Loos_2018] Obviously, this latter point deserves further investigations. However, if confirmed, this would be a strong argument in favor of fully self-consistent schemes. Also, for extended systems, these issues might be mitigated by the plasmon modes that dominate the high-energy spectrum of the screened Coulomb interaction. PFL would like to thank Xavier Blase and Fabien Bruneval for valuable discussions. The authors would like to thank Valerio Olevano for stimulating discussions during his sabbatical stay at IRSAMC. MV thanks Université Paul Sabatier (Toulouse, France) for a PhD scholarship. Appendix: DIIS implementation for GW {#appendix-diis-implementation-for-gw .unnumbered} ==================================== @ifundefined [107]{} Hedin, L. New Method for Calculating the One-Particle [[Green]{}]{}’s Function with Application to the Electron-Gas Problem. Phys. Rev. 1965, 139, A796 Onida, G.; Reining, L.; and, A. R. 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--- abstract: 'We report on a search for host galaxies of a subset of Rotating Radio Transients (RRATs) that possess a dispersion measure (DM) near or above the maximum Galactic value in their direction. These RRATs could have an extragalactic origin and therefore be Fast Radio Bursts (FRBs). The sizes of related galaxies on the sky at such short distances are comparable to the beam size of a single-dish telescope (for example, the $7.0''$ radius of the Parkes beam). Hence the association, if found, could be more definitive as compared to finding host galaxies for more distant FRBs. We did not find any host galaxy associated with six RRATs near the maximum Galactic DM. This result is consistent with the fact that the probability of finding an FRB host galaxy within this volume is also very small. We propose that future follow-up observations of such RRATs be carried out in searching for local host galaxies as well as the sources of FRBs.' author: - \| A. Rane$^{1}$[^1] and A. Loeb$^{2}$[^2]\ $^{1}$Department of Physics & Astronomy, West Virginia University, Morgantown, WV, 26506 USA\ $^{2}$Astronomy Department, Harvard University, Cambridge, MA 02138, USA\ title: A novel approach for identifying host galaxies of nearby FRBs --- \[firstpage\] (stars:) pulsars: general; (galaxies:) intergalactic medium Introduction {#sec:intro} ============ Fast Radio Bursts (FRBs) are millisecond-duration radio signals, first reported in 2007 [@lorimerburst2007] and since then detected by the Parkes, Arecibo, and Green Bank telescopes between $0.7-1.5~ \mathrm{GHz}$, primarily through processing of pulsar surveys and a few in the real-time detection pipeline. To date, 17 FRBs have been published[^3] (@lorimerburst2007; @keanefrb2012; @thorntonfrb2013; @spitlerfrb2014; @burkespolaorfrb2014; @petrofffrb; @ravifrb2015; @championfrb2016; @masui2015; @keanehost2016) and two of them had been observed to repeat (@scholzrepeat2016; @maoz2015). Fourteen of these sources have been detected at high Galactic latitudes ($\|b\|>5\degree$) and their large dispersion measures (DM $\sim 375-1700~ \mathrm{pc~ cm^{-3}}$) exceed the expected Galactic contribution predicted by the NE2001 model in the direction of the bursts [@cordesne2001], suggesting an extragalactic origin. However, no host galaxy has yet been confirmed for any of these events. @keanehost2016 proposed association of FRB 150418 with an elliptical galaxy at redshift 0.5; however, @wb2016 and @vedantham2016 have demonstrated that the radio afterglow emission is instead due to AGN variability. Since the telescopes that are efficient in detecting these bursts are single-dish, the beam sizes are typically a few arcminutes which makes the position uncertainty large, limiting the ability to identify the host galaxy. The excess DM suggests redshifts in the range $\sim 0.3-1.3$ for FRBs if it originates from the intergalactic medium. Given these circumstances, it is extremely difficult to associate an FRB with a particular host galaxy unless there is some extraordinary evidence for coincident, transient multiwavelength emission within the beam. Another class of transients, known as RRATs are a group of Galactic pulsars that emit sporadic pulses [@maurarrats2006]. There are about 112 RRATs discovered so far and some of them have been observed at only one epoch or have been observed to emit only one pulse[^4]. Before it was discovered that FRB 121102 is repeating, @keane2016 discussed the uncertainty in the RRAT/FRB classification by analysing RRATs from which only one pulse has been detected so far. But since we know of repeating FRBs, there is no distinction between pulses from RRATs and FRBs. The only clear distinction is associated with their DM values. So, finding a host galaxy is really important at this point in order to constrain emission models for these bursts. Since pulses from RRATs are essentially indistinguishable from FRBs, as explained in detail in § 2, we decided to find and confirm the origin of a subset of RRATs that are close to the edge of the galaxy. Taking into account the uncertainty in the free electron density model, these RRATs could possibly have an extragalactic origin. The excess DM for our sample is not as high as seen in FRBs, corresponding to very low redshifts at which the size of the host galaxy on the sky is comparable to the beam size of a single-dish telescope (for example, the Parkes beam has a HPBW of 14’). This similarity of scales would make the association, if found, more definitive than at cosmological distances. In § 2 we compare the most commonly used model for the Galactic distribution with the newly available model to derive the DM distribution of radio transients on the sky. In § 3 we outline the basic criterion used to choose our RRAT sample. The results for each RRAT candidate are discussed in § 4 . In § 5 we discuss the non-detection of host galaxies and related probabilities. Finally, in § 6 we summarize our results and present our conclusions. Galactic free electron density models {#sec:models} ===================================== The NE2001 model [@cordesne2001] describes the structure of ionised gas in the Galaxy and is widely used to estimate distances to radio pulsars for which DM is the only distance indicator. In the case of FRBs for which the DMs are too high, this model is used to estimate the DM contribution in the direction of FRB from the Galaxy. This model is based on the observed DMs of Galactic radio pulsars and includes contributions from the thin disk associated with low-latitude HII regions, the thick-disk, the spiral arms, small-scale features corresponding to local ISM, individual high-density clumps and voids. However, the uncertainty in the NE2001 model is about $\sim 20\%$, particularly at higher latitudes, as also discussed in @gaensler2008. Recently, a new model by Yao, Manchester and Wang[^5], called as YMW16 has been proposed for the distribution of free electrons in the Galaxy, the Magellanic Clouds, and the inter-galactic medium (IGM). This model is based on measurements from 189 pulsars with independently determined distances as well DMs. We compared the two models by integrating both models to the edge of the Galaxy for each radio transient’s direction. The list of pulsars in the Milky Way, SMC and the LMC is obtained from @atnf2005. The ratio ($r$) of the measured DM to the maximum Galactic DM versus the measured DM is plotted for the NE2001 model in @spitlerfrb2014. We show a similar plot for the YMW16 model in Figure \[fig:dm-ymw16\]. We can see that the galactic DM contribution is lowered along certain lines of sight towards the galactic center (GC) pulsars, minimizing the gap between GC pulsars and the overall pulsar population. Pulsars in the LMC and SMC and FRBs, all have $r>1$, consistent with the NE2001 model. A small fraction of pulsars have $r>1$ in both models, possibly due to uncertainties or them being in the Galactic halo but we will not discuss these pulsars here. Another promising difference between the two models is that all RRATs had $r<1$ according to the NE2001 model, thus confirming their Galactic origin; however, according to the YMW16 model, some RRATs now have shifted to having $r>1$. Assuming this model is closer to the true values than the NE2001 model, these RRATs are very similar to FRBs. This RRAT sample provides a promising opportunity to find host galaxies that are close to us since their excess DM is much lower than most of the FRBs. We also compared the pulse width and flux distributions for both RRATs and FRBs, as seen in Figure \[fig:hist\]. The two sample Kolmogorov-Smirnov test gives $\mathrm{D=0.37}$ and a p-value of 0.03 for the pulse widths, thus indicating that the pulse width distributions of the two populations are not significantly different. For the flux distribution, $\mathrm{D=0.80}$ and the p-value is small ($9.1\times10^{-9}$). This is not surprising since the peak fluxes of known FRBs are higher than RRATs. ![The ratio of measured DM to maximum Galactic DM versus the measured DM (in $\mathrm{pc~ cm^{-3}}$) for all radio transients. The maximum Galactic DM is calculated by integrating the YMW16 model to the edge of the Galaxy for each transient’s direction. The dashed line shows the maximum ratio expected for Galactic objects if the electron density is accurate for all lines of sight.[]{data-label="fig:dm-ymw16"}](dmexcess-ymw16v2.eps) ![image](hist_wf1.eps) Our sample ========== In an attempt to find host galaxies as discussed in § 1, we have selected a sample of RRATs which have $r$ greater than 0.9 and for which $\rm{DM_{diff}= DM-DM_{MW}}$ is less than 10 $\rm{pc~ cm^{-3}}$ as seen in Figure \[fig:dmzoom\]. These RRATs are at the edge of our galaxy and if the uncertainties in the electron density model are taken into account, they could be Galactic or extragalactic. ![A zoomed in plot of the ratio between the measured DM and the maximum Galactic DM versus measured DM subtracted from maximum Galactic DM for RRATs.[]{data-label="fig:dmzoom"}](dmzoomv2.eps) Results ======= Next, we discuss the individual RRAT candidates which could possibly be FRBs. If the DM due to the host galaxy is neglected and if the excess DM$_{\rm diff}$ is assumed to be entirely due to the intergalactic medium, then we infer a redshift $z \sim 0.005$ and distances up to $\sim 20~\rm{Mpc}$. But since these RRATs have low DM and the Galactic $\rm{DM_{MW}}$ uncertainties might be within this excess, we search for galaxies within 120 Mpc (corresponding to a DM $\sim 30 \mathrm{pc~ cm^{-3}}$ for the local density of the intergalactic medium) at the corresponding beam size based on the redshift information provided on NED[^6]. The summary is given in Table \[table:summaryt\]. The $\rm{DM_{halo}}$ contribution is determined from the free electron density profile as a function of galecto-centric radius obtained by the latest model that fits best O VIII observations (see, Figure 8 of @mb2015). The number of objects found in NED within this search radius are listed in each subsection, and if the spectrum is available then their redshifts are determined by cross-correlating the spectrum against template spectra using the IRAF task <span style="font-variant:small-caps;">xvsao</span> in the <span style="font-variant:small-caps;">rvsao</span> package. We also determined the variation in $\rm{DM_{MW}}$ within the beam uncertainty using the YMW16 model for each RRAT, as seen in Figure \[fig:dmall\]. The $\rm{DM_{MW}}$ variation is within $12\%$ for all six RRATs. ![image](dmvariationallv2.eps) ------------ -------- ---------- ------ ------- ------- ------ ------ --- Name RRAT J1332$-$03 322.25 57.91 19.4 27.1 24.23 1.12 0.5 4 J0156$+$04 152.00 $-$55.00 7.5 27.5 25.18 1.09 0.5 1 J1603$+$18 32.85 45.28 7.5 29.7 29.21 1.02 0.5 0 J1354$+$24 27.43 75.78 19.4 20.0 20.48 0.98 0.07 0 J0837$-$24 247.45 9.80 7.0 142.8 147.3 0.97 0.22 1 J1433$+$00 349.75 53.79 7.5 23.5 26.02 0.90 0.06 1 ------------ -------- ---------- ------ ------- ------- ------ ------ --- \[table:summaryt\] RRAT J1332$-$03 --------------- This RRAT was discovered in the 350-MHz Drift-scan pulsar survey with the GBT and was confirmed in follow-up observation. The uncertainty in the beam position is 19.4’ at 350 MHz. The four extra-galactic source galaxies found within this search radius are listed in Table \[table:summaryt2\]. For LCRS B133012.2-031854, we did not find any spectrum from online literature. All of these galaxies have higher redshifts than what we would require to account for the intergalactic medium contribution, $\rm{DM_{IGM}}$. Hence these galaxies are most probably not related to this RRAT. ------------ ------------------------------ -------- ---------------- ---------- -- RRAT Name J1332$-$03 LCRS B133012.2-031854 14.522 0.022482 - GALEXASC J133140.75-030956.2 16.692 $-$0.000076 0.063 2dFGRS N138Z073 17.365 $-$0.000200860 0.063 2dFGRS N138Z028 18.214 0.000200 0.055 J0156$+$04 IC 1750 5.44 0.018860 0.018838 J1603$+$18 - - - - J1354$+$24 - - - - J0837$-$24 2MASX J08374183-2451356 0.313 - 0.15 J1433$+$00 2dFGRS N346Z227 3.052 $-$0.000500 0.07 ------------ ------------------------------ -------- ---------------- ---------- -- \[table:summaryt2\] RRAT J0156$+$04 --------------- This RRAT was discovered in the single-pulse search of the data obtained in the Arecibo Drift Pulsar survey at 327 MHz and two pulses were observed from it at only one epoch [@deneva16]. Follow-up observations detected no pulses from this RRAT. The uncertainties in both the coordinates are $7.5'$, the 327 MHz beam radius. We found one galaxy within this beam with a too high redshift of 0.18, hence indicating no association with this RRAT (Table \[table:summaryt2\]). RRAT J1603$+$18 --------------- This RRAT was also discovered in the Arecibo Drift Pulsar survey at 327 MHz and was confirmed in follow-up observation [@deneva16]. We did not find any galaxy within a search radius of $7.5'$ from the beam center up to 120 Mpc. RRAT J1354$+$24 --------------- This RRAT was discovered in the Green Bank North Celestial Cap survey (GBNCC) at 350 MHz[^7]. We did not find any galaxy within a search radius of $19.4'$ up to 120 Mpc. RRAT J0837$-$24 --------------- This RRAT was discovered in the single-pulse search of the High Time Resolution Universe (HTRU) pulsar survey carried out with the Parkes telescope at $\sim 1.4$ GHZ [@burkeIII2011]. We could not find a spectrum for this galaxy and hence the redshift is determined using the luminosity-size relation (see, equation 4 of @mcintosh2005). The inferred redshift of 0.15 is too high, indicating no association with this RRAT (Table \[table:summaryt2\]). RRAT J1433$+$00 --------------- This RRAT was discovered in the Arecibo Drift Pulsar survey at 327 MHz and was confirmed in follow-up observation [@deneva16]. The 2dFGRS source found within the search radius of $7.5'$ was at an inferred redshift of 0.07, too high than expected, so it is unrelated to the RRAT (Table \[table:summaryt2\]). Discussion ========== Since we did not find any possible host galaxy in § 4 that might be associated with any of the six RRATs in our sample, these most likely do not have an extragalactic origin. For RRAT J1603$+$18, the DM$_{\rm halo}$ contribution adds to match the measured DM and hence this RRAT might be within the halo of our galaxy. We have also determined the DM associated with the local group by checking the direction of each of these RRATs based on the right panel of Figure 3 in @rl2014, which yields a DM $\sim 5 \rm{pc~ cm^{-3}}$. RRATs J1332$-$03 and J0156$+$04 could reside within the local group.The probability of finding a galaxy within the volume out to 120 Mpc by chance for a beam radius of $7.5'$ is $\sim 0.043$, whereas for a beam radius of $19.4'$ it is $\sim 0.28$, based on the average number density of galaxies within the search volume, using NED. Our null result is thus consistent with this estimate. Conclusions =========== We have presented a search for host galaxies in a subset of RRATs that are at the edge of our Galaxy. These RRATs are interesting since they could either have Galactic or extragalactic origin. In the latter case, the sizes of the host galaxies on the sky at such distances would be comparable to the beam size of a single-dish telescope. We did not find any nearby host galaxy associated with the six RRATs in our sample. Although finding nearby galaxies for such RRATs that could possibly be FRBs is a novel approach, the probability of actually finding a nearby host galaxy is low. Nevertheless, we suggest applying this search strategy to new discoveries of RRATs that will be in the uncertainty zone of the electron density model. Follow-up observations could determine if these RRATs are of a Galactic origin or are extragalactic FRBs, and help us pin down the mysterious origin of FRBs. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Dr. Duncan Lorimer for insightful comments and careful reading of the paper. A. Rane would like to thank the hospitality of the Institute for Theory and Computation (ITC). This work was supported in part by NSF grant AST-1312034. Burke-Spolaor, S., Bailes, M., Johnston, S., et al. 2011, MNRAS, 416, 2465 S., [Bannister]{} K. W., 2014, ApJ, 792, 19 Champion, D. 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G., & Berger, E. 2016, ApJL, 821, L22 \[lastpage\] [^1]: email: arane@mix.wvu.edu [^2]: email: aloeb@cfa.harvard.edu [^3]: See http://www.astronomy.swin.edu.au/pulsar/frbcat/ [^4]: See RRATalog:\ http://astro.phys.wvu.edu/rratalog/ [^5]: http://www.atnf.csiro.au/research/pulsar/ymw16/ [^6]: The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. [^7]: http://www.hep.physics.mcgill.ca/ karakoc/GBNCC.html
--- author: - Alberto Buzzoni --- [Osservatorio Astronomico di Brera]{} [Via Bianchi 46 23807 Merate (Lc), Italy]{} The problem of UV properties of primeval galaxies is briefly assessed from the theoretical point of view discussing its impact on the definition of the cosmological model. Introduction ============ The recent major improvements in the observation of high-redshift galaxies, coming from HST (Williams [et al.]{} 1996; Madau [et al.]{} 1996) and other ground-based telescopes of the new generation (Koo [et al.]{} 1996; Lilly [et al.]{} 1996; Cowie and Hu 1998), have urgently called for a better understanding of galaxy UV spectrophotometric properties. Optical and infrared observations of the most distant objects are in facts inspecting their restframe ultraviolet features, so that any chance of successful detection of primeval galaxies should eventually rely on their fair recognition in this photometric range. A theoretical approach has to be preferred in this regard to match the data since ground-based observations prevent us to explore local galaxy templates in the extreme UV band with comparable detail. In this note we would briefly complement Buzzoni’s (1998a,b) full analysis assessing the general problem of the UV properties of primeval galaxies and its impact on the cosmological model. UV Luminosity and Star Formation Rate ===================================== In Fig. 1 we show a synthetic c-m diagram in the $U$ vs. $U-B$ plane for a 4 Gyr old simple stellar population (SSP) of solar metallicity generating from a burst star formation according to a Salpeter IMF (Buzzoni 1989). This diagram might probably match the real case of a young elliptical galaxy as seen about $z \sim 2$. It is evident from the figure that the SSP UV luminosity is largely dominated by the stars around the main sequence turn off (TO) region (note, on the contrary, that red giant stars are definitely too cool to give any major contribution at short wavelength). This tight dependence still holds even in a more complex evolutionary scenario dealing with a continuous star formation rate (SFR) so that a direct link exists between total UV luminosity of the composite stellar population and the relative number of young stars of higher mass (that is those with the hottest TO). One could therefore envisage a straightforward relationship between galaxy UV luminosity and [actual]{} SFR, as outlined in Fig. 2 according to Buzzoni’s (1998b) calibration. The Cosmic SFR and UV Luminosity Density ======================================== As the prevailing characteristics of the UV radiation is to track galaxy SFR, this leads to a quite different interpretative approach to the galaxy luminosity function as observed in the restframe UV range. In this case in facts galaxy luminosity is not tracking the object size but rather its actual star formation activity. A study of the luminosity function at low redshift clearly displays a change in the Schechter fitting parameters in the sense of a steepening in the faint-end tail of the function as far as we move from optical to ultraviolet wavelengths (cf. Fig. 3). If this trend is maintained also at high redshift, then undetected “quiescent” galaxies might provide a major contribution to the cosmic UV background. Once trying to account for this large fraction of faint objects and correct accordingly the current estimates of the cosmic UV luminosity density, according for instance to Madau (1997b), the final result would lead to a measure of the cosmic SFR like in Fig. 4. No evident signs of enhanced star formation at $z \sim 1.5$ appear from the figure, contrary to Madau’s (1997b) original results. The inferred SFR is a flat or decreasing function of the cosmic age, depending on the assumed cosmological model. For an Einstein-De Sitter model, a power law such as ${\rm SFR} \propto t^{-1}$ seems to consistently match the observations, while a low-density open Universe suggests a possibly constant ${\rm SFR} \sim 0.02~[H_o/50]~M_\odot/yr$. This calls for a prevailing population of quiescent star-forming galaxies at high-redshift, although it does eventually not imply for those objects to be also [bona fide]{} “primeval” galaxies. Buzzoni, A., 1989, Buzzoni, A., 1998a, in [Proceedings of the IAU Symp. No. 183, Cosmological Parameters and Evolution of the Universe. ASP Conf. Series]{} in press ed. K. Sato ([astro-ph/9711072]{}) Buzzoni, A., 1998b, submitted Cowie, L.L., Hu, E.M., 1998, in press ([astro-ph/9801003]{}) Koo, D.C., [et al.]{}, 1996, Lilly, S.J., Le Fèvre, O., Hammer, F., Crampton, D., 1996, Madau, P., 1997a in [Proceedings of The Hubble Deep Field, STScI Symp Series, Baltimore]{} in press eds. M. Livio, S.M. Fall and P. Madau ([astro-ph/9709147]{}) Madau, P., 1997b in [Proceedings of the 7th International Origins Conf., Estes Park CO,]{} in press ([astro-ph/9707141]{}) Madau, P., Ferguson, H.C., Dickinson, M.E., Giavalisco, M., Steidel, C.C., Fruchter, A., 1996, Williams, R.E. [et al.]{} (the HDF project), 1996, , [112]{} [1335]{}
--- author: - 'O. Golubov' - 'D. J. Scheeres' - 'Yu. N. Krugly' title: 'A 3-dimensional model of tangential YORP' --- Introduction ============ The YORP effect is a torque acting on an asteroid, created by the recoil force of the light reflected or reemitted by the surface ([@rubincam00], [@bottke06]). In simulations of YORP the heat conductivity in the asteroid used to be considered as 1-dimensional, and the local curvature of the asteroid’s surface was neglected [@rozitis13]. Under such assumptions, the YORP torque was only due to non-symmetries of the asteroid’s shape, while a symmetric asteroid could possess no YORP. Heat conductivity through the asteroid’s body was accounted only for small asteroids measuring some metres in diameter [@breiter10]. Still, surfaces of asteroids can be covered with stones, making the heat conductivity problem at the surface 3-dimensional and allowing heat fluxes through the stones. Recently [@golubov12] demonstrated that accounting for these effects can substantially change the picture of YORP. Heat fluxes through stones on the asteroid’s surface can cause their western sides to be slightly warmer than their eastern sides, thus causing them to experience a net drag parallel to the global surface of the asteroid, and to create a torque increasing the rotation rate of the asteroid. This torque was called the tangential YORP (or TYORP), in contrast to the normal YORP (or NYORP), which had been considered previously and is produced by forces normal to the global surface. Even a perfectly symmetric asteroid was demonstrated to experience TYORP, while its NYORP is nill. For reallistic moderately asymmetric asteroids the strength of TYORP torque was estimated as comparable to the one of NYORP. Still, these estimates were very rough, as [@golubov12] did all their simulations in a simple 1-dimensional model, substituting stones with high thin walls standing on the asteroid’s surface in the meridional direction. Thus their consideration gave just an order-of-magnitude estimate of the strength of TYORP, and the question of a more precise description of the effect remained. In this article we construct a more realistic model of TYORP, and study an asteroid’s surface covered with spherical stones. In Section 2 we describe our model and review the methods used for its simulations. Results of the simulations are presented in Section 3. In Section 4 we discuss the results and their implications. In Appendix A discuss our numeric algorithm in more detail, and in Appendix B we derive formulae for the integrated TYORP torque of an ellipsoidal asteroid. Model ===== We consider a flat patch of the asteroid’s surface covered with spherical stones, as shown in Figure \[spheres\]. Each stone has radius $R$, and its center is situated at a height $hR$ above the surface ($-1<h<1$). Stones are arranged in a periodic square grid of size $aR$ ($a \ge 2$), with the sides of the squares going in the directions south-north and east-west. The size of the patch under consideration is assumed to be much smaller than the size of the asteroid, so that we disregard curvature of the surface. We describe the entire patch with the same latitude $\psi$, determined as the angle between the normal of the patch and the asteroid’s equatorial plane. The entire semispace below the surface ($z_3 < 0$) is filled with regolith, so that there is regolith everywhere between and under the stones. The heat conductivity of the regolith is assumed to be much smaller than the heat conductivity of stones, so that no heat conductivity between a stone and the surrounding regolith occurs, and the reemission of the absorbed heat by the regolith is instantaneous. The heat conductivity of the stones is $\kappa$, the heat conductivity of the regolith is 0. The albedo of both the stones and the regolith is $A$. The heat capacity of a stone is $C$ and its density is $\rho$. Then the temperature distribution in a stone obeys the heat conductivity equation $$C\rho\frac{\partial T}{\partial t}=\kappa\sum_{i=1}^3\frac{\partial^2 T}{\partial x_i^2}. \label{conductivity}$$ The boundary condition for this equation above the ground is $$\kappa\frac{\partial T}{\partial x_i}=\left\{ \begin{array}{c c} n_i((1-A)\alpha\Phi-\epsilon\sigma T^4), & x_3>0,\\ 0, & x_3 \le 0. \end{array} \right.\ \label{boundary}$$ Here $\sigma$ is Stefan–Boltzmann’s constant, $\epsilon$ is emissivity of stone, $\boldmath{n}$ is the normal vector of the surface, and $\alpha\Phi$ is the incoming light power per unit surface of the stone, with $\Phi$ being the solar constant and some variable coefficient $\alpha \la 1$. The characteristic scale of the temperature is the equilibrium temperature of the subsolar point, $$T_0=\sqrt[4]{\frac{(1-A)\Phi}{\epsilon\sigma}}.$$ Distance has two important scales, namely the wavelength of the heat conductivity wave $L_\mathrm{wave}$, and the heat conductivity length $L_\mathrm{cond}$, expressed by formulae $$L_\mathrm{wave}=\sqrt{\frac{\kappa}{C\rho\omega}}, \label{L_cond}$$ $$L_\mathrm{cond}=\frac{\kappa}{\left((1-A)\Phi\right)^{3/4}\left(\epsilon \sigma \right)^{1/4}}.$$ The physical meaning of $L_\mathrm{cond}$ is that it is the distance at which temperature difference equal to $T_0$ causes heat flux equal to $\Phi$. Usually at distance scales much bigger than $L_\mathrm{cond}$, heat conductivity can be neglected. The ratio of $L_\mathrm{cond}$ and $L_\mathrm{wave}$ is called the thermal parameter $$\theta=\frac{\left(C\rho\kappa\omega\right)^{1/2}}{\left((1-A)\Phi\right)^{3/4}\left(\epsilon \sigma \right)^{1/4}}. \label{theta}$$ The thermal parameter characterizes the relative importance of heat conductivity with respect to heat absorption and emission. We non-dimensionalize all the variables. We introduce dimensionless variables $\xi=x/L_\mathrm{cond}$ and $\tau=T/T_0$. Instead of time $t$ we use the rotation phase $\phi=\omega t$, with $\omega$ being the angular velocity of the asteroid. In these terms Equations \[conductivity\] and \[boundary\] transform into $$\frac{\partial \tau}{\partial \phi}=\frac{1}{\theta^2}\sum_{i=1}^3\frac{\partial^2 \tau}{\partial \xi_i^2}. \label{conductivity_nonD}$$ The boundary condition for this equation above the ground is $$\frac{\partial \tau}{\partial \xi_i}=\left\{ \begin{array}{c c} n_i(\alpha-\tau^4), & \xi_3>0\\ 0, & \xi_3 \le 0 \end{array} \right.\ \label{boundary_nonD}$$ To simplify the analysis we keep the number of free parameters to the minimum. In our dimensionless simulations we assume albedo $A=0$, emissivity $\epsilon=1$, and Lambert’s law for scattered and emitted light. Allowing for different $A$, different $\epsilon$, different scattering and emission laws, would make the problem too difficult to tackle. If no instances of self-illuminations occur, i.e. a ray emitted by the asteroid never falls back onto the asteroid, the non-dimensionalized results can be easily rescaled to different $A$ and $\epsilon$ with the aid of Equations \[L\_cond\] and \[theta\]. And there is a good reason to believe that this rescaling gives relatively accurate results even in the presence of some self-illumination, as on the one hand usually only a minor portion of emitted or reflected light falls back onto the surface, on the other hand $A$ is usually only slightly deviates from 0 and $\epsilon$ only slifhtly deviates from 1, so accounting for both these effects simultaneously should only give a second order correction. Without investigating this question in more detail, we non-dimensionalize our model with Equations \[L\_cond\] and \[theta\], and then in the non-dimensional model assume $A=0$ and $\epsilon=1$. This treatment is precisely correct if in the initial model really $A=0$ and $\epsilon=1$, and if not then this treatment presumably gives a good approximation. We simulate the heat conductivity in stones and the ray tracing numerically using Monte Carlo technique. The stone is modelled with the aid of a finite difference method on a cubic mesh. We assume all stones to be the same, thus posing periodic boundary conditions and assuming that a ray leaving through the left boundary reappears on the right boundary. We cast rays from the Sun onto the asteroid and trace each of them. If a ray is absorbed by the regolith on the surface, it is instantly reemitted with the same energy and with a random direction determined in accordance with Lambert’s law. If a ray hits a stone, it is absorbed and its energy is deposited to the closest node of the mesh within the stone. Then the stone emits rays according to the Stefan–Boltzmann’s law. The directions of the rays are chosen randomly according to Lambert’s law, their initiation points are randomly chosen on the stone’s open surface, and their energies are determined by temperatures in the closest nodes of the mesh. The energy of each ray is subtracted from the neighbouring node. Then the ray is traced, and is either absorbed by another stone (and then its energy is returned back) or goes into space. It can be also scattered by the regolith. We repeat the procedure many times, as the Sun follows its diurnal path. The simulation continues for several asteroidal days to make the system forget its initial conditions. Then the momentum in $x$ direction given to asteroid by the emitted rays is calculated and averaged over several days. Our numeric algorithm is explained in more detail in Appendix A. The momentum is expressed in dimensionless units as the force acting on a stone divided over the area occupied by the stone and over the solar light momentum flux, $$p_x=\frac{F_x c}{\pi R^2 \Phi}. \label{px_normalization}$$ If we want go from $p_x$ to the force per unit area $P$, we must multiply $p_x$ by the solar light momentum flux $\Phi/c$ and by the fraction of the surface area occupied by the stones. The trade-off between parameters governing the simulation (the size of the spatial mesh, the timestep, the number of rays cast from the Sun and of rays emitted by the stone in each step, the number of asteroidal days simulated) is adjusted so that we reach the highest accuracy for a given computation time. Then $p_x$ is studied as a function of physical parameters. Origination of the effect can be explained with the aid of Figures \[tau\] and \[tau1D\]. Figure \[tau\] shows the temperature distribution inside the stone at six different moments. At sunrise (6am) the temperature of the stone is the lowest, as the asteroid was cooled all night long. In the morning the stone is heated from the East, at noon from the top, and in the afternoon from the West, that is illustrated in the next 3 panels. At 6pm sun sets, heating stops and the stone slowly cools down. In the lower left panel the surface of the stone is already relatively cool as for some time before 6pm the stone has been largely shadowed by the next stone to the West. The western side of the stone is still significantly warmer than its eastern side, but this difference vanishes before midnight. In Figure \[tau1D\] we plot the temperatures of the three most characteristic points of the stone: the eastmost, the top, and the westmost points. We see that the temperature in the East is the first to start rising. It pulls up the temperatures of the other points due to heat conductivity. When the afternoon sun starts heating the western part of the stone, its temperature rises even more, and reaches levels never attained by the eastern part of the stone. Even though the mean temperature for the two parts is nearly the same, the temperature in the East is more uniform, while the temperature in the West has a sharper maximum. As a result the mean fourth power of the temperature is bigger in the West, so is the recoil force due to the Stefan-Boltzmann’s law. In the lower part of the figure we plot the TYORP force integrated over time (starting at midnight), $$\tilde{p}_x(t)=\frac{1}{t_{rot}}\int_0^t\frac{F_x(t_1) c}{\pi R^2 \Phi}\, \mathrm{d}t_1.$$ We see that in the morning the integral gets negative, as the eastern side of the stone is warmer, emits more infrared light, and slows down rotation of the asteroid. In the afternoon the western side of the asteroid gets warmer, causes the integrated TYORP force to increase, and in the end prevails. The integral of the TYORP force over the whole rotation period is positive, which means a positive $p_x=\tilde{p}_x(t_\mathrm{rot})$. Results ======= In Figure \[p\_x\] we plot the dimensionless TYORP force $p_x$ as a function of different parameters. There are 5 relevant parameters: dimensionless radius of the spherical stones $r$, relative distance between the stones $a$, relative height of the center of a stone above the ground $h$, latitude $\psi$, and thermal parameter $\theta$. Five panels of the plot show dependencies on one variable each, and only the first panel shows a 2-dimensional dependence on $r$ and $\theta$ colour-coded. From the first panel we see that the TYORP acceleration is significant only inside an ellipsis stretching from the upper left to the lower right. The maximum is attained around $r=0.3$ and $\theta=2$, where we have $p_x \approx 0.003$. The function decreases from this point to the lower left and the upper right very steeply, while the decrease in the upper left and the lower right is much slower. The middle and the lower panel in the left show cross-sections of the upper left panel in the horizontal and vertical lines respectively. We see maxima in each line, situated in the area where the line crosses the red ellipsis in the upper left panel. The maximum is lower if the intersection point is farther from the centre of the ellipsis. The general appearance of the upper and the lower left panels is similar to the lower two panels in Figure 2 in [@golubov12]. The physical interpretation is also similar; the TYORP effect originates as a result of the heat conductivity lag on the scale of the stone, therefore for any substantial effect, the heat conductivity length, the thermal wavelength, and the size of the stone must be comparable. In the upper right panel Figure \[p\_x\] we plot the dimensionless TYORP force $p_x$ as a function of the latitude $\psi$. We see that $p_x$ is biggest at the equator of the asteroid ($\psi=0$), and goes to 0 at the poles ($\psi=90^\circ$). Dashed lines showing a cosine function are overplotted for comparison. The middle right panel shows the dependence of $p_x$ on the distance between the stones $a$. The left limit of the plot $a=2$ corresponds to stones touching each other. The last panel shows $p_x$ as a function of the height of the center of a stone above the ground $h$. When $h$ is close to $-1$, the stones are almost entirely below the ground, and the effect vanishes. Then when $h$ increases to 0, $p_x$ also increases. Finally, when $h$ goes on increasing to 1, $p_x$ does not demonstrate any more significant increase, due to shadowing of the lower parts of the spheres. Discussion ========== To estimate the relative importance of TYORP and NYORP we follow [@golubov12] and use the normalized YORP torque, determined as $$\tau_z=\frac{T_z c}{\Phi r_\mathrm{eq}^3}, \label{tau_z}$$ with $T_z$ being the YORP torque, $r_\mathrm\mathrm{eq}$ being the equivalent radius of the asteroid (the radius of the sphere of the same volume), $c$ being the speed of light, and $\Phi$ being the solar radiation flux at the position of the asteroid. Estimate of dimensionless NYORP give $\tau_z=0.008$ for 1620 Geographos [@durech08geo] and $\tau_z=0.002$ for 54509 YORP [@lowry07]. To estimate $\tau_z$ we need to integrate the TYORP force over the surface of the asteroid. As the latitude dependence of $p_x$ (upper right panel of Fugure \[p\_x\]) is relatively complicated, we try two limiting cases, a sinusoidal dependence and a constant (see Appendix B). Both estimates give nearly the same result, $$\tau \approx 9 p_0 f,$$ where $p_0$ is $p_x$ at equator and $f$ is the fraction of the surface occupied by stones. It implies that in our model the dimensionless TYORP can reach up to about 0.01 (for $p_0$ about 0.003 and $f$ close to 0.5). This is an order of magnitude less than the value obtained by [@golubov12], which makes sense as now the presence of the upper boundary of stones as an emitter and absorber of heat must decrease the temperature contrast in the stone, and the presence of underground part of the stone acting as a heat reservoir must also level temperature gradients. Still, this value is comparable to the strength of NYORP. If we have spherical stones of different sizes lying on the surface of an asteroid, then the TYORP force acting on a patch of the surface should be integrated over all sizes of stones. The problem is especially complicated because of effects of shadowing and self-illumination, so that stones can not be considered independently, but only in toto. It is evident from the middle right panel of Figure \[p\_x\]. If stones were not influencing each other, the total force would be proportional to their number, and $p_x$ would be constant. It eventually happens at $a\rightarrow \infty$, where $p_x$ reaches saturation. For such low surface densities of stones they stop influencing each other, and the overall TYORP force can be obtained as the integral over all sizes of stones of particular TYORP force for each size calculated in the absence of any other stones. But for $a \approx 2$, where stones lie close to each other, significant deviation from the saturated limit are observed. Moreover, $p_x$ in this case is sensitive to the arrangement of stones, so that our results obtained for a square grid are only an estimate of what happens if stones are positioned more randomly. In any case, significant contribution to TYORP is given only by stones of some particular sizes, belonging to the maximum in the upper left panel of Figure \[p\_x\]. The observed and predicted TYORP acceleration of Itokawa is compared in Table \[tab-itokawa\]. Despite a large variation between different predictions, they are systematically smaller than the observations by few $10^{-3}$. [@golubov12] assumed that this descrepancy could be due to TYORP, but could support this claim only with rough estimates. Now we are capable of a much more quantitative analysis. We estimate TYORP for Itokawa, assuming rotation period 12.1 hr, semimajor axis 1.324 AU, $\kappa = 2.65$ W m$^{-1}$K$^{-1}$, $C = 680$ J kg$^{-1}$K$^{-1}$, $\rho = 3500$ kg m$^{-3}$, $A=0.23$, $\epsilon = 0.7$. For such parameters we get $\theta = 18$, $L_\mathrm{cond}=1.5$ m, $L_\mathrm{wave}=9$ cm. The dimensiomless TYORP drag $p_x$ reaches the maximal value of 0.00025 for stones with radius $R\approx 4$ cm, This maximum is very broad, so that $p_x > 0.00015$ for $R=1\div 15$ cm. The corresponding dimensionless TYORP torque is $\tau_z\approx 0.002 f$, where $f$ is the fraction of the surface occupied by stones with radii of 1 to 15 cm. If these stones are really abundant on the surface ($f\approx 0.5$), TYORP can suffice to account for the descrepancy between the theory and the observations. But even if the stones of these sizes are relatively rare ($f\approx 0.1$), TYORP still must provide a major contribution to the observed YORP acceleration. If $\theta$ is not 18 as for Itokawa, but an order of magnitude less, TYORP can be an order of magnitude bigger. This could happen for slow rotators, so that the slower the rotatation becomes the bigger is TYORP to speed up the rotation. It could be the reason why slow rotators are rarely observed. If we have any asteroid of known shape, for each patch of whose surface the matelial properties of stones and their size distribution is known, the results of this article can be used to reliably estimate the TYORP torque for this asteroid. We must only add up TYORP forces produced by stones of different sizes of each patch of the surface, and then integrate these forces over the whole surface of the asteroid to get the torque. Limitatios of this method include: non-sphericity of stones; non-convexity of the overall shape of the asteroid, which will alter illuminations conditions of some patches; mutual shadowing and self-illumination of stones, which can not be precisely accounted for if the stones are not arranged in a regular pattern. Still, methods similar to the ones used in this article can be applied for any shapes and mutual distributions of stones, and illmination conditions can also be adjusted to account for shadowing of one part of the asteroid by others. This problem will depend on a very big number of free parameters. It is hard to tackle this problem in a general case, but it can be solved individually for each asteroid of interest and with enough data. And even if such detailed data are available and such a sophisticated simulation is performed, the results of this article can be useful as a simple and robust estimate of the TYORP torque. Acknowledgements ================ OG is very greateful to Dr. Anton Tkachuk for helpful discussion of numerical methods and revising Appendix A, to Dr. Glib Ivashkevych for speeding up the program, and to Prof. Cornelis P. Dullemond for discussing the algorithm implemented in the program. OG and DJS acknowledge support from NASA Grant NNX11AP24G. Numerical methods ================= In this appendix we describe in more detail the numerical algorithms used to solve Equations \[conductivity\_nonD\] and \[boundary\_nonD\]. We separate each stone into small cubes of the size $\mathrm{d}r=r/N_r$, and discretize dimensionless time $\phi$ into intervals $\mathrm{d}\phi=1/(sN_r^2)$. Here $N_r$ and $s$ are some constants, which have to be big enough to provide a good accuracy of the solution. In each timestep $\mathrm{d}\phi$ we 1) trace the incoming rays and add the energy brought by them to the surface of the stone, 2) do one step of the heat conductivity equation, 3) subtract the emitted energy from the surface and trace the outcoming rays. All simulations are done within an area of $ar\times ar$ assuming that all neighbouring stones have the same properties. 1\) We run $N_\mathrm{vis}$ rays. All rays come from the sun and thus have the same direction, but initial coordinates are random so that the rays uniformly cover the area $ar\times ar$. Each ray brings in the energy $\mathrm{d}E=-a^2r^2\sin \phi\cos \Psi\,\mathrm{d}\phi/N_{rays}$. If the ray is absorbed by the stone, the closest node to the absorption point is found, and its temperature is increased by $\mathrm{d}E/(\theta^2\mathrm{d}r^{3})$. If it is absorbed by regolith, it is instantly re-emitted from the same point, with the direction determined by Lambert’s law. If the ray leaves through the side of the simulated volume, it reappears on the other side with the same direction. These periodic boundary conditions imply periodic arrangement of similar stones in a square grid on the surface. If the ray leaves the simulated volume through the top, it stops being calculated. 2\) The heat conductivity equation is solved using the first order explicit finite-difference scheme in all three directions consequently. The scheme is chosen for its simplicity of realization and fast performance. As computation errors and shot noise from the ray tracing deteriorate the accuracy, it makes no sense to implement a more sophisticated scheme. 3\) We run $N_\mathrm{IR}$ infrared rays emitted by the stone. The emission point of each ray is chosen at random on the open surface of the stone, the direction of the ray is chosen at random in accordance with Lambert’s law, and the energy is proportional to the fourth power of the temperature of the nearest node. The temperature of this nearest node is decreased in accordance with the energy taken away by the ray. The ray is traced in the manner similar to the step (1), with the possibility of being returned back to the stone. Simultaneously with each ray we trace the ray symmetric to it with respect to the vertical axis crossing the centre of the stone. (This allows us to reduce the shot noise produced by the limited number of rays.) As the initial condition we set a uniform temperature distribution inside the asteroid, with the temperature being equal to the mean temperature at the latitude of the stone. Then we study the evolution of the temperature for $2t_\mathrm{eq}$ rotation periods. The first $t_\mathrm{eq}$ periods are not used to compute the TYORP force. They are introduced only to give enough time to the stone to forget the initial conditions. The TYORP force is computed as the average over the last $t_\mathrm{eq}$ rotation periods. We implement this algorithm in a program written in C++. The program uses only standard libraries $cmath$ and $cstdlib$. All procedures related to the ray tracing and integration of the heat conductivity equations are written from scratch. The program requires 5 physical parameters of the stones ($r$, $\theta$, $h$, $a$, $\psi$), and also 5 simulation parameters of the algorithm ($N_r$, $N_\mathrm{vis}$, $N_\mathrm{IR}$, $s$, $t_\mathrm{eq}$), which must be adjusted depending on the physical parameters and the available computation time to provide the best possible accuracy. The computation time is roughly proportional to $$t_\mathrm{comp} = C_1 t_\mathrm{eq}sN_r^2(N_r^3+0.46N_\mathrm{vis}+1.9N_\mathrm{IR}), \label{t_comp}$$ where $C_1$ is a machine-dependent constant. The first term in the brackets corresponds to the time spent to solve the heat conductivity problem, the second and the third terms correspond to tracing of incoming and outcoming light rays respectively. Decreasing any of these terms separately from the other two terms does not give any significant gain in the performance time, but often leads to significant loss of accuracy of the problem. Therefore we decide to dedicate a comparable amount of computation time to all three parts of the algorithm, and to take $$N_\mathrm{vis}=N_\mathrm{IR}=N_r^3. \label{N}$$ Then the computation time is $$t_\mathrm{comp} = C_2 t_\mathrm{eq}s N_r^5, \label{t_comp2}$$ with $C_2$ being another constant. The error of $p_x$ due to the shot noise is inversely proportional to the square root of the total number of rays emitted, $$\Delta p_x \propto \frac{1}{\sqrt{sN_r^2 N_\mathrm{IR}}}. \label{delta_p_x}$$ Equation \[N\] provides $\Delta p_x \propto t_\mathrm{comp}$. In addition to the random error $\Delta p_x$, the simulation has some systematic error due to limited resolution, and several conditions must be fulfilled for stability and good convergence of the simulation. We use these conditions to determine the best parameters for each simulation, and there is always a trade-off between all these conditions. First of all, the spatial discretization used for our simulation must be relatively fine, $$N_r \gg 1 \label{condition_N_r}.$$ Otherwise the model is too rough and gives a bad approximation of the real temperature distribution and thus the real TYORP force. Secondly, our method of solving the heat conductivity equation requires $$s \theta^2 r^2 \gg 1. \label{condition_step}$$ Our explicit time integration scheme is only conditionally stable, and when the left-hand side of Equation \[condition\_step\] is of order of unity it loses stability. Even when the numeric scheme is stable, smaller values of this left-hand side lead to loss of accuracy, that is why it is warranted to have this as big as possible. Thirdly, the time allowed for equilibrization of the temperature distribution inside the stone must be large enough to allow for the heat wave to cross the stone, $$\frac{t_\mathrm{eq}}{\theta r} \gg 1. \label{condition_t}$$ If this condition is not fulfilled, the temperature distribution inside the stone does not have enough time to reach its periodic diurnal cycle, and the obtained TYORP force can differ significantly from reality. It is also necessary that the total energy absorbed or emitted in one time step is smaller than the heat energy of the outer shell of volume elements. $$s N_r \theta^2 r \gg 1. \label{condition_T}$$ If this condition is not met, then the temperatures of the outmost volume elements oscillate largely during each timestep, deteriorating the accuracy of the obtained solution. The total number of rays is $N_\mathrm{vis} \approx N_\mathrm{IR} \approx N_r^3$, implying that each volume element at the surface of the stone absorbs and emits of order of $N_r$ rays at each timestep. As $N_r \gg 1$ from Equation \[condition\_N\_r\] we do not expect any great deviations from the mean energy absorbed or emitted, and if Equation \[condition\_T\] provides moderate temperature changes for all volume elements on average, it also provides moderate temperature differences for each volume element. But in the whole area of interest condition Equation \[condition\_T\] is weaker than Equation \[condition\_step\] and thus can be neglected. The three inequalities in Equations \[condition\_N\_r\], \[condition\_step\], and \[condition\_t\] must hold simultaneously. It is impossible to select one set of free parameters that provides these conditions in the whole area where we simulate the effect, and even if this could be done it would be overkill. So we select the simulation parameters separately for each set of physical parameters. If the computation time given by Equation \[t\_comp2\] is constant, then an increase in the left-hand side of any of the three inequalities can be attained only at the cost of a decrease in the left-hand sides of the others. We consider these inequalities to have roughly equal importance for the accuracy of the final result, and thus require the left-hand sides of Equations \[condition\_N\_r\], \[condition\_step\], and \[condition\_t\] to be equal, $$N_r = s \theta^2 r^2 = \frac{t_\mathrm{eq}}{\theta r}. \label{equal_separation_of_resources}$$ We fix the computation time, and then Equations \[N\], \[equal\_separation\_of\_resources\], and \[t\_comp2\] provide us with a system of 5 equations, from which we express the 5 simulation parameters $N_r$, $N_\mathrm{vis}$, $N_\mathrm{IR}$, $s$, and $t_\mathrm{eq}$. Thus for each point of the plot our program automatically determines the simulation parameters from the given simulation time. We prescribe the same computation time $t_\mathrm{comp}$ to all points in the plot, so that the shot noise for all the points is limited to nearly the same amount. The left-hand sides of Equations \[condition\_N\_r\], \[condition\_step\], \[condition\_t\], and \[condition\_T\] are shown in Figure \[error\]. We see that even though we try to choose the values in the first 3 panels (left-hand sides of Equations \[condition\_N\_r\], \[condition\_step\], \[condition\_t\]) equal, they are actually not. The most important reason is that $t_\mathrm{eq}$ must be integer, and thus not less than 1. This limitations causes us to allow a bigger value in the 2nd panel, and thus lower values in the 1st and 3rd panels. Almost everywhere in the plots all the parameters are bigger than 10, and go down to about 3 only in the lower left corner of the plot, where both TYORP and Yarkovsky are negligible anyway. This figure validates the applicability of our program to the simulations performed. TYORP of a 3-axial ellipsoid ============================ In this appendix we compute TYORP dimensionless torque for sinusoidal dependence of $p_x$ on latitude and then estimate the torque for $p_x$ independent of latitude. The two results are relatively close to each other, and the correct TYORP should lie somewhere between them. Let us first assume a sinusoidal latitude dependence, so that the TYORP stress at each point of the surface of an asteroid is expressed by the formula $$P=\frac{\Phi}{c} p_0 f \cos \psi, \label{app_sigma}$$ where $p_0$ is $p_x$ at the equator and $f$ is the fraction of the surface occupied by stones. Let us compute the TYORP torque experienced by the asteroid. First we consider a surface element $\mathbf{dS}=(dS_x,dS_y,dS_z)$ that has the radius-vector $\mathbf{r}=(r_x,r_y,r_z)$. The TYORP force acting on the element is tangential to the surface and perpendicular to the rotation axis $\mathbf{e}_z$ of the asteroid, therefore it is parallel to $\mathbf{e}_z\times\mathbf{dS}=(-dS_y,dS_x,0)$. (We assume the following sign convention: $P$ is positive if the force accelerates the asteroid’s rotation, and $\mathbf{e}_z$ is co-directional with the angular velocity of the asteroid.) Therefore the TYORP force acting on the surface element is $$\mathbf{dF}=P\,dS\frac{(-dS_y,dS_x,0)}{\sqrt{dS_x^2+dS_y^2}}.$$ The TYORP torque acting on the asteroid is $$\begin{aligned} T_z&=&\oint\,\mathbf{e}_z\cdot[\mathbf{r}\times\mathbf{dF}]= \nonumber \\ &=&\oint\,(0,0,1)\cdot\left[(r_x,r_y,r_z)\times P\,dS\frac{(-dS_y,dS_x,0)}{\sqrt{dS_x^2+dS_y^2}}\right]= \nonumber \\ &=&\frac{\Phi}{c} p_0 f \oint\,\cos \psi\,\frac{dS}{\sqrt{dS_x^2+dS_y^2}}(r_x dS_x+r_y dS_y)= \nonumber \\ &=&\frac{\Phi}{c} p_0 f \oint(r_x dS_x+r_y dS_y).\end{aligned}$$ The corresponding non-dimensional torque is $$\tau_z=\frac{cT_z}{\Phi r_\mathrm\mathrm{eq}^3}=\frac{p_0 f}{r_\mathrm\mathrm{eq}^3}\oint(r_x dS_x+r_y dS_y), \label{app_tau}$$ where $r_\mathrm\mathrm{eq}$ is volume-equivalent radius of the asteroid. An important consequence of this formula is that whenever we stretch the asteroid in either polar or equatorial direction, its dimensionless TYORP stays unchanged. Indeed, let say we apply the transformation $(r_x,r_y,r_z)\rightarrow(ar_x,br_y,cr_z)$. Then $(dS_x,dS_y,dS_z)\rightarrow(bc\,dS_x,ac\,dS_y,ab\,dS_z)$, the integral gets multiplied by the factor $abc$, but $r_\mathrm\mathrm{eq}^3$ gets multiplied by the same factor, and the two factors cancel. In particular, $\tau_z$ for a triaxial ellipsoid rotating around one of its major axes is the same as for a sphere. The latter is $$\begin{aligned} \tau_z&=&p_0 f\int_0^{2\pi}\,d\phi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\,d\theta(\cos \theta\,\cos \phi\cdot\cos ^2\theta\,\cos \phi+ \nonumber \\ &+&\cos \theta\,\sin \phi\cdot\cos ^2\theta\,\sin \phi)= \nonumber \\ &=&2\pi p_0 f\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\,d\theta\,\cos ^3\theta=\frac{8\pi}{3}p_0 f \approx 8.38 p_0 f. \label{app_sphere}\end{aligned}$$ From Figure \[p\_x\] we see that Equation \[app\_sigma\] is only a rough approximation. Another extreme could be to say that $P$ does not depend on the latitude at all, $$P=\frac{\Phi}{c} p_0 f, \label{app_sigma_2}$$ In this approximation the integration for a 3-axial ellipsoid can not be done that easily, but we also do not expect dimensionless TYORP $\tau_z$ to significantly depend on the shape. However, calculation of the torque for a sphere is easy, and it gives $$\begin{aligned} \tau_z=\pi^2 p_0 f \approx 9.87 p_0 f. \label{app_sphere_2}\end{aligned}$$ The coefficients in Equations \[app\_sphere\] and \[app\_sphere\_2\] are close to each other, that gives us a good reason to believe that for an asteroid of any ellipsoidal shape $\tau_z \approx 9 p_0 f$. The closeness of the results obtained for such different latitude dependencies of $P$ is not surprizing, as high latitudes give only a minor contribution to the total torque, firstly because of their small surface area, and secondly because their small lever arm. $\tau_z$ is predominantly detemined by low latitudes, where Equations \[app\_sigma\_2\] and \[app\_sigma\] are close to each other. [99]{} Bottke, W. F., Vokrouhlický, D., Rubincam, D. P., Nesvorný, D. 2006, AREPS, 34, 157 Breiter, S., Vokrouhlický, D., Nesvorný, D. 2010, MNRAS, 401, 1933 Breiter, S., Bartczak, P., Czekaj, M., Oczujda, B., Vokrouhlický, D., 2009, A&A 507, 1073 Ďurech J., Vokrouhlický D., Kaasalainen M., et al. 2008a, A&A 489, 25 Ďurech, J., Vokrouhlický, D., Kaasalainen, M., et al. 2008b, A&A 488, 345 Golubov, O., Krugly, Yu. N. 2012, ApJL, 752, 11 Fujiwara, A., Kawaguchi, J., Yeomans, D. K., et al. 2006, Science, 312, 1330 Lowry, S. C., Weissman, P. R., Duddy, S. R., Rozitis, B., Fitzsimmons, A. 2014, A&A, 562, 48 Lowry, S. C., Fitzsimmons, A., Pravec, P., et al. 2007, Science, 316, 272 Rozitis, B., Green, S. F. 2013, MNRAS, 433, 603 Rubincam, D. P., 2000, Icarus, 148, 2. Scheeres, D. J., Gaskell, R. W., 2008, Icarus, 198, 125 Scheeres, D. J., Abe, M., Yoshikawa, M., et al. 2007, Icarus, 188, 425 ![Model studied in the article. Spherical stones of radius $R$ lie on regolith. Centers of stones are a height $hR$ above the level of regolith. The distance between the stones is $aR$. Heat conductivity of the stones is characterized by the heat parameter $\theta$, the regolith is a perfect heat insulator.[]{data-label="spheres"}](spheres){width="150mm"} ![image](tau) ![Temperature in the stone and TYORP drag force as functions of time. Time is expressed in “asteroid hours”. The upper panel shows temperatures in the eastmost, top, and westmost points of the stone. Temperatures in the eastmost and westmost points differ significantly. The lower panel shows different scale we plot the time-integrated $p_x$. When the eastern part of the stone is warmer than the western part $p_x$ decreases, when it is cooler $p_x$ rises, and in the end $p_x$ reaches a positive value.[]{data-label="tau1D"}](tau1D) ![image](px) ![image](error) [c\|l]{} Notation & Meaning\ \ \[-0.1cm\] $R$ & radius of the stones\ $r$ & dimensiomless radius of the stones $R/L_\mathrm{cond}$\ $a$ & relative distance between the stones expressed in terms of the stone’s radius\ $h$ & relative height of the center of a stone above the ground in terms of the stone’s radius\ $\psi$ & latitude on the surface\ \ \[-0.1cm\] $C$ & heat capacity of the stone\ $\rho$ & density of the stone\ $\kappa$ & heat conductivity of the stone\ $\sigma$ & Stefan–Boltzmann’s constant\ $\epsilon$ & heat emissivity\ $\Phi$ & solar energy flux\ $A$ & albedo of the stone\ $\theta$ & thermal parameter\ \ \[-0.1cm\] $x_i$ & coordinates ($i$=1,2,3)\ $\xi_i$ & normalized coordinates ($i$=1,2,3)\ $t$ & time\ $\phi$ & rotation phase $\omega t$\ $T$ & temperature\ $\tau$ & normalized temperature\ \ \[-0.1cm\] $N_r$ & number of nodes along the radius\ $N_\mathrm{vis}$ & number of incoming rays per step\ $N_\mathrm{IR}$ & number of outcoming rays per step\ $s$ & parameter determining the duration of the step\ $t_\mathrm{eq}$ & number of rotation periods for equilibrization\ \[tab-var\] Source $d\omega /dt$ $\tau_z$ ----------------- ---------------------------------------------------- ------------------------- Theory: [@scheeres07] $-(2.5 \div 4.5)\times 10^{-17}$ rad s$^{-2}$ $-(0.0015\div 0.0028)$ [@durech08ito] $-(0.730 \div 3.097)\times 10^{-7}$ rad day$^{-2}$ $-(0.0006 \div 0.0026)$ [@breiter09] $-(2.5 \div 5.5)\times 10^{-7}$ rad day$^{-2}$ $-(0.0021 \div 0.0046)$ Observations: [@lowry14] $(3.54 \pm 0.38)\times 10^{-8}$ rad day$^{-2}$ $0.00029 \pm 0.00003$ TYORP (maximum) 0.002 : Predicted and observed YORP acceleration of Itokawa. The normalized YORP torque $\tau_z$ is calculated with Equation \[tau\_z\] assuming Itokawa’s principal axes to be 535, 294, and 209 metres, the mass to be $3.51\times 10^{10}$ kilograms @fujiwara06, the solar constant $\Phi=1360$ W m$^{-2}$, and the semimajor axis 1.324 AU. \[tab-itokawa\]
--- abstract: 'Microwave experiments in dilution refrigerators are a central tool in the field of superconducting quantum circuits and other research areas. This type of experiments relied so far on attaching a device to the mixing chamber of a dilution refrigerator. The minimum turnaround time in this case is a few days as required by cooling down and warming up the entire refrigerator. We developed a new approach, in which a suitable sample holder is attached to a cold-insertable probe and brought in contact with transmission lines permanently mounted inside the cryostat. The total turnaround time is 8 hours if the target temperature is 80 mK. The lowest attainable temperature is 30 mK. Our system can accommodate up to six transmission lines, with a measurement bandwidth tested between DC and 12 GHz. This bandwidth is limited by low pass components in the setup; we expect the intrinsic bandwidth to be at least 18 GHz. We present our setup, discuss the experimental procedure, and give examples of experiments enabled by this system. This new measurement method will have a major impact on systematic ultra-low temperature studies using microwave signals, including those requiring quantum coherence.' author: - 'Florian R. Ong' - 'Jean-Luc Orgiazzi' - Arlette de Waard - Giorgio Frossati - Adrian Lupascu title: Insertable system for fast turnaround time microwave experiments in a dilution refrigerator --- Introduction ============ A growing variety of experiments requires the combination of ultra-low temperatures (below 100 mK) and the application and detection of electrical signals with bandwidth as large as tens of GHz. These experiments cover a wide area of research, including quantum computing with solid-state devices [@clarke-wilhelm; @nori-review2011-2], quantum optics on chip [@nori-review2011], development of quantum limited amplifiers [@clerk2010; @muck_2001_1; @lehnert2008; @devoret-JPC2010], nanoelectromechanical resonators [@vandersant2009; @oconnell2010; @teufel_2011_Cooling], fundamental transport phenomena in mesoscopic devices [@nazarov-QT; @schoelkopf_1997_freqdepshotnoise; @gabelli_2004_HBTMeso], and broadband microwave spectroscopy in Corbino geometry [@steinbergRSI2012]. The combination of millikelvin temperatures and microwave frequencies arises naturally when studying mesoscopic physics in solid-state systems. On the one hand quantum effects typically become relevant at low temperatures. Examples include collective behaviour in superconductors [@tinkham], or the increase of the electronic coherence length beyond system size in mesoscopic systems [@imry]. On the other hand microwaves are needed to probe relevant energy scales, such as the plasma frequency in Josephson junctions [@clarke-wilhelm], or the charging energy and Zeeman splitting in semiconducting nanostructures [@nazarov-QT; @nadjperge2010]. Furthermore, lowering the temperature $T$ down to a regime where $k_{\rm B} T \ll h \nu$, where $k_B$ is the Bolzmann constant, $h$ is the Planck constant and $h \nu$ is the energy gap from the ground to the first excited state of the quantum system, enables preparation of the ground state and results in optimal coherence [@walls_1995_1]. The research areas enumerated above require that a mesoscopic device is placed in a dilution refrigerator. Preparation of the cryostat for cooldown (installation of vacuum cans and radiation shields, pumping of large volumes) and the actual cooldown to millikelvin temperature take, depending on the configuration of the system, a time varying between half a day and three days. To warm up the system and then replace the device, an additional time of a few hours to a day is needed. Combining a cold insertable probe developed by Leiden Cryogenics [@leiden-website] and a new type of sample connection system we were able to reduce these overhead times dramatically. The turnaround time, defined here as the minimum time needed to measure two successive devices at 80 mK, is reduced to 8 hours. This method has a significant impact on experiments which require multiple device testing. Experimental setup {#section-setup} ================== In this section we present the experimental setup, which builds on a cryogen-free dilution refrigerator type CF-650 and a cold-insertable probe, both available from Leiden Cryogenics [@leiden-website]. Dilution refrigerator with cold-insertable probe ------------------------------------------------ The dilution refrigerator CF-650 has the following basic characteristics. The cooling power is 650 $\mu$W for an operation temperature of 120 mK. The base temperature, without experimental wiring installed, is 12 mK. After the installation of wiring for our experiments, the lowest temperature reached at the mixing chamber is 20 mK. Three line-of-sight access ports with a 50 mm diameter run through the inner vacuum can (IVC) from the top plate down to below the mixing chamber. One of them can be used to fit a 2 meter long cold insertable probe (see Fig. \[fig1\]). The thermalization of the probe is achieved by a mechanism in which anchoring clamps attached to the successive probe stages are brought into contact with the fixed plates of the refrigerator at different temperatures. To establish contact, the anchoring clamps are moved sideways using a knob at the top of the probe. ![ \[fig1\] Section view of the CF-650 refrigerator and its insertable probe. a) General view of the system. b) Close-up view of the low-temperature parts of the probe, corresponding to the red rectangle in panel a).](fig1.pdf) The Leiden Cryogenics insertable probe comes with a loadlock chamber that can be clamped on any of the three IVC access ports (Fig. \[fig1\].a). Apart from thermometry related wires, the probe is fitted in its standard configuration with a set of twelve twisted pair wires which can be used for low-frequency electrical measurements. It is possible to add more wires. However, the addition of transmission lines is only possible to a limited extent. The reason is that a coaxial cable has a diameter of the order of millimiters for reasonably low attenuation at high frequencies. Coaxial connectors, filters and attenuators occupy an even larger space. An even more severe problem occurs for experiments involving low-noise microwave measurements, which require the installation of circulators/isolators and amplifiers. These packaged components have a bulky profile which could not fit in the space allowed by the probe. While a larger probe diameter is possible, such a design would make manipulation more difficult and also would lead to increased heat leakage. An additional problem is the fact that microwave amplifiers dissipate a significant amount of heat (typically 1-100 mW), and therefore are less efficiently thermalized when mounted on the insertable probe than when thermally anchored to one of the cold plates in the refrigerator. Motivated by the difficulties enumerated above with adding transmission lines on the cold insertable probe we introduce a method for sample insertion explained in the next subsection. Fixture for guided insertion of sample holder and coupling to microwave lines ------------------------------------------------------------------------------ ![ \[fig2\] Drawings of the mechanical assemblies, seen from two point of views. The top assembly (a) is attached to the end of the insertable probe, while the bottom assembly (b) is permanently fixed to the mixing chamber of the refrigerator. 1: cold end of probe. 2: sample holder. 3: stopper. 4: teflon guiding rods. 5: rails, height adjustable with respect to the mixing chamber. 6: SMP-SMP adapter (“bullet”). 7: guiding hole. 8: berylium copper strip. 9: SMP-SMA adapter.](fig2.pdf) Figure \[fig2\] shows a drawing of the insertion system, consisting of two mechanical assemblies. The top assembly is attached to the cold end of the insertable probe (Fig. \[fig1\].b) and thus is mobile. It holds the sample holder to be cooled down. The bottom assembly is attached to the mixing chamber of the refrigerator. The top and bottom assemblies are electrically interconnected by an arrangement of SMP adapters[@MW-components] (cf Fig.\[fig3\].a) including spring connectors (so-called bullets). The latter allow for slight misalignment without impairing microwave properties. All the metallic parts are machined from Oxygen-Free High-Conductivity (OFHC) copper for optimal thermalization. The temperature of the inserted device is measured using a calibrated 100 Ohm SPEER carbon resistor thermometer thermally anchored to the cold end of the probe. The sample holder and its fixture to the cold end are machined out of OFHC copper, ensuring good thermalization. The transmission lines used in the experiments run from the top plate of the refrigerator (where they are fed into the IVC using vacuum feedthrougs) down to the mixing chamber plate. They are mechanically attached to all the refrigerator plates for proper thermal anchoring. The lines are terminated by SMA microwave connectors plugged to the bottom mechanical assembly. ![ \[fig3\] Interfacing between insertable devices and the microwave setup. a) Close-up section of the interconnected part of the setup presented on Fig.\[fig2\], in the situation where top and bottom assemblies mate. b) Alternative to assembly used to connect sample holders not specifically designed to be mated directly onto the bottom assembly.](fig3.pdf) The top mechanical assembly is attached to the lowest stage of the cold-insertable plate, which is thermally clamped to the mixing chamber after the probe is fully inserted. This assembly carries the sample holder. The substrate (typically a silicon chip) which contains the device to be measured is connected by wire bondings to a printed circuit board (PCB) (Fig. \[fig3\].a). Both the chip and the PCB are enclosed in the sample holder. The free volume inside the sample holder is kept as small as possible to move any parasitic resonance above our measurement bandwidth (typically 20 GHz). Launching SMP connectors are soldered on the PCB. Microwave connections to the outside of the sample holder are realized with threaded SMP-SMP adapters which prevent RF leakage in and out the device’s space. The threaded SMP adapters directly mate with bullet adapters on the bottom assembly. The bullets are designed to accommodate relatively large axial and longitudinal misalignements without a significant degradation of the microwave transmission up to 40 GHz. This is important for this system where significant misalignement may occur during insertion and cooldown. The alignment of the two assemblies prior to connector mating is done using two teflon rods (diameter 7 mm) in the top assembly sliding through guiding holes (diameter 8 mm) in the bottom assembly. The holes are tapered and the teflon rods are terminated in a conical profile to enable easy reach. Coarse angular alignment is done simply using visual marks on the room temperature parts of the probe. A stopper system is designed with a screw protruding from the top assembly, whose head lies on a beryllium copper strip attached to the bottom assembly. This arrangement prevents all the weight of the probe from being entirely supported by the RF connectors when the probe is in the fully inserted position. Magnetic shielding of the experiment, which is very critical eg for work with superconducting flux qubits or squid devices, is done in the following way. A magnetic shielding system, formed of three concentric cylindrical high-magnetic permeability layers, is permanently attached to the mixing chamber plate of the dilution refrigerator. This shield surrounds the bottom mechanical assembly and enables reaching shielding factors between typically 100 and 1000, depending on the insertion depth. This configuration allows for good magnetic shielding without requiring the shielding element to be part of the sample holder, which simplifies the setup. We note that the connection between the fixed RF lines and the sample can be made in a more indirect but more versatile way than presented above, where the sample holder had to be specifically designed to mate the bottom assembly. Fig.\[fig3\].b shows the top assembly of this alternative arrangement (the bottom assembly is the same as above). The sample holder of Fig.\[fig2\] and Fig.\[fig3\].a is replaced here by a simple plate hosting threaded SMP-SMP adapters mating the bottom assembly. In contrast with the previous arrangement the upper side of these adapters is now free to be connected via coaxial cables to any sample holder that fits in the experimental space. This configuration adds more flexibility to the setup, and allows for wiring sample holders whose design prevents direct connector mating. In the following we will focus our discussion on the direct configuration (Figs. \[fig2\] and \[fig3\].a), but the results hold for the alternative configuration as well. Operation {#section-operation} ========= We describe in this section the experimental procedure to cooldown and connect a device with the insertable probe, indicating the duration of each step. We start in a state where the IVC contains exchange gas and all the plates of the refrigerator have a temperature of approximately 4 K. This temperature is maintained by running the pulse tube cooler. The dilution circuit is under vacuum and the $^3$He-$^4$He mixture is stored at room temperature. The IVC port used for probe insertion (in this case the central 50 mm line of sight port) is isolated by a manual gate valve. Once the device is wire-bonded to the PCB and enclosed in the sample holder, we attach the top assembly to the end plate of the probe using a threaded rod. The probe is attached to the IVC port and its loadlock is pumped for typically 30 min to reach a few $10^{-2}$ mBar. Then the gate valve to the IVC is opened, the sliding seal is loosened, and the probe is inserted in the IVC. Visual markers on the load lock are used to roughly align the teflon guides of the top assembly with the guiding holes of the bottom assembly, before inserting the probe all the way down and mating the RF connectors. Finally the sliding seal is tightened and the probe is brought in thermal contact with the refrigerator plates. Note that since room temperature parts are brought in contact with the plates at $\approx$ 4 K, the insertion procedure has to be done slowly and carefully, which takes approximately 15 min. Due to the considerable heat transferred to the refrigerator the temperatures of the cold plates increase up to approximately 50 K. This heat is extracted by the pulse tube cooler. After 3 hours all the IVC volume is thermalized at $\approx$ 3.8 K. Once the exchange gas in the IVC has been adsorbed by charcoals ($\approx 10$ min) the mixture can be condensed. After an additional 2h30 the coldest stage of the probe reaches 80 mK. 2h30 are further needed to reach a temperature of 40 mK. In stationary regime, the probe temperature settles typically to 10 mK above the dilution refrigerator base temperature[@note-thermalization]. Removing the probe proceeds as follows. First the $^3$He-$^4$He mixture is recovered. This step is optional, however it has the following advantages: the risk of sudden pressure increase in the dilution unit is removed and it leaves the system in a state compatible with the next use of the insertable probe. After the thermal clamps are released and the sliding seal slightly loosened, the probe is retracted in the loadlock, the IVC is isolated, and exchange gas is introduced to speed up thermalization to room temperature. The overall time to remove the probe and bring it to room temperature is 1h to 1h30. In summary it takes less than 6h30 between sample mounting and performing measurements at 80 mK, which is a temperature low enough to characterize superconducting devices involving aluminium Josephson junctions in a regime where quasiparticle poisoning is negligible [@tinkham]. The critical time is the thermalization to 4 K after insertion from room temperature, since all the cooling power is provided by a pulse tube rather than Helium vapors as in a regular dip-stick configuration. At the time we write this manuscript this operating mode has been repeated over 40 times in 8 months and the design has proven extremely robust. Actually not a single mechanical part or RF connector has needed replacement yet. As discussed in detail in the next section, the RF properties have also proven reliable and stable over time. We note that it is possible to insert/remove the probe without extracting the mixture. Maintaining the circulation has the advantage that the temperature increase during probe insertion/removal is significantly lower, which is advantageous if other experiments are being done on devices attached to the refrigerator mixing chamber plate. The disadvantage of this alternative method is the fact that probe insertion requires a more controlled thermalization procedure, by successively clamping to all the plates during insertion, which requires significantly more care. In addition, there is a more significant risk of uncontrolled pressure increase in the dilution circuit. Examples of measurements ======================== In this section we present examples of microwave measurements performed with the insertable system. We focus here on experiments related to the field of superconducting quantum devices, however we emphasize that this approach can be applied to any experiment requiring microwave frequencies and dilution temperatures. All the RF components used to build the mobile connections are rated up to at least 40 GHz. However the fixed parts of the measurement lines (SMA connectors) are specified to 18 GHz, and our microwave setup contains circulators whose maximum working frequency is 12 GHz. So in this work the tested bandwidth is DC to 12 GHz, but we expect the performances of the design to be similar up to at least 18 GHz. We first present the characterization of the assembly by measuring the transmission using a wideband through connection. Then we present transmission measurements of a coplanar waveguide (CPW) resonator. Finally we show measurements performed on a superconducting qubit coupled to a microwave resonator. Measurement of transmission using a through transmission line {#through-meas} ------------------------------------------------------------- A first characterization of the setup consists in measuring the microwave transmission of the RF interconnections, to ensure that the stacking up of RF connectors described in Section \[section-setup\] does not introduce unacceptable insertion loss or spurious resonances. For that purpose a PCB which contains a coplanar wave guide (CPW) is mounted in the sample holder. ![ \[fig4\] Characterization of the assembly. a) Setup used to measure the microwave transmission of a device. The rectangles are attenuators, the circles with arrows are isolators. LNA = Low Noise Amplifier. b) Modulus of the transmission through a coplanar waveguide PCB at $T=$ 50 mK.](fig4.pdf) The measurement setup is sketched on Fig.\[fig4\].a. A vector network analyser (VNA) sends a microwave tone of frequency $f$ from its Port 1. The signal travels through coaxial lines and attenuators down to the bottom assembly where it is connected to the sample holder. The output signal passes through two isolators, is amplified by a cryogenic Low Noise Amplifier (LNA), and reaches Port 2 of the VNA. The VNA measures the complex transmission $S_{21}$ from Port 1 to Port 2 as a function of $f$. In a preliminary characterization at room temperature, with access to the mixing chamber, we measured the transmission as sketched on Fig.\[fig4\].a and compared it to a reference measurement where the whole assembly was replaced with a coaxial cable (data not shown). In the former case $\|S_{21}(f)\|$ lies 0.1 to 1 dB below the reference and does not exhibit sudden variations or modulations. This shows that the interconnecting scheme of the assembly does not act as a significant source of loss or reflection at room temperature. Fig.\[fig4\].b shows the amplitude of $S_{21}$ measured at $T=$ 50 mK after the insertion procedure described in Section \[section-operation\]. $S_{21}$ is a smooth function of $f$, which is remarkable given the multiple interconnections involved in the setup. By calculating the attenuation and gain of the measurement line at a few frequencies spanning our measurement bandwidth (2-10 GHz), we estimate the losses added by the insertable assembly to lie within the uncertainty range ($\pm$ 2 dB) of the overall expected transmission. The connecting of the microwave connectors is very reproducible and was tested by applying the following protocol: connect the coplanar wave guide at low temperature, measure transmission, release, warm up, cool down, reconnect, remeasure. Within the uncertainty of the VNA (0.2 dB) we observe no difference on the transmission between two successive measurements of the same device. Measurement of a coplanar waveguide resonator --------------------------------------------- We now turn to the characterization of a superconducting microwave resonator, a model system in microwave engineering [@pozar] as well as a building block for various areas of physics, including photon detection devices for astronomy [@zmuidzinas2003], circuit quantum electrodynamics (circuit QED) [@blais2004; @wallraff2004], or quantum limited amplifiers [@clerk2010; @lehnert2008; @devoret-JPC2010]. The device presented here is a distributed element resonator made of a CPW whose central line is interrupted by two gaps forming capacitors and defining a cavity [@pozar], as sketched in the insert of Fig.\[fig5\].a. The device is made from aluminium (thickness 200 nm) on a silicon substrate, using a liftoff process. ![ \[fig5\] Transmission measurement of a CPW resonator. a) Full line: amplitude of $S_{21}$ over the full measurement bandwidth. Dashed line: reference transmission measured with a through PCB. Inset: sketch of the device. b) Close-up view of the complex transmission on the resonance probed at the single photon level. The red thick lines show Eq.\[S21-trans\] for the best fitted values of circuit parameters.](fig5.pdf) The measurement setup is identical to the one used in \[through-meas\] and sketched on Fig.\[fig4\].a. The complex transmission $S_{21}$ for this device is shown in Fig.\[fig5\].a over the full measurement bandwidth (full line). We observe a transmission peak around 7 GHz corresponding to the first mode of the resonator. Apart from this peak, $\|S_{21}\|$ lies well below the transmission measured previously with a through PCB and reproduced in Fig.\[fig5\].a as the dashed line. In particular $\|S_{21}\|$ does not exhibit parasitic features like box resonances or spurious capacitive coupling between ports. The absence of parasitic transmission combined to the insertion loss below 2 dB stated previously can be seen as a proof of the robustness of the design from the microwave point of view. We show on Fig.\[fig5\].b a close-up view of the first mode probed at the level of one photon populating the cavity on average when the drive is resonant. To separate the external and internal quality factors (respectively $Q_{\rm e}$ and $Q_{\rm i}$, yielding a total quality factor $Q_{\rm t} = (1/Q_{\rm e} + 1/Q_{\rm i})^{-1}$) we fit the complex transmission $S_{21}(f)$ with the transfer function: $$\tau(f) = \frac{A}{Q_{\rm e}} \frac{1}{ \frac{1}{Q_{\rm t}} + 2 j \frac{f-f_0}{f_0} } e^{j \varphi_0} \label{S21-trans}$$ where $A=1/\|S_{21}^{\rm ref}\|$ is a normalization factor obtained directly from the through measurement (dotted line), $f_0$ is the loaded resonance frequency, and $\varphi_0$ is a global phase factor. We obtain $f_0$ = 7.029 GHz, $Q_{\rm e}$ = 53,000 and $Q_{\rm i}$ = 77,000. $f_0$ and $Q_{\rm e}$ are in good agreement with the designed values, and the internal quality factor $Q_{\rm i}$ reaches a state of the art value for non epitaxial aluminium on silicon resonators at the single photon level [@sage2011]. In this subsection we only presented measurements of a CPW type resonator measured in a transmission configuration. However, the versatility of the setup allows for other types of resonators (eg lumped elements circuits, multiplexed notch resonators) measured in either reflexion or transmission configuration. Measurement of a circuit-QED device using a superconducting flux qubit ---------------------------------------------------------------------- We conclude this section with results obtained on a circuit-QED experiment [@blais2004; @wallraff2004] with a flux qubit. The device is sketched on Fig.\[fig6\].a and consists of a flux qubit [@mooij99] inductively coupled to a coplanar waveguide resonator [@abdumalikov08]. A superconducting coil attached to the sample holder enables biasing of the qubit with a DC flux $\Phi_{\rm ext}$, and a local wideband flux line placed close to the qubit is used for fast bias and drive with RF signals. ![ \[fig6\] Circuit-QED experiment a) Sketch of the device. b) Spectroscopy of the qubit. The dashed line is a fit of the qubit transition frequency. c) Relaxation (left) and Rabi oscillations (right) measurements. Dots are data, red lines are fits.](fig6.pdf) We only present here the main results and focus on typical figures of merit evaluating the quality of the electromagnetic environment that couples to the device in our setup. For the details we refer the reader to the literature covering the topic [@haroche-book; @blais2004; @wallraff2004]. Circuit-QED focuses on the coherent interaction between light and matter at the single excitation level. Coherence is extremeley sensitive to electromagnetic noise coupling to the device, which can significantly enhance the relaxation and pure dephasing rates of the qubit. Probing the coherence of a qubit is thus a way to measure the electromagnetic isolation of the whole device. First, qubit spectroscopy is performed as a function of the flux bias applied to the qubit ring. The qubit state is readout using a dispersive measurement scheme [@blais2004]: the resonator is driven at its resonance frequency and the qubit-state-dependent phase $\phi$ of the transmitted signal is measured by homodyne detection (Fig. \[fig6\].b). The spectroscopic data can be used to extract the parameters describing the flux qubit-resonator coupled system. The dashed line on Fig. \[fig6\].b is plotted for a tunneling energy $\Delta= \hbar \times$ 6.90 GHz, a persistent current $I_{\rm p}$ = 150 nA and a qubit-resonator coupling $g = \hbar \times$ 95 MHz. Next the coherence of the flux qubit is probed. In superconducting qubits, decoherence has two components of comparable importance: relaxation and pure dephasing. Fig. \[fig6\].c shows two kinds of time domain experiments allowing to estimate the coherence of a qubit: energy relaxation (left panel), and Rabi oscillations (right panel). Both experiments are performed at $\Phi/\Phi_0=0.507$. The relaxation is exponential with a time constant $T_1=(2.0 \pm 0.2)$ $\mu$s, whereas the characteristic damping time of the Rabi oscillations is $T_{\rm R} = (180 \pm 25)$ ns. These values of $T_1$ and $T_{\rm R}$ are consistent with state of the art experiments involving flux qubits away from the symmetry point [@working-point]. We thus conclude that our insertable system can be used for sensitive experiments where preserving coherence is challenging. Conclusion ========== We designed and tested a mechanical system for connection of devices to high frequency transmission lines into a running dilution refrigerator. This system is used in combination with a cold insertable probe to perform fast turnaround experiments using low noise large bandwidth electrical measurements at temperatures below 100 mK. The total time to cool a device down to 80 mK and warm it up to room temperature using this system is 8 hours, which is a major improvement over the regular mode of operation used so far in similar experiments. This system is robust: it was used over 40 times over a time period of 8 months. We showed two examples of measurements enabled by this method. The first example is a transmission measurement of a superconducting cavity. The second experiment is a study of a persistent current qubit, in which the qubit has coherence time comparable with state of the art at this time in the field. This method will be highly relevant to various experiments in the field of low-temperature physics, in particular for quantum coherence studies. We acknowledge Harmen Vander Heide, Andrew Dube and Michael Lang from Science Technical Services at University of Waterloo for help with design and realization of the mechanical assembly. We also acknowledge Mustafa Bal and Chunqing Deng for acquiring the data on qubit measurements using this setup. This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Waterloo Institute for Nanotechnology, and the Alfred Sloan Fundation. The infrastructure used for this work would not be possible without the significant contributions of the Canada Foundation for Innovation, the Ontario Ministry of Research and Innovation and Industry Canada. 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--- abstract: 'The black hole (BH) mass in the centre of AGN has been estimated for a sample of radio-selected flat-spectrum quasars to investigate the relationship between BH mass and radio properties of quasars. We have used the virial assumption with measurements of the H$\beta$ FWHM and luminosity to estimate the central BH mass. In contrast to previous studies we find no correlation between BH mass and radio power in these AGN. We find a range in BH mass similar to that seen in radio-quiet quasars from previous studies. We believe the reason that the low BH mass radio-loud quasars have not been measured in previous studies is due to optical selection effects which tend to miss the less optically luminous radio-loud sources.' author: - 'A. Y. K. N. Oshlack,' - 'R. L. Webster' - 'and M. T. Whiting' title: Black Hole Mass Estimates of Radio Selected Quasars --- Introduction ============ Although it is commonly accepted that quasars harbor a super-massive black hole (BH) in their core, neither the distribution of the physical characteristics of these BHs, nor their relationship to other properties of quasars is clear. In this paper we further explore the connection between BH mass and radio luminosity in a quasar. Recently a range of methods have been used to determine the BH masses in galaxies and quasars. For nearby galaxies, stellar velocity dispersions and high resolution optical images are used to determine the BH and bulge masses. Recently a tight correlation between BH mass and velocity dispersion in the bulge of galaxies has been found [$M_{BH} - \sigma$ relation: @geb00; @fer00]. In addition there is evidence that the mass of the bulge component of the host galaxy correlates with BH mass [@kor95; @mag98]. In AGN, it is difficult to determine BH mass using host galaxy dynamics or structure as the nucleus swamps the light from the host galaxy. Also since the redshifts are large, the projected sizes of the host galaxies are relatively small. Therefore indirect methods have been developed to measure BH mass in quasars. The mass of the BH in a quasar can be estimated using the virial technique. @lao98 assumed that the H$\beta$-emitting clouds in the broad line region (BLR) are virialized, and then required only estimates of the radius of the H$\beta$-emitting region and the rotational velocity of that region to determine the mass. In quasars, the radius of the BLR has also been estimated using reverberation techniques. This method tracks intensity variations in the continuum luminosity which, a certain time later, are reflected in the emission line flux. This time lag is interpreted as the light travel time between the core of the quasar and the broad emission line region and therefore gives the radius of the broad line clouds from the central continuum source. Reverberation mapping measurements have been restricted to a few dozen Seyferts and quasars as they require intensive regular monitoring over periods of years and highly variable quasars. The quasars so far studied may not represent the broader AGN population. Regardless of this, these observations give extremely interesting results. The time delay for different ionization species provides strong support for photoionization of the broad line region and evidence that the kinematics of the broad line region is dominated by the central BH [@pet00a]. The BLR clouds are assumed to have velocities related to the widths of the broad emission lines so if the velocities of the clouds are assumed to be Keplerian, the mass of the central BH can be deduced. Several low luminosity quasars have been studied using both reverberation techniques and galactic velocity dispersion to measure their BH mass [@fer01]. These results show that both these methods give consistent BH masses and that the M$-\sigma$ relation holds for both active and non-active galaxies. Estimates of the BH masses of radio-loud quasars have been published for several samples. [@lao00] used the Palomar-Green sample of quasars [PG; @sch83] with spectra provided by @bor92 to deduce that a relatively powerful jet requires a BH with mass $M_{BH} > 3 \times 10^8 $M$_{\odot}$. The key point Laor makes is that the BH mass does not depend on radio luminosity, but on the radio luminosity [relative]{} to the optical luminosity (denoted by the ratio $\cal R$). The PG sample comprises quasars selected for their UV-excess, and a limiting magnitude $B \lesssim 16.6$ giving a population biased towards low redshift quasars. @gu01 consider a sample of very radio-loud quasars taken from the 1Jy, S4 and S5 catalogues. They estimate the BH masses for these quasars also using the virial technique. They are able to divide their sample into steep spectrum and flat spectrum sources and find that the distribution of properties is similar for both subsets. They find 7 out of their 86 sources have BH masses $<10^8$M$_{\odot}$ unlike previous authors [eg, @lao00]. In another carefully matched sample of radio-galaxies, radio-loud and radio-quiet quasars, @dun01 show that the host galaxies of each class of AGN are nearly indistinguishable. However these authors confirm the @lao00 result that radio-loud quasars have BH masses $ \gtrsim 10^9$M$_{\odot}$. A second line of enquiry relates radio continuum emission to the mass of the BH. @lac01 used the First Bright Quasar Survey [FBQS; @whi00] in their analysis. These quasars are the optically brightest ($R\leq 17.8$), blue ($B-R \leq 2.0$) quasars in a radio sample selected at $\lambda = 20cm$. These authors obtained a best fit relation $$log L_{5GHz} = 1.9 Log M_{BH} + 1.0 log(\frac{L}{L_{Edd}}) +7.9$$ giving a continuous, monotonic, dependence of radio luminosity on BH mass. These studies used the virial estimator for BH mass. This is similar to the result by @fra98 who found a very steep dependence of BH mass on radio flux, $P_{5GHz} \propto M_{BH}^{2.5}$, and attributed this to BH accretion in an advection dominated accretion flow. Most recently @ho01 has compiled a complete list of BH measurements which use robust methods to determine mass (stellar dynamics, reverberation mapping or maser dynamics). He finds that the BH mass shows little dependence on $\cal R$ and that the radio luminosity correlates poorly with BH mass. We have been studying the properties of the Parkes Half-Jansky flat-spectrum Sample [PHFS; @dri97], to better understand the emission mechanisms particularly at optical wavelengths. We find that the optical emission in radio quasars is a mixture of several different components: a blue component similar to the ‘big blue bump’ in optically-selected quasars, which is presumed to photoionize the BLR, plus a synchrotron component which turns over somewhere in the optical-to-IR and is a significant contributor to the optical emission in a large fraction of sources [@whi01]. Two major differences between our sample and those already published are (1) our sample is completely identified in the optical, and we have not selected optically bright sources, and (2) our sources are flat spectrum, which suggests that the radio flux may be be boosted by orientation effects. In section \[meth\], we discuss the method used for estimation of BH mass from measured optical parameters. In section \[data\], we describe our application of these techniques to the PHFS dataset. Results and discussion are given in section \[disc\]. Where required, a cosmology with $H_0 =75 $ km s$^{-1}$ Mpc$^{-1}$ and q$_0=0.5$ is assumed. Method {#meth} ====== When the motions of the gas moving around the BH are dominated by the gravitational forces, virial estimates of the BH mass can be made based on the radius and velocity of this gas using $M_{BH}=rv^{2}G^{-1}$. There is evidence that the BLR gas emitting the H$\beta$ line is virialized [@pet00a]. To estimate BH masses for this sample, we follow the work of @kas00 who used reverberation mapping to derive an empirical relationship between radius and luminosity. To determine $v$, the rotational velocity, we correct full-width half-maximum $v_{FWHM}$ of the H$\beta$ emission line by a factor of $\sqrt{3}/2$ to account for random orientation affecting the width of emission lines. The assumption of random orientation is discussed further in Section \[bias\]. The mass is then $$M = 1.464 \times 10^{5} \left(\frac{R_{BLR}}{\text{ lt-days}}\right)\left(\frac{v_{FWHM}}{10^{3}{\rm km~s}^{-1}}\right)^{2} M_{\odot}$$ The empirical relation of @kas00 relates the radius of the emission lines to the continuum luminosity of the source: $$R_{BLR} = (32.9)\left[\frac{\lambda L_{\lambda}(5100 {\text \AA})}{10^{44} {\text erg~s}^{-1}}\right]^{0.700} \text{lt-days}$$ with an error in the fitted slope of this relation of $<5\%$. This correlation holds above a luminosity of $10^{44}$ erg s$^{-1}$. Below this value the correlation is less certain and at low luminosity most values of the emission line region radius lie well below the linear relation, which would lead to an even lower estimate of the BH mass. Although the slope of this relation cannot be explained theoretically, the radius of the BLR is approximately linear with the strength of the radiation field ionizing the BLR gas. Therefore, to use the H$\beta$ BH mass estimator, two measurements are involved. Firstly, the luminosity at 5100Å and secondly, the H$\beta$ FWHM. PHFS Data {#data} ========= The data used in the analysis are taken from the PHFS [@dri97]. The selection criteria for the sample are: (1) Radio loud: 2.7GHz flux $>$ 0.5 Jy; (2) Flat spectrum taken from the 2.7 GHz and 5 GHz fluxes: $\alpha > -0.5$, where $S_\nu \propto \nu^\alpha$; (3) Dec range +10 to $-$45 and galactic latitude $\|$b$\| > 10^{\circ}$. The selection criteria produced 323 radio sources where optical identification has been made for 321 sources. Therefore the sample has no optical selection criteria. The flat spectrum criterion may mean that sources where the radio jets are predominantly oriented towards us are preferentially selected. Most of the radio morphologies are compact. For the PHFS, estimating the optical luminosity of the quasar is difficult as there is a very large range in colour for the sources [@fra00]. This means we cannot use a general K-correction for all the objects. In the initial plots we will use the B$_j$ magnitude to calculate the luminosity. The error in the B$_j$ magnitudes from the COSMOS catalogue is quoted as being $\pm 0.5$ magnitudes, but @dri97 have found that some fields seem to be in error by more than one magnitude. This produces an error in the BH mass estimates of about a factor of 2. Given that the range of masses covers 4 orders of magnitude, these errors are not critical. In section \[bias\] we only consider the subset of quasars which have more accurate magnitudes. The radio flux of the sources given in @dri97 comes from the Parkes catalogue (PKSCAT90, @wri90) at 5GHz. It should also be noted that there is a large range in B$_j$ magnitudes, including very faint sources which have luminosities in the Seyfert regime. 187 sources in the sample have low to moderate resolution spectra (discussed in @fra01). Of these 101 were obtained at the AAT and Siding Spring 2.3m telescopes [@dri97], and 86 are from the compilation of @wil83. The quality and wavelength coverage of the spectra is diverse; many have low resolution and signal-to-noise. Estimates of the likely errors are further discussed in the next section. All spectra with a wavelength range covering H$\beta$ were used in the analysis. This gave a list of 39 quasars. Each H$\beta$ emission line was fitted by a Gaussian and Lorentzian using the [splot]{} package in IRAF. If H$\beta$ was outside the observed wavelength range but H$\alpha$ was observed then the fit was done for H$\alpha$. The Lorentzian fits were better approximations to the line shapes by eye. The Gaussian fits consistently underestimated the peak flux; therefore Lorentzian fits had consistently lower FWHMs. In the analysis we used the Lorentzian fits for measurements of the line widths. The errors in the estimation of $v_{FWHM}$ are of the order of 10-20%. Where there was both the H$\beta$ and H$\alpha$ lines measured we calculated the BH mass using both values for the FWHM. The results of this is shown in Figure \[HaHb\]. It can be seen that the H$\alpha$ widths give consistently lower BH masses, on average, 0.157 dex lower. Therefore a correction of $10^{0.157}$ was made to all BH masses estimated using H$\alpha$. The radio loudness parameter $\cal R$ is defined by ${f_{\nu}(5GHz)}/ f_{\nu}$(5100Å). A value of ${\cal R} = 10$ is usually taken as the break in a bimodal distribution, dividing radio-loud and radio-quiet. All the PHFS sources are clearly radio-loud with $10^2 < {\cal R} < 10^5$. Table 1 gives the observed parameters for each source. Results and Discussion {#disc} ====================== Considerable care must be exercised in searching for correlations between observed and derived quantities, particularly when the plotted quantities both depend in a similar way on distance. As an example, in a radio flux-limited sample, most quasars have a flux near the sample limit. Therefore two quantities which depend on distance will be correlated, including BH mass calculated from the virial estimator described above, which uses the absolute luminosity. Correlations with Black Hole Mass --------------------------------- Figure \[bh\_R\] plots $\cal R$, the ratio of radio to optical flux, against the BH mass of each quasar. Contrary to some previous results but consistent with @ho01, we find a large range in derived BH masses. The apparent anti-correlation here is a consequence of using the optical flux in the measurement of BH mass and also using it in the calculation of $\cal R$. A significant fraction of the masses are well below limit of $3\times 10^{8} M_{\odot}$ suggested by @lao00 as a necessary condition for radio-loud quasars. In fact, the range in BH masses is similar to the radio-quiet population of the PG quasar sample [@lao00]. For 8 of the AGN the host galaxies are visible but it is clear from figure \[bh\_R\], where these galaxies are marked with a ‘g’ or ‘G’, that these are not the only low BH mass. In these cases we expect the quasar luminosity to be over-estimated, due to the inclusion of galaxy flux. The likely increase in flux is $<50\%$ [@masci98]. Only two of the sources have redshifts $<0.1$. A histogram of the BH mass is shown in Figure \[bhmass\]. The low values for the BH mass are a direct consequence of the low optical luminosities, with the range in the width of the H$\beta$ emission line introducing some scatter. This dependence is clearly shown in Figure \[bh\_opt\] where the optical luminosity is plotted against the BH mass, and there is no evidence of a lower limit in either optical luminosity or BH mass for any of these radio-loud quasars. Figure \[bh\_rad\] shows the relationship between radio power and BH mass. There is no clear relationship between these variables unlike the results of @fra98 and @lac01. The fact that we are not seeing a relationship between BH mass and radio power is a consequence of the fact that there isn’t a tight correlation between optical luminosity and radio luminosity in quasars [@ste00; @wad99]. This has been investigated by @hoo96 for the LBQS who find that the distribution of radio luminosity does not depend on absolute magnitude over most of the range of M$_B$. Furthermore the PG survey used by @lao00 differs from this and other optically selected samples in its population of radio-loud quasars with respect to optical luminosity [@hoo96]. The PG sample contains a higher fraction of radio-loud quasars which have bright absolute magnitudes compared to other surveys like the LBQS [@hoo96]. If we consider Figure 2 in @ho01, then our data points lie in the underpopulated upper left region of this plot. This confirms Ho’s conclusion that there is no obvious relationship between these two variables. However we note that our sources are flat spectrum and the radio-flux may be boosted by beaming which would tend to increase the value of $\cal R$. Also included in Figure \[bh\_rad\] is a line of constant observed flux. This is included to demonstrate the correlation that would be observed for objects with the same measured flux (eg: the flux limit of the sample where the majority of object will be selected) at the different redshifts of the sources. Possible Biases in Estimates of Parameters {#bias} ------------------------------------------ There are two measurements involved in deriving the BH mass for any object while using this method: firstly the optical flux, to derive the radius to the BLR and secondly the velocity width, $v_{FWHM}$ used for the velocity component in the virial assumption. There are several factors which might affect the applicability of the virial method to the quasars we are considering. 1. All the sources in this sample are flat spectrum. One might speculate that the radio emission in particular is beamed, increasing the radio flux relative to the optical. The effect of the beaming would be that the points in Figure \[bh\_R\] are at higher $\cal R$ values than their non-beamed values, but this should be independent of BH mass unless the latter also effects the Lorentz factor of the jet. Interestingly, in a recent preprint, @gu01 have shown using a similar sample to the PHFS, that steep and flat spectrum quasars have similar distributions of BH mass compared to radio luminosity and radio loudness $\cal R$. 2. The spectral energy distributions (SEDs) for radio-loud quasars vary greatly, making a uniform K-correction invalid. This means that for quasars at different redshifts we are sampling different parts of the spectrum by using the $B_j$ magnitude. To investigate this effect, we can use 24 of the quasars with measured velocity profiles which also have quasi-simultaneous photometry in seven bands covering the optical and near IR [@fra00]. Four of these have evidence for dust reddening which is discussed further in item 4. The original BH mass distribution of these 20 unreddened quasars is shown in Figure \[all\_hist\]a. Using the simultaneous photometry we can now linearly interpolate between these SED data points to obtain an accurate flux at a rest wavelength of 5100Å. The histogram of the results is shown in Figure \[all\_hist\]b. The range in BH masses has decreased, and the lower limit proposed by @lao00 is even more strongly violated. 3. It has been shown that the synchrotron radiation from the jet can extend into the near IR and optical [@whi01]. This will cause an increase in the observed continuum flux but, since the jet emission is beamed, would not contribute to the ionizing flux seen by the BLR. @whi01 have fitted models to the quasi-simultaneous photometry that consist of both a synchrotron component and a blue powerlaw (representing the ionizing flux from the accretion disk). From these fits we can calculate the fraction of the total emission that comes from the accretion disk. The accretion disk flux will be less than the observed flux if there is a significant synchrotron contribution present in the optical. The results of this correction are shown in Figure \[all\_hist\]c where some of the objects with a large synchrotron component now have much lower BH masses. Nearly all the quasars are below $10^9$M$_\odot$. 4. Dust may affect the observed luminosity of the quasar. Dust will tend to redden the spectrum of the quasar and reduce the observed luminosity. The dust extinction is modeled as an exponential function of wavelength with flux at the blue wavelengths depleted significantly more than the red. For quasars where we have simultaneous photometry, evidence of dust is shown by a steep decrease in flux at higher energies. For the subset of quasars that have photometry we see evidence of dust in the SEDs of four of the quasars. We have not used these quasars in the previous analysis as their flux will be underestimated. After removing these, we still see the very large range in BH masses including low mass BHs. 5. The H$\beta$ line might be emitted from a disk-like structure, with its line width reflecting the rotation of that disk [@wil86]. Then, since it is expected that flat spectrum sources are viewed face-on, the width of the H$\beta$ line may be underestimated. In the @gu01 sample the median and mean values for their flat spectrum sample are lower than the steep spectrum sample. This data partially supports the flattened disk scenario (see Table \[tab2\]). However it cannot be the whole story as the widths of the emission lines of the flat spectrum quasars are still a significant fraction of the steep spectrum ones and, the flat-spectrum sources need only to be oriented at an average angle of 53$^{\circ}$ to the line-of-sight to reconcile to difference in the median values of FWHM. If we use the @gu01 difference, this introduces a mean difference of a factor of 2 in the BH mass. 6. Contamination of the AGN by the host galaxy light will effect all AGN to some extent, however it is obviously more significant for the objects where the host galaxy is resolved optically. The flux measured is the integrated flux over the entire galaxy. The central source flux which is required to calculate the BH mass, is much smaller. This effect will be present for all objects, but where the galaxy is resolved it will be the most pronounced as the host galaxy is a bigger percentage of the total observed flux. The result will be that the actual BH mass, for the AGN, is quite a bit smaller than calculated, so these masses are upper limits. The FWHM of the emission lines may also be effected by the host galaxy if it has strong H$\alpha$ or H$\beta$ emission lines. As we haven’t resolved the broad and narrow components of the emission, the FWHM may be under-estimated due to a higher peak coming from the host galaxy emission. This is more likely to be a factor in the H$\alpha$ emission lines. The resolution of the spectra was not sufficient to consider contamination by weaker nearby emission lines such as and . It is noted that the FWHM velocity of the H$\beta$ line, after deconvolution, would need to increase by a factor of 10 to be consistent with the @lao00 finding - an unlikely result. Optical Properties of Quasars ----------------------------- The results suggest that BH mass is not connected to the radio-loudness of the objects. The question then arises: Why have studies in the past produced a different result? Figure \[bh\_opt\] shows that the range in values that we calculate for the BH mass is largely a consequence of using the optical luminosity to estimate the radius of the BLR. The velocity width of the lines has a much smaller effect on the BH mass distribution due to its smaller dynamic range. Therefore, the reason that previous studies using this method have missed the low BH mass radio-loud objects is because they have missed the low luminosity radio-loud objects. We believe that optical selection is biased against selecting low optical luminosity, radio-loud quasars as shown in the PG sample [@hoo96]. The selection criterion for the low redshift PG quasars used by @lao00 are: (1) UV excess (U$-$B$ < -0.44$). (2) morphological criteria where the objects must have a ‘dominant star-like appearance’. They are also bright with a limiting magnitude $\sim$16.6. Therefore we obtain a sample of bright, blue, point-like objects. The key difference in the PHFS and the sample used by @lao00 is that the PHFS has a large range in optical luminosity which produces the large distribution in BH mass whereas the radio-loud quasars selected on UV excess tend to have the characteristic of being optically luminous which therefore gives large BH masses. From the PHFS it is evident that not all radio-loud quasars are highly optically luminous. Figure \[L\_colour\] provides evidence of an additional correlation which might effect selection into quasar samples. In figure \[L\_colour\], which plots optical luminosity against colour (B-I), low luminosity quasars are shown to be redder. Thus any additional selection criterion based on colour will further bias against low luminosity quasars. The photometry of these red quasars indicates that there are three types of reddening taking place. We find that a proportion of these are red due to synchrotron extending into the optical part of the spectrum and therefore boosting the red end of the continuum. This has been discussed and corrected for in section \[bias\]. This is an effect that will only be evident in radio-loud quasars with high energy synchrotron jets and will not be observed in radio-quiet quasars. Another subset of the redder objects in the PHFS have a substantial fraction of galactic light included in the photometry so the 4000Å break contributes to the red colour. This occurs mainly in the low redshift population which is a small subsample of our estimated BH masses. Most of the sources classified as ‘g’ or ‘G’ in Figure \[bh\_R\] do not have a significant galactic contribution in their spectra (for example, a significant 4000Å break), and are obviously AGN due to their strong radio loudness and broad emission lines. A smaller proportion of the sample are very faint and show some evidence of dust reddening where the blue end of the continuum is strongly suppressed. In this case the BH masses will be underestimated but the previous two effects lead to overestimation of the BH mass. Using the simultaneous photometry we have identified 4 out of 24 dust-reddened objects in the sample and have excluded them in the analysis shown in Figure \[all\_hist\]: we still get a population of quasars with relatively low BH mass. Why do red quasars have low optical luminosity, and therefore (under our assumptions), low BH mass? We have argued that a significant fraction of red quasars have synchrotron turning over at optical wavelengths [@whi01]. This seems to be correlated with lower optical emission which could be interpreted in two ways. Either, the synchrotron emission remains unchanged but appears dominant only when the blue disk emission is low, or secondly, when the blue disk emission is low, relatively more energy goes into the jet and the synchrotron is increased. We do not have a large enough sample to establish the correlation between low luminosity and redness unequivocally, but if the correlation could be substantiated it might provide physically interesting constraints on the jet formation mechanism. Theoretical calculations considered by @mei99 demonstrate explicitly that it is not necessary to have a relatively massive BH to produce powerful radio jets. In his model, based on current accretion and jet-production theory, the strength of the radio emission is tied to the rotation rate of the central BH. Therefore, it is quite possible to have highly powered radio jets with small BH masses if the BH is spinning rapidly. Conclusion ========== The virial method for estimating BH mass in conjunction with the empirical relation between luminosity and radius [@kas00] has been applied to the PHFS sample of flat spectrum radio sources. This is the first time the analysis has been applied to an optically complete radio-selected sample. The analysis is complimentary to the @ho01 analysis which concentrates on optically luminous sources used in longterm monitoring programs. The errors and applicability of this method have been discussed. The main result are summarized as follows: 1. We find no evidence for a lower cut off in the masses of radio-loud quasars as previously suggested in the literature. 2. The previous lower boundary on mass of radio-loud quasars was an optical selection effect. Low luminosity radio-loud AGN have been shown to have redder optical colours. These red quasars will be selected against in optical surveys. 3. We do not find any evidence for a relationship between radio properties and the BH mass of quasars. Further investigation of this relationship is hampered by unknown orientation effects. To obtain more accurate BH estimates and improve our understanding of the relationship between the mass of the central BH and other properties of quasars, we need large samples of BH masses measured using the reverberation technique. This is now observationally possible using multi-fibre spectrographs. We wish to acknowledge the anonymous referee for careful reading of the manuscript and valuable comments which lead to the improvement of this paper. Boroson, T.A. & Green, R.F. 1992, , 80, 109 Dietrich, M. et al. 1993, , 408, 416 Drinkwater, M. J., Webster, R. L., Francis, P. J., Condon, J. J., Ellison, S. L., Jauncey, D. L., Lovell, J., Peterson, B. A., & Savage, A. 1997, , 284, 85 Dunlop, J. S., McLure, R. J., Kukula, M. J., Baum, S. A., O’Dea, C. P. & Hughes, D. H., submitted to , astro-ph/0108397 Ferrarese, L. & Merritt, D. 2000, , 539, L9 Ferrarese, L., Pogge R. W., Peterson B. M., Merrit D., Wandel A. & Joseph C. L. 2001, , 555, L79 Franceschini, A., Vercellone, S., & Fabian, A.C. 1998, , 297, 817 Francis, P. J., Whiting, M. T., & Webster, R. L. 2000, Publications of the Astronomical Society of Australia, 17, 56 Francis, P. J., Drake, C. L., Whiting, M. T., Drinkwater, M. J., & Webster, R. L. 2001, Publications of the Astronomical Society of Australia, 18, 221 Gebhardt, K. et al. 2000, , 539, L13 Gelderman, R., & Whittle, M., 1994, , 91, 491 Gu, M., Cao, X., & Jiang, D. R. 2001, , 327, 1111 Ho, L. C. 2002, , 564, 120. Hooper, E. J., Impey, C. D., Foltz, C. B., & Hewett, P. C. 1996, , 473, 746 Kaspi, S., Smith, P. S., Netzer, H., Maoz, D., Jannuzi, B. T., & Giveon, U. 2000, , 533, 631 Kellerman, K. I., Sramek, R., Schmidr, M., Shaffer, D. B., Green, R., 1989, , 98, 1195 Kormendy, J. & Richstone, D. 1995, , 33, 581 Lacy, M., Laurent-Muehleisen, S. A., Ridgway, S. E., Becker, R. H., & White, R. L. 2001, , 551, L17 Laor, A. 1998, , 505, L83 Laor, A. 2000, , 543, L111 Magorrian, J. et al. 1998, , 115, 2285 Masci, F. J., Webster, R. L., & Francis, P. J. 1998, , 301, 975 Meier, D. L. 1999, , 522, 753. Peterson, B. M. & Wandel, A. 2000, , 540, L13 Remillard, R. A., Grossan, B., Bradt, H. V., Ohashi, T. & Hayashida, K., 1991 , 350, 589 Schmit, M., & Green, R.F., 1983 , 269,352 Stern, D., Djorgovski, S. G., Perley, R. A., de Carvalho, R. R., & Wall, J. V. 2000, , 119, 1526 Wadadekar, Y. & Kembhavi, A. 1999, , 118, 1435 White, R. L. et al. 2000, , 126, 133 Whiting, M. T., Webster, R. L., & Francis, P. J. 2001, , 323, 718 Wilkes, B. J., Wright, A. E., Jauncey, D. L., & Peterson, B. A. 1983, Proceedings of the Astronomical Society of Australia, 5, 2 Wills, B. J. & Browne, I. W. A. 1986, , 302, 56 Wright, A. & Otrupcek, R. 1990, Parkes Catalog. Astralia Telescope National Facility, Epping (PKSCAT90) (113) \[tab1\] Name z $B_j$ $\lambda L_{\lambda}$ (ergs s$^{-1}$) 5GHz flux (Jy) P$_{rad}$ (5GHz) (W/Hz) $\cal R$ H$\beta$ FWHM ($km s^{-1}$) H$\alpha$ FWHM ($km s^{-1}$) BH mass (M$_\odot$) ---------------- ------- ------- --------------------------------------- ---------------- ------------------------- ------------------- ----------------------------- ------------------------------ --------------------- PKS 0114$+$074 0.343 22.14 1.05$\times 10^{43}$ 0.67 1.74$\times 10^{26}$ 1.11$\times 10^5$ 2515 ... 6.30$\times 10^6$ PKS 0153$-$410 0.226 19.41 5.41$\times 10^{43}$ 0.94 1.02$\times 10^{26}$ 1.26$\times 10^4$ 3385 1932 3.59$\times 10^7$ PKS 0221$+$067 0.510 20.76 8.75$\times 10^{43}$ 0.77 4.66$\times 10^{26}$ 3.59$\times 10^4$ 2118 ... 1.97$\times 10^7$ PKS 0327$-$241 0.888 19.39 1.03$\times 10^{45}$ 0.73 1.48$\times 10^{27}$ 9.63$\times 10^3$ 4007 ... 3.96$\times 10^8$ PKS 0454$+$066 0.405 19.79 1.30$\times 10^{44}$ 0.44 1.63$\times 10^{26}$ 8.39$\times 10^3$ ... 2128 3.77$\times 10^7$ PKS 0502$+$049 0.954 18.7 2.28$\times 10^{45}$ 0.82 1.94$\times 10^{27}$ 5.73$\times 10^3$ 4184 ... 7.53$\times 10^8$ PKS 0912$+$029 0.427 19.56 1.80$\times 10^{44}$ 0.46 1.90$\times 10^{26}$ 7.10$\times 10^3$ 2678 3152 5.23$\times 10^7$ PKS 0921$-$213 0.052 16.4 4.25$\times 10^{43}$ 0.42 2.23$\times 10^{24}$ 3.53$\times 10^2$ ... 7238 1.99$\times 10^8$ PKS 0925$-$203 0.348 16.35 2.25$\times 10^{45}$ 0.70 1.87$\times 10^{26}$ 5.61$\times 10^2$ 2611 2217 2.90$\times 10^8$ PKS 1016$-$311 0.794 17.58 4.28$\times 10^{45}$ 0.65 1.03$\times 10^{27}$ 1.62$\times 10^3$ 3416 ... 7.79$\times 10^8$ PKS 1020$-$103 0.196 15.07 2.19$\times 10^{45}$ 0.49 3.93$\times 10^{25}$ 1.21$\times 10^2$ 7920 6138 2.62$\times 10^9$ PKS 1034$-$293 0.312 15.94 2.60$\times 10^{45}$ 1.51 3.21$\times 10^{26}$ 8.30$\times 10^2$ 3463 ... 5.65$\times 10^8$ PKS 1036$-$154 0.525 21.8 3.57$\times 10^{43}$ 0.78 5.03$\times 10^{26}$ 9.47$\times 10^4$ 5216 3671 6.38$\times 10^7$ PKS 1101$-$325 0.355 16.45 2.14$\times 10^{45}$ 0.73 2.04$\times 10^{26}$ 6.42$\times 10^2$ 3135 2949 4.04$\times 10^8$ PKS 1106$+$023 0.157 18.01 9.22$\times 10^{43}$ 0.50 2.54$\times 10^{25}$ 1.85$\times 10^3$ ... 2632 4.52$\times 10^7$ PKS 1107$-$187 0.497 22.44 1.76$\times 10^{43}$ 0.50 2.86$\times 10^{26}$ 1.09$\times 10^5$ ... 2361 1.14$\times 10^7$ PKS 1128$-$047 0.266 21.41 1.21$\times 10^{43}$ 0.90 1.37$\times 10^{26}$ 7.63$\times 10^4$ ... 2186 7.52$\times 10^6$ PKS 1136$-$135 0.557 16.3 6.43$\times 10^{45}$ 2.22 1.63$\times 10^{27}$ 1.70$\times 10^3$ 2275 2731 4.60$\times 10^8$ PKS 1200$-$051 0.381 16.42 2.55$\times 10^{45}$ 0.46 1.49$\times 10^{26}$ 3.94$\times 10^2$ 2361 1803 2.59$\times 10^8$ PKS 1226$+$023 0.158 12.93 1.01$\times 10^{46}$ 40.0 2.05$\times 10^{27}$ 1.37$\times 10^3$ 4040 ... 1.98$\times 10^9$ PKS 1237$-$101 0.751 17.46 4.23$\times 10^{45}$ 1.13 1.58$\times 10^{27}$ 2.52$\times 10^3$ 5392 ... 1.93$\times 10^9$ PKS 1254$-$333 0.190 17.05 3.31$\times 10^{44}$ 0.54 4.06$\times 10^{25}$ 8.25$\times 10^2$ 7800 ... 6.78$\times 10^8$ PKS 1302$-$102 0.286 15.71 2.68$\times 10^{45}$ 1.0 1.77$\times 10^{26}$ 4.45$\times 10^2$ 3417 2439 5.61$\times 10^8$ PKS 1352$-$104 0.332 17.6 6.43$\times 10^{44}$ 0.98 2.37$\times 10^{26}$ 2.49$\times 10^3$ 2814 ... 1.40$\times 10^8$ PKS 1359$-$281 0.803 18.71 1.55$\times 10^{45}$ 0.67 1.09$\times 10^{27}$ 4.72$\times 10^3$ 1889 ... 1.17$\times 10^8$ PKS 1450$-$338 0.368 22.52 8.61$\times 10^{42}$ 0.54 1.63$\times 10^{26}$ 1.27$\times 10^5$ ... 1824 4.14$\times 10^6$ PKS 1509$+$022 0.219 19.83 3.44$\times 10^{43}$ 0.54 5.46$\times 10^{25}$ 1.07$\times 10^4$ 6551 ... 9.80$\times 10^7$ PKS 1510$-$089 0.362 16.21 2.78$\times 10^{45}$ 3.25 9.46$\times 10^{26}$ 2.29$\times 10^3$ 2796 ... 3.86$\times 10^8$ PKS 1546$+$027 0.415 18.54 4.34$\times 10^{44}$ 1.42 5.53$\times 10^{26}$ 8.56$\times 10^3$ 3893 2831 2.04$\times 10^8$ PKS 1555$-$140 0.097 16.99 8.77$\times 10^{43}$ 0.83 1.57$\times 10^{25}$ 1.20$\times 10^3$ ... 2001 2.53$\times 10^7$ PKS 1706$+$006 0.449 22.8 1.02$\times 10^{43}$ 0.38 1.75$\times 10^{26}$ 1.16$\times 10^5$ ... 2086 6.08$\times 10^6$ PKS 1725$+$044 0.296 18.2 2.90$\times 10^{44}$ 1.21 2.30$\times 10^{26}$ 5.33$\times 10^3$ 2638 2399 7.07$\times 10^7$ PKS 1954$-$388 0.626 17.82 2.04$\times 10^{45}$ 2.0 1.89$\times 10^{27}$ 6.21$\times 10^3$ 3293 ... 4.32$\times 10^8$ PKS 2004$-$447 0.240 18.09 2.07$\times 10^{44}$ 0.65 7.96$\times 10^{25}$ 2.59$\times 10^3$ 1939 1602 3.01$\times 10^7$ PKS 2059$+$034 1.012 17.64 6.90$\times 10^{45}$ 0.75 2.02$\times 10^{27}$ 1.97$\times 10^3$ 3815 ... 1.36$\times 10^9$ PKS 2120$+$099 0.932 20.16 5.65$\times 10^{44}$ 0.5 1.13$\times 10^{27}$ 1.34$\times 10^4$ 3083 ... 1.54$\times 10^8$ PKS 2128$-$123 0.499 15.97 6.88$\times 10^{45}$ 2.0 1.16$\times 10^{27}$ 1.13$\times 10^3$ 4652 4220 2.02$\times 10^9$ PKS 2143$-$156 0.698 17.24 4.41$\times 10^{45}$ 0.82 9.80$\times 10^{26}$ 1.49$\times 10^3$ 836 ... 4.78$\times 10^7$ PKS 2329$-$415 0.671 18.2 1.67$\times 10^{45}$ 0.47 5.15$\times 10^{26}$ 2.07$\times 10^3$ 4952 ... 8.49$\times 10^8$ : Observational Data for PHFS Quasars and Estimated BH Mass [p[3.3cm]{}p[2.9cm]{}p[2.9cm]{}]{} & mean (km s$^{-1}$) & median (km s$^{-1}$)\ Flat Spectrum & 3632 & 3600\ Steep Spectrum & 5240 & 4500\
--- author: - 'Helen Au-Yang and Jacques H.H. Perk${}^{1,2}$' date: title: 'Wavevector-Dependent Susceptibility in $\myb Z$-Invariant Pentagrid Ising Model ' --- by 7pt Introduction {#sect1} ============ In an experiment [@SBGC] done in 1984, Shechtman and his coworkers found fivefold symmetry in the diffraction patterns of some rapidly cooled alloys. As such a symmetry is incompatible with lattice periodicity, it was concluded that the crystalline structures of these alloys, if any, must necessarily be quasiperiodic. This theoretical explanation came forward almost immediately, as Penrose, de Bruijn, and Mackay [@Penrose; @Pen0; @Pen1; @Bruijn1; @Mackay1; @Mackay2] had already studied tilings that have fivefold symmetry, well before this experimental discovery. Quasiperiodic tilings are types of almost periodic structures that permit sharp peaks in the diffraction patterns, but have normally forbidden symmetries [@Mackay2; @LSt0; @BGM; @CJ; @Henley; @Baake]. Already in 1986, Korepin [@K1] introduced a $Z$-invariant eight-vertex model on Penrose tiles. The $Z$-invariant inhomogeneous models are completely integrable[@BaxZI] even on irregular lattices and their critical exponents are known to be the same as those of homogeneous systems on regular lattices. Thus, the critical behaviors of these quasiperiodic $Z$-invariant models[@BaxZI; @K2; @AK1; @AK2; @BGB; @GBS; @GB; @Choy] have to be the same,[^1] independent of the lattice structure.[^2] In such models, the order parameter is the same [@BaxZI; @K1] for all sites and it vanishes towards the critical point. Therefore, the Fourier transform of the one-point function of a $Z$-invariant Ising model is the product of this order parameter and the lattice sum $\sum{\rm e}^{{\rm i}{\bf q}\cdot{\bf r}}$. Experiments that probe the resulting “magnetic" Bragg peaks are restricted to the low-temperature phase and the corresponding theory is essentially the zero-temperature theory, well-studied in the literature. [@Mackay2; @LSt; @SoSt] The aforementioned Bragg peaks will broaden, if we allow the underlying quasicrystalline lattice to become distorted by lattice vibrations.[@LSSBH; @Hof] However, we shall not consider this possibility in this paper, as we assume the underlying lattice to be a perfect and rigid Penrose tiling, restricting our attention solely to the ordering of the spins at the lattice sites under thermal fluctuations. Contrary to the above theory for the Bragg scattering, the situation is far more complicated for scattering experiments that probe the pair-correlation function $\langle\sigma_{\bf r}\sigma_{\bf r'}\rangle$ via the wavevector-dependent susceptibility $\chi({\bf q})$. This last quantity is defined as $$k_{\rm B}T\chi({\bf q})\equiv\bar\chi({\bf q})= \lim_{{\cal L}\to\infty}{\frac 1 {{\cal L}}} \sum_{\bf r}\sum_{\bf r'} {\rm e}^{{\rm i}{\bf q}\cdot({\bf r'}-{\bf r})} \big[{\langle\sigma_{\bf r}\sigma_{\bf r'}\rangle}- \langle\sigma_{\bf r}\rangle\langle\sigma_{\bf r'}\rangle\big], \label{chi}$$ where ${\cal L}$ is the number of lattice sites, ${\bf r}$ and ${\bf r}'$ run through all these sites, and ${\bf q}=(q_x,q_y)$. It is the Fourier transform of the connected pair correlation function, defined by what is inside the square brackets. Furthermore, $\bar\chi({\bf q})$ is the reduced ${\bf q}$-dependent susceptibility, taking out a trivial factor involving the absolute temperature $T$. In the well-known lattice-gas language it becomes proportional to the structure function, the Fourier transform of the density-density correlation function, which is also measurable in diffraction experiments, revealing the symmetry of the lattice. Thus, the $\chi({\bf q})$ in $Z$-invariant models can indeed be used to show the difference between quasiperiodic and regular lattices and we expect it to provide a diffuse scattering pattern both above and below the critical temperature $T_{\rm c}$, with more structure closer to $T_{\rm c}$. There is good reason to pick a $Z$-invariant Ising model for our present study of $\chi({\bf q})$. It is taken from the foremost class of models with short-range interactions allowing exact computations. Moreover, comprehensive extensive studies on spin-spin correlation functions in nontrivial models with short-range interactions have been done only for Ising models[@KO49; @Fisher59a; @MPW63; @MWbk; @WMTB76; @Ab78b; @Perkd; @BaxZI; @AP-ZI]. Cited here is just a fraction of the literature. The accumulative knowledge of these studies has made the calculations in $Z$-invariant quasi-periodic Ising models possible. Correlations in other nontrivial cases are still mostly inaccessible to exact methods of evaluation. For instance, in 1988, Tracy [@Tr1; @Tr2] introduced the layered Fibonacci Ising model, which is not $Z$-invariant. The row correlation functions in the layered Ising model are known to be block Toeplitz determinants[@AYM], of which we still have no idea how to evaluate them exactly in general, except in a few simpler cases[@AYM; @McWu67a]. Tracy has shown that the critical exponent of the specific heat remains unchanged in such quasiperiodic layered models. In our previous papers [@AJPq; @APmc1], we have studied Fibonacci Ising models whose spins are on regular lattices, but whose nearest-neighbor interactions are quasi-periodic. They are special cases of inhomogeneous Ising models whose Hamiltonians are given by $$-\beta{\cal H}\,=\sum_{m,n}\,({\bar K}_{m,n}\sigma_{m,n}\sigma_{m,n+1} + K_{m,n}\sigma_{m,n}\sigma_{m+1,n}), \label{energy}$$ with $\beta\equiv1/k_{\rm B}T$. When the system is periodic, the pair correlations are translationally invariant. Thus one of the sums in (\[chi\]) can be carried out. For $T\ne T_{\rm c}$, the connected correlations decay exponentially as functions of distance.[^3] Therefore, only a finite number of short-distance correlation functions are needed for the calculation of the $\chi({\bf q})$ in Eq. (\[chi\]). For the Ising models considered previously[@AJPq; @APmc1], the couplings between specific nearest-neighbor spin pairs form Fibonacci sequences $\{S_{n}\}$ defined recursively [@Tr1] by $$S_{n+1}=S_nS_{n-1},\quad S_0={\rm B},\quad S_1={\rm A},$$ so that $S_2={\rm AB}$, $S_3={\rm ABA}$, $S_4={\rm ABAAB}$ and so on. Since this sequence is quasiperiodic, as arbitrary long subsequences are repeated infinitely often, the model is also aperiodic. Consequently, the correlations are no longer translationally invariant. However, the averages of the correlations for two spins at fixed distance can be evaluated by using a theorem of Tracy[@Tr1]. In our previous works,[@AJPq; @APmc1] we have found that the various ${\bf q}$-dependent susceptibilities $\chi({\bf q})$ of our Fibonacci Ising models are always periodic. They can have multiple incommensurate everywhere-dense peaks in each unit cell only if the aperiodic oscillations in the average correlation functions are not negligibly small. This is true in the mixed case when the interactions are aperiodic sequences of ferromagnetic and antiferromagnetic couplings. The number of visible peaks in $\chi({\bf q})$ increases as the correlation length increases. In contrast, in the periodic Fibonacci Ising lattices with mixed bonds, the visible peaks in the ${\bf q}$-dependent susceptibility are at commensurate positions and their number has a finite maximum.[@AJPq] The ferromagnetic aperiodic Fibonacci Ising lattice, on the other hand, behaves almost like the regular Ising model—one peak per unit cell, located at the commensurate position—because the aperiodic oscillations in its average correlation functions are negligibly small at all temperatures.[@AJPq] The ${\bf q}$-dependent susceptibility $\chi({\bf q})$ is found to have almost the same behaviors for both $T<T_{\rm c}$ and $T>T_{\rm c}$.[@AJPq] This is in agreement with the result of Peter Stephens, who showed that the randomized (disordered) icosahedral system [@Stephens] gives almost the same diffraction pattern as a quasicrystal—which is in the solid (ordered) phase. Even though both ferromagnetic and antiferromagnetic edge interactions are present in the mixed case, the mixed systems in Refs. and are not frustrated. In fact, their partition functions are equal to the partition functions of the ferromagnetic models, from which they differ by gauge transformations of signs. Also, the $\chi({\bf q})$ of a fully-frustrated model may not show incommensurate peaks. [@Kong] In this paper, we turn our attention to systems with a quasiperiodic lattice structure[@BGB; @GBS; @GB]. More specifically we study a $Z$-invariant Ising model whose spins are on vertices of a Penrose fat-and-skinny rhombus tiling with the pentagrid as its rapidity lines[@Bruijn1; @K1; @K2; @AK1; @AK2]. Outline ------- This paper is organized as follows. In Section \[sect2\], we introduce a $Z$-invariant Ising model on Penrose tiles constructed from a pentagrid[@Bruijn1], with spins on either the odd or even sublattice. We show how the pair-correlation functions can be evaluated in Section \[sect3\]. Section \[sect4\] is rather lengthy. In it, a new method of counting all the spin sites and evaluating the joint probabilities of the occurrence of two neighborhoods of two spins is given. In subsection \[sect51\], we describe in detail the calculation for the ${\bf q}$-dependent susceptibility for the odd lattice. The results are given in subsection \[sect52\]. In subsection \[sect53\] we present a mapping between the odd and the even sublattices. Finally, in Section \[sect6\] we present our conclusions. Pentagrid and Penrose Tiles {#sect2} =========================== In two ingenious papers by the famous Dutch mathematician N.G. de Bruijn, he relates the non-periodic Penrose tilings in a plane to a pentagrid[@Bruijn1], which is a superposition of five grids. Each grid consists of parallel lines with equal spacings between the lines; the grids may be obtained from one another by rotations of angles which are multiples of $2\pi/5$. This is shown in Fig. \[fig1\]. Here some grid lines of the pentagrid are shown; the arrows on the lines shown in Fig. \[fig1\] should be ignored for the moment, as they will define the direction of the “rapidities" which we define later. =0.88 0.2in To describe the pentagrid in mathematical formulas[@Bruijn1], let $$\zeta={\rm e}^{2{\rm i}\pi/5},\qquad \zeta+\zeta^{-1}=2\cos(2\pi/5)=p^{-1}={{\textstyle \frac 1 2}}(\sqrt 5-1), \label{goldratio}$$ in which $p$ is the golden ratio. Then, choose $\gamma_0,\gamma_1,\gamma_2,\gamma_3,\gamma_4$ to be five real numbers, satisfying $$\gamma_{0}+\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}=0. \label{shift}$$ Now the $j$th grid in the pentagrid consists of lines given by $$G_j=\{z\in \mbox{\mymsbm C}\| {\rm Re} (z\zeta^{-j}) +\gamma_j=k_j, k_j\in \mbox{\mymsbm Z}\},\qquad j=0,\cdots,4. \label{grid}$$ The pentagrid is called regular, if there is no point in the complex plane $\mbox{\mymsbm C}$ belonging to more than two of the five grids. This also means, every vertex of the regular pentagrid is an intersection of no more than two lines. Each vertex is surrounded by four meshes (which are often called faces in physics). Now to every point $z$ in the complex plane $\mbox{\mymsbm C}$, de Bruijn associates an integer vector ${\vec K}(z)=(K_0(z),\cdots,K_4(z))$ whose five elements are integers given by $$K_j(z)=\lceil{\rm Re} (z\zeta^{-j}) +\gamma_j\rceil, \label{mesh}$$ in which $\lceil x\rceil$ denotes the “roof of $x$", which is the smallest integer $\geqslant x$. It is easily seen from (\[mesh\]) and (\[grid\]) that whenever $z$ moves across a line of the $j$th grid, $K_j(z)$ changes by 1. All points in the same mesh (face) have the same integer vector and the integer vectors of different meshes are different. From Fig. \[fig1\], we may already see that some fraction of the meshes (or faces) becomes infinitesimally small in size as the number of lines in each grid becomes infinite. Since every vertex of the regular pentagrid is surrounded by four meshes (faces), by assigning to each of their four corresponding integer vectors ${\vec K}(z)$ $=$ $(K_0(z),\cdots,K_4(z))$ a complex number $$f(z)=\sum_{j=0}^4 K_j(z)\zeta^j, \label{penrose}$$ these four meshes are now mapped to the vertices of a rhombus. More specifically, to the intersection of two grid lines $k_r$ and $k_s$, ($r\ne s$), one assigns a rhombus in $\mbox{\mymsbm C}$ whose vertices are the four complex numbers $f(z)$, $f(z)+\zeta^r$, $f(z)+\zeta^s$ and $f(z)+\zeta^r+\zeta^s$ assigned to the four surrounding meshes. Clearly, there are two different kinds of rhombuses: the thick one having angles $72^{\circ}$ and $108^{\circ}$ for $r=s\pm1$, and the thin one having angles $36^{\circ}$ and $144^{\circ}$ for $r=s\pm2$. In both rhombuses, all sides have length 1. They are shown in Fig. \[fig2\]. =0.8 to 0.3in In these two papers [@Bruijn1], de Bruijn also showed that, even though there are many different choices of $\gamma_j$ in (\[shift\]), many of the resulting pentagrids are shift-equivalent, that is, they can be obtained from each other by a parallel shift.[^4] We now assign to each grid line in the grid $j$, $j=0,\cdots,4$, of the pentagrid a rapidity $u_j$ pointing into the upper half plane, as is shown in Figs. \[fig1\] and \[fig3\]. In Fig. \[fig3\], it is also shown how the five grid lines $k_j=0$, for $j=0,\cdots,4$, are shifted to make the grid regular, again following de Bruijn [@Bruijn1]. The usual $Z$-invariant Ising model [@BaxZI; @AP-ZI] is formed by putting spins inside the meshes, but here, however, the Ising spins are on the vertices of the rhombuses. There is an one-to-one mapping given by (\[penrose\]) relating a mesh in the pentagrid to a vertex of the Penrose tiling. After the mapping, the grid lines in the pentagrid—which are also the rapidity lines—become “Conway worms" (no longer straight) in the Penrose tiling [@Gardner]. Since the rapidity lines are used to define commuting transfer matrices, they do not have to be straight lines. =0.5 -0.0in The model so defined is a special case of the inhomogeneous $Z$-invariant eight-vertex model proposed by Korepin[@K1; @K2; @AK1; @AK2]. More specifically, the four-spin couplings are identically zero, and the eight-vertex model decomposes into two independent Ising models. The interactions of the Ising spins are along the diagonals of the rhombuses. The odd and even sublattices are therefore decoupled.[@Wu; @KW] As in the earlier works [@BaxZI; @AP-ZI], the coupling ${K}(u_i,v_j)$ between two spins is represented by a line connecting these two spins, with the arrows of the two rapidity lines $u_{i}$ and $v_{j}$ on the same side of this line, as shown in Fig. \[fig4\]$\,$(a), while the line representing the coupling ${\bar{K}}(u_i,v_j)$ has the arrows of the two rapidity lines $u_{i}$ and $v_{j}$ on opposite sides, as shown in Fig. \[fig4\]$\,$(b). The edge interactions are parametrized by $$\begin{aligned} &&\sinh\big(2{K}(u_i,v_j)\big)= k\,{\rm sc}(u_i-v_j,k')={\rm cs}\big(\lambda+v_j-u_i,k'\big), \nonumber\\ &&\sinh\big(2{\bar{K}}(u_i,v_j)\big) ={\rm cs}(u_i-v_j,k')=k\,{\rm sc}\big(\lambda+v_j-u_i,k'\big). \label{couplings}\end{aligned}$$ Here $\lambda={\rm K}(k')$ is the elliptic integral of the first kind, and $k$ and $k'=\sqrt{1-k^2}$ are the elliptic moduli; these are convenient temperature variables, assumed to be the same for all sites. 0.05in=0.50 0.1in From Fig. \[fig2\], we can see that the lengths of the four diagonals of the two rhombuses are different. The interactions between the spins are chosen to depend on the interparticle spacings only, but not on the orientations. Consequently, we must have $$u_{0}-u_{1}=u_{2}-u_{3}=u_{4}-u_{0} =\lambda+u_{1}-u_{2}=\lambda+u_{3}-u_{4}.$$ From this, we find $$u_{4}-u_{1}=\frac{4\lambda}{5},\quad u_{2}-u_{1}=\frac{3\lambda}{5},\quad u_{0}-u_{1}=\frac{2\lambda}{5},\quad u_{3}-u_{1}=\frac{\lambda}{5}. \label{rapidity}$$ If we let $$s_{j}=k\,{\rm sc}(j\lambda/5,k'),$$ then for the thick rhombus in Fig. \[fig2\]$\,$(a), we assign $s_2$ to the longer diagonal and $s_3$ to the shorter diagonal, while for the thin rhombus in Fig. \[fig2\]$\,$(b), $s_4$ to the shorter diagonal and $s_1$ to the longer one. Thus, to the four types of diagonals are assigned four kinds of couplings according to their lengths, with a stronger coupling for a shorter interparticle distance. The two Ising sublattices on the Penrose tiling are indicated in Fig. \[fig5\]. The edges in the even sublattice are omitted. There are eight types of vertices S, K, Q, D, J, S3, S4, S5 in the Penrose tiling, which are shown in Fig. 7 of Ref. . The coordination numbers of spins in the Penrose Ising model are 3 for types Q and D; 4 for K; 5 for S, J and S5; 6 for S4; 7 for S5. 0.1in=0.76 0.15in Correlations {#sect3} ============ Two spins $\sigma_{\bf r}$ and $\sigma_{\bf r'}$ at two different vertices of the Ising lattice just defined have different integer vectors ${\vec K}=(K_0,\cdots, K_4)$ and ${\vec K'}=(K'_0,\cdots, K'_4)$. Since there is a one-to-one mapping between the vertices of the Penrose tiles and the meshes of the pentagrid, the integer vectors can be used to denote the positions of the spins: ${\bf r}\leftrightarrow{\vec K}$. Since each grid consists of parallel lines with equal spacings, the absolute value of the difference $\ell_{j}=K'_j-K_j$ is actually the number of the $j$th kind of rapidity lines sandwiched between these two spins. The correlation functions were shown[@BaxZI; @AP-ZI] to be $$\begin{aligned} &&\hspace{-3.7em}\langle\sigma_{\vec K}\sigma_{\vec K'}\rangle= \langle\sigma\sigma'\rangle_{[\ell_0,\cdots,\ell_4]}\nonumber\\ &&=g(\overbrace{u'_0,\ldots,u'_0}^{\|\ell_0\|}, \overbrace{u'_1,\ldots,u'_1}^{\|\ell_1\|}, \overbrace{u'_2,\ldots,u'_2}^{\|\ell_2\|}, \overbrace{u'_3,\ldots,u'_3}^{\|\ell_3\|}, \overbrace{u'_4,\ldots,u'_4}^{\|\ell_4\|}), \label{cor}\end{aligned}$$ where $u'_j=u_j$ for rapidity lines of type $j$ with arrows pointing to the same side of the line joining the two spins, and $u'_j=u_j\pm\lambda$ for rapidities with arrows pointing to opposite sides of the line. It is as if the rapidities of lines pointing to the opposite side need to be flipped by adding $\pm\lambda$, i.e. adding $\pm\pi$ to the angle variable $u_j\pi/\lambda$.[@AP-ZI] The functions $g$ have both the “permutation symmetry" (which means that they are invariant under all permutations of the rapidities) and the “difference property" (which implies a translation invariance under shifting all the rapidities by the same amount).[@BaxZI] We next examine in more detail when we have to choose $u'_j=u_j$ or $u'_j=u_j\pm\lambda$ in (\[cor\]). Through a point in the mesh where spin $\sigma$ sits, as shown in Fig. \[fig1\], we draw five dashed lines parallel to each of the five grids. As the choice of the point in the mesh is rather arbitrary, the dashed lines should really have been drawn with a finite thickness, i.e. open strips without their boundaries between two consecutive grid lines of the pentagrid. Five arrows are also drawn perpendicular to the dashed lines to indicate the directions in which the integers $k_j$ increase. If the other spin $\sigma'$ is in a mesh crossed by the $j$th dashed line, i.e. $\sigma$ and $\sigma'$ lie within the same strip, then the two spins have the same $K_j$ ($\ell_j=0$) and their pair correlation function does not depend on the value of $u'_j$. When $\sigma'$ moves away from this dashed line in the direction of the arrow, we have $\ell_j\geqslant 0$, whereas $\ell_j\leqslant 0$ if $\sigma'$ moves away in the direction opposite to the arrow. These dashed lines (or more precisely strips) divide the entire plane into ten regions, and we numbered them from I to X. From Fig. \[fig1\], we can see that $(\ell_0,\cdots,\ell_4)$ have the same signs inside each region. If $\sigma'$ is in regions III or VIII, we find from Fig. \[fig1\] that the arrows of all the rapidity lines point to the same side of the line joining the two spins. Thus $u'_j=u_j$ for all five $j$-values. When $\sigma'$ is in regions II or VII, then the rapidity lines with $u_4$ and the other rapidity lines are pointing to opposite sides of the line joining the spins, so that $u'_4=u_4-\lambda$ and $u'_j=u_j$ for $j\ne 4$. For regions IV and IX the $u_1$ rapidity lines point in the other direction with respect to the other rapidity lines, implying $u'_1=u_1+\lambda$. If $\sigma'$ is in regions I and VI, the arrows of the $u_2$ and $u_4$ rapidity lines are on the opposite side and, therefore, $u'_4=u_4-\lambda$ and $u'_2=u_2-\lambda$. Similarly, for regimes V and X, $u'_1=u_1+\lambda$ and $u'_3=u_3+\lambda$. For all other $j$-values, $u'_j=u_j$. In summary, our choices for $u'_j$ are listed in Table \[raptab\], Regions $u'_4$ $u'_2$ $\;u'_0\;$ $u'_3$ $u'_1$ -------------- --------------- --------------- ------------ --------------- --------------- I and VI $u_4-\lambda$ $u_2-\lambda$ $u_0$ $u_3$ $u_1$ II and VII $u_4-\lambda$ $u_2$ $u_0$ $u_3$ $u_1$ III and VIII $u_4$ $u_2$ $u_0$ $u_3$ $u_1$ IV and IX $u_4$ $u_2$ $u_0$ $u_3$ $u_1+\lambda$ V and X $u_4$ $u_2$ $u_0$ $u_3+\lambda$ $u_1+\lambda$ : Rapidities \[raptab\] with the $u_j$’s given in (\[rapidity\]), where we may set $u_1=0$ without loss of generality in view of the difference property of the pair correlation function.[@BaxZI] We may even shift the five rapidity values $u'_0,\ldots,u'_4$ in (\[cor\]) by the same amount, depending on the choice of region, such that $\min_j u'_j=0$. We can then also use the permutation property[@BaxZI] of the pair-correlation function $g$, given in (\[cor\]), to rearrange the five resulting rapidity values $u'_j$ in decreasing order as $\frac{4}{5}\lambda,\frac{3}{5}\lambda,\frac{2}{5}\lambda, \frac{1}{5}\lambda,0$. Therefore, it suffices to calculate the quantity $$\begin{aligned} &&\hspace{-3.7em}g[m_4,m_3,m_2,m_1,m_0]\equiv\nonumber\\ &&g\bigg( \overbrace{\frac{4\lambda}{5},\!\ldots\!,\frac{4\lambda}{5}}^{m_4}, \overbrace{\frac{3\lambda}{5},\!\ldots\!,\frac{3\lambda}{5}}^{m_3}, \overbrace{\frac{2\lambda}{5},\!\ldots\!,\frac{2\lambda}{5}}^{m_2}, \overbrace{\frac{\lambda}{5},\!\ldots\!,\frac{\lambda}{5}}^{m_1}, \overbrace{0,\ldots,0\vphantom{\frac{\lambda}{5}}}^{m_0}\bigg), \label{corems}\end{aligned}$$ where the $m_j$’s are nonnegative integers depending on the $\ell_j$’s and the choice of region. To determine this dependence, we can first use (\[rapidity\]), from which we find $u_3+\lambda>u_1+\lambda>u_4>u_2>u_0>u_3>u_1>u_4-\lambda>u_2-\lambda$. Comparing (\[cor\]), Table \[raptab\] and (\[rapidity\]) we can then express all pair correlations in the form (\[corems\]). We list the results for the ten different regions in Table \[corrtab\]. This completes, more or less, the calculation of the pair correlation function, as we can refer to our previous papers[@APmc1; @APmc2] for further details on how to evaluate the $g[m_4,m_3,m_2,m_1,m_0]$ defined in (\[corems\]). Regions Signs of $(\ell_0,\ell_1,\ell_2,\ell_3,\ell_4)$ $\langle\sigma\sigma'\rangle_{[\ell_0,\cdots,\ell_4]}=$ ------------ ------------------------------------------------- --------------------------------------------------------- I & VI $(+,+,+,-,-)$ & $(-,-,-,+,+)$ $g[\|\ell_0\|,\|\ell_3\|,\|\ell_1\|,\|\ell_4\|,\|\ell_2\|]$ II & VII $(+,+,-,-,-)$ & $(-,-,+,+,+)$ $g[\|\ell_2\|,\|\ell_0\|,\|\ell_3\|,\|\ell_1\|,\|\ell_4\|]$ III & VIII $(+,+,-,-,+)$ & $(-,-,+,+,-)$ $g[\|\ell_4\|,\|\ell_2\|,\|\ell_0\|,\|\ell_3\|,\|\ell_1\|]$ IV & IX $(+,-,-,-,+)$ & $(-,+,+,+,-)$ $g[\|\ell_1\|,\|\ell_4\|,\|\ell_2\|,\|\ell_0\|,\|\ell_3\|]$ V & X $(+,-,-,+,+)$ & $(-,+,+,-,-)$ $g[\|\ell_3\|,\|\ell_1\|,\|\ell_4\|,\|\ell_2\|,\|\ell_0\|]$ : Pair-Correlation Function \[corrtab\] Since the pentagrid is invariant under rotations by angles that are integer multiples of $2\pi/5$, the grids may be relabeled $m \to m+j$, (mod 5). Then the differences of the five integer vectors of the two spins are also relabeled $\ell_{m}\to\ell_{m+j}$. This shows that the pair correlation function must have the cyclic property $$\langle\sigma\sigma'\rangle_{[\ell_0,\ell_1,\ell_2,\ell_3,\ell_4]}= \langle\sigma\sigma'\rangle_{[\ell_j,\ell_{j+1},\ell_{j+2}, \ell_{j+3},\ell_{j+4}]},\quad(\mbox{mod 5}). \label{cyclic}$$ From Table \[corrtab\], we indeed find that this property holds. Enumeration of sites {#sect4} ==================== As mentioned in the introduction in the context of the Fibonacci Ising model,[@AJPq; @APmc1] to calculate the ${\bf q}$-dependent susceptibility (\[chi\]) for lattices for which the correlations are not translationally invariant, one needs to find a way to calculate suitable averages of the pair-correlation function. Since the meshes in the pentagrid—and even the distances between them—can be infinitesimally small in the thermodynamic limit, the problem of counting all the spin sites must be first solved. We proceed by considering in detail the parallelograms bounded by two sets of parallel grid lines in the pentagrid and examining all possible spin sites in each of these parallelograms. Let $P(k_j,k_{j+1})$ denote the parallelogram sandwiched between four grid lines $k_j-1$, $k_j$, $k_{j+1}-1$ and $k_{j+1}$ for any $j$. (Throughout the entire paper, we let $k_{j+5}\equiv k_j$, i.e. the index $j$ is considered mod 5.) Obviously, for all points $z\in P(k_j,k_{j+1})$, we have $K_j(z)=k_j$ and $K_{j+1}(z)=k_{j+1}$. The different choices of $j$ give the different orientations of the parallelograms. Next, we determine how many spin sites a parallelogram may have, what are the integer vectors for these spins, etc. Since this section is rather lengthy, we have subdivided it into many parts, and put the main conclusion at the end. Reference vector for $\myb {P}$ ------------------------------- The vertices of parallelogram $P(k_j,k_{j+1})$ can be calculated from (\[grid\]) as the intersections of grid lines in grids $G_j$ and $G_{j+1}$, i.e. $$G_j\cap G_{j+1}=\biggl\{z\in \mbox{\mymsbm C}\biggm\| z=\frac{\,{\rm i}\,[\zeta^j(k_{j+1}\!-\!\gamma_{j+1})- \zeta^{j+1}(k_j\!-\!\gamma_j)]} {\sin(2\pi/5)}\biggr\}, \label{inters}$$ for $k_{j},k_{j+1}\in\mbox{\mymsbm Z}$. Moreover, any point $z$ in the interior of $P(k_j,k_{j+1})$ may be expressed in terms of $\myb{\epsilon}=(\epsilon_j,\epsilon_{j+1})$, with $0\leqslant\epsilon_j,\epsilon_{j+1}\leqslant 1$, as $$z=\frac{\,{\rm i}\,[\zeta^j(k_{j+1}\!-\!\gamma_{j+1}\!-\!\epsilon_{j+1})- \zeta^{j+1}(k_j\!-\!\gamma_j\!-\!\epsilon_j)]}{\sin(2\pi/5)} \equiv z(\myb{\epsilon}), \label{zinP}$$ allowing us a change of notation $K_{j+{\rm m}}(\myb{\epsilon})\equiv K_{j+{\rm m}}(z(\myb{\epsilon}))$ for $z\in P$. The four corners of parallelogram $P(k_j,k_{j+1})$ are given by $\myb{\epsilon}\!=\!(0,0),$ $(0,1),$ $(1,0),$ or $(1,1)$ as can be seen from (\[inters\]). Now for each $P(k_j,k_{j+1})$, we pick a reference integer vector $(k_0,\cdots,k_4)$, which is related to the integer vector of the corner of $P(k_j,k_{j+1})$ with $\myb{\epsilon}\!=\!(0,0)$. Apart from the obvious identities $k_j=K_{j}(\myb{0})$ and $k_{j+1}=K_{j+1}(\myb{0})$, we have $$\begin{aligned} k_{j+2}&=&K_{j+2}(\myb{0})\nonumber\\ &=&\lceil p^{-1}(k_{j+1}-\gamma_{j+1})-k_j+\gamma_j+\gamma_{j+2}\rceil =\lceil\alpha\rceil-k_j,\nonumber\\ k_{j+4}&=&K_{j+4}\left(\myb{0}\right)\nonumber\\ &=&\lceil p^{-1}(k_j-\gamma_j)-k_{j+1}+\gamma_{j+1}+\gamma_{j+4}\rceil =\lceil\beta\rceil-k_{j+1}, \label{kj12}\end{aligned}$$ in which $$\begin{aligned} &&\alpha\equiv\hat\alpha(k_{j+1})\equiv p^{-1}(k_{j+1}-\gamma_{j+1})+\gamma_j+\gamma_{j+2},\nonumber\\ &&\beta\equiv\hat\beta(k_j)\equiv p^{-1}(k_j-\gamma_j)+\gamma_{j+1}+\gamma_{j+4}. \label{alpha}\end{aligned}$$ However, for the last component of the reference integer vector we choose $$k_{j+3}=2-\lceil\alpha\rceil-\lceil\beta\rceil =-\lfloor\alpha\rfloor-\lfloor\beta\rfloor\not\equiv K_{j+3}(\myb{0}), \label{kj3}$$ where $\lfloor x\rfloor$ denotes the “floor of $x$", which is the largest integer $\leqslant x$ and $\lfloor x\rfloor=\lceil x\rceil$ if and only if $x\in \mbox{\mymsbm Z}$. Since the pentagrid is regular, we find $\alpha,\beta\notin\mbox{\mymsbm Z}$ and the second equality in the above equation holds. The index of any mesh, whose integer vector is ${\vec K}(z)$, is defined as $\sum_j K_j(z)$. It is shown by de Bruijn[@Bruijn1] that it has one of the four possible values, 1, 2, 3, or 4. We associate odd spins to meshes with index 1 or 3, and even spins to meshes with index 2 or 4. The index of the reference integer vector is $\sum_j k_j=2$. From (\[mesh\]) and (\[zinP\]) we find $$K_{j+3}(\myb{0})= \lceil-p^{-1}(k_{j+1}-\gamma_{j+1}+k_j-\gamma_j)+\gamma_{j+3})\rceil =\lceil-\alpha-\beta\rceil. \label{k3}$$ Using $$\{x\}=x-\lfloor x\rfloor,\quad \lceil-x-y\rceil= -\lfloor x\rfloor-\lfloor y\rfloor+\lceil-\{x\}-\{y\}\rceil \label{frac}$$ and comparing (\[kj3\]) with (\[k3\]), we find $$K_{j+3}(\myb{0})=\cases{\begin{array}{llr} k_{j+3}-1 &\hbox{for}& \{\alpha\}+\{\beta\}\geqslant 1,\\ k_{j+3} &\hbox{for}& \{\alpha\}+\{\beta\}<1.\upstrut\end{array}} \label{dkj3}$$ This shows the mesh below the upper right corner with $\myb{\epsilon}=(0,0)$ belongs to the even sublattice and its integer vector is the reference vector of $P$ only for $\{\alpha\}+\{\beta\}<1$, but not for $\{\alpha\}+\{\beta\}\geqslant 1$. It would be more natural to choose this corner as our reference and to compare the integer vectors of other spins inside $P\equiv P(k_j,k_{j+1})$ with it. This was what we did originally. However, we find that the rather odd choice of the reference vector given by (\[kj12\]) and (\[kj3\]), which may not even be the integer vector of a mesh, has made calculations much simpler. We next examine the differences between the integer vectors of the other spins in $P(k_j,k_{j+1})$ with respect to this reference vector. Integer vectors for $\myb {z\in P}$ ----------------------------------- Substituting (\[zinP\]) into (\[mesh\]) and using (\[kj12\]) and (\[kj3\]), we evaluate the integer vectors $\vec{K}(\myb{\epsilon})$ for every point in $P$ that is not on a grid line of the pentagrid. We find that its components for ${\rm m}=2,3,4$ are given by $$\begin{aligned} &&K_{j+{\rm m}}(\myb{\epsilon})= k_{j+{\rm m}}+\partial K_{j+{\rm m}}(\myb{\epsilon}),\nonumber\\ &&\partial K_{j+{\rm m}}(\myb{\epsilon})= \lceil\lambda_{j+{\rm m}}(\myb{\epsilon})-1\rceil= \lfloor\lambda_{j+{\rm m}}(\myb{\epsilon})\rfloor, \label{kz}\end{aligned}$$ with $$\lambda_{j+2}(\myb{\epsilon}) =\{\alpha\}+\epsilon_j-p^{-1}\epsilon_{j+1},\quad \lambda_{j+4}(\myb{\epsilon})=\{\beta\}+\epsilon_{j+1}-p^{-1}\epsilon_j. \label{dk2}$$ and $$\lambda_{j+3}(\myb{\epsilon})=p^{-1}(\epsilon_j+\epsilon_{j+1})-\{\alpha\} -\{\beta\}+1. \label{dk3}$$ The last equality in (\[kz\]) does not hold if $\lambda_{j+{\rm m}}(\myb{\epsilon})$ is an integer, i.e. if the point is on a grid line of grid $G_{j+{\rm m}}$. If we define the difference vector $\partial\vec{K}(\myb{\epsilon})$ for each mesh in $P$ as[^5] $$\partial\vec{K}(\myb{\epsilon})=\bigl[\partial K_{j+2}(\myb{\epsilon}),\partial K_{j+3}(\myb{\epsilon}),\partial K_{j+4}(\myb{\epsilon})\bigr], \label{dv}$$ then for $\{\alpha\}+\{\beta\}>1$, we have $\partial\vec{K}(\myb{0})=[0,-1,0]$, and for $\{\alpha\}+\{\beta\}<1$, $\partial\vec{K}(\myb{0})=[0,0,0]$. For fixed $\gamma_j$’s with $j=0,\cdots,4$, which are the shifts of the pentagrid, the $\hat\alpha(k_{j+1})$ and $\hat\beta(k_{j})$ in (\[alpha\]) are uniquely determined for each $P(k_j,k_{j+1})$. Consequently, the number of meshes in $P$ and the difference vectors $\partial\vec{K}$ for each mesh are uniquely determined by (\[kz\]) to (\[dk3\]). The configurations of two parallelograms $P$ and $P'$ are the same, if they have the same number of meshes (spin sites), and the same sets of difference vectors. The difference in the configurations does not depend on the exact locations of the relevant grid lines or their intersections. However, whenever a grid line or an intersection moves in or out of the parallelogram $P(k_j,k_{j+1})$, the configuration changes. Relevant grid lines ------------------- It is easy to see from (\[mesh\]) that $\partial K_{j+{\rm m}}(\myb{\epsilon})$ changes its value whenever lines in the $(j\!+\!{\rm m})$th grid are crossed. Because $0\leqslant\{x\},\epsilon_j,\epsilon_{j+1}<1$, we find from (\[dk2\]) and (\[dk3\]) that $$-1<\lambda_{j+2}(\myb{\epsilon}),\lambda_{j+4}(\myb{\epsilon})<2, \quad -1<\lambda_{j+3}(\myb{\epsilon})<3. \label{ineq}$$ Consequently, the only relevant grid lines for the parallelogram $P(k_j,k_{j+1})$ are those having integer labels $k_{j+2}-1+n'$, $k_{j+3}-1+m$, or $k_{j+4}-1+n$, with $n,n'=0,1$ and $m=0,1,2$. Indeed, these are the only integer values that the $K_{j+{\rm m}}(\myb{\epsilon})$ in (\[kz\]), or $k_{j+{\rm m}}-1+\lceil\lambda_{j+{\rm m}}(\myb{\epsilon})\rceil$, can assume. The loci of these lines are given by linear equations in $\epsilon_j$ and $\epsilon_{j+1}$ as $$\lambda_{j+2}(\myb{\epsilon})=n',\quad \lambda_{j+3}(\myb{\epsilon})=m,\quad \lambda_{j+4}(\myb{\epsilon})=n. \label{linear}$$ From (\[dk2\]), we find $$\begin{aligned} &0<\lambda_{j+2}(\myb{\epsilon})<2 &\quad\hbox{if}\quad \{\alpha\}>p^{-1},\nonumber\\ &0<\lambda_{j+4}(\myb{\epsilon})<2 &\quad\hbox{if}\quad \{\beta\}>p^{-1}. \label{line12}\end{aligned}$$ Therefore the equation $\lambda_{j+2}=0$ ($\lambda_{j+4}=0$) cannot be satisfied if $\{\alpha\}>p^{-1}$ ($\{\beta\}>p^{-1}$), while equations $\lambda_{j+2}=1$ and $\lambda_{j+4}=1$ always have solutions in $P$. This means that the only grid lines in grids $G_{j+2}$ and $G_{j+4}$ crossing $P$ are given by $$\begin{aligned} &k_{j+2},k_{j+4}\in P &\quad\hbox{always},\nonumber\\ &k_{j+2}-1\in P &\quad\hbox{if}\quad\{\alpha\}<p^{-1},\nonumber\\ &k_{j+4}-1\in P &\quad\hbox{if}\quad\{\beta\}<p^{-1}. \label{lcon12}\end{aligned}$$ From (\[dk3\]) we find that the grid lines $k_{j+3}+m$ are parallel to the diagonal $\epsilon_j+\epsilon_{j+1}=1$, and $$\begin{aligned} &1\!-\!p^{-3}<\lambda_{j+3}(\myb{\epsilon})<2p^{-1}\!+\!1 &\quad\hbox{if}\quad 0<\{\alpha\}+\{\beta\}<p^{-3},\nonumber\\ &0<\lambda_{j+3}(\myb{\epsilon})<2 &\quad\hbox{if}\quad p^{-3}<\{\alpha\}+\{\beta\}<1,\nonumber\\ &-p^{-3}<\lambda_{j+3}(\myb{\epsilon})<2p^{-1} &\quad\hbox{if}\quad 1<\{\alpha\}+\{\beta\}<2p^{-1},\nonumber\\ &-1<\lambda_{j+3}(\myb{\epsilon})<1 &\quad\hbox{if}\quad 2p^{-1}<\{\alpha\}+\{\beta\}<2, \label{line3}\end{aligned}$$ where we used the identity $p^{-3}=2p^{-1}-1$. This means, there can be at most two lines of grid $G_{j+3}$ going through the inside of the parallelogram. We find that $\lambda_{j+3}=2$ has a solution in $P$ if $0<\{\alpha\}+\{\beta\}<p^{-3}$; $\lambda_{j+3}=1$ has a solution in $P$ for $0<\{\alpha\}+\{\beta\}<2p^{-1}$; $\lambda_{j+3}=0$ has a solution in $P$ for $1<\{\alpha\}+\{\beta\}$. These facts can be summarized as $$\begin{aligned} &k_{j+3}+1\in P\quad&\hbox{if}\quad\{\alpha\}+\{\beta\}<p^{-3} =\sqrt{5}-2,\nonumber\\ &k_{j+3}\in P\phantom{\;\,+1}\quad&\hbox{if}\quad\{\alpha\}+\{\beta\}<2p^{-1} =\sqrt{5}-1,\nonumber\\ &k_{j+3}-1\in P\quad&\hbox{if}\quad\{\alpha\}+\{\beta\}>1. \label{lcon3}\end{aligned}$$ At this point one may note the symmetry under $k_{j+2}\leftrightarrow k_{j+4}$ and $\alpha\leftrightarrow\beta$ in conditions (\[lcon12\]) and (\[lcon3\]). Intersections of the grid lines ------------------------------- Next, we need to calculate the positions of the intersections. We let ${\myb a}^{n,m}$ denote the intersection of a pair of grid lines in $G_{j+3}$ and $G_{j+4}$ numbered $k_{j+3}\!-\!1\!+\!m$ and $k_{j+4}\!-\!1\!+\!n$, ${\myb b}^{n'\!,m}$ the intersection of lines $k_{j+3}\!-\!1\!+\!m$ and $k_{j+2}\!-\!1\!+\!n'$ in $G_{j+3}$ and $G_{j+2}$, while ${\myb c}^{n,n'}$ the intersection of lines $k_{j+4}\!-\!1\!+\!n$ and $k_{j+2}\!-\!1\!+\!n'$ in $G_{j+4}$ and $G_{j+2}$. The locations of these intersections are found by solving the corresponding linear equations given in (\[linear\]). We find $$\begin{aligned} &&{\myb a}^{n,m}= \big(p(\{\beta\}\!-\!n)\!+\!m\!+\!n\!-\!1\!+\!\{\alpha\},\, p^{-1}(\{\alpha\}\!+\!n\!+\!m\!-\!1)\big),\hspace{0.5in}\label{pa}\\ &&{\myb b}^{n'\!,m}= \big(p^{-1}(\{\beta\}\!+\!m\!+\!n'\!\!-\!1),\, p(\{\alpha\}\!-\!n')\!+\!m\!+\!n'\!\!-\!1\!+\!\{\beta\}\big),\label{pb}\\ &&{\myb c}^{n,n'}=\big(pn'\!+\!n\!-\!p\{\alpha\}\!-\!\{\beta\},\, pn\!+\!n'\!-\!\{\alpha\}\!-\!p\{\beta\}\big). \label{pc}\end{aligned}$$ Clearly, whenever both components ($\epsilon_j,\epsilon_{j+1}$) of an intersection are positive and less than 1, it is inside the parallelogram $P(k_j,k_{j+1})$. This way, we find from (\[pa\]) the conditions for the three possible cases, namely $$\begin{aligned} &{\myb a}^{0,1}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \{\beta\}<p^{-1}-p^{-1}\{\alpha\},\qquad\qquad\nonumber\\ &{\myb a}^{1,0}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \{\beta\}>1-p^{-1}\{\alpha\},\qquad\qquad\nonumber\\ &{\myb a}^{1,1}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \cases{\begin{array}{l} 0<\{\alpha\}<p^{-1}\quad\hbox{and}\\ p^{-2}\!-\!p^{-1}\{\alpha\}\!<\!\{\beta\}\!<\!1\!-\!p^{-1} \{\alpha\}\upstrut. \end{array}} \label{cona}\end{aligned}$$ Similarly, from (\[pb\]) we obtain $$\begin{aligned} &{\myb b}^{0,1}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \{\beta\}<1-p\{\alpha\},\nonumber\\ &{\myb b}^{1,0}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \{\beta\}>p(1-\{\alpha\}),\nonumber\\ &{\myb b}^{1,1}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \cases{\begin{array}{l} 0<\{\beta\}<p^{-1}\quad\hbox{and}\\ p^{-2}\!-\!p^{-1}\{\beta\}\!<\!\{\alpha\}\!<\!1\!-\!p^{-1} \{\beta\}\upstrut, \end{array}} \label{conb}\end{aligned}$$ and from (\[pc\]) we get $$\begin{aligned} &{\myb c}^{0,1}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad p^{-1}\!-\!p\{\alpha\}\!<\!\{\beta\}\!<\!p^{-1}(1\!-\!\{\alpha\}), \nonumber\\ &{\myb c}^{1,0}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad p^{-2}\!-\!p^{-1}\{\alpha\}\!<\!\{\beta\}\!<\!1\!-\!p\{\alpha\}, \nonumber\\ &{\myb c}^{1,1}\in P(k_j,k_{j+1})\quad&\hbox{if}\quad \{\beta\}>\max(1\!-\!p^{-1}\{\alpha\},p\!-\!p\{\alpha\}). \label{conc}\end{aligned}$$ Note the symmetry between (\[cona\]) and (\[conb\]) and between the first two lines of (\[conc\]) under $\alpha\leftrightarrow\beta$ and implicitly $k_{j+2}\leftrightarrow k_{j+4}$. Three lines cannot exactly meet in a common intersection, as was shown by de Bruijn [@Bruijn1] for a regular pentagrid. However, they can meet arbitrarily close and the theoretical limiting conditions of triple intersection $$\begin{aligned} &``\,{\myb a}^{1,0}={\myb b}^{1,0}={\myb c}^{1,1}\,"\quad &\Longleftrightarrow\quad \{\alpha\}+\{\beta\}=p,\nonumber\\ &``\,{\myb a}^{0,1}={\myb b}^{1,1}={\myb c}^{0,1}\,"\quad &\Longleftrightarrow\quad \{\alpha\}+\{\beta\}=p^{-1},\nonumber\\ &``\,{\myb a}^{1,1}={\myb b}^{0,1}={\myb c}^{1,0}\,"\quad &\Longleftrightarrow\quad \{\alpha\}+\{\beta\}=p^{-1}, \label{contri}\end{aligned}$$ play a role in the following subsection. The twenty-four allowed configurations -------------------------------------- Next, we use (\[lcon12\]), (\[lcon3\]), (\[cona\])–(\[contri\]) to study how the configuration $C(m)$ of parallelogram $P(k_j,k_{j+1})$ depends on the values of $\{\alpha\}=\{\hat\alpha(k_{j+1})\}$ and $\{\beta\}=\{\hat\beta(k_j)\}$. We show the various cases in Fig. \[fig6\] for $j\!=\!0$. For $j\!\ne\!0$ we need to rotate each picture $j$ times $72^\circ$. =0.22 =0.22 =0.22 =0.22 to 0.1in =0.22 =0.22 =0.22 =0.22 to 0.1in =0.22 =0.22 =0.22 =0.22 to 0.1in 0.1in=0.22 =0.22 =0.22 =0.22 to 0.1in =0.22 =0.22 =0.22 =0.22 to 0.1in =0.22 =0.22 =0.22 =0.22 to 0.1in For $\{\alpha\},\{\beta\}>p^{-1}$, the three grid lines $k_{j+2}$, $k_{j+3}\!-\!1$, $k_{j+4}$ and their three intersections ${\myb a}^{1,0}$, ${\myb b}^{1,0}$, ${\myb c}^{1,1}$ are inside the parallelogram, producing seven meshes in $P(k_j,k_{j+1})$, as shown in Figs. \[fig6\]$\,$(a) and \[fig6\]$\,$(b). The difference between the two cases is that the intersection ${\myb c}^{1,1}$ is on opposite sides of the grid line $k_{j+3}\!-\!1$, as the sign of $\{\alpha\}+\{\beta\}-p$ changes, cf. (\[contri\]). The grid line $k_{j+3}\!-\!1$ moves upward toward the upper right corner as $\{\beta\}$ decreases. It is below the diagonal $\epsilon_j+\epsilon_{j+1}=1$ for $\{\alpha\}+\{\beta\}>p$, corresponding to Fig. \[fig6\]$\,$(a), and above the diagonal for $\{\alpha\}+\{\beta\}<p$, as shown in Fig. \[fig6\]$\,$(b). The index of the inner triangle changes from odd to even in view of (\[dk2\]), (\[dk3\]) and (\[linear\]). Hence, the spin configurations for the two cases are different: C(1) in Fig. \[fig6\]$\,$(a) has 4 odd sites and 3 even sites, and C(2) shown in Fig. \[fig6\]$\,$(b) has 3 odd sites and 4 even sites. Also, C(1) is the only configuration for which the reference integer vector does not correspond to an actual mesh. For $\{\alpha\}>p^{-1}$ and $0<\{\beta\}<p^{-1}$, we have six cases denoted by C(3) to C(8) arranged in the decreasing order of $\{\beta\}$. For $1-p^{-1}\{\alpha\}<\{\beta\}$, the configuration C(3) is almost the same as C(2) except having one more even site because line $k_{j+4}\!-\!1$ is now inside the parallelogram, as shown in Fig. \[fig6\]$\,$(c). As $\{\beta\}$ decreases, line $k_{j+4}$ moves downward and for $2p^{-1}\!-\!\{\alpha\}<\{\beta\}<1-p^{-1}\{\alpha\}$, the intersections of line $k_{j+4}$ with lines $k_{j+3}\!-\!1$ (${\myb a}^{1,0}$) and $k_{j+2}$ (${\myb c}^{1,1}$) are seen from (\[cona\]) and (\[conc\]) to be outside of $P(k_j,k_{j+1})$. As a result, two of the even sites are now outside, and C(4), shown in Fig. \[fig6\]$\,$(d), has only six sites, three of which are odd, and three even. As $\{\beta\}$ decreases further to $p(1\!-\!\{\alpha\})<\{\beta\}<2p^{-1}\!-\!\{\alpha\}$, line $k_{j+3}$ is now inside $P(k_j,k_{j+1})$, as seen from (\[lcon3\]). Thus, configuration C(5), shown in Fig. \[fig6\]$\,$(e), has one more even site than C(4). Both lines $k_{j+3}$ and $k_{j+3}\!-\!1$ move upward as $\{\beta\}$ decreases. For $1\!-\!\{\alpha\}<\{\beta\}<p(1-\{\alpha\})$, the intersection ${\myb b}^{1,1}$ of line $k_{j+3}$ with $k_{j+2}$ moves inside of $P(k_j,k_{j+1})$, while ${\myb b}^{1,0}$, which is the intersection of $k_{j+3}\!-\!1$ and $k_{j+2}$, moves out, giving rise to configuration C(6) shown in Fig. \[fig6\]$\,$(f), with three even sites and four odd sites. For $p^{-1}(1\!-\!\{\alpha\})<\{\beta\}<1\!-\!\{\alpha\}$, line $k_{j+3}\!-\!1$ moves out of $P$, as shown in Fig. \[fig6\]$\,$(g). Its configuration C(7) has 3 odd sites and 3 even sites differing from C(6) in that the odd site with $\partial{\vec K}({\myb 0})=(0,-1,0)$ is now outside $P(k_j,k_{j+1})$. For $0<\{\beta\}<p^{-1}(1\!-\!\{\alpha\})$, the intersections ${\myb a}^{0,1}$ and ${\myb c}^{0,1}$ of line $k_{j+4}-1$ with $k_{j+3}$ and $k_{j+2}$ are now seen from (\[cona\]) and (\[conc\]) to be inside of $P(k_j,k_{j+1})$ adding 2 more odd sites to C(8), which is shown in Fig. \[fig6\]$\,$(h). In Fig. \[fig6\]$\,$(i) through Fig. \[fig6\]$\,$(n), the six cases C(9) through C(14) are shown for $\{\alpha\}<p^{-1}$ and $\{\beta\}>p^{-1}$. Because of (\[kj12\]) and (\[alpha\]) we can use the reflection symmetry $\{\alpha\}\leftrightarrow\{\beta\}$, $k_{j}\leftrightarrow k_{j+1}$, $k_{j+2}\leftrightarrow k_{j+4}$, which was noted also in the previous subsections. Thus these cases are similar to the configurations C(3) to C(8), and obtainable simply by replacing $k_{j+4}+n$ by $k_{j+2}+n$ and vice versa. To summarize, we find C(9) has 3 odd sites and 5 even sites; C(10) 3 odd sites and 3 even sites; C(11) 3 odd sites and 4 even sites; C(12) 4 odd sites and 3 even sites; C(13) has 3 odd sites and 3 even sites, and C(14) 5 odd sites and 3 even sites. For $\{\alpha\},\{\beta\}<p^{-1}$, at least five grid lines $k_{j+2}$, $k_{j+2}\!-\!1$, $k_{j+3}$, $k_{j+4}$ and $k_{j+4}\!-\!1$ are inside $P(k_j,k_{j+1})$. In Fig. \[fig6\]$\,$(o), we show configuration C(15) valid for $1<\{\alpha\}+\{\beta\}$, when both lines $k_{j+3}-1$ and $k_{j+3}$ and the intersections ${\myb b}^{1,1}$ and ${\myb a}^{1,1}$ are inside $P(k_j,k_{j+1})$, as seen from (\[cona\]) and (\[conb\]). Configuration C(15) has 4 odd sites and 5 even sites. For the region satisfying the three inequalities $\{\beta\}<1\!-\!\{\alpha\}$, $\{\beta\}>1\!-\!p\{\alpha\}$ and $\{\beta\}>p^{-1}(1\!-\!\{\alpha\})$, grid line $k_{j+3}$ and the corresponding odd site with $\partial{\vec K}=(0,-1,0)$ are now outside $P(k_j,k_{j+1})$, such that configuration C(16) shown in Fig. \[fig6\]$\,$(p) has one site less than C(15). It has 3 odd and 5 even sites. For $1\!-\!p\{\alpha\}<\{\beta\}<p^{-1}(1\!-\!\{\alpha\})$, the intersections ${\myb a}^{0,1}$ and ${\myb c}^{0,1}$ of line $k_{j+4}\!-\!1$ are both also inside $P(k_j,k_{j+1})$, as seen from (\[cona\]) and (\[conc\]), adding two odd sites to C(16). Thus, C(17) in Fig. \[fig6\]$\,$(q) has 5 odd sites and 5 even. However, for $p^{-1}(1\!-\!\{\alpha\})<\{\beta\}<1\!-\!p\{\alpha\}$ the intersections ${\myb b}^{0,1}$ and ${\myb c}^{1,0}$ of lines $k_{j+2}\!-\!1$ are now inside instead, adding two different odd sites to configuration C(16). The resulting configuration C(18) shown in Fig. \[fig6\]$\,$(r) also has ten sites and relates to C(17) by the above reflection symmetry. When both conditions $p^{-1}\!-\!p\{\alpha\}<\{\beta\}<1\!-\!p\{\alpha\}$ and $p^{-2}\!-\!p^{-1}\{\alpha\}<\{\beta\}<p^{-1}(1\!-\!\{\alpha\})$ are satisfied, we find six intersections ${\myb a}^{1,1}$, ${\myb b}^{1,1}$, ${\myb a}^{0,1}$, ${\myb b}^{0,1}$, ${\myb c}^{0,1}$ and ${\myb c}^{1,0}$ inside $P(k_j,k_{j+1})$. As a result, there are 12 sites for the two cases C(19) and C(20) shown in Fig. \[fig6\]$\,$(s) and Fig. \[fig6\]$\,$(t) respectively. The difference between the two cases is that the intersections ${\myb c}^{0,1}$ and ${\myb c}^{1,0}$ are on opposite sides of the grid line $k_{j+3}$, as the sign of $\{\alpha\}+\{\beta\}-p^{-1}$ changes, cf. (\[contri\]). For $\{\alpha\}+\{\beta\}>p^{-1}$, line $k_{j+3}$ lies below the diagonal, as can be seen from (\[dk3\]) and (\[linear\]), so that its configuration C(19) has 7 odd sites and 5 even sites; for $\{\alpha\}+\{\beta\}<p^{-1}$, $k_{j+3}$ is above the diagonal, and C(20) has 5 odd sites and 7 even sites. For $p^{-1}\!-\!p\{\alpha\}<\{\beta\}<p^{-2}\!-\!p^{-1}\{\alpha\}$, intersections ${\myb a}^{1,1}$ and ${\myb c}^{1,0}$ are no longer inside $P(k_j,k_{j+1})$. As a consequence two of the even sites are now outside, leaving C(21) shown in Fig. \[fig6\]$\,$(u) with 5 odd sites and 5 even. For $p^{-2}\!-\!p^{-1}\{\alpha\}<\{\beta\}<p^{-1}\!-\!p\{\alpha\}$, however, intersections ${\myb b}^{1,1}$ and ${\myb c}^{0,1}$ are outside $P(k_j,k_{j+1})$ instead, such that two different even sites are now outside. The resulting configuration C(22) shown in Fig. \[fig6\]$\,$(v) has the same 5 odd sites as C(21), but a different set of 5 even sites. Again, C(21) and C(22) are related by the aforementioned reflection symmetry. For $\{\beta\}<p^{-2}\!-\!p^{-1}\{\alpha\}$ and $\{\beta\}<p^{-1}\!-\!p\{\alpha\}$, only two intersections ${\myb a}^{0,1}$ and ${\myb b}^{0,1}$ are still inside $P(k_j,k_{j+1})$. These are the cases C(23) and C(24) shown in Fig. \[fig6\]$\,$(w) and Fig. \[fig6\]$\,$(x). For $\{\beta\}+\{\alpha\}>p^{-3}$, configuration C(23), shown in Fig. \[fig6\]$\,$(w), has eight sites: 5 odd and 3 even. Finally, for $0<\{\beta\}<p^{-3}\!-\!\{\alpha\}$, the grid line $k_{j+3}+1$ is inside $P$, so that C(24) shown in Fig. \[fig6\]$\,$(x) has nine sites, of which the 5 odd sites are identical to those of C(20) through C(23). In summary, the boundaries for the above 24 regions are the 13 lines given by $$\begin{aligned} &&\{\alpha\}=p^{-1},\quad \{\beta\}=p^{-1},\nonumber\\ &&\{\beta\}+\{\alpha\}=p^{-3},\;p^{-1},\;1,\;2p^{-1}, \hbox{ or }p,\nonumber\\ &&\{\beta\}+p^{-1}\{\alpha\}=p^{-2},\;p^{-1},\hbox{ or }1,\nonumber\\ &&\{\beta\}+p\{\alpha\}=p^{-1},\;1,\hbox{ or }p, \label{bound}\end{aligned}$$ which are also the boundaries of the inequalities in (\[lcon12\]), (\[lcon3\]), (\[cona\]), (\[conb\]) and (\[conc\]), together with the two conditions in (\[contri\]). These are exactly all conditions for three grid lines to meet at a corner of $P$, on an edge of $P$, or inside of $P$, respectively, i.e. the only conditions under which some mesh can appear or disappear in $P$ under shifts of grid lines. In Fig. \[fig7\]$\,$(a), we plot the boundaries lines given by (\[bound\]) in the unit square with $\{\alpha\}$ and $\{\beta\}$ along the horizontal and vertical axes. These lines indeed divide the unit square into 24 regions. Each of these 24 regions corresponds to a different configuration of the parallelogram $P(k_j,k_{j+1})$. The above analysis shows that the parallelograms can only have 6, 7, 8, 9, 10 or 12 sites inside. The position of $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ in the unit square shown in Fig. \[fig7\]$\,$(a) determines which configuration the parallelogram $P(k_j,k_{j+1})$ is in. 0.03in0.1in =0.45=0.45 to -0.2in The areas of the 24 regions in Fig. \[fig7\]$\,$(a) can be easily calculated. We find, after using the formula of the area of a triangle in terms of the coordinates of its three vertices, $$\begin{aligned} &&A(1)=A(2)={\textstyle\frac{5}{2}}-{\textstyle\frac{3}{2}}p= {\textstyle\frac{1}{2}}p^{-4},\nonumber\\ &&A(3)=A(5)=A(6)=A(8)=A(9)=A(11)=A(12)\nonumber\\ &&\qquad=A(14)=A(19)=A(20)={\textstyle\frac{5}{2}}p-4= {\textstyle\frac{1}{2}}p^{-5},\nonumber\\ &&A(4)=A(7)=A(10)=A(13)=A(15)=A(17)=A(18)\nonumber\\ &&\qquad=A(21)=A(22)=A(24)={\textstyle\frac{13}{2}}-4p= {\textstyle\frac{1}{2}}p^{-6},\nonumber\\ &&A(16)=A(23)=9p-{\textstyle\frac{29}{2}}= {\textstyle\frac{1}{2}}p^{-3}-p^{-6}. \label{areacalc}\end{aligned}$$ The areas of all 22 triangular areas differ by powers of the golden ratio $p$. For later reference, we display in Fig. \[fig7\]$\,$(b) the eight different regions with equivalent odd configurations. Two odd configurations are equivalent, if they have an equal number of odd sites with the same sets of integer vectors for these odd sites. This identification can most easily be made using the information in Tables \[conftab\] and \[conftabb\] below. Probability ----------- From the definition (\[alpha\]), we find that the fractional parts $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ are related to the golden ratio $p$ which is irrational. From the well-known theorem of Kronecker [@HW], we conclude that $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ are everywhere dense and uniformly distributed in the interval $(0,1)$, as the integers $k_j$ and $k_{j+1}$ vary from $-\infty$ to $\infty$. As a consequence, every point in the unit square in Fig. \[fig7\] is equally probable. Therefore, the frequency or probability for a parallelogram to be in one of the twenty-four configurations, say $m$, is given by the area $A(m)$ of the $m$th region. Although the pentagrids are different for different choices of the shifts $\gamma_j$, and the values of $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ are also different for different $\gamma_j$’s, this does not change the probability distributions of $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ in the thermodynamic limit of $k_j$ and $k_{j+1}$ varying from $-\infty$ to $\infty$. In other words, the area $A(m)$ for the $m$th configuration is independent of the $\gamma_j$’s, and is the same for all regular pentagrids. Difference vectors ------------------ For $\{\alpha\}+\{\beta\}>1$ the grid line $k_{j+3}\!-\!1$ is seen from (\[lcon3\]) to be inside $P(k_j,k_{j+1})$. Consequently, the mesh below corner ${\myb \epsilon}=(0,0)$ is odd and $\partial{\vec K}=(0,-1,0)$. For $\{\alpha\}+\{\beta\}<1$, line $k_{j+3}-1$ is outside $P(k_j,k_{j+1})$, and the mesh at corner ${\myb \epsilon}=(0,0)$ is even. In this case, its integer vector is identical to the reference vector for $P(k_j,k_{j+1})$ with $\partial{\vec K}({\myb 0})=(0,0,0)$. The difference vectors $\partial{\vec K}$ for all other meshes in the parallelograms in each of the twenty-four configurations can be easily obtained from Fig. \[fig6\], as $\partial K_{j+{\rm m}}$ changes its values only when a lines in the $(j+{\rm m})$th grid is crossed. In Tables \[conftab\] and \[conftabb\], we list for each of the 24 configurations all the difference vectors $\partial{\vec K}=(\partial K_{j+2},\partial K_{j+3},\partial K_{j+4})$, with Table \[conftab\] for $\{\alpha\}+\{\beta\}>1$ and Table \[conftabb\] for $\{\alpha\}+\{\beta\}<1$. From these tables one can also immediately read off which regions have equivalent odd (or even) configurations. [\|p[0.18in]{}\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|]{} &C(1)&C(2)&C(3)&C(4)&C(5)&C(6)&C(9)&C(10)&C(11)&C(12)&C(15)&\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\] &\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\]&\[0,-1,0\] ${\rm o}$&\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\] &\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\] ${\rm d}$&\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&\[0,0,1\] &\[0,0,1\] &\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&\[0,0,1\] ${\rm d}$&\[1,-1,1\]&&&&&&&&&& &&&&&&\[0,1,0\]&&&&\[0,1,0\]&\[0,1,0\] &\[1,-1,0\]&\[1,-1,0\]&\[1,-1,0\]&\[1,-1,0\]&\[1,-1,0\]&&\[1,-1,0\] &&&& ${\rm e}$&&\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\] &\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\] ${\rm v}$&\[0,-1,1\]&\[0,-1,1\]&\[0,-1,1\]&&&&\[0,-1,1\]&\[0,-1,1\] &\[0,-1,1\]&& ${\rm e}$&\[1,0,1\]&\[1,0,1\]&\[1,0,1\]&&&&\[1,0,1\]&&&& ${\rm n}$&&&\[1,0,-1\]&\[1,0,-1\]&\[1,0,-1\]&\[1,0,-1\]&&&&&\[1,0,-1\] &&&&&&&\[-1,0,1\]&\[-1,0,1\]&\[-1,0,1\]&\[-1,0,1\]&\[-1,0,1\] &&&&&\[1,1,0\]&\[1,1,0\]&&&&&\[1,1,0\] &&&&&&&&&\[0,1,1\]&\[0,1,1\]&\[0,1,1\] \[conftab\] [\|p[0.085in]{}\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|]{} &C(7)&C(8)&C(13)&C(14)&C(16)&C(17)&C(18)&C(19)&C(20) &C(21)&C(22)&C(23)&C(24)&\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&\[1,0,0\] &\[1,0,0\]&\[1,0,0\]&\[1,0,0\]&&&&& &\[0,1,0\]&\[0,1,0\]&\[0,1,0\]&\[0,1,0\]&\[0,1,0\] &\[0,1,0\]&\[0,1,0\]&\[0,1,0\]&\[0,1,0\]&\[0,1,0\] &\[0,1,0\]&\[0,1,0\]&\[0,1,0\] ${\rm o}$&\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&\[0,0,1\] &\[0,0,1\]&\[0,0,1\]&\[0,0,1\]&&&&& ${\rm d}$&&\[0,0,-1\]&&&&\[0,0,-1\]&&\[0,0,-1\]&\[0,0,-1\]& \[0,0,-1\]&\[0,0,-1\]&\[0,0,-1\]&\[0,0,-1\] ${\rm d}$&&\[1,1,-1\]&&&&\[1,1,-1\]&&\[1,1,-1\]&\[1,1,-1\] &\[1,1,-1\]&\[1,1,-1\]&\[1,1,-1\]&\[1,1,-1\] &&&&\[-1,0,0\]&&&\[-1,0,0\]&\[-1,0,0\]&\[-1,0,0\] &\[-1,0,0\]&\[-1,0,0\]&\[-1,0,0\]&\[-1,0,0\] &&&&\[-1,1,1\]&&&\[-1,1,1\]&\[-1,1,1\]&\[-1,1,1\] &\[-1,1,1\]&\[-1,1,1\]&\[-1,1,1\]&\[-1,1,1\] &\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\] &\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\]&\[0,0,0\] &\[0,0,0\]&\[0,0,0\]&\[0,0,0\] &&&\[0,1,1\]&\[0,1,1\]&\[0,1,1\]&\[0,1,1\] &\[0,1,1\]&\[0,1,1\]&\[0,1,1\]&&\[0,1,1\]&& ${\rm e}$&\[1,1,0\]&\[1,1,0\]&&&\[1,1,0\]&\[1,1,0\] &\[1,1,0\]&\[1,1,0\]&\[1,1,0\]&\[1,1,0\]&&& ${\rm v}$&&&&&&&&&\[-1,1,0\]&\[-1,1,0\]&\[-1,1,0\]&\[-1,1,0\] &\[-1,1,0\] ${\rm e}$&&&&&&&&&\[0,1,-1\]&\[0,1,-1\]&\[0,1,-1\]&\[0,1,-1\] &\[0,1,-1\] ${\rm n}$&\[1,0,-1\]&\[1,0,-1\]&&&\[1,0,-1\]&\[1,0,-1\] &\[1,0,-1\]&\[1,0,-1\]&\[1,0,-1\]&\[1,0,-1\]&&& &&&\[-1,0,1\]&\[-1,0,1\]&\[-1,0,1\]&\[-1,0,1\] &\[-1,0,1\]&\[-1,0,1\]&\[-1,0,1\]&&\[-1,0,1\] &&&&&&&&&&&&&&&\[0,2,0\] \[conftabb\] Average Number $\myb{\cal N}$ ----------------------------- We shall now calculate ${\cal N}$, which is the average number of spin sites in a parallelogram. The total number of sites can be evaluated by counting all the sites in each parallelograms $P(k_j,k_{j+1})$, and then adding all of them together for all the $P$’s. This is equivalent to splitting the summation over all sites into two parts—first summing over all sites in $P$ represented by their integer vectors ${\vec K}(\myb{\epsilon})$ and then adding all of them for all the parallelograms. Let there be ${\cal M}$ lines in each of the five grids, so that there are ${\cal M}^2$ parallelograms, ignoring boundary effects that cancel in the thermodynamic limit. The average ${\cal N}$ then equals the total number of lattice sites divided by ${\cal M}^2$. We have already shown that each parallelogram $P(k_j,k_{j+1})$ is in one of 24 configurations $C(m)$ uniquely determined by the values of $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$. The allowed configurations $C(m)$ have $N(m)=6$, 7, 8, 9, 10 or 12 sites inside $P$. The frequency or probability $A(m)$ is defined as the number of parallelograms in the $m$th configuration divided by the total number of parallelograms. If we let $k_j$ and $k_{j+1}$ in $P(k_j,k_{j+1})$ run over the ${\cal M}$ values, each of the parallelograms is counted once. As ${\cal M}$ approaches $\infty$, so that $-\infty<k_j,k_{j+1}<\infty$, the values of $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ are everywhere dense and uniformly distributed [@HW] between $0$ and $1$. The frequency $A(m)$ is the area of the $m$th region in the unit square in Fig. \[fig7\]$\,$(a). The values of these $A(m)$’s are listed in (\[areacalc\]). Denoting the number of sites in the $m$th configuration by $N(m)$, with values also given in the captions of Fig. \[fig6\], then the average number of sites per parallelogram is $$\begin{aligned} {\cal N}&=&\frac 1 {{\cal M}^2}\! \sum_{{\vec K}(z\in\mbox{\mymsbms C})}1=\frac 1 {{\cal M}^2} \sum_{{\rm all}\,{\vphantom{{\vec K}}} P}\,\sum_{{\vec K}(z\in P)}1 \label{split}\\ &&\stackrel{{\cal M}\to\infty}{\longrightarrow} \sum_{m=1}^{24} A(m)\sum_{n=1}^{N(m)}1=\sum_{m=1}^{24} A(m)N(m)=5p. \label{spsum}\end{aligned}$$ where we have used the notation ${\vec K}(z\!\in\!\mbox{\mymsbm C})$ to denote the integer vectors of all the meshes in the pentagrid, while ${\vec K}(z\!\in\!P)$ denotes only those meshes in parallelogram $P$. Penrose Tiles ------------- The above method provides an alternative way to draw the Penrose tiles. To illustrate this, we let $j=0$. For some fixed shifts $\gamma_j$, we let $-J\leqslant k_0,k_1\leqslant J$ for some positive integer $J$. For any values $k_0$ and $k_1$ in this set, $\{\hat\alpha(k_1)\}$, $\{\hat\beta(k_0)\}$ are uniquely determined from (\[alpha\]). The elements $k_{\rm m}$, for ${\rm m}=2$, 3 ,4, of the reference vector in $P(k_0,k_1)$ are given by (\[kj12\]) and (\[kj3\]). From the values of $\{\hat\alpha(k_1)\}$ and $\{\hat\beta(k_0)\}$, we can determine from Fig. \[fig7\]$\,$(a), what configuration $C(m)$ the parallelogram $P(k_0,k_1)$ is in. Then we can use Tables \[conftab\] and \[conftabb\] to obtain the difference vectors $\partial{\vec K}$ for the $N(m)$ sites inside $P(k_0,k_1)$. The integer vectors for these different sites in $P(k_0,k_1)$ are then given by $${\vec K}(\myb\epsilon)= ( k_0, k_1, k_2+\partial K_2, k_3+\partial K_3, k_4+\partial K_4 ). \label{intvec}$$ We use (\[penrose\]) and (\[intvec\]) to obtain the positions of the spins in the complex plane for the meshes in $P(k_0,k_1)$. Hence, as $k_0$ and $k_1$ run over the values from $-J$ to $J$ we obtain the positions of the spins in both odd and even sublattices shown in Fig. \[fig5\]. This figure has been plotted using Maple. Summary ------- Consider the parallelograms $P(k_j,k_{j+1})$ which contains all points $z\in\mbox{\mymsbm C}$ such that $K_j(z)=k_j$ and $K_{j+1}(z)=k_{j+1}$, cf. (\[mesh\]). The configurations of two such parallelograms are considered to be the same, if they contain the same number of spin sites and the corresponding sites have the same difference vectors. The different configurations do not depend on the exact locations of the relevant grid lines or their intersections. However, whenever a grid line or an intersection moves in or out of the parallelogram $P$, the configuration changes. The above analysis shows that there are only 24 allowed configurations, with 6, 7, 8, 9, 10 and 12 sites inside $P$. The configuration of $P(k_j,k_{j+1})$ is uniquely determined by the values of $\{\hat\alpha(k_{j+1})\}$ and $\{\hat\beta(k_j)\}$ defined in (\[alpha\]). By examining the locations of the relevant grid lines and their intersections, we find that the unit square, with $\{\alpha\}$ and $\{\beta\}$ along the horizontal and vertical axes, is divided into 24 regions, corresponding to the 24 possible configurations of the parallelogram $P$. This is shown Fig. \[fig7\]$\,$(a), and the 24 configurations of $P$ are shown in Fig. \[fig6\]. Using the theorem of Kronecker [@HW], we find that the area $A(m)$ of the $m$th region is actually proportional to the probability for the $m$th configuration to occur. Even though the pentagrids and configurations of the $P(k_j,k_{j+1})$’s are different for different choices of the shifts $\gamma_j$, the area $A(m)$ is independent of these shifts and is the same for all regular pentagrids. In each parallelogram $P(k_j,k_{j+1})$, a reference integer vector is chosen whose components are given by (\[kj12\]) and (\[kj3\]). The difference vectors with respect to this reference vector are defined in (\[kz\]) and calculated for all of the sites inside $P$. The number of sites $N(m)$, the area $A(m)$, and the difference vectors $\partial{\vec K}$ for the $N(m)$ spin sites for $m=1...24$ are listed in Fig. \[fig6\], Eq. (\[areacalc\]) and Tables \[conftab\] and \[conftabb\], respectively. Susceptibility {#sect5} ============== There are three $Z$-invariant Ising models that can be defined on the vertices of the Penrose rhombus tiles using the prescriptions of Section \[sect2\]. Model 1 has spins on all odd sites only, interacting along the diagonals of the tiles, as is illustrated in Fig. \[fig5\]. Model 2 is defined similarly with spins only on the even sites. Model 3 has all sites of the Penrose tiling, but the even and odd sites are decoupled, with the odd spins interacting as in model 1 and the even spins as in model 2. We will see that the three models have the identical wavevector-dependent susceptibility $\chi({\bf q})$ per spin site in the thermodynamic limit. The physical positions of the spins have been expressed in (\[penrose\]) as complex numbers depending on the integer vectors ${\vec K}(z)$ of the meshes. Let $q$ be a complex number and $q=q_x+{\rm i}q_y$ so that $q^$ denotes its complex conjugate (while ${\bf q}=(q_x,q_y)$), then the ${\bf q}$-dependent susceptibility is $$\begin{aligned} k_{\rm B}T\chi({\bf q})\!=\!\! \lim_{{\cal M}\to\infty}{\frac{1}{{\cal N}{\cal M}^{2}}}\! \sum_{{\vec K}(z\in\mbox{\mymsbms C})} \sum_{{\vec K}(z'\in\mbox{\mymsbms C})} {\rm cos}\,{\rm Re}\Bigl\{q^\sum_{j=0}^4[K_j(z')-K_j(z)]\zeta^j\Bigr\} \nonumber\\ \times\bigr[{\langle\sigma_{{\vec K}(z)}\sigma_{{\vec K}(z')}\rangle}- \langle\sigma_{{\vec K}(z)}\rangle\langle\sigma_{{\vec K}(z')} \rangle\bigl]= \left\{\begin{array}{ll} 2{\hat\chi}^{\rm o}({\bf q}),&\text{(model 1)},\quad\cr 2{\hat\chi}^{\rm e}({\bf q}),&\text{(model 2)},\upstrut\quad\cr {\hat\chi}^{\rm o}({\bf q})+{\hat\chi}^{\rm e}({\bf q}),\upstrut &\text{(model 3)}.\quad \end{array}\right. \label{chipt}\end{aligned}$$ Here the double sums denoted by ${\vec K}(z\in\mbox{\mymsbm C})$ are over all the odd spin sites in model 1, over all the even sites in model 2, and over all sites in model 3. In the last case, since the spins on the odd sublattice do not interact with those on the even sublattice, the ${\bf q}$-dependent susceptibility $\chi({\bf q})$ becomes the sum of two parts: ${\hat\chi}^{\rm o}({\bf q})$ denoting the contribution from the odd sublattice and ${\hat\chi}^{\rm e}({\bf q})$ from the even sublattice. For model 3 the average number of spin sites per parallelogram was given in (\[spsum\]) as ${\cal N}=5p$. For models 1 and 2 this number becomes ${\cal N}=5p/2$, explaining the extra factors 2 in the last member of (\[chipt\]). We shall first consider ${\hat\chi}^{\rm o}({\bf q})$ and show later that ${\hat\chi}^{\rm e}({\bf q})$ is equal to it, implying that the susceptibilities of the three models are indeed equal. Calculation of $\myb{\chi^{\rm o}({\bf q})}$ {#sect51} -------------------------------------------- We again split the sum over all odd spin sites into two parts as in (\[split\]), and let ${\vec K}^{\rm o}(z\in P)$ run over all odd spins in parallelogram $P$. Consequently, $${\hat\chi}^{\rm o}({\bf q})= \lim_{{\cal M}\to\infty}{\frac{1}{{\cal N}{\cal M}^{2}}} \sum_{{\rm all}\,{\vphantom{{\vec K}}} P}\, {\sum_{{\vec K}\vbox to 1.1ex{}^{\rm o}(z\in P)}} \sum_{{\rm all}\,{\vphantom{{\vec K}}} P'} {\sum_{{\vec K}\vbox to 1.1ex{}^{\rm o}({z'\in P'})}} U({\vec K}^{\rm o}\!(z'),{\vec K}^{\rm o}\!({z})), \label{chipto}$$ where $$\begin{aligned} U({\vec K}^{\rm o}\!(z'),{\vec K}^{\rm o}\!({z})) &=&\langle\sigma_{{\vec K}\vbox to 1.1ex{}^{\rm o}(z)} \sigma_{{\vec K}\vbox to 1.1ex{}^{\rm o}(z')}\rangle^{\rm c}\nonumber\\ &\times&{\rm cos}\,{\rm Re}\Bigl\{q^\sum_{n=0}^4 [K^{\rm o}_n\!(z')-K^{\rm o}_n\!({z})]\zeta^n\Bigr\}, \label{sumd}\end{aligned}$$ with $\langle\sigma\sigma'\rangle^{\rm c}$ denoting the connected pair correlation function, subtracting the contribution from the spontaneous magnetization. Now we let $P=P(k_j,k_{j+1})$ and $P'=P(k_j\!+\!\ell,k_{j+1}\!+\!\ell')$. As ${\cal M}\to\infty$, $\ell$ and $\ell'$ are kept fixed, and $k_j$ and $k_{j+1}$ vary from $-\infty$ to $\infty$, all the $P$’s and $P'$’s are counted once. It is also evident that the different choices of $j$ correspond to choosing one of five orientations for the parallelograms, and they should give the same ${\bf q}$-dependent susceptibility. In (\[zinP\]) we defined $z=z({\myb\epsilon})$ for $z\in P$. Similarly, for $z'\in P'$, we let $$z'=\frac{\,{\rm i}\,[\zeta^j(k_{j+1}\!+\!\ell'\!- \!\gamma_{j+1}\!-\!\epsilon'_{j+1})- \zeta^{j+1}(k_j\!+\!\ell\!-\!\gamma_j\!-\!\epsilon'_j)]}{\sin(2\pi/5)} \equiv z'({\myb\epsilon'}), \label{parallelp}$$ so that $z'\leftrightarrow{\myb\epsilon'}=(\epsilon'_j,\epsilon'_{j+1})$ and $0<\epsilon'_j,\epsilon'_{j+1}<1$. The corresponding integer vectors ${\vec K}(z')={\vec K}(z'({\myb\epsilon'})) \equiv{\vec K}'({\myb\epsilon'})$ also have two fixed components $K_j'({\myb\epsilon'})=k_j\!+\!\ell\equiv k'_j$ and $K_{j+1}'({\myb\epsilon'})=k_{j+1}\!+\!\ell'\equiv k'_{j+1}$. Now, following (\[alpha\]), we let $$\begin{aligned} &&\alpha'\equiv\hat\alpha(k'_{j+1})= p^{-1}(k_{j+1}\!+\!\ell'\!-\!\gamma_{j+1})+\gamma_j+\gamma_{j+2} =\alpha+p^{-1}\ell',\nonumber\\ &&\beta'\equiv\hat\beta(k'_j)= p^{-1}(k_j\!+\!\ell\!-\!\gamma_j)+\gamma_{j+4}+\gamma_{j+1} =\beta+p^{-1}\ell. \label{alphap}\end{aligned}$$ According to (\[kj12\]) and (\[kj3\]) the reference integer vector $\vec k'$ for $P'$ is chosen to have components, $$\begin{aligned} &&k'_{j+2}=\lceil\alpha'\rceil-k_j-\ell =k_{j+2}+\delta_{j+2} +\lfloor p^{-1}\ell'\rfloor-\ell,\nonumber\\ &&k'_{j+3}=-\lfloor\alpha'\rfloor-\lfloor\beta'\rfloor =k_{j+3}-\delta_{j+2}-\lfloor p^{-1}\ell'\rfloor -\delta_{j+4}-\lfloor p^{-1}\ell\rfloor,\nonumber\\ &&k'_{j+4}=\lceil\beta'\rceil-k_{j+1}-\ell' =k_{j+4}+\delta_{j+4}+\lfloor p^{-1}\ell\rfloor-\ell', \label{kjp}\end{aligned}$$ in which[^6] $$\delta_{j+2}=\lfloor\{\alpha\}+\{p^{-1}\ell'\}\rfloor,\quad \quad\delta_{j+4}=\lfloor\{\beta\}+\{p^{-1}\ell\}\rfloor. \label{delta24}$$ Substituting (\[parallelp\]) into (\[mesh\]) we obtain for ${\rm m}=2,3,4$ $$K_{j+{\rm m}}'(\myb{\epsilon}')=k'_{j+{\rm m}}+\partial K'_{j+{\rm m}}(\myb{\epsilon}'),\quad \partial K'_{j+{\rm m}}(\myb{\epsilon}')=\lfloor \lambda'_{j+{\rm m}}(\myb{\epsilon}')\rfloor, \label{kzp}$$ where $$\lambda'_{j+2}(\myb{\epsilon}') =\{\alpha'\}+\epsilon'_j-p^{-1}\epsilon'_{j+1},\quad \lambda'_{j+4}(\myb{\epsilon}')= \{\beta'\}+\epsilon'_{j+1}-p^{-1}\epsilon'_j. \label{dkp2}$$ and $$\lambda'_{j+3}(\myb{\epsilon}')= p^{-1}(\epsilon'_j+\epsilon'_{j+1})-\{\alpha'\} -\{\beta'\}+1. \label{dkp3}$$ Comparing (\[kzp\]) to (\[dkp3\]) with (\[kz\]) to (\[dk3\]), we find that the dependence of $\lambda'_{j+{\rm m}}(\myb{\epsilon}')$ on $\{\alpha'\}$ and $\{\beta'\}$ is the same as the dependence of $\lambda_{j+{\rm m}}(\myb{\epsilon})$ on $\{\alpha\}$ and $\{\beta\}$. Consequently, the configurations of $P'$ depend on $\{\alpha'\}$ and $\{\beta'\}$ in the same way as the configurations of $P$ on $\{\alpha\}$ and $\{\beta\}$. Therefore, the position of $\{\alpha'\}$ and $\{\beta'\}$ in the unit square in Fig. \[fig7\]$\,$(a) with $\{\alpha'\}$ and $\{\beta'\}$ along the horizontal and vertical axes uniquely determines the configuration of $P'$. The difference vectors $\partial {\vec K}'(\myb{\epsilon}')$ for the sites in $P'$, which is in one of the 24 configurations, are again given in Tables \[conftab\] and \[conftabb\]. The above results are valid for all spin configurations in $P$ and $P'$, but we shall consider only the odd spins at first. By examining Tables \[conftab\] and \[conftabb\], we can find that the odd spin configurations of the parallelograms are simpler, because several connected regions—see C(2) to C(5), or C(9) to C(11) in Fig. \[fig7\]$\,$(a) as examples—have the same odd spin configurations. In fact, there are only eight distinct odd spin configurations. In Fig. \[fig7\]$\,$(b), the regions for these 8 odd spin configurations are shown in the unit square whose axes are the $\{\alpha\}$ and $\{\beta\}$ directions. Listed in Table \[oddtab\] are the number of odd sites ${\hat N}(m)$ in $P$, the region of validity $R(m)$, and the area ${\hat A}(m)$ of the $m$th odd configuration for all $m=1,\cdots,8$. In the $m$th odd configuration, the difference vectors of the ${\hat N}(m)$ spins are denoted by $\partial {\vec K}^{[m,n]}$ for $n=1,\cdots,{\hat N}(m)$. They are equal to the difference vectors $\partial{\vec K}^{\rm o}$ of the odd sites in some configuration C($l$), listed in Tables \[conftab\] and \[conftabb\] for $l=1,\cdots,24$. In the last column of Table \[oddtab\] it is indicated which C($l$)’s correspond to a given $m$. [\|p[0.14in]{}\|c\|c\|p[3.66in]{}\|p[1.55in]{}\|]{} $m$ &${\hat N}(m)$&${\hat A}(m)$& & $\begin{array}{c}\partial{\vec K}^{[m,n]}=\partial{\vec K}^{\rm o}\\ 1\leqslant n\leqslant{\hat N}(m)\end{array}$ 1&4&${\textstyle\frac{1}{2}}p^{-4}$&$ p^{-1}<\{\alpha\}<1\;\&\; p-\{\alpha\}<\{\beta\}<1$& $\partial{\vec K}^{\rm o}$ in C(1)2&3&${\textstyle\frac{1}{2}}p^{-1}$& $\begin{array}{l}0<\{\alpha\}\leqslant p^{-1}\;\&\; 1-p^{-1}\{\alpha\}<\{\beta\}<1;\\ p^{-1}\leqslant\{\alpha\}<1\;\&\; p(1-\{\alpha\})<\{\beta\}<p-\{\alpha\}\end{array}$& $\begin{array}{l}\hbox{$\partial{\vec K}^{\rm o}$ in C(2) or}\\ \hbox{C($l$), $l$=3,4,5,9,10,11}\end{array}$3&4&${\textstyle\frac{1}{2}}p^{-3}$& $\begin{array}{l}0<\{\alpha\}\leqslant p^{-1}\;\&\; 1-\{\alpha\}<\{\beta\}<1-p^{-1}\{\alpha\};\\ p^{-1}\leqslant\{\alpha\}<1\;\&\; 1-\{\alpha\}<\{\beta\}<p(1-\{\alpha\})\end{array}$& $\begin{array}{l}\hbox{$\partial{\vec K}^{\rm o}$ in C(6),}\\ \hbox{C(12) or C(15)}\end{array}$4&3&${\textstyle\frac{1}{2}}p^{-3}$& $\begin{array}{l}0<\{\alpha\}\leqslant p^{-2}\;\&\; 1-p\{\alpha\}<\{\beta\}<1-\{\alpha\};\\ p^{-2}\leqslant\{\alpha\}<1\;\&\; p^{-1}(1-\{\alpha\})<\{\beta\}<1-\{\alpha\}\end{array}$& $\begin{array}{l}\hbox{$\partial{\vec K}^{\rm o}$ in C(7),}\\ \hbox{C(13) or C(16)}\end{array}$5&5&${\textstyle\frac{1}{2}}p^{-4}$&$0<\{\beta\}<p^{-2}\;\&\; p^{-1}(1-\{\beta\})<\{\alpha\}<1-p\{\beta\}$& $\partial{\vec K}^{\rm o}$ in C(8) or C(17)6&5&${\textstyle\frac{1}{2}}p^{-4}$&$0<\{\alpha\}<p^{-2}\;\&\; p^{-1}(1-\{\alpha\})<\{\beta\}<1-p\{\alpha\}$& $\partial{\vec K}^{\rm o}$ in C(14) or C(18)7&7&${\textstyle\frac{1}{2}}p^{-5}$& $\begin{array}{l}0<\{\alpha\}\leqslant p^{-2}\;\&\; p^{-1}-\{\alpha\}<\{\beta\}<p^{-1}(1-\{\alpha\});\hspace{-3pt}\\ p^{-2}\leqslant\{\alpha\}<p^{-1}\;\&\; p^{-1}-\{\alpha\}<\{\beta\}<1-p\{\alpha\}\end{array}$& $\partial{\vec K}^{\rm o}$ in C(19)8&5&${\textstyle\frac{1}{2}}p^{-2}$&$0<\{\alpha\}<p^{-1}\;\&\; 0<\{\beta\}<p^{-1}-\{\alpha\}$& $\partial{\vec K}^{\rm o}$ in C(20) to C(24) \[oddtab\] Let the distances $\ell$ and $\ell'$ between the two parallelograms $P$ and $P'$ be fixed, but $k_j$ and $k_{j+1}$ vary from $-\infty$ to $\infty$. Then $\{\alpha'\}$ and $\{\beta'\}$ given by (\[alphap\]) are also everywhere dense and uniformly distributed in the interval $(0,1)$. The area ${\hat A}(m)$ is again the probability or frequency of the $m$th configuration. From (\[alphap\]) we find $$\begin{aligned} &\{\alpha'\}=&\left\{ \begin{array}{lcr}\{\alpha\}+a&\hbox{for}&\{\alpha\}+a<1\\ \{\alpha\}+a-1&\hbox{for}&\{\alpha\}+a\geqslant1\upstrut\end{array} \right\},\quad a=\{p^{-1}\ell'\},\nonumber\\ &\{\beta'\}=&\left\{ \begin{array}{lcr}\{\beta\}+b&\hbox{for}&\{\beta\}+b<1 \\ \{\beta\}+b-1&\hbox{for}&\{\beta\}+b\geqslant1\upstrut\end{array} \right\},\quad b=\{p^{-1}\ell\}. \label{betap}\end{aligned}$$ 0.01in0.1in =0.45=0.45 to -0.2in The plot of the eight regions for the odd configurations of $P=P(k_j,k_{j+1})$ has been given in Fig. \[fig7\]$\,$(b). The plot for $P'=P(k_j\!+\!\ell,k_{j+1}\!+\!\ell')$ is the same, only now with $\{\alpha'\}$ and $\{\beta'\}$ along the axes. However, if the eight regions are replotted with $\{\alpha\}$ and $\{\beta\}$ along the horizontal and vertical axes, then we obtain unit squares as shown in Fig. \[fig8\]$\,$(a) \[for the special case of $\ell=2$ and $\ell'=3$\]. Because of the relation (\[betap\]), we find that Fig. \[fig8\]$\,$(a) can be obtained from Fig. \[fig7\]$\,$(b) by cutting a horizontal slice with width $b$ from the bottom of Fig. \[fig7\]$\,$(b), and pasting it on the top; then cutting a vertical slice of width $a$ from the left and pasting it to the right. After the cutting and pasting, the connected region $R(m)$ for $P'$ in Fig. \[fig7\]$\,$(b) becomes $R'(m)$ in Fig. \[fig8\]$\,$(a), which—for $\ell\ne 0$ or $\ell'\ne 0$—may consist of disjointed pieces pasted in up to four different sections of the unit square. The probability for $P'$ to be in the $m$th configuration is still the area ${\hat A}(m)={\rm area}(R(m))={\rm area}(R'(m))$, which in the latter case, could be a sum of areas of disjointed pieces. For fixed $\ell$ and $\ell'$ \[chosen to be $\ell=2$ and $\ell'=3$ in Fig. \[fig8\]$\,$(a)\], the position of $\alpha=\hat\alpha(k_{j+1})$ and $\beta=\hat\beta(k_{j})$ in the unit square shown in Fig. \[fig7\]$\,$(b) uniquely determines the configuration of $P(k_j,k_{j+1})$, while the position of $\alpha'=\hat\alpha(k_{j+1}\!+\!\ell')$ and $\beta'=\hat\beta(k_{j}\!+\!\ell)$ in Fig. \[fig8\]$\,$(a) determines the configuration of $P(k_j\!+\!\ell,k_{j+1}\!+\!\ell')$. As $k_j$ and $k_{j+1}$ run over all the values from $-\infty$ to $\infty$, we find from Kronecker’s theorem [@HW] that every point in either Fig. \[fig7\]$\,$(b) or Fig. \[fig8\]$\,$(a) is equally probable. However, the positions of $(\alpha,\beta)$ and $(\alpha',\beta')$ are completely correlated by the shift $(a,b)$ in (\[betap\]), which is fixed as long as $\ell$ and $\ell'$ are unchanged. Thus, the joint probability for $P(k_j,k_{j+1})$ to be in the $m$th configuration and $P(k_j\!+\!\ell,k_{j+1}\!+\!\ell')$ to be in the $m'$th configuration is the area of the intersection of the two regions $R(m)$ and $R'(m')$, and is denoted by $$A_{m,m'}(\ell,\ell')={\rm area}(R(m)\cap R'(m')). \label{overlap}$$ By superimposing Fig. \[fig7\]$\,$(b) on top of Fig. \[fig8\]$\,$(a), we obtain Fig. \[fig8\]$\,$(b). This figure gives the intersections of all the regions of Fig. \[fig7\]$\,$(b) with all the regions of Fig. \[fig8\]$\,$(a), and the joint probabilities can be read off as the areas of these intersections. For different values of $\ell$ and $\ell'$, we get different values of the width $a$ given in (\[betap\]) of the vertical slice in Fig. \[fig8\]$\,$(a) and also of the width $b$ of the horizontal slice. After cutting and pasting the difference slices, the resulting figures are very different, so are the superimposed figures. Thus, the area of intersection $A_{m,m'}(\ell,\ell')$ depends on the choice of $\ell$ and $\ell'$. However, just as ${\hat A}(m)$ and ${\hat A}(m')$ do not depend on the shifts $\gamma_j$, $A_{m,m'}(\ell,\ell')$ also does not depend on these shifts. Moreover, it is easily seen that this joint probability $A_{m,m'}(\ell,\ell')$ is not only the same for all the different regular pentagrids, but also the same for the different orientations of the parallelograms (i.e. different choices of $j=0,\cdots,4$). We let ${\vec K^{[m,n]}}$ denote the $n$th integer vector in the $m$th odd configuration for odd spins inside $P$, where $n=1,\cdots,N(m)$ and $m=1,\cdots,8$. Similarly ${\vec K^{[m',n']}}$ denotes the $n'$th integer vector of the $m'$th odd configuration of odd spins inside $P'$, with $n'=1,\cdots,N(m')$ and $m=1,\cdots,8$. From Tables \[conftab\], \[conftabb\] and \[oddtab\], the possible difference vectors $\partial\vec K^{[m,n]}$ and $\partial\vec K^{[m',n']}$ for two spins in $P$ and $P'$ may be found. Adding, as in (\[kz\]) and (\[kzp\]), these to their corresponding reference integer vectors, with three of their components given in (\[kj12\]), (\[kj3\]) and (\[kjp\]), we obtain the two integer vectors $\vec K^{\rm o}(z)={\vec K^{[m,n]}}$ for the spin in $P$ and $\vec K^{\rm o}(z')={\vec K^{[m',n']}}$ for the spin in $P'$. In (\[chipt\]) and (\[sumd\]) we only need their difference $${\vec K^{\rm o}}(z')-{\vec K^{\rm o}}(z)= {\vec K^{[m',n']}}-{\vec K^{[m,n]}} \equiv(\ell_0,\ell_1,\cdots,\ell_4). \label{dfell}$$ We find from (\[kz\]), (\[kjp\]) and (\[kzp\]) that $$\begin{aligned} &&\ell_j=\ell, \qquad \ell_{j+1}=\ell',\nonumber\\ &&\ell_{j+2}=\delta_{j+2}+\lfloor p^{-1}\ell'\rfloor-\ell +\partial K^{[m'\!,n']}_{j+2}-\partial K^{[m,n]}_{j+2} \equiv\ell'',\nonumber\\ &&\ell_{j+3}=-\delta_{j+2}-\delta_{j+4} -\lfloor p^{-1}\ell'\rfloor-\lfloor p^{-1}\ell\rfloor +\partial K^{[m'\!,n']}_{j+3}-\partial K^{[m,n]}_{j+3} \equiv\ell''',\nonumber\\ &&\ell_{j+4}=\delta_{j+4}+\lfloor p^{-1}\ell\rfloor-\ell' +\partial K^{[m'\!,n']}_{j+4}-\partial K^{[m,n]}_{j+4}\equiv\ell''''. \label{ell234}\end{aligned}$$ It is easy to see from (\[delta24\]) and (\[betap\]) that $$\begin{aligned} &&\delta_{j+2}=\cases{\begin{array}{lcr} 0&\hbox{if}&0\leqslant\{\alpha\}<1-a,\\ 1&\hbox{if}&1-a\leqslant\{\alpha\}<1,\upstrut \end{array}}\,\nonumber\\ &&\delta_{j+4}\!=\!\cases{\begin{array}{lcr} 0&\hbox{if}&0\leqslant\{\beta\}<1-b,\\ 1&\hbox{if}&1-b\leqslant\{\beta\}<1,\upstrut \end{array}}\end{aligned}$$ which shows that in the four sectors of the unit square shown in Fig. \[fig8\]$\,$(b), the $\delta_{i}$ pairs are different. As $a$ and $b$ are functions of $\ell$ and $\ell'$ only, and the entries in Tables \[conftab\], \[conftabb\] and \[oddtab\] are also independent of $j$, the results in (\[ell234\]) are easily seen to be functions of $\ell$ and $\ell'$, and of $\{\alpha\}$ and $\{\beta\}$, but they are [independent]{} of $j$. Therefore, we use the primed variables $\ell,\cdots,\ell''''$ to denote these $j$-independent values of $\ell_j,\cdots,\ell_{j+4}$, i.e.$$\tilde{\myb\ell}\equiv [\ell,\ell',\cdots,\ell'''']=[\ell_j,\ell_{j+1},\cdots,\ell_{j+4}]. \label{sumd3}$$ For different choices of $j$, $[\ell_0,\ell_1,\ell_2,\ell_3,\ell_4]$ is just a cyclic permutation of $\tilde{\myb\ell}$. If we let the distance vector $(\ell,\ell')$ between the two parallelograms $P=P(k_j,k_{j+1})$ and $P'=P(k_j\!+\!\ell,k_{j+1}\!+\!\ell')$ in (\[chipto\]) be fixed, while letting both $k_j$ and $k_{j+1}$ vary from $-\infty$ to $\infty$ (which is equivalent to ${\cal M}\to\infty$), then the parallelograms $P$ and $P'$ each are in one of eight different odd configurations. Since the joint probability for $P$ being in the $m$th configuration and $P'$ in the $m'$th configuration is $A_{m,m'}(\ell,\ell')$ given in (\[overlap\]), the double sum in (\[chipto\]) can be rewritten as $$\begin{aligned} &&{\hat\chi}^{\rm o}({\bf q})= \lim_{{\cal M}\to\infty}{\frac{1}{{\cal N}{\cal M}^2}} \sum_{\ell,\ell'}\sum_{k_j, k_{j+1}}\, \sum_{{\vec K}\vbox to 1.1ex{}^{\rm o}({\myb{\epsilon}})} {\sum_{{\vec K}{\vbox to 1.1ex{}^{\rm o}}'({\myb{\epsilon}'})}} U({\vec K^{\rm o}}({\myb{\epsilon}}),{\vec K^{\rm o}}\vphantom{K}' ({\myb{\epsilon}'}))\nonumber\\ &&\quad={\frac{1}{\cal N}}\sum_{\ell,\ell'}\sum_{m=1}^{8}\sum_{m'=1}^{8} A_{m,m'}(\ell,\ell') \sum_{n=1}^{N(m)}\sum_{n'=1}^{N(m')} U({\vec K^{[m,n]}},{\vec K^{[m',n']}}).\hspace{3em} \label{dsum}\end{aligned}$$ with ${\cal N}=5p$, cf. (\[spsum\]). Using (\[cor\]) and (\[dfell\]), we find (\[sumd\]) becomes $$U({\vec K^{[m,n]}},{\vec K^{[m',n']}})={\rm cos} \bigl[{\rm Re}(q^{\textstyle{\sum_{k=0}^4}}\ell_k\zeta^k)\bigr]\, \langle\sigma\sigma'\rangle_{[\ell_0,\ell_1,\cdots,\ell_4]}^{\rm c}, \label{sumd2}$$ which is different for different $j$ in view of (\[sumd3\]). Since the correlation functions have the cyclic property shown in (\[cyclic\]), and ${\hat\chi}^{\rm o}({\bf q})$ can be evaluated by choosing parallelograms $P$ and $P'$ in (\[chipto\]) oriented in any one of the five directions (any choice of $j$), we can rewrite ${\hat\chi}^{\rm o}(q_x,q_y)$ in a more symmetric way by expressing it as the sum over the five different orientations $j$, and then dividing the result by 5. This means, $${\hat\chi}^{\rm o}({\bf q})= \sum_{\ell=-\infty}^{\infty}\sum_{\ell'=-\infty}^{\infty} {\hat\chi}^{\rm o}({\bf q})_{\ell,\ell'} \label{ochi2}$$ where (\[dsum\]) to (\[sumd3\]) and (\[cyclic\]) are used to find $$\begin{aligned} &&{\hat\chi}^{\rm o}({\bf q})_{\ell,\ell'}={\frac 1{\cal N}} \sum_{m=1}^{8}\sum_{m'=1}^{8} A_{m,m'}(\ell,\ell') \sum_{n=1}^{N(m)}\sum_{n'=1}^{N(m')}c({\bf q},\tilde{\myb\ell})\,\langle \sigma\sigma'\rangle_{[\ell,\ell',\ell'',\ell''',\ell'''']},\nonumber\\ &&c({\bf q},\tilde{\myb\ell})={\frac 1 5}\sum_{j=1}^{5} \cos {\rm Re}\bigl[q^\zeta^{j} (\ell+\ell'\zeta+\ell''\zeta^2+\ell'''\zeta^3+\ell''''\zeta^4)\bigr].\quad \label{ochiell}\end{aligned}$$ It satisfies the following identities, $${\hat\chi}^{\rm o}({\bf q})_{-\ell,-\ell'}= {\hat\chi}^{\rm o}({\bf q})_{\ell,\ell'},\quad {\hat\chi}^{\rm o}({\bf q^})_{\ell',\ell}= {\hat\chi}^{\rm o}({\bf q})_{\ell,\ell'}, \label{qsymm}$$ in which ${\bf q^}\leftrightarrow(q_x,-q_y)$. The former identity in (\[qsymm\]) is easily seen as a consequence of the reflection symmetry in the correlation function $\langle\sigma\sigma'\rangle=\langle\sigma'\sigma\rangle$; the latter one is due to five-fold rotation and reflection symmetry.[^7] In the actual calculation, because $\delta_2$ and $\delta_4$ generally differ in four sectors of the unit square, the contributions to the susceptibility from these different sectors are evaluated separately. Results {#sect52} ------- To evaluate the wavevector-dependent susceptibility (\[ochi2\]), (\[ochiell\]), we can compute the $A_{m,m'}(\ell,\ell')$ as the overlap area (\[overlap\]) and the $\ell$’s from (\[ell234\]). In Table \[corrtab\] and Eq. (\[corems\]) in Section \[sect3\] we have expressed the pair-correlation function $\langle\sigma\sigma'\rangle_{[\ell_0,\cdots,\ell_4]}$ in terms of Baxter’s universal functions $g$ [@BaxZI], which can be evaluated using methods in our earlier work [@AJPq; @APmc1; @APmc2]. Near the critical point $k=1$ the leading asymptotic behavior of the pair-correlation function is the same Painlevé III or V scaling function [@AJPq; @APmc2] as in the uniform rectangular lattice [@WMTB76]. Therefore, the scaling behavior of the central peak of $\chi({\bf q})$ in our Penrose Ising model is also known [@APmc2; @APsus1; @APsus2] to be the same as for the regular Ising model. More interesting is the incommensurate behavior of $\chi({\bf q})$ as a function of wavevector ${\bf q}$ and how it changes with temperature or $k$. At the critical point we expect $\chi({\bf q})$ to be a function that has everywhere dense $7/4$-th power divergencies, but is locally integrable. It is nontrivial to show this directly, but this conclusion seems to impose itself as one approaches the critical point from either side. Since the correlation functions decay exponentially away from the critical point, we find that the ${\hat\chi}^{\rm o}({\bf q})_{\ell,\ell'}$ are rapidly decreasing functions of $\ell$ and $\ell'$. Putting terms of about the same order of magnitude together, we find $${\hat\chi}^{\rm o}({\bf q})={\hat\chi}^{\rm o}({\bf q})_{0,0}+ \sum_{\ell=1}^{\infty}{\cal S}_{\ell},\quad {\cal S}_{\ell}=2\!\sum_{n=-\ell+1}^{\ell}[{\hat\chi}^{\rm o}({\bf q})_{\ell,n} +{\hat\chi}^{\rm o}({\bf q^})_{\ell,n-1}], \label{sl}$$ using both identities in (\[qsymm\]). We shall give density plots of several cases next, displaying the temperature dependence more clearly. However, it must be said that we can calculate ${\hat\chi}^{\rm o}({\bf q})$ to very high precision in the cases shown, which fact is not clear from looking at these density plots. We shall give plots both above the critical temperature $T_{\rm c}$, ($k_>\!\equiv\!k\!<\!1$), and below $T_{\rm c}$, ($k_<\!\equiv\!1/k\!<\!1$). The value of modulus $k$ corresponds to the row correlation length[@MWbk] $$\xi=1/\|\hbox{arsinh}(1/\sqrt{k})-\hbox{arsinh}(\sqrt{k})\| \label{onscorr}$$ of the symmetric square-lattice Ising model for $T>T_{\rm c}$. For $T<T_{\rm c}$ the true value of this row correlation length is $\xi/2$ with $\xi$ again given by (\[onscorr\]).[@MWbk; @WMTB76] At very low temperature, we only need to consider ${\cal S}_{\ell}$ for very small $\ell$. For $\ell,\ell'\leqslant2$ the joint probabilities $A_{m,m'}(\ell,\ell')$ can be easily evaluated by hand as it only involves the calculation of areas of triangles and rectangles. For $k_<=.04847302$, which corresponds to $\xi\approx1/2$, we find ${\cal S}_{\ell}<10^{-10}$ for $\ell>2$. The density plot for $1/{\hat\chi}^{\rm o}({\bf q})$ is shown in Fig. \[fig9\]$\,$(a) for $-4\pi\leqslant q_x,q_y\leqslant 4\pi$ where ${\bf q}=(q_x,q_y)$. We find ten-fold symmetry, corresponding to the five-fold symmetry of the Penrose tiling. =0.475=0.370=0.475=0.370 to -0.2in For $k_<=.2363562$ ($\xi\approx1$), we find it necessary to consider all ${\cal S}_{\ell}$ for $\ell\leqslant4$. As the temperature increases, larger and larger $\ell$’s are needed. To evaluate the joint probability by hand is no longer feasible. To symbolically program the calculation for any values of $\ell$ and $\ell'$ is highly nontrivial, as there are many different situations to take into account. It took us several months to sort out all cases, programming the calculation using Maple. =0.475=0.370=0.475=0.370 to 0.1in =0.475=0.370=0.475=0.370 to -0.2in A density plot for $k_<=.2363562$ is shown in Fig. \[fig9\]$\,$(b). Plots of $1/{\hat\chi}^{\rm o}({\bf q})$ for $\xi\approx 4$ and $\xi\approx 8$ are shown in Fig. \[fig10\]$\,$(a) and Fig. \[fig10\]$\,$(b), together with corresponding plots for the dual cases with $T>T_{\rm c}$ in Fig. \[fig10\]$\,$(c) and Fig. \[fig10\]$\,$(d). We can see clearly that the number of visible peaks increases as $T\to T_{\rm c}$ and that this effect is more pronounced as $T_{\rm c}$ is approached from above. In Fig. \[fig11\], a density plot for $1/{\hat\chi}^{\rm o}({\bf q})$ at $\xi=2$ is given for $-16\pi\leqslant q_x,q_y\leqslant 16\pi$, and we can already see some evidence for the quasiperiodic pattern of the ${\bf q}$-dependent susceptibility in the full ($q_x,q_y$)-plane. -1in=1.0=0.779 0.2in The q-dependent susceptibility $\myb{\chi^{\rm e}({\bf q})}$ {#sect53} ------------------------------------------------------------ Finally, we show that the ${\bf q}$-dependent susceptibility ${\hat\chi}^{\rm e}({\bf q})$ of the even sublattice is identical to ${\hat\chi}^{\rm o}({\bf q})$ of the odd sublattice. From Tables \[conftab\] and \[conftabb\], we find there are 16 different even configurations for $P(k_j,k_{j+1})$. The corresponding 16 regions are plotted in the unit square with $\{\alpha\}$ and $\{\beta\}$ along the axes in Fig. \[fig12\]$\,$(a). The regions for the 8 odd spin configurations of $P(k_j\!+\!1,k_{j+1}\!+\!1)$ are also plotted with $\{\alpha\}$ and $\{\beta\}$ as axes in Fig. \[fig12\]$\,$(b). It is easy to see that after inverting one of the squares, the two figures are identical up to labeling. Furthermore, looking at Fig. \[fig12\]$\,$(b), we see that the five disjoint regions $2,\cdots,6$ become connected, if we impose periodic boundary conditions on the square. This “wrapping on a torus" is consistent with moving the slices as discussed below Eq. (\[betap\]). We next compare how the even regions in Fig. \[fig12\]$\,$(a) relate under the same periodic boundary conditions, to see if this is somehow true here also. As first examples we look at the two even configurations $\mathrm{e}(7)$ and $\mathrm{e}(13)$ and find that they have the same number of sites, and their difference vectors are related by $$\delta{\vec K^{\mathrm{e},7}}=(0,-1,1)+\delta{\vec K^{\mathrm{e},13}}. \label{eveneven1}$$ For the other regions, we similarly find equal numbers of sites and $$\begin{aligned} &&\delta{\vec K^{\mathrm{e},10}}=(1,-1,0)+\delta{\vec K^{\mathrm{e},1}},\quad \delta{\vec K^{\mathrm{e},3+m}}=(1,-1,0)+\delta{\vec K^{\mathrm{e},14+m}}, \hspace{2em}\nonumber\\ &&\delta{\vec K^{\mathrm{e},6}}=(0,-1,1)+\delta{\vec K^{\mathrm{e},1}},\quad \delta{\vec K^{\mathrm{e},8+n}}=(0,-1,1)+\delta{\vec K^{\mathrm{e},15+n}}, \label{eveneven2}\end{aligned}$$ where $m=0,1,2$ and $n=0,1$. The difference vectors of the odd configurations in Fig. \[fig12\]$\,$(b) are also related to those of the even configurations in Fig. \[fig12\]$\,$(a), i.e. $$\begin{aligned} &&\delta{\vec K^{\mathrm{e},1}}=(1,-1,1)-\delta{\vec K^{\mathrm{o},2}},\quad \delta{\vec K^{\mathrm{e},2}}=(1,-1,1)-\delta{\vec K^{\mathrm{o},1}}, \hspace{2em}\nonumber\\ &&\delta{\vec K^{\mathrm{e},11+n}}=(0,1,0)-\delta{\vec K^{\mathrm{o},8-n}},\quad n=0,\cdots,5. \label{evenodd}\end{aligned}$$ If the dependences of difference vectors on the values of $\{\alpha\}$ and $\{\beta\}$ are included in the equations, we may relate the even spins in $P(k_j,k_{j+1})$ with the odd spins in $P(k_j\!+\!1,k_{j+1}\!+\!1)$ by $$\delta{\vec K^{\rm e}}[\{\alpha\},\{\beta\}]=(1,-1,1)-(\delta_2, -\delta_2\!-\!\delta_4, \delta_4)-\delta{\vec K^{\rm o}}[\{-\alpha\},\{-\beta\}], \label{devenodd}$$ where $1-\{x\}=\{-x\}$, for $x$ not an integer, and $$\delta_{2}=\lfloor{1-\{\alpha\}+p^{-1}}\rfloor,\quad \delta_{4}=\lfloor{1-\{\beta\}+p^{-1}}\rfloor.$$ Consider the pentagrid with $\gamma_j\to -\gamma_j$ and denote its parallelograms by ${\bar P}(k_j,k_{j+1})$ such that $$\bar\alpha(-k_{j+1})=-\alpha(k_{j+1}),\quad \bar\beta(-k_{j})=-\beta(k_{j}). \label{balpha}$$ Then it is easy to show that $$\delta_{2}+\lfloor{-\alpha(k_{j+1})}\rfloor =\lfloor{\bar\alpha(1\!-\!k_{j+1})}\rfloor,\quad \delta_{4}+\lfloor{-\beta(k_j)}\rfloor =\lfloor{\bar\beta(1\!-\!k_j)}\rfloor.$$ Now, the integer vectors of even spins in $P(k_j,k_{j+1})$ given by (\[kz\]), (\[kj12\]) and (\[kj3\]) can be shown to relate to the integer vectors of the odd spins in ${\bar P}(1\!-\!k_j,1\!-\!k_{j+1})$ by $${\vec K^{\rm e}}({\myb{\epsilon'}}) =(1,1,1,1,1)-{\vec K^{\rm o}}({\myb{\epsilon}}). \label{ievenodd}$$ This equation is consistent with the fact that the index of the odd spins is either 1 or 3, while the index of the even spins is either 2 or 4. Since the even spins in the original pentagrid are related to the odd spins in a different pentagrid, and the joint probabilities are independent of shifts, we have shown that the susceptibility of the even sublattice is identically the same as the one of the odd sublattice. 0.01in0.1in =0.45=0.45 to -0.2in Conclusions and Final Remarks {#sect6} ============================= In this paper we have presented a systematic way of evaluating the averaged pair-correlation function of a $Z$-invariant ferromagnetic Ising model with spins on half the sites of a Penrose tiling and Ising interactions across the diagonals of the rhombuses. Next, we have found that the ${\bf q}$-dependent susceptibility of this model is a superposition of incommensurate everywhere-dense peaks, though not many peaks are visible at temperatures very far away from $T_{\rm c}$. For $T<T_{\rm c}$ these peaks add a diffuse background to the Bragg peaks due to the spontaneous magnetization. Since the $S_{\ell}$ in (\[sl\]) consists of $4\ell$ terms of the same order of magnitude, we compare their contributions. We find that the number of peaks of $S_{\ell}$ increases as $\ell$ increases, but that the numbers at fixed $\ell$ are almost independent of temperature, even though the magnitudes of the peaks change as the temperature varies. As $T\to T_{\rm c}$, the correlations decay more and more slowly, so the $S_{\ell}$’s increase, and more and more of these $S_{\ell}$’s are to be included in the numerical evaluation of the ${\bf q}$-dependent susceptibility, which accounts for the ever-increasing number of peaks. This is unlike the behavior of the Fibonacci Ising models, considered earlier,[@AJPq; @APmc1] where the ferromagnetic aperiodic Fibonacci lattice behaves almost like the regular Ising model. Moreover, the ${\bf q}$-dependent susceptibility is not a periodic function of $q_x$ or $q_y$. This behavior is different from that of aperiodic models defined on regular lattices.[@AJPq] This is because, when the lattice is aperiodic, we cannot separate the average of the correlation functions from the exponential (or cosine) terms, which contain the information about the lattice structure, as can be seen from (\[ochi2\]) and (\[ochiell\]). At $T_{\rm c}$, the ${\bf q}$-dependent susceptibility has everywhere-dense divergences with the Ising exponent $7/4$, but is still locally integrable. Away from $T_{\rm c}$ the $\chi({\bf q})$ is a continuous function. The Bragg peaks below $T_{\rm c}$ form a set of everywhere-dense Dirac delta functions of various strengths, but their sum is also locally integrable. These are strange objects and de Bruijn initiated their mathematical study[@Bruijn3]. One of the main results of the current paper is that it provides a new method for doing calculations of probabilities on Penrose tilings. In Section \[sect4\], the calculation of the joint probability of the configurations of two parallelograms on the pentagrid is reduced to linear programming. Penrose tilings may be obtained by projecting certain subsets of the $\mbox{\mymsbm Z}^5$ lattice into the plane[@Bruijn1]. The frequencies of the different types of vertices are then given as areas in the orthogonal spaces[@Bruijn1]. This method, which is known as the cut-and-project method, has been applied and generalized by many authors [@Mackay2; @CN; @GRh; @KGR; @DK; @KKL; @LSt0; @Jar; @Elser; @RGS; @RRG; @BKSZ; @BJKS]. The equivalence of the projection method and a generalized grid method has been demonstrated[@CN; @GRh; @KGR]. The positions of the Bragg peaks have been worked out [@Mackay2; @DK; @KKL; @LSt0; @Jar; @Elser], together with the values of probabilities of local configurations. The Penrose tilings can even be obtained from projections in a four-dimensional root lattice[@BKSZ; @BJKS]. It would be interesting to obtain results for joint probabilities similar to ours also by cut-and-project methods. This would generalize the windowing method of Baake and Grimm[@BG]. We have not pursued this here, as our model is defined in terms of rapidity lines on the pentagrid. Another possible generalization of our work is to consider Penrose tilings that are periodic in either one or both directions. Finite approximants to the aperiodic Penrose tiling have been constructed through periodic pentagrids or projection methods.[@TFUT; @TFUT2; @TFUT3; @RGS] The mathematician Robinson has brought to our attention an exercise in the book by Grünbaum and Shephard [@GrSh] where a tiling with Penrose rhombuses can be cut into patches, and then converted into an aperiodic set of 24 Wang’s tiles. We have found that each patch in the Penrose tiling described in the exercise is in fact the image of a parallelogram under the mapping (\[penrose\]). Thus these 24 configurations of the parallelogram can be easily converted into Wang’s tiles. Finally, in our previous studies [@AJPq] we have examined the $q$-dependent susceptibility $\chi({\bf q})$ of some quasiperiodic Ising models on the square lattice defined in terms of Fibonacci sequences. It may be of interest to study models based on other sequences such as the aperiodic sequences studied by de Bruijn[@Bruijn2] or Tracy[@Tr2]. We may ask what effect this has on the mixed interaction cases and also if it makes any difference for purely ferromagnetic models. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =================== An earlier version of this work has been presented to Professor M. E. Fisher at a conference in honor of his seventieth birthday. We are most thankful to Dr. M. Widom for his interest in using exactly solvable models to study quasicrystals, which led us to start this work. We also thank Dr. M. Baake, Dr. U. Grimm, and Dr. E. A.Robinson for providing us with many useful references. This work has been supported by NSF Grant PHY 01-00041. ‘=11 ‘=12 [000]{} =cmssi10 [^1]: Universality of the critical exponents of ferromagnetic Ising models on quasiperiodic lattices has been confirmed for non-$Z$-invariant cases also using real-space renormalization group techniques,[@AO1] Monte Carlo simulations,[@AAA; @BHJ; @ON1; @ON2; @SJR; @TJ; @RB] series expansion methods,[@AbeDot; @DotAbe; @Repet] and the study of Yang–Lee zeros.[@SBG; @SB; @RGS2] [^2]: Unlike the ${\bf q}$-dependent susceptibility, thermodynamic quantities like the free energy, the specific heat, and the bulk susceptibility do not probe the lattice structure, although subtle lattice effects do show up in corrections to scaling.[@APsc] [^3]: For $T=T_{\rm c}$, the correlations decay algebraically, so that we need the Epstein-Ewald summation formula[@Kong] to take into account the long-distance behavior. [^4]: He also proved that if $\xi\equiv\sum_j\gamma_j\zeta^{2j}$ times a power of $\zeta$ is not purely imaginary (modulo the principal ideal of $1-\zeta$ given by all complex numbers of the form $\sum_j n_j\zeta^j$ with integers $n_0,\cdots,n_4$ satisfying $\sum_j n_j=0$), then the corresponding pentagrid is regular [@Bruijn1]. [^5]: In our notation we suppress the two trivial components $\partial K_{j}(\myb{\epsilon})\equiv \partial K_{j+1}(\myb{\epsilon})\equiv0$, since in parallelogram $P$ we have by definition $K_{j}(\myb{\epsilon})\equiv k_j$, $K_{j+1}(\myb{\epsilon})\equiv k_{j+1}$. [^6]: From the first members of (\[kjp\]) and (\[kj12\]) we find that $\delta_{j+2}= \lceil\alpha'\rceil-\lceil\alpha\rceil-\lfloor p^{-1}\ell'\rfloor= \lceil\alpha+p^{-1}\ell'\rceil-\lceil\alpha\rceil- \lfloor p^{-1}\ell'\rfloor=\lceil\{\alpha\}-1+\{p^{-1}\ell'\}\rceil= \lfloor\{\alpha\}+\{p^{-1}\ell'\}\rfloor$. For the last two steps here and the remaining steps in the derivation of (\[kjp\]) and (\[delta24\]) we must make explicit use of the fact that $\alpha$, $\alpha'$, $\beta$ and $\beta'$ are not integer for a regular pentagrid. [^7]: In particular, one can start with the reflection symmetry about the direction of the $k_{j+3}$ grid-line and its action on the parallelograms $P(k_j,k_{j+1})$ and $P(k'_j,k'_{j+1})$. One arrives at $\langle\sigma\sigma'\rangle_{[\ell,\ell',\ell'',\ell''',\ell'''']}= \langle\sigma\sigma'\rangle_{[\ell',\ell,\ell'''',\ell''',\ell'']}$. Replacing $j\rightarrow1-j$ in (\[ochiell\]) then completes the proof of the second identity in (\[qsymm\]).
--- abstract: 'Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson–Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine–Tristram signatures to compute these bounds and produce explicit examples.' address: - 'Rényi Institute of Mathematics, Budapest, Hungary' - 'Department of Mathematics, Uppsala University, Sweden' - 'Aix Marseille Univ., CNRS, Centrale Marseille, I2M, Marseille, France' author: - Paolo Aceto - Marco Golla - 'Ana G. Lecuona' bibliography: - 'casson-gordon.bib' title: 'Handle decompositions of rational homology balls and Casson–Gordon invariants' ---
--- address: - 'University of Warwick, Mathematics Institute, Coventry CV4 7AL, UK' - 'Caltech, Department of Mathematics, MC 253-37, 362 Sloan Laboratory, Pasadena, CA 91125, USA' - 'JGU Mainz, Institut für Mathematik, Staudingerweg 9, 55128 Mainz, Germany' author: - 'Michel van Garrel, Tom Graber, Helge Ruddat' title: 'Local Gromov-Witten Invariants are Log Invariants' --- [^1] Introduction ============ Let $X$ be a smooth projective variety, $D$ a smooth nef divisor on $X$, and $\beta$ a curve class on $X$ with $d:=\beta\cdot D >0$. The goal of this note is to prove a simple equivalence between two virtual counts of rational curves on $X$ that can be associated to this situation. First there are the [local invariants]{}, the Gromov-Witten invariants which virtually count curves in the total space of ${\mathcal{O}}_X(-D)$. These can be defined as integrals against the virtual fundamental class of ${\overline{\mathcal{M}}}_{0,0}({\operatorname{Tot}}({\mathcal{O}}_X(-D)))$. The conditions on $D$ and $\beta$ are exactly the ones under which these are well-defined. Second, we can consider the relative or logarithmic invariants which virtually count rational curves in $X$ which intersect $D$ in a single point, necessarily with multiplicity $d$. These could also be thought of as a virtual count of maps from ${\mathbb{A}}^1$ to the open variety $X\backslash D$. There exist several moduli spaces that can be used to define these counts. We will use the space of logarithmic stable maps to the log smooth space $X(\log D)$ associated to the pair $X$ and $D$. This space parametrizes many types of maps with different specified contact orders, but the one directly relevant to this problem can be denoted ${\overline{\mathcal{M}}}_{0,(d)}(X(\log D),\beta)$, where the $(d)$ is meant to denote a single marked point having maximal contact order with $D$. Our main result is a comparison of the two virtual fundamental classes here. \[mainthm-intro\] $F_[{\overline{\mathcal{M}}}_{0,(d)}(X(\log D),\beta)]^{{\rm vir}}=(-1)^{d+1}d[{\overline{\mathcal{M}}}_{0,0}({\operatorname{Tot}}({\mathcal{O}}_X(-D)),\beta)]^{{\rm vir}}$. Here $F$ denotes the map that takes a logarithmic map to $X(\log D)$ and forgets the logarithmic structure as well as the marked point. By the negativity of ${\mathcal{O}}_X(-D)$, the space of maps to ${\mathcal{O}}_X(-D)$ is just the same as the space of maps to $X$, so both sides of the formula are being considered as elements of $A_({\overline{\mathcal{M}}}_{0,0}(X,\beta))$. We remark that we show the same formula holds if we add $n$ marked points to the moduli problems on each side (with 0 contact order with $D$ in the case of left hand side). This equality of virtual cycles leads to analogous equalities of certain numerical invariants after capping with constraints and integrating. Precisely, let $\gamma=(\gamma_1,....,\gamma_n)$ denote a collection of insertions, with each $\gamma_i$ either a primary invariant (i.e. no descendants) or a descendant of a cycle class which does not meet $D$. Let $N_\beta(\gamma)$ denote the genus zero local Gromov-Witten invariant for these insertions and $R_\beta(\gamma)$ the genus zero relative Gromov-Witten invariant with the same insertions (and one maximal contact order marking as before), then as a direct corollary of Theorem \[mainthm-intro\], we find: \[maincor-intro\] $R_\beta(\gamma)=(-1)^{d+1}dN_\beta(\gamma)$. This follows from Theorem \[mainthm-intro\] and the projection formula, since the restriction on the insertions corresponds exactly to the condition that the constraints on the space of log stable maps be pulled back from classes on ${\overline{\mathcal{M}}}_{0,n}(X,\beta)$. This formula was first conjectured by Takahashi [@Ta01] for $X={\mathbb{P}}^2$ and $D$ a smooth cubic. It was proven in this case by Gathmann via explicit calculation of both sides [@Ga03]. In an unpublished note, Graber and Hassett gave a simple proof of the formula when $X$ is any smooth del Pezzo surface and $D$ is a smooth anti-canonical divisor. The proof we will give here follows the main idea of that argument which is to apply the degeneration formula for Gromov-Witten invariants [@LR01; @Li02; @IP04; @AF11; @Ch14; @KLR17] to a twist of the degeneration to the normal cone of the preimage of $D$ in ${\operatorname{Tot}}({\mathcal{O}}(-D))$. In the general setting though, we need to take into account the existence of rational curves in $D$. In order to control these, we need to study the following situation which may be of independent interest, comparing the virtual geometry of genus zero stable log maps to certain projective bundles to the geometry of stable maps to the base. Let ${\mathcal{A}}$ be a nef line bundle on a smooth projective variety $D$, $$Y=\mathbb{P}({\mathcal{O}}_D\oplus{\mathcal{A}}) \stackrel{p}{\longrightarrow} D$$ the ${\mathbb{P}}^1$-bundle over $D$ and $\beta$ a curve class on $Y$. Let $D_0$ be the section of $p$ with normal bundle ${\mathcal{A}}^\vee$ and let ${\overline{\mathcal{M}}}_{0,\Gamma}(Y(\log D_0),\beta)$ be the space of genus zero basic stable log maps to $Y$ where $\Gamma$ is a list of $n$ points with prescribed contact orders to $D_0$. We have a pushforward map $$w: {\overline{\mathcal{M}}}_{0,\Gamma}(Y(\log D_0),\beta) \to {\overline{\mathcal{M}}}_{0,n}(D,p_\beta)$$ which forgets the log structures, composes with the projection to $D$ and stabilizes. We would like to say that the virtual class on the space of log stable maps to $Y(\log D_0)$ is the pullback of the virtual class of the space of stable maps to $D$, but since the map $w$ need not be flat, there is no well-defined pullback in general. Nevertheless, we have the following result. \[thm-P1-bundle-pullback\] The map $w$ factors through an intermediate space ${\mathcal{M}}$, i.e. ${\overline{\mathcal{M}}}_{0,\Gamma}(Y(\log D_0),\beta)\stackrel{u}{{\longrightarrow}}{\mathcal{M}}\stackrel{v}{{\longrightarrow}} {\overline{\mathcal{M}}}_{0,n}(D,p_\beta)$ where $u$ is smooth and $v$ a pull-back of a map $\nu$ to a smooth stack, see for details. Then $$[{\overline{\mathcal{M}}}_{0,\Gamma}(Y(\log D_0),\beta)]^{{\rm vir}}= u^\nu^![{\overline{\mathcal{M}}}_{0,n}(D,p_\beta)]^{{\rm vir}}.$$ In fact, this theorem is proven in more generality below – the only feature needed of the map $p:Y(\log D_0) \to D$ is that it is a log smooth projective morphism whose relative log tangent bundle satisfies a positivity condition. We believe that our main result generalizes as follows. \[conjecture\] Let $D\subset X$ be a normal crossing divisor with smooth nef components $D_1,...,D_k$ and $\beta$ a curve class such that $d_i:=\beta\cdot D_i>0$ for all $i$. Let $$F:{\overline{\mathcal{M}}}_{0,(d_1),...,(d_k)}(X(\log D),\beta){\to}{\overline{\mathcal{M}}}_{0,0}(X,\beta)$$ be the forgetful map from the moduli space of genus zero basic stable log maps with one marking for each $D_i$ requiring maximal order of contact $d_i$ at $D_i$, then $$F_[{\overline{\mathcal{M}}}_{0,(d_1),...,(d_k)}(X(\log D))]^{{\rm vir}}=\left(\prod_{i=1}^k(-1)^{d_i+1}d_i\right)[{\overline{\mathcal{M}}}_{0,0}({\operatorname{Tot}}({\mathcal{O}}_X(-D_1)\oplus....\oplus {\mathcal{O}}_X(-D_k)))]^{{\rm vir}}.$$ We conjecture also the similar statement where further markings are added on both sides (with zero contact orders to the $D_i$ for the left hand side). As evidence for this conjecture, note that it allows computing the local invariant $\frac{1}{d^3}$ of ${\mathcal{O}}_{{\mathbb{P}}^1}(-1,-1)$ for degree $d$ curves from the unique Hurwitz cover of ${\mathbb{P}}^1$ with maximal branching at $0$ and $\infty$ and cyclic order $d$ automorphism, so the log invariant is $\frac1{d}$, see also [@MR16 Remark 4.17]. We also checked Conjecture \[conjecture\] for ${\mathbb{P}}^n$ with $D$ the toric boundary in the case of a single insertion of $\psi^{n-1}[pt]$: the log invariant can easily be computed from [@MR16 Theorem 1.1]+[@MR19] and equals $1$ for all $n$ and $d$. We are grateful to Andrea Brini for computing for us the local invariant for this situation confirming the conjecture in this case. We thank Dhruv Ranganathan for pointing out that [@PZ08 Lemma 3.1] computes the virtual count of degree $d$ genus zero curves in ${\operatorname{Tot}}({\mathcal{O}}_{{\mathbb{P}}^2}(-1)^{\oplus 3})$ passing through two points as $(-1)^{d-1}/d$ and using [@MR16 Theorem 1.1] for the computation of the log invariant confirms the conjecture also in this case. Further evidence will appear in [@BBvG]. Once the technology of punctured Gromov-Witten invariants and a more general degeneration formula is fully developed, we expect that the proof we give in this paper could be iterated to imply Conjecture \[conjecture\]. If the $D_i$ are disjoint (as in the first example above) the existing technology is already enough to establish the conjecture. We are grateful to Dan Abramovich, Mark Gross, Davesh Maulik, Rahul Pandharipande and James Pascaleff for helpful and inspiring discussions, and to the anonymous referee for valuable suggestions. Deducing the main result from the degeneration formula {#section-setup} ====================================================== First we briefly describe the strategy of proof. We have a smooth projective variety $X$ containing a nef divisor $D$. If we let ${\mathcal{X}}= {\operatorname{Bl}}_{D\times\{0\}}(X\times {\mathbb{A}}^1){\to}{\mathbb{A}}^1$ be the degeneration to the normal cone of $D$ in $X$, then we get a family over ${\mathbb{A}}^1$ whose general fiber is $X$ and whose special fiber is a union of a copy of $X$, which we denote by $X_0$, and a ${\mathbb{P}}^1$-bundle over $D$, which we will denote by $Y$. Precisely, if we let ${\mathcal{A}}={\mathcal{N}}_{D/X}$, then $Y=\mathbb{P}({\mathcal{O}}_D\oplus{\mathcal{A}})\stackrel{p}{{\to}} D$. This ${\mathbb{P}}^1$ bundle comes with two obvious sections whose images we denote by $D_0$ and $D_\infty$ which have normal bundles ${\mathcal{A}}^\vee$ and ${\mathcal{A}}$ respectively. The intersection of $X_0$ with $Y$ is given by $D$ in $X_0$ and by $D_0$ in $Y$. To construct a degeneration of the total space of ${\mathcal{O}}_X(-D)$, we can just look at the total space of any line bundle over ${\mathcal{X}}$ whose restriction to a general fiber is ${\mathcal{O}}(-D)$. Our choice, which we denote by ${\mathcal{L}}$, is ${\operatorname{Tot}}({\mathcal{O}}(-{\mathcal{D}}))$ where ${\mathcal{D}}$ is the proper transform of $D \times {\mathbb{A}}^1$. Then, since ${\mathcal{D}}\cap X_0 = \emptyset$ and ${\mathcal{D}}\cap Y = D_\infty$ we find that the projection ${\mathcal{L}}\to {\mathbb{A}}^1$ gives us a family whose general fiber is ${\operatorname{Tot}}{\mathcal{O}}_X(-D)$ and whose special fiber is a union of two components $L_X$, and $L_Y$ where $L_X \cong X \times {\mathbb{A}}^1$ and $L_Y \cong {\operatorname{Tot}}({\mathcal{O}}_Y(-D_\infty)$. Applying the degeneration formula to this family will then relate Gromov-Witten theory of ${\operatorname{Tot}}({\mathcal{O}}_X(-D)$ to the relative or log invariants of the two pairs $(X \times {\mathbb{A}}^1, D\times {\mathbb{A}}^1)$ and $(L_Y, D_0 \times {\mathbb{A}}^1)$. While the degeneration formula in general is quite complicated and given by a sum over combinatorial types of curve degenerations, we will see that for this degeneration in genus zero, every term but one in that sum vanishes, and we can find explicitly the contribution from $L_Y$, so we are left with an expression for the genus zero Gromov-Witten theory of ${\mathcal{O}}_X(-D)$ in terms of that of $(X\times {\mathbb{A}}^1, D\times {\mathbb{A}}^1)$ which is just the same as that of $(X,D)$. In order to carefully write down a proof of that vanishing, we will need to establish our conventions about notations for the degeneration formula. Because we will eventually need to make use of the theory of logarithmic stable maps, we will state the version in that setting, although it is worth noting that thanks to the comparison theorems of [@AMW14], we could equally well use the better known formalism for degeneration in terms of relative invariants. Degeneration formula {#sec-degenformula} -------------------- We recall the degeneration formula where for us the version in [@KLR17 Theorem 1.4] is most convenient, further details on this subsection can be found there. The input is a space ${\mathcal{X}}_0$ that is log smooth over the standard log point with the additional assumption that ${\mathcal{X}}_0= X\sqcup_D Y$ is a union of two smooth components that meet along $D$, a smooth divisor in each component. We are interested in basic stable log maps to ${\mathcal{X}}_0$. The domain curve components map into either $X$ or $Y$ (or both) and the degeneration formula uses this fact effectively to decompose the moduli space of stable maps and hence the virtual fundamental class and Gromov-Witten invariants as a sum of contributions from each type of stable map. We next give the details for the situation of genus zero with $n\ge 0$ markings. We recall [@KLR17 §2]: let a curve class $\beta$ in ${\mathcal{X}}_0$ and an integer $n$ be given, and let $\Omega({\mathcal{X}}_0)$ denote the set of graphs $\Gamma$ with the following decorations and properties. The vertices of $\Gamma$ are partitioned into two sets indexed by $X$ and $Y$ and no edge has vertices labeled by only $X$ or only $Y$, this property for a graph is typically called bipartite. We will henceforth speak of $X$- and $Y$-vertices, referring to the partition membership. The edges of $\Gamma$ are enumerated $e_1,...,e_r$ (where $r$ may vary), each edge $e$ is decorated with a positive integer $w_e$, each vertex $V$ is decorated with a set $n_V$ and a class $\beta_V$ that is an effective curve class in $X$ or $Y$ depending on the bipartition type of $V$. The set $n_V$ is a subset of $\{1,...,n\}$ to be thought of as the set of marking labels attached to $V$. Every $\Gamma\in \Omega({\mathcal{X}}_0)$ is subject to the following stability condition: if $\beta_V=0$, then the valency of $V$ is at least $3$. Furthermore, $\beta =\sum_V\beta_V$ and $\beta_V\cdot D=\sum_{V\in e} w_e$ and $\{1,...,n\}=\coprod_V n_V$. These conditions make $\Omega({\mathcal{X}}_0)$ a finite set. Given $\Gamma\in\Omega({\mathcal{X}}_0)$ and a vertex $V$ of $\Gamma$, we define $\Gamma_V$ as the “subgraph” of $\Gamma$ at the vertex $V$. That is, $\Gamma_V$ has a single vertex $V$ that is decorated with the set $n_V$ and curve class $\beta_V$ and has as adjacent half-edges the edges adjacent to $V$ in $\Gamma$ with their weights. If $V$ is an $X$-vertex, we define ${\overline{\mathcal{M}}}_V:={\overline{\mathcal{M}}}_{\Gamma_V}(X(\log D),\beta_V)$, that is, the moduli space of genus zero stable maps to $X(\log D)$ of class $\beta_V$, with edges of $\Gamma_V$ indexing the markings with contact order to $D$ given by the weight of an edge and $n_V$ enumerating additional markings (with contact order zero to $D$). Analogously, if $V$ is a $Y$-vertex, we set ${\overline{\mathcal{M}}}_V:={\overline{\mathcal{M}}}_{\Gamma_V}(Y(\log D),\beta_V)$. We define $\bigodot _V {\overline{\mathcal{M}}}_V $ by the Cartesian square $$\label{gluediag} \vcenter{ \xymatrix{ \bigodot _V {\overline{\mathcal{M}}}_V \ar[r]\ar_{\operatorname{ev}}[d]& \prod_V {\overline{\mathcal{M}}}_V\ar^{\operatorname{ev}}[d]\\ \prod_{e} D \ar^(.4)\Delta[r]& \prod_V\prod_{e\ni V} D.\\ } }$$ For each $\Gamma\in\Omega({\mathcal{X}}_0)$, consider the moduli space ${\overline{\mathcal{M}}}_\Gamma$ of basic stable log maps to ${\mathcal{X}}_0$ where the curves are marked by $\Gamma$, i.e. a subset of the nodes is marked by $e_1,...,e_r$ and the dual intersection graph collapses to $\Gamma$, for details see [@KLR17 §4]. There is an étale map that partially forgets log structure $\Phi:{\overline{\mathcal{M}}}_\Gamma{\to}\bigodot _V {\overline{\mathcal{M}}}_V$ and another finite map that forgets the graph-marking $G:{\overline{\mathcal{M}}}_\Gamma{\to}{\overline{\mathcal{M}}}_{0,n}({\mathcal{X}}_0,\beta)$ where the latter refers to the moduli space of $n$-marked basic stable log maps to the log space ${\mathcal{X}}_0$ that is log smooth over the standard log point. \[degen-formula\] We have $$[{\overline{\mathcal{M}}}_{0,n}({\mathcal{X}}_0,\beta)]^{{\rm vir}}= \sum_{\Gamma\in \Omega({\mathcal{X}}_0)} \frac{{\operatorname{lcm}}(w_{e_1},...,w_{e_r})}{r!} G_\Phi^\Delta^! \prod_V[{\overline{\mathcal{M}}}_V]^{{\rm vir}}.$$ Setup {#sec-degen-formula} ----- Instead of writing ${\overline{\mathcal{M}}}_{0,n}(..)$ to refer to moduli spaces of genus zero basic stable log maps with no markings, we simply write ${\overline{\mathcal{M}}}(..)$ in the following. Since ${\mathcal{L}}{\to}{\mathbb{A}}^1$ and ${\mathcal{X}}{\to}{\mathbb{A}}^1$ are log smooth when given the divisorial log structure from the central fiber respectively, by [@GS13 Theorem 0.2 and Theorem 0.3], [@Ch11 Theorem 1.2.1], we obtain moduli spaces of basic stable log maps ${\overline{\mathcal{M}}}({\mathcal{X}}/{\mathbb{A}}^1,\beta)$ and ${\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta)$ that are proper over ${\mathbb{A}}^1$ and whose formation commutes with base change. These carry virtual fundamental classes $[{\overline{\mathcal{M}}}({\mathcal{X}}/{\mathbb{A}}^1,\beta)]^{{\rm vir}},[{\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta)]^{{\rm vir}}$ that are compatible with base change. Moreover, since $\beta\cdot c_1({\mathcal{O}}_{\mathcal{X}}(-{\mathcal{D}}))<0$, we get that ${\overline{\mathcal{M}}}({\mathcal{X}}/{\mathbb{A}}^1,\beta)={\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta)$ and ${\overline{\mathcal{M}}}({\mathcal{X}}_0,\beta)={\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta)$. We furthermore consider the projection $\tilde p:{\mathcal{X}}_0=Y\sqcup_D X_0{\to}X$ induced by $p$ via the universal property of the co-product. \[lemma-Tot-is-p-pushforward\] Let $P:{\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta){\to}{\overline{\mathcal{M}}}(X,\beta)$ be the map that takes a basic stable log map to ${\mathcal{L}}_0$, forgets the log structure, composes with $\tilde p$ and stabilizes. Then $$[{\overline{\mathcal{M}}}({\operatorname{Tot}}({\mathcal{O}}_X(-D)),\beta)]^{{\rm vir}}= P_[{\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta)]^{{\rm vir}}.$$ Consider the four Cartesian squares of proper morphisms $$\label{eq-glue-square} \xymatrix@C=30pt { {\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta)\ar@{^{(}->}[r]\ar_P[d] & {\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta)\ar_Q[d] & \ar@{_{(}->}[l]{\overline{\mathcal{M}}}({\operatorname{Tot}}({\mathcal{O}}_X(-D)),\beta)\ar@{=}[d] \\ {\overline{\mathcal{M}}}(X,\beta)\ar@{^{(}->}[r]\ar[d] & {\overline{\mathcal{M}}}(X\times{\mathbb{A}}^1/{\mathbb{A}}^1,\beta)\ar^p[d] & \ar@{_{(}->}[l]{\overline{\mathcal{M}}}(X,\beta)\ar[d] \\ \{0\}\ar^{i_0}@{^{(}->}[r]&{\mathbb{A}}^1&\ar_{i_1}@{_{(}->}[l]\{1\} }$$ and notice that $[{\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta)]^{{\rm vir}}$ is a class in the top left corner that is the Gysin pullback of $[{\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta)]^{{\rm vir}}$ from the top middle which in turn Gysin pulls back to $[{\overline{\mathcal{M}}}({\operatorname{Tot}}({\mathcal{O}}_X(-D)),\beta)]^{{\rm vir}}$ in the top right. The statement now follows from the commuting of Gysin pullbacks with proper pushforward applied to the top two squares, since we see that $$P_[{\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta)]^{{\rm vir}}= i_0^!Q_ {\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta) = i_1^!Q_* {\overline{\mathcal{M}}}({\mathcal{L}}/{\mathbb{A}}^1,\beta) = [{\overline{\mathcal{M}}}({\operatorname{Tot}}({\mathcal{O}}_X(-D)),\beta)]^{{\rm vir}}$$ where the middle equality follows since $p$ is a trivial family and the last equality is a consequence of the family $p\circ Q$ being trivial and equal to $p$ in a neighborhood of 1. Applying the degeneration formula to ${\mathcal{L}}_0$ ------------------------------------------------------ We are going to apply the degeneration formula Theorem \[degen-formula\] to the log smooth space ${\mathcal{L}}_0$ over the standard log point which is the central fiber of the log smooth family ${\mathcal{L}}{\to}{\mathbb{A}}^1$ from the previous subsection. The following theorem will be proved in the next chapters. \[prop-vanishing-pushforward\] Given $\Gamma\in\Omega({\mathcal{L}}_0)$, we have $P_G_\Phi^\Delta^! \prod_V[{\overline{\mathcal{M}}}_V]^{{\rm vir}}=0$ unless $\Gamma$ is the graph with $V_1$ an $X$-vertex and $V_2$ a $Y$-vertex and furthermore, $w_e=\beta\cdot D$, $\beta_{V_1}=\beta$, $n_{V_1}=\{1,...,n\}$, $n_{V_2}=\emptyset$ and $\beta_{V_2}$ is $w_e$ times the class of a fiber of $p:Y{\to}D$. For the remainder of this section, we deduce the main Theorem \[mainthm-intro\] from Theorem \[prop-vanishing-pushforward\]. Set $L_D:=D\times{\mathbb{A}}^1$ which we view as the intersection of the components $L_X$ and $L_Y$ of ${\mathcal{L}}_0$ and thus as a divisor in $L_X$ as well as in $L_Y$. Set $d=\beta\cdot D$ and let $\Gamma$ in the following denote the exceptional graph given in Theorem \[prop-vanishing-pushforward\]. In light of Lemma \[lemma-Tot-is-p-pushforward\], we conclude from Theorem \[degen-formula\] and Theorem \[prop-vanishing-pushforward\] that $$\label{class-pulled-thru-degenform} \begin{array}{l} [{\overline{\mathcal{M}}}({\operatorname{Tot}}({\mathcal{O}}_X(-D)),\beta)]^{{\rm vir}}\\ \ = d\cdot P_ G_\Phi^\Delta^!\left([{\overline{\mathcal{M}}}_{(d)}(L_X(\log L_D),\beta_{V_1})]^{{\rm vir}}\times [{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})]^{{\rm vir}}\right). \end{array}$$ Note that $\bigodot _V {\overline{\mathcal{M}}}_V$ can be identified with the top left corner in the diagram[^2] $$\xymatrix{ {\overline{\mathcal{M}}}_{(d)}(L_X(\log L_D),\beta)\times_{L_D} {\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})\ar_{{\operatorname{pr}}_1}[d]&\ar_(.13){\Phi}[l] {\overline{\mathcal{M}}}_\Gamma\ar^(.4)G[r]&{\overline{\mathcal{M}}}({\mathcal{L}}_0,\beta)\ar^P[d]\\ {\overline{\mathcal{M}}}_{(d)}(X(\log D),\beta)\ar^F[rr]&&{\overline{\mathcal{M}}}(X,\beta) }$$ and this diagram is commutative because the curves in $${\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})={\overline{\mathcal{M}}}_{(d)}(Y(\log D),\beta_{V_2})$$ become entirely unstable when composing with $p:Y{\to}D$. Therefore, the right hand side in equals $$d\cdot F_({\operatorname{pr}}_1)_\Phi_\Phi^\Delta^!\left([{\overline{\mathcal{M}}}_{(d)}(L_X(\log L_D),\beta)]^{{\rm vir}}\times [{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})]^{{\rm vir}}\right)$$ $$=d\deg(\Phi)\cdot F_({\operatorname{pr}}_1)_\Delta^!\left([{\overline{\mathcal{M}}}_{(d)}(L_X(\log L_D),\beta)]^{{\rm vir}}\times [{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})]^{{\rm vir}}\right).$$ We note that $\deg(\Phi)=1$ by [@KLR17 Equation (1.4)]. So, in order to identify the last equation with the left hand side in Theorem \[mainthm-intro\] (up to moving the factor $(-1)^{d+1}d$ to the other side) and thereby via deducing the Theorem, it suffices to observe that $[{\overline{\mathcal{M}}}_{(d)}(L_X(\log L_D))]^{{\rm vir}}$ Gysin-restricts to $[{\overline{\mathcal{M}}}_{(d)}(X(\log D))]^{{\rm vir}}$ when confining the evaluation to be in $D\times\{0\}$ and then to prove the following result. Under the evaluation map ${\operatorname{ev}}:{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2}){\to}D$, we find $${\operatorname{ev}}_[{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})]^{{\rm vir}}= \frac{(-1)^{d+1}}{d^2}[D].$$ The maps $L_Y(\log L_D){\to}Y(\log D_0)\stackrel{p}{\longrightarrow}D{\to}{\operatorname{pt}}$ are log smooth and $\beta_{V_2}$ is the $d$’th multiple of a fiber class of $p$, so we find $$[{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D)/{\operatorname{pt}},\beta_{V_2})]^{{\rm vir}}=[{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D)/D,\beta_{V_2})]^{{\rm vir}}.$$ Since $\operatorname{vdim}{\overline{\mathcal{M}}}_{(d)}(Y(\log D_0)/D,\beta_{V_2})=\dim D$ (e.g. by inserting $h=0$ in [@BP08 §6.3]), necessarily ${\operatorname{ev}}_[{\overline{\mathcal{M}}}_{(d)}(L_Y(\log L_D),\beta_{V_2})]^{{\rm vir}}$ is a multiple of $[D]$. We can compute the degree by Gysin pulling back to a point in $D$ and the formation of the virtual fundamental class is compatible with this pullback by [@BF97 Prop. 7.3]. Hence, it remains to show that $$\deg \big([{\overline{\mathcal{M}}}_{(d)}({\operatorname{Tot}}({\mathcal{O}}_{{\mathbb{P}}^1}(-1))(\log (\{0\}\times{\mathbb{A}}^1)),d[{\mathbb{P}}^1])]^{{\rm vir}}\big)=\frac{(-1)^{d+1}}{d^2}.$$ The exact sequence $$0 {\to}{\mathcal{T}}_{{\mathbb{P}}^1(\log\{0\})}{\to}{\mathcal{T}}_{{\operatorname{Tot}}\big({\mathcal{O}}_{{\mathbb{P}}^1}(-1)\big)(\log (\{0\}\times{\mathbb{A}}^1))}\|_{{\mathbb{P}}^1}{\to}{\mathcal{O}}_{{\mathbb{P}}^1}(-1) {\to}0$$ relates the local part of this moduli space to the twist by the obstruction bundle, hence we need to show that $$\label{BP-result-eqn} \deg \big(e(O)\cap[{\overline{\mathcal{M}}}_{1}({\mathbb{P}}^1(\log \{0\}),d[{\mathbb{P}}^1])]^{{\rm vir}}\big)=\frac{(-1)^{d+1}}{d^2}$$ where $O=R^1\pi_f^{\mathcal{O}}_{{\mathbb{P}}^1}(-1)$ for ${\overline{\mathcal{M}}}_{1}({\mathbb{P}}^1(\log \{0\}))\stackrel{\pi}{{\longleftarrow}}{\mathcal{C}}\stackrel{f}{{\longrightarrow}}{\mathbb{P}}^1$ the maps from the universal curve to moduli space and target. By [@BP08 Lemma 6.3][^3], specializing the equivariant parameter $t_2$ in loc.cit. to $1$, the left hand side of equals the coefficient of $1/u$ in $$\operatorname{GW}(0\|-1,0)_{(d)} = \frac{(-1)^{d+1}}{d}\left(2\sin\frac{du}{2}\right)^{-1}$$ which is readily seen to be $\frac{(-1)^{d+1}}{d^2}$. Excluding multiple gluing points of curves in $L_X$ {#sec-exclude-mult-glue-X} =================================================== In this section, we prove the statement of Theorem \[prop-vanishing-pushforward\] for graphs $\Gamma\in\Omega({\mathcal{L}}_0)$ that have a vertex with bipartition membership in $X$ that has at least two adjacent edges. \[lemma-no-mult-edges-X\] Let $\Gamma\in\Omega({\mathcal{L}}_0)$ be a graph with an $X$-vertex $V$ with $r>1$ adjacent edges, then $[{\overline{\mathcal{M}}}_\Gamma]^{{\rm vir}}=0$. Let $r+s$ be the number of edges of $\Gamma$. Since maps from compact curves to ${\mathbb{A}}^1$ are constant, the evaluation map ${\overline{\mathcal{M}}}_V \to (D\times{\mathbb{A}}^1)^r$ factors through $D^r \times {\mathbb{A}}^1$ where ${\mathbb{A}}^1$ is embedded diagonally in ${\mathbb{A}}^r$. The same is true for vertices corresponding to components in $L_Y$, since the bundle ${\mathcal{O}}_Y(D_\infty)$ is nef. Using this, we rewrite the diagram by separating out the factors for $V$ to find Cartesian squares. $$\xymatrix@C=30pt { {\overline{\mathcal{M}}}_V\times_{L_D^r} \bigodot_{V'\neq V}{\overline{\mathcal{M}}}_{V'} \ar[r] \ar^{{\operatorname{ev}}}[d] & {\overline{\mathcal{M}}}_V\times \prod_{V'\neq V}{\overline{\mathcal{M}}}_{V'} \ar^{{\operatorname{ev}}}[d]\\ (D^r\times{\mathbb{A}}^1)\times (D\times{\mathbb{A}}^1)^{s} \ar^{({\operatorname{id}}\times{\operatorname{diag}})\times{\operatorname{id}}=:\delta}[d]\ar^{\Delta'}[r] & (D^r\times{\mathbb{A}}^1)^2\times (D\times{\mathbb{A}}^1)^{2s}\ar^{({\operatorname{id}}\times{\operatorname{diag}})\times{\operatorname{id}}}[d]\\ (D\times{\mathbb{A}}^1)^{r}\times (D\times{\mathbb{A}}^1)^{s}\ar^{\Delta}[r]& (D\times{\mathbb{A}}^1)^{2r}\times (D\times{\mathbb{A}}^1)^{2s}. }$$ Let $N$ denote the normal bundle of the embedding $\Delta$ which has rank $(r+s)(\dim D+1)$ and $N'$ denote that of $\Delta'$ which has rank $r\dim D+1+s(\dim D+1)$. Set $E=(\delta^N)/N'$ which is of rank $r-1$, let $c_{r-1}(E)$ be its top Chern class. For any $k$ and $\alpha\in A_k\big({\overline{\mathcal{M}}}_V\times \prod_{V'\neq V}{\overline{\mathcal{M}}}_{V'}\big)$, the excess intersection formula says $$\Delta^!\alpha= c_{r-1}(E)\cap(\Delta')^!\alpha.$$ Note that the normal bundle of the bottom right vertical map is trivial and, by Cartesianness of the lower square, its pullback under $\Delta'$ is isomorphic to $E$, so $c_{r-1}(E)=0$ because $r>1$ by assumption. Applying this to the virtual fundamental class $\alpha=[{\overline{\mathcal{M}}}_V]^{{\rm vir}}\times \prod_{V'\neq V}[{\overline{\mathcal{M}}}_{V'}]^{{\rm vir}}$ proves the Lemma. Comparing stable maps to $Y(\log D_0)$ and $D$ ============================================== We let $D$ be an arbitrary smooth projective variety (with the trivial log structure), and $Z$ a log scheme[^4] with a log smooth and projective morphism $p:Z{\to}D$. This induces a morphism of spaces of (log) stable maps. The example relevant to our main theorem is given by $Z=Y(\log D_0)$ as in Section \[section-setup\]. In this case ${\mathcal{T}}_{Z/D} = {\mathcal{O}}_Y(D_\infty)$ which is a nef line bundle, since $D_\infty$ is an effective divisor with nef normal bundle. This implies that the relative log tangent bundle has no higher cohomology when pulled back under any morphism ${\mathbb{P}}^1 \to Z$, or more generally any morphism from a genus zero curve. This property is useful in studying the induced morphism on spaces of stable maps. In this section it is not necessary that the underlying scheme of $Z$ is smooth and we can state our main result in arbitrary genus (although most examples will be in genus zero). Consequently, our notation for the space of log maps will just be ${\overline{\mathcal{M}}}_{g,n}(Z,\beta)$ where we do not try to describe the type of contact at the points. This will just be the disjoint union over all possible conditions to impose at the markings. Let ${{H_2(Z)^+}}$ denote the submonoid of $H_2(Z,{\mathbb{Z}})$ spanned by effective curve classes (including zero). Fixing a class $\beta\in {{H_2(Z)^+}}$ we would like to compare the virtual fundamental classes of ${\overline{\mathcal{M}}}_{g,n}(Z, \beta)$ and ${\overline{\mathcal{M}}}_{g,n}(D,p_\beta)$. The natural map between them is induced by forgetting the log structures, composing the maps, and stabilizing. We want to factor that map via an intermediate space in order to deal with these steps separately. To this end we need to discuss certain stacks of curves with extra structure. First, we have ${\mathfrak{M}}_{g,n}$ which parametrizes prestable curves of genus $g$ with $n$ markings. The stack ${\overline{\mathcal{M}}}_{g,n}(D,p_\beta)$ has a natural map to ${\mathfrak{M}}_{g,n}$, given by remembering the underlying curve, but forgetting the map to $D$. We would like to forget the map to $D$, but remember the homology class of each irreducible component of the source curve, so we want to factor this map through a stack $\mathfrak M_{g,n,{{H_2(D)^+}}}$ which parametrizes prestable genus $g$ curves together with an effective curve class on each irreducible component and satisfying the stability condition that components of degree and genus zero must contain three special points. Families of such curves are required to satisfy the obvious continuity condition that when a component degenerates, the sum of the classes on the degenerations is equal to the class of the original component. This stack was introduced by Costello in [@costello] and used for a similar purpose by Manolache in [@Ma12b]. The crucial fact for us will be that the deformation theory of such a decorated curve is identical to that of the undecorated curve, i.e. $\mathfrak M_{g,n,{{H_2(D)^+}}}$ is étale over $\mathfrak M_{g,n}$, see [@costello Proposition 2.0.2]. Therefore, instead of thinking of the standard obstruction theory on ${\overline{\mathcal{M}}}_{g,n}(D,p_\beta)$ as being a relative obstruction theory over ${\mathfrak{M}}_{g,n}$, we can think of it as a relative obstruction theory over ${\mathfrak{M}}_{g,n,{{H_2(D)^+}}}$. Similarly, we can factor the structure map from ${\overline{\mathcal{M}}}_{g,n}(Z,\beta)$ to ${\mathfrak{M}}_{g,n}^{\log}$ (the stack parametrizing log smooth families of curves over log schemes) through a scheme ${\mathfrak{M}}^{\log}_{g,n,{{H_2(Z)^+}}}$ which again records an effective curve class on each irreducible component. We arrive at the following commutative diagram, where ${\mathcal{M}}$ is taken to make the right hand square Cartesian. (We suppress some indices in the subscripts for clarity both here and in what follows.) $$\label{main-diagram-Y-D} \begin{aligned} \xymatrix@C=30pt { {\overline{\mathcal{M}}}(Z,\beta)\ar^-u[r]\ar[d]& {\mathcal{M}}\ar^-v[r]\ar[d]& {\overline{\mathcal{M}}}(D,p_\beta)\ar[d]\\ {\mathfrak{M}}^{\log}_{{{H_2(Z)^+}}} \ar^{id}[r]& {\mathfrak{M}}^{\log}_{{{H_2(Z)^+}}} \ar^\nu[r]& {\mathfrak{M}}_{{{H_2(D)^+}}}. } \end{aligned}$$ The main reason for introducing the labeled curves, is that the stabilization map is now defined on the bottom row, since we can tell which components become unstable just from the discrete data of the homology classes. (We remark that the existence of the relevant stabilization maps depends crucially on the fact that 0 is indecomposable in ${{H_2(D)^+}}$.) The mapping $\nu$ is given by forgetting the log structures, applying $p_$ to the homology markings, and then stabilizing as necessary, so we have a natural morphism from the universal curve ${\mathfrak{C}}^{\rm log}_{{{H_2(Z)^+}}} $ to the pullback under $\nu$ of the universal curve ${\mathfrak{C}}_{{{H_2(D)^+}}}$ over ${\mathfrak{M}}_{{{H_2(D)^+}}}$. This map contracts rational curves that are destabilized by forgetting the log structures and pushing forward the homology class. In particular, any positive-dimensional fiber of this map is a union of ${\mathbb{P}}^1$’s whose labelling becomes zero in ${{H_2(D)^+}}$. Now we are in a position to state the main theorem of this section. \[thm-pullback\] Let $p:Z \to D$ be a log smooth morphism where $D$ has trivial log structure. Suppose that for every log stable morphism $f:C \to Z$ of genus $g$ and class $\beta$ we have $H^1(C,f^{\mathcal{T}}_{Z/D}) = 0$, then $$[{\overline{\mathcal{M}}}_{g,n}(Z,\beta)]^{{\rm vir}}= u^ \nu^! [{\overline{\mathcal{M}}}_{g,n}(D,p_\beta)]^{{\rm vir}}$$ provided that ${\overline{\mathcal{M}}}_{g,n}(D,p_\beta) \neq \emptyset$. In particular, if $[{\overline{\mathcal{M}}}_{g,n}(D,p_\beta)]^{{\rm vir}}$ can be represented by a cycle supported on some locus $W\subset {\overline{\mathcal{M}}}_{g,n}(D,p_\beta)$, then $[{\overline{\mathcal{M}}}_{g,n}(Z,\beta)]^{{\rm vir}}$ can be represented by a cycle supported on $w^{-1}(W)$ where $w = v\circ u$ is the natural map between these stacks. Note that while the last statement involves only the standard spaces of (log) stable maps, we do not know how to formulate the precise relationship without passing through ${\mathcal{M}}$. Since $w$ does not seem to be flat in general, it is not apparent how to pull back classes under $w$. To prove this result, the point will be to show that ${\mathcal{M}}$ itself has a relative perfect obstruction theory such that the associated virtual class $[{\mathcal{M}}]^{{\rm vir}}$ satisfies the equations $$\label{first} \nu^![{\overline{\mathcal{M}}}_{g,n}(D,p_\beta)]^{{\rm vir}}= [{\mathcal{M}}]^{{\rm vir}},$$ $$\label{second} u^[{\mathcal{M}}]^{{\rm vir}}= [{\overline{\mathcal{M}}}_{g,n}(Z,\beta)]^{{\rm vir}}.$$ To obtain a relative perfect obstruction theory on ${\mathcal{M}}$ we can just pull back the obstruction theory using $\nu$. The fact that Equation \[first\] holds follows from the fundamental base change property of virtual fundamental classes which we will recall here for the reader’s convenience. Assume that we are given a fiber diagram of Artin stacks $$\begin{aligned} \xymatrix@C=30pt { {\mathcal{M}}\ar[r]^m\ar[d]& {\mathcal{N}}\ar[d]^f\\ G \ar^\mu[r]& H. } \end{aligned}$$ with $G$ and $H$ pure dimensional, $f$ of DM type, and so that ${\mathcal{M}}$ admits a stratification by quotient stacks. If there is a relative perfect obstruction theory with virtual tangent bundle $\mathfrak E$ for ${\mathcal{N}}$ over $H$, then there is an induced relative perfect obstruction theory with virtual tangent bundle $m^{\mathfrak E}$ for ${\mathcal{M}}$ over $G$, and the associated virtual fundamental classes satisfy $\mu^!([{\mathcal{N}}]^{{\rm vir}}) = [{\mathcal{M}}]^{{\rm vir}}$ if $\mu$ is either flat or an l.c.i. morphism. This all follows immediately from results in [@Ma12a] using the definition that $[{\mathcal{N}}]^{{\rm vir}}= f_{\mathfrak E}^!([H])$. The case of $\mu$ flat is a special case of Theorem 4.1 (ii) and the case where $\mu$ is lci is a special case of Theorem 4.3. To apply the Proposition to our situation, we factor the morphism $\nu$ as the composition of the graph followed by the projection and use the fact that the latter is flat and the former is lci (since ${\mathfrak{M}}_{{{H_2(D)^+}}}$ is smooth). This is also how we define $\nu^!$ on Chow groups. To obtain Equation \[second\] we want to use that this pulled back obstruction theory has a geometric interpretation, which we describe now. The obstruction theory on ${\overline{\mathcal{M}}}_{g,n}(D,p_\beta)$ arises from the fact that it is an open subset of the relative Hom stack ${\operatorname{Hom}}({\mathfrak{C}}_{g,n}/{\mathfrak{M}}_{g,n},D)$. In keeping with our diagram above, we want to consider it instead as an open substack of ${\operatorname{Hom}}({\mathfrak{C}}_{{{H_2(D)^+}}}/{\mathfrak{M}}_{{{H_2(D)^+}}}, D)$. For what follows, it will be convenient to notice that it is clearly contained in the open set where the labelling of the components of the fibers of the universal curve coming from the universal property of ${\mathfrak{M}}_{{{H_2(D)^+}}}$ agrees with the labelling coming from the homology of the image under the morphism. We denote this open subset by ${\operatorname{Hom}}^0({\mathfrak{C}}_{{H_2(D)^+}}/ {\mathfrak{M}}_{{H_2(D)^+}}, D)$. This stack parametrizes morphisms from homology labeled curves satisfying the condition that the pushforward of the homology class of a component is given by the label of that component. Since formation of the relative Hom scheme is compatible with base change, we know that ${\mathcal{M}}$ is an open subset of ${\operatorname{Hom}}(\nu^{-1}{\mathfrak{C}}_{{H_2(D)^+}}/ {\mathfrak{M}}^{\rm log}_{{H_2(Z)^+}}, D)$, and the obstruction theory obtained by pullback under $\nu$ is simply the natural obstruction theory for this Hom scheme. The key observation for proving Equation \[second\] is that ${\mathcal{M}}$ can also be thought of as an open subset of ${\operatorname{Hom}}({\mathfrak{C}}^{\rm log}_{{H_2(Z)^+}}/ {\mathfrak{M}}^{\rm log}_{{H_2(Z)^+}}, D)$. Inside this space there is an analogous open ${\operatorname{Hom}}^0$ where we demand that the labelling obtained from the morphism to $D$ agrees with $p_$ of the labelling coming from the universal property. The natural morphism $$\Psi:{\operatorname{Hom}}^0(\nu^{-1}{\mathfrak{C}}_{{H_2(D)^+}}/ {\mathfrak{M}}_{{H_2(Z)^+}}^{\rm log} , D) \to {\operatorname{Hom}}^0({\mathfrak{C}}^{\rm log}_{{H_2(Z)^+}}/ {\mathfrak{M}}^{\rm log}_{{H_2(Z)^+}}, D)$$ induced by the morphism $f: {\mathfrak{C}}^{\rm log}_{{H_2(Z)^+}}\to \nu^{-1}{\mathfrak{C}}_{{H_2(D)^+}}$ is an isomorphism. The map $f$ just contracts some destabilized ${\mathbb{P}}^1$’s. The superscripts 0 imply that the rational curves contracted by $f$ are necessarily contracted by every morphism parametrized by the ${\operatorname{Hom}}^0$ schemes we are considering. Given an $S$-valued point of ${\mathfrak{M}}_{{H_2(Z)^+}}^{\rm log}$ corresponding to a marked log curve with underlying nodal curve $C \to S$, denote by $C' \to S$ the family obtained by applying $\nu$ and $c:C \to C'$ the associated contraction mapping (which corresponds to $f$). A morphism $g: C' \to D$ corresponding to a point of ${\operatorname{Hom}}^0(\nu^{-1}{\mathfrak{C}}_{{H_2(D)^+}}/ {\mathfrak{M}}_{{H_2(Z)^+}}^{\rm log} , D)$ is taken by $\Psi$ to $\Psi(g) = g\circ c : C \to D$. The statement that this morphism $\Psi$ gives an isomorphism amounts to the statement that any morphism $\tilde g : C \to D$ which is constant on those components of $C$ which are contracted by $c$ factors uniquely through $c$. This is obvious when $S$ is the spectrum of an algebraically closed field. To see that this holds over an arbitrary base, one needs to use the standard fact about contraction maps between families of curves that $c_{\mathcal{O}}_C = {\mathcal{O}}_{C'}$. The result then follows from Lemma 2.2 of [@BM96]. In addition to being isomorphic stacks, the natural obstruction theories on ${\mathcal{M}}$ induced by these two descriptions agree, because the contracted genus zero components make no contribution to the cohomology of $f^({\mathcal{T}}_D)$. To prove Equation \[second\], we note that we now have that ${\overline{\mathcal{M}}}(Z,\beta)$ and ${\mathcal{M}}$ are two stacks with relative perfect obstruction theories over the same base, ${\mathfrak{M}}^{\rm log}_{{H_2(Z)^+}}$. They are both open subsets of stacks of morphisms from the same family of curves. In the case of the space of log stable maps to $Z$, we have that ${\overline{\mathcal{M}}}_{g,n}(Z,\beta)$ is an open subset of the logarithmic ${\operatorname{Hom}}$ stack ${\operatorname{Hom}}^{\rm log}({\mathfrak{C}}^{\rm log}_{g,n}/{\mathfrak{M}}^{\rm log}_{g,n}, Z)$ parametrizing logarithmic maps from fibers of the universal family to $Z$. The obstruction theory for this scheme is given by the cohomology of the pullback of the logarithmic tangent bundle of $Z$. In particular, this obstruction complex depends only on the underlying morphism of schemes, not on the log structures. The obstruction theory of ${\mathcal{M}}$ comes from the cohomology of $f^({\mathcal{T}}_D)$. In order to compare them, we use the short exact sequence $$0\to {\mathcal{T}}_{Z/D} \to {\mathcal{T}}_Z\to {\mathcal{T}}_D \to 0.$$ Given that we have $H^1(C,f^{\mathcal{T}}_{Z/D})=0$ for all stable maps $f:C \to Z$, it follows from the associated long exact sequence in cohomology that $u$ is smooth with relative cotangent complex supported in one term, given by $H^0(f^{\mathcal{T}}_{Z/D})$ and we have a very special case of a compatibility datum, implying via [@Ma12a] Corollary 4.9 that $[{\overline{\mathcal{M}}}(Z)]^{{\rm vir}}= u^[{\mathcal{M}}]^{{\rm vir}}$ where $u^$ denotes smooth pullback. Excluding nontrivial curves in $L_Y$ ==================================== For given $\Gamma\in\Omega({\mathcal{L}}_0)$, recall the maps $G$ and $P$ from §\[sec-degen-formula\]. Let $r_V$ denote the number of edges of $\Gamma_V$, i.e. edges in $\Gamma$ adjacent to $V$. In the following, we will use abusive notation by referring by $n_V$ not only to the set introduced in §\[sec-degenformula\] but also its size. Note that $P\circ G:{\overline{\mathcal{M}}}_\Gamma{\to}{\overline{\mathcal{M}}}_{0,n}(X,\beta)$ factors through the étale map $\Phi$, say $P\circ G=P'\circ\Phi$ for $P':\bigodot_V{\overline{\mathcal{M}}}_V{\to}{\overline{\mathcal{M}}}_{0,n}(X,\beta)$. Now $P'$ is induced by maps $P_V:{\overline{\mathcal{M}}}_V{\to}{\overline{\mathcal{M}}}_{0,n_V+r_V}(X,\tilde p_\beta_V)$ on the factors of $\bigodot_V{\overline{\mathcal{M}}}_V$ which is on each factor the composition of a stable map with $\tilde p$ which of course only has an effect for ${\overline{\mathcal{M}}}_V$ with $V$ associated to $Y$. For a $Y$-vertex $V$, recall the forgetful morphism $w:{\overline{\mathcal{M}}}_{\Gamma_V}(Y(\log D_0),\beta_V){\to}{\overline{\mathcal{M}}}_{0,n_V+r_V}(D,p_\beta_V)$ that we studied in the previous section. \[lem-zero-pushforward\] If $\beta_V \in H^+_2(Y)=H_2^+(L_Y)$ is a curve class such that $p_\beta_V \in H_2^+(D)$ is non-zero, then the class $[{\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\gamma)]^{{\rm vir}}$ pushes forward to zero under the natural map to ${\overline{\mathcal{M}}}_{0,n_V+r_V}(D,p_\gamma)$. As we said in the previous section, ${\mathcal{T}}_{Y(\log D_0)/D}\cong{\mathcal{O}}_Y(D_\infty)$. Thus, for any genus zero stable map $f:C{\to}Y(\log D_0)$ of type $\Gamma_V$, we have ${\operatorname{rk}}H^0(C,f^{\mathcal{O}}_Y(D_\infty))=\beta_V\cdot D_\infty+1$ and this number is the relative virtual and actual dimension of the map $u$ in . This coincides with the relative dimension of $w$ because the map that forgets the log structure is of relative dimension zero. Denoting by ${\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)\stackrel{\pi}{{\leftarrow}}{\mathcal{C}}\stackrel{f}{{\to}} Y(\log D_0)$ the maps from the universal curve and setting ${\mathcal{E}}=R^1\pi_f^{\mathcal{O}}_Y(-D_\infty)$, we also have that $$[{\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)]^{{\rm vir}}= e({\mathcal{E}}) \cap [{\overline{\mathcal{M}}}_{\Gamma_V}(Y(\log D_0),\beta_V)]^{{\rm vir}}.$$ Since the rank of ${\mathcal{E}}$ is $\beta_V \cdot D_\infty - 1$, we find the virtual dimension of ${\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)$ to be strictly greater than the virtual dimension of ${\overline{\mathcal{M}}}_{0,n_V+r_V}(D,p_\beta_V)$. But Theorem \[thm-pullback\] implies that the pushforward of $[{\overline{\mathcal{M}}}_{\Gamma_V}(Y(\log D_0),\beta_V)]^{{\rm vir}}$ can be supported on any cycle that supports $[{\overline{\mathcal{M}}}_{0,n_V+r_V}(D,p_\beta_V)]^{{\rm vir}}$. Since ${\overline{\mathcal{M}}}_{0,n_V+r_V}(D,p_\beta_V)$ is a Deligne-Mumford stack and we are working in Chow groups with rational coefficients, this implies the desired vanishing. It was pointed out to us by Feng Qu that a similar result to Lemma \[lem-zero-pushforward\] was proved in [@LLQW16 Prop. 3.2.2.] which would imply this result when ${\mathcal{E}}= 0$. \[prop-nontrivial-in-Y-pushes-to-zero\] If $\Gamma\in \Omega({\mathcal{L}}_0)$ has a vertex $V$ with bipartition membership in $Y$ and $\beta_V$ is not a multiple of the class of a fiber of $p$, then $P_G_\Phi^\Delta^!\prod_V[{\overline{\mathcal{M}}}_V]^{{\rm vir}}=0$. Let $\Gamma$ and $V$ be as in the assertion. Set $r:=r_V$. The argument here is simple, but we include a large diagram to remind the reader of the names of the maps. The obvious, but important, point is that the evaluation maps from ${\overline{\mathcal{M}}}_V = {\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)$ to $L_D$ factors through the map $P_V$ to ${\overline{\mathcal{M}}}_{0,n_V+r}(L_D, p_\beta_V)$. The stack $M'$ below is defined to make the bottom right square Cartesian. $$\xymatrix@C=30pt { {\overline{\mathcal{M}}}_\Gamma \ar[d]_G\ar[r]^(.3){\Phi}& {\overline{\mathcal{M}}}_V\times_{L_D^r} \bigodot_{V'\neq V}{\overline{\mathcal{M}}}_{V'} \ar[r] \ar[d] & {\overline{\mathcal{M}}}_V\times \prod_{V'\neq V}{\overline{\mathcal{M}}}_{V'} \ar^{P_V\times id}[d]\\ {\overline{\mathcal{M}}}(X\cup Y)\ar[d]_P & M' \ar[ld]_\tau \ar[d]\ar[r] & {\overline{\mathcal{M}}}_{0,n_V+r}(L_D,p_\beta_V) \times \prod_{V'\neq V}{\overline{\mathcal{M}}}_{V'} \ar^{{\operatorname{ev}}}[d]\\ {\overline{\mathcal{M}}}(X)&L_D^r\ar^{\Delta}[r]&( L_D)^{2r}. }$$ The only point not yet explained is the existence of the map $\tau$ but this is just the usual clutching construction for boundary strata. Since $p_\beta_V$ is nonzero, Lemma \[lem-zero-pushforward\] applies and we have $$P_{V}[{\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)]^{{\rm vir}}=0.$$ Rewriting the class in the proposition as $$\deg(\Phi) \cdot \tau_\Delta^! \big(P_{V}[{\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)]^{{\rm vir}}\times \prod_{V'\neq V}[{\overline{\mathcal{M}}}_{V'}]^{{\rm vir}}\big)$$ the result follows immediately. Let us collect what is implied for the graph $\Gamma\in\Omega({\mathcal{L}}_0)$ if the pushforward to ${\overline{\mathcal{M}}}_{0,n}(X,\beta)$ of the corresponding virtual fundamental class is non-trivial: by Lemma \[lemma-no-mult-edges-X\], the X-vertices of the graph have no more than one adjacent edge and by Proposition \[prop-nontrivial-in-Y-pushes-to-zero\], every curve component in $Y$ needs to be a multiple of a fiber. If a $P_G_[{\overline{\mathcal{M}}}_\Gamma]^{{\rm vir}}$ is non-trivial, it already follows that $\Gamma$ has a unique $Y$-vertex $V$ and we are only left with showing that this has only a single adjacent edge to it. If $\beta_V$ is a multiple of a fiber class, borrowing notation from the proof of Proposition \[prop-nontrivial-in-Y-pushes-to-zero\] and setting ${\overline{\mathcal{M}}}^\circ:=\prod_{V'\neq V}[{\overline{\mathcal{M}}}_{V'}]^{{\rm vir}}$, the map from ${\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V) \times_{(D\times{\mathbb{A}}^1)^{r_V}} {\overline{\mathcal{M}}}^\circ$ to ${\overline{\mathcal{M}}}_{0,n}(X)$ factors through $D \times_{(D \times {\mathbb{A}}^1)^{r_V}}{\overline{\mathcal{M}}}^\circ$. Hence, it suffices to check that the pushforward of the virtual class from ${\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)$ to $D$ vanishes which is precisely what the next lemma achieves. If $\beta_V$ is a multiple of the class of a fiber of $p:Y{\to}D$, then the pushforward of $[{\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)]^{{\rm vir}}$ under the evaluation map to $(D \times {\mathbb{A}}^1)^{r_V}$ is trivial if $n_V+r_V>1$. Set $r:=r_V$. The evaluation map from ${\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)$ to $D^r\times {\mathbb{A}}^r$ factors through the embedding $D \to D^r\times {\mathbb{A}}^r$ given by ${\operatorname{diag}}\times \{0\}$. However, the virtual dimension of ${\overline{\mathcal{M}}}_{\Gamma_V}(L_Y(\log L_D),\beta_V)$ is $\dim(D) + n_V+r-1$, so for $n_V+r>1$ this vanishing is immediate. [77]{} Abramovich, D. and Fantechi, B.: “Orbifold techniques in degeneration formulas", Annali della SNS XVI (2016), no. 2, p. 519–579. Abramovich, D., Marcus, S. and Wise, J.: “Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations”, Annales de l’institut Fourier [64]{} (2014), no. 4, p. 1611–1667. 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Takahashi, N.: “Log Mirror Symmetry and Local Mirror Symmetry”, [Comm. Math. Phys.]{} [220*]{} (2001), no. 2, p. 293–299. [^1]: This work was supported by HR’s DFG Emmy-Noether grant RU1629/4-1 and by MvG’s affiliation with the Korea Institute for Advanced Study. [^2]: Despite the notation, ${\overline{\mathcal{M}}}_{(d)}(L_X(\log L_D),\beta_{V_1})$ isn’t proper but this doesn’t affect us. [^3]: This is also Theorem 5.1 of [@BP05]. [^4]: We assume the log structure is in the Zariski topology to satisfy the assumptions of [@GS13].
--- abstract: 'We prove that the complex of free factors of a free group of finite rank is hyperbolic.' author: - 'Mladen Bestvina and Mark Feighn[^1]' bibliography: - './ref.bib' title: Hyperbolicity of the complex of free factors --- Introduction ============ The [complex of free factors]{} of a free group $\FF$ of rank $n$ is the simplicial complex $\F$ whose vertices are conjugacy classes of proper free factors $A$ of $\FF$, and simplices are determined by chains $A_1<A_2<\cdots<A_k$. The outer automorphism group $Out(\FF)$ acts naturally on $\F$, which can be thought of as an analog of the Bruhat-Tits building associated with $GL_n(\Z)$. This complex was introduced by Hatcher and Vogtmann in [@HV] where it is shown that it has the homotopy type of the wedge of spheres of dimension $n-2$. They defined this complex in terms of sphere systems in $\#_1^n S^1\times S^2$ and used variants in their work on homological stability [@HV2; @HV3; @HV4]. There is a very useful analogy between $\F$ and the curve complex $\mathcal C$ associated with a compact surface (with punctures) $\Sigma$. The vertices of $\mathcal C$ are isotopy classes of essential simple closed curves in $\Sigma$, and simplices are determined by pairwise disjoint curves. The curve complex was introduced by Harvey [@harvey] and was classically used by Harer in his work on duality and homological stability of mapping class groups [@harer1; @harer2]. The key result here is that the curve complex is homotopy equivalent to the wedge of spheres. More recently, the curve complex has been used in the study of the geometry of mapping class groups and of ends of hyperbolic 3-manifolds. The fundamental result on which this work is based is the theorem of Masur and Minsky [@MM] that the curve complex is hyperbolic. In the low complexity cases when $\mathcal C$ is a discrete set one modifies the definition of $\C$ by adding an edge when the two curves intersect minimally. In the same way, we modify the definition of $\F$ when the rank is $n=2$ by adding an edge when the two free factors (necessarily of rank 1) are determined by a basis of $\FF$, i.e. whenever $\FF=\langle a,b\rangle$, then $\langle a\rangle$ and $\langle b\rangle$ span an edge. In this way $\F$ becomes the standard Farey graph. The main result in this paper is: The complex $\F$ of free factors is hyperbolic. The statement simply means that when the 1-skeleton of $\F$ is equipped with the path metric in which every edge has length 1, the resulting graph is hyperbolic. There are variants of the definition that give rise to quasi-isometric complexes. For example, one can take the complex of partial bases, where vertices are conjugacy classes of elements that are part of a basis, and simplices correspond to compatibility, i.e. subsets of a basis. The $Aut(\FF)$-version of this complex was used in [@day-putman] to study the Torelli group. As another example, $\F$ is quasi-isometric to the nerve of the cover $\{U(A)\}$ of the thin part of Outer space, where for a conjugacy class of proper free factors $A$, the set $U(A)$ consists of those marked graphs whose $A$-cover has core of volume $<\epsilon$, for a fixed small $\epsilon>0$. Our proof is very much inspired by the Masur-Minsky argument, which uses Teichmüller theory. Bowditch [@bo:hyp] gave a somewhat simpler argument. In the remainder of the introduction, we give an outline of the hyperbolicity of the curve complex which follows [@MM] and [@bo:hyp], and where we take a certain poetic license. The proof starts by defining a coarse projection $\pi:\mathcal T\to\mathcal C$ from Teichmüller space. To a marked Riemann surface $X$ one associates a curve with smallest extremal length. To see that this is well defined one must argue that short curves intersect a bounded number of times (in fact, at most once), and then one uses the inequality $$d_{\mathcal C}(\alpha,\beta)\leq i(\alpha,\beta)+1$$ where $i$ denotes the intersection number. Interestingly, the entire argument uses only this inequality to estimate distances in $\mathcal C$ (and only for bounded intersection numbers). Teichmüller space carries the Teichmüller metric, and any two points are joined by a unique Teichmüller geodesic. If $t\mapsto X_t$ is a Teichmüller geodesic, consider the (coarse) path $\pi(X_t)$ in $\mathcal C$. One observes: (i) The collection of paths $\pi(X_t)$ is (coarsely) transitive, i.e. for any two curves $\alpha,\beta$ there is a path $\pi(X_t)$ that connects $\alpha$ to $\beta$ (to within a bounded distance). Next, for any Teichmüller geodesic $\{X_t\}$ one defines a projection $\mathcal C\to \{X_t\}$. Essentially, for a curve $\alpha$ the projection assigns the Riemann surface $X_t$ on the path in which $\alpha$ has the smallest length. The key lemma is the following (see [@MM Lemma 5.8]), proved using the intersection number estimate above: (i) If $\alpha,\beta$ are adjacent in $\mathcal C$ and $X_\alpha,X_\beta$ are their projections to $X_t$, then $\pi(X_\alpha)$ and $\pi(X_\beta)$ are at uniformly bounded distance in $\mathcal C$. Consequently, one has a (coarse) Lipschitz retraction $\mathcal C\to\pi(X_t)$ for every Teichmüller geodesic $X_t$. It quickly follows that the paths $\pi(X_t)$ are reparametrized quasi-geodesics (this means that they could spend a long time in a bounded set, but after removing the corresponding subintervals the resulting coarse path is a quasi-geodesic with uniform constants, after possibly reparametrizing). The final step is: (i) Triangles formed by three projected Teichmüller geodesics are uniformly thin. Hyperbolicity of $\mathcal C$ now follows by an argument involving an isoperimetric inequality (see [@bo:hyp Proposition 3.1] and our Proposition \[hyperbolicity\]). Our argument follows the same outline. In place of Teichmüller metric and Teichmüller geodesics we use the Lipschitz metric on Outer space and folding paths. There are technical complications arising from the non-symmetry of the Lipschitz metric and the non-uniqueness of folding paths between a pair of points in Outer space. Our projection from $\F$ to a folding path comes in two flavors, left and right, and we have to work to show that the two are at bounded distance from each other when projected to $\F$. Similarly, we have to prove directly that projections of folding paths fellow travel, even when the two have opposite orientations. The role of simple closed curves is played by [simple conjugacy classes]{} in $\FF$, i.e. nontrivial conjugacy classes contained in some proper free factor. The first two hints that Outer space has some hyperbolic features was provided by Yael Algom-Kfir’s thesis [@yael] and by [@BF2]. Algom-Kfir showed that axes of fully irreducible automorphisms are strongly contracting. In the course of our proof we will generalize this result (see Proposition \[contraction\]) which states that all folding paths are contracting provided their projections to $\F$ travel with definite speed. In [@BF2] a certain non-canonical hyperbolic $Out(\FF)$ complex was constructed. It is also known that fully irreducible automorphisms act on $\F$ with positive translation length [@ilya-lustig; @BF2]. Below is a partial dictionary between Teichmüller space and Outer space relevant to this work. .5cm Surfaces Free groups ---------------------------- -------------------------------------------- curve complex $\mathcal C$ complex of free factors $\F$ simple closed curve free factor, or a simple conjugacy class intersection number number of times $i(\alpha,\beta)$ a loop in a graph crosses an edge Teichmüller space Outer space Teichmüller distance Lipschitz distance Teichmüller geodesic folding path Teichmüller map optimal map quadratic differential train track structure on the tension graph horizontal curve legal loop vertical curve illegal loop The paper is organized as follows. In Section \[s:review\] we review the basic notions about Outer space, including the Lipschitz metric, train tracks, and folding paths. Section \[s:whitehead\] proves the analog of the inequality $d_{\mathcal C}(\alpha,\beta)\leq i(\alpha,\beta)+1$, using the Whitehead algorithm. Sections \[s:folding\] and \[s:gafa\] contain some additional material on folding paths, including the formula for the derivative of a length function, as well as the key fact that a simple class which is largely illegal must lose a fraction of its illegal turns after a definite progress in $\F$. In Section \[s:projection\] we define the (left and right) projections of a free factor to a folding path and establish that images in $\F$ of folding paths are reparametrized quasi-geodesics. The key technical lemma in that section is Proposition \[left-right\](legal and illegal) establishing that the two projections are at bounded distance when measured in $\F$. We end this section with a very useful method of estimating where the projections lie in Lemma \[criterion\]. In Section \[s:hyperbolicity\] we recall the argument that for hyperbolicity it suffices to establish the Thin Triangles condition, and we also derive the contraction property of folding paths, measured in $\F$. In Section \[s:FT\] we prove the Fellow Travelers property (which of course follows from the Thin Triangles property), in both parallel and anti-parallel setting. Finally, in Section \[s:thin\] we establish the Thin Triangles property. The proofs of the three main technical statements in the paper, namely Proposition \[gafa\](surviving illegal turns), Proposition \[left-right\](legal and illegal), and Proposition \[general\](closing up to a simple class), should be omitted on the first reading. 0.5cm [Acknowledgments.]{} We thank the American Institute of Mathematics and the organizers and participants of the Workshop on Outer space in October 2010 for a fruitful exchange of ideas. We particularly thank Michael Handel for telling us a proof of Lemma \[michael2\]. We thank Saul Schleimer for an inspiring conversation and Yael Algom-Kfir for her comments on an earlier version of this paper. We heartily thank the anonymous referee for a very careful reading and many suggestions we feel greatly improved the exposition. We also thank the referee for pointing out an error in one of two arguments we gave to finish the proof of Proposition \[gafa\]. Since the first version of this paper appeared on the arXiv, there have been some very interesting developments. Lee Mosher and Michael Handel [@hm:hyperbolicity] have proved that the free splitting complex $\mathcal S$ for $\FF$ is hyperbolic. Ilya Kapovich and Kasra Rafi [@kr:hyperbolicity] have shown that the hyperbolicity of the complex of free factors follows from the hyperbolicity of the free splitting complex. Arnaud Hilion and Camille Horbez [@hh:hyperbolicity] have proved that $\mathcal S$ is hyperbolic from the point of view of the sphere complex. Review {#s:review} ====== In this section we review some definitions and collect standard facts about Outer space, Lipschitz metric, train tracks, and folding paths. 0.5cm [ Outer space.]{} A [graph]{} is a cell complex $\gG$ of dimension $\leq 1$. The [ rose]{} $R_n$ of rank $n$ is the graph with one 0-cell (vertex) and $n$ 1-cells (edges). A vertex of $\gG$ not of valence 2 is [topological]{}. The closure in $\gG$ of a component of the complement of the set of topological vertices is a [ topological edge]{}. In particular, a topological edge may be a circle. A [marking]{} of a graph $\gG$ is a homotopy equivalence $g:R_n\to \gG$. A [metric]{} on $\gG$ is a function $\ell$ that to each edge $e$ assigns a positive number $\ell(e)$. We often view the graph $\gG$ as the path metric space in which each edge $e$ has length $\ell(e)$. The Unprojectivized Outer space $\hX$ is the space of equivalence classes of triples $(\gG,g,\ell)$ where $\gG$ is a finite graph with no vertices of valence $\leq 2$, $g$ is a marking of $\gG$, and $\ell$ is a metric on $\gG$. Two triples $(\gG,g,\ell)$ and $(\gG',g',\ell')$ are equivalent if there is a homeomorphism $h:\gG\to\gG'$ that preserves edge-lengths and commutes with the markings up to homotopy. Outer space $\X$ is the space of projective classes of such (equivalence classes of) triples, i.e. modulo scaling the metric. Equivalently, $\X$ is the space of triples as above where the metric is normalized so that the volume $vol(\gG):=\sum\ell(e)=1$. Assigning length 0 to an edge is interpreted as a metric on the graph with that edge collapsed, and in this way $\X$ becomes a complex of simplices with missing faces (the missing faces correspond to collapsing nontrivial loops), which then induces the simplicial topology on $\X$. Outer space was introduced by Culler and Vogtmann [@CV], who showed that $\X$ is contractible. We will usually suppress markings and metrics and talk about $\gG\in\X$. It is sometimes convenient to pass to the universal cover and regard $\gG\in\hX$ as an action of $\FF$ on the tree $\tilde\bG$. We find the following notation useful. If $z$ is a nontrivial conjugacy class, it can be viewed as a loop in the rose $R_n$, and via the marking can be transported to a unique immersed loop in any marked graph $\gG$. This loop will be denoted by $z\|\gG$. The length of this loop, i.e. the sum of the lengths of edges crossed by the loop, counting multiplicities, is denoted by $\ell(z\|\gG)$. Note that if $z$ is not simple, then $\ell(z\|\gG)\geq 2~vol(\gG)$ and in fact $z\|\gG$ must cross every edge at least twice. 0.5cm [Morphisms between trees and train tracks.]{}\[morphisms0\] Recall that a [morphism]{} between two $\R$-trees $S,T$ is a map $\tilde\phi:S\to T$ such that every segment $[x,y]\subset S$ can be partitioned into finitely many subintervals on which $\tilde\phi$ is an isometric embedding. For simplicity, in this paper we work only with simplicial metric trees. A [direction]{} at $x\in S$ is a germ of nondegenerate segments $[x,y]$ with $y\neq x$. The set $D_x$ of directions at $x$ can be thought of as the unit tangent space; a morphism $\tilde\phi:S\to T$ determines a map $D\tilde\phi_x:D_x\to D_{\tilde\phi(x)}$, thought of as the derivative. A [turn]{} at $x$ is an unordered pair of distinct directions at $x$. A turn $\{d,d'\}$ at $x$ is [illegal]{} (with respect to $\tilde\phi$) if $D\tilde\phi_x(d)=D\tilde\phi_x(d')$. Otherwise the turn is [ legal]{}. There is an equivalence relation on $D_x$ where $d\sim d'$ if and only if $d=d'$ or $\{d,d'\}$ is illegal. The equivalence classes are [gates]{}. The collection of equivalence classes at each $x$ is called the [illegal turn structure on $S$ induced by $\tilde\phi$.]{} If at each $x\in S$ there are at least two gates, the illegal turn structure is called a [train track structure on $S$]{}. This is equivalent to the requirement that $\tilde\phi$ embeds each edge of $S$ and has at least two gates at every vertex. A path in $S$ is [legal]{} if it makes only legal turns. If $S\overset{\tilde\phi}{\to}T{\to}U$ is a composition of morphisms then there are two illegal turn structures of interest on $S$: one induced by $\tilde\phi$ and the other induced by the composition. In this situation, we will sometimes refer to the second of these as the [pullback illegal turn structure on $S$ via $\tilde\phi$]{}. Note that an illegal turn in the first structure is also illegal in the second. In particular, if the second structure is a train track then so is the first. If $S$ and $T$ are equipped with abstract train track structures (equivalence relation on $D_x$ for every vertex $x$ with at least two gates), we say that a morphism $\tilde\phi:S\to T$ is a [train track map]{} if on each edge $\tilde\phi$ is an embedding and legal turns are sent to legal turns[^2]. In particular, legal paths map to legal paths. We also extend this terminology to maps between graphs. If $\phi:\Delta\to\Sigma$ is a map between connected metric graphs such that the lift $\tilde\phi:\tilde\Delta\to\tilde\Sigma$ is a morphism of trees, then we also say that $\phi$ is a morphism and we can define the notion of legal and illegal turns on $\tilde\Delta$, which descends to $\Delta$. If there are at least two gates at each point, we have a train track structure on $\Delta$. If $\Delta$ and $\Sigma$ are equipped with abstract train track structures, the map $\phi$ is a train track map if it sends edges to legal paths and legal turns to legal turns. When the graphs $\Delta$ or $\Sigma$ are not connected we work with components separately. 0.5cm [Lipschitz metric and optimal maps.]{} Let $\gG$ and $\gG'$ be two points in $\hX$. The homotopy class of maps $h:\gG\to\gG'$ such that $hg\simeq g'$ (with $g,g'$ markings for $\gG,\gG'$) is called the [ difference of markings]{}. If $\gG$ and $\gG'$ are in $\X$, i.e. if they have volume 1, the [Lipschitz distance]{} $d_\X(\gG,\gG')$ between $\gG$ and $\gG'$ is the log of the minimal Lipschitz constant over all difference of markings maps $\gG\to\gG'$. The Lipschitz distance is an asymmetric metric on $\X$ inducing the correct topology. For more information, see [@francaviglia-martino; @ab; @b:bers]. The basic fact, that plays the role of Teichmüller’s theorem in Teichmüller theory, is the following statement, due to Tad White. For a proof see [@francaviglia-martino] or [@b:bers]. \[greengraph\] Let $\gG,\gG'\in\hX$. There is a difference of markings map $\phi:\gG\to\gG'$ with the following properties. - $\phi$ sends each edge of $\gG$ to an immersed path (or a point) with constant speed (called the [slope]{} of $\phi$ on that edge). - The union of all edges of $\gG$ on which $\phi$ has the maximal slope $\lambda=\lambda(\phi)$, is a subgraph of $\gG$ with no vertices of valence 1. This subgraph is called the [tension graph]{}, denoted $\Delta=\Delta(\phi)$. - $\phi$ induces a train track structure on $\Delta$. The last bullet says that the map $\lambda\Delta\to \gG'$ induced by $\phi$ is a morphism immersing edges and each vertex has at least two gates. Note that the Lipschitz constant of $\phi$ is $\lambda$ and so, in the case that $\gG, \gG'\in\X$, we have that $d_\X(\gG,\gG')\leq \log\lambda$. In fact equality holds since the presence of at least two gates at each vertex of $\Delta$ guarantees that it contains legal loops, whose length gets stretched by precisely $\lambda$ so there can be no better map homotopic to $\phi$. We call any $\phi$ satisfying Proposition \[greengraph\] an [optimal map]{}. A morphism that induces a train track structure is an example of an optimal map. Unfortunately, unlike Teichmüller maps, optimal maps are not uniquely determined by $\gG,\gG'$. A legal loop can be constructed in $\Delta$ by starting with an edge and extending it inductively to longer legal edge paths until some oriented edge is repeated. In fact, this guarantees the existence of a “short” legal path. We will say a loop in $\Delta$ is a [ candidate]{} if it is either embedded, or it forms the figure 8, or it forms a “dumbbell”. Every candidate determines a conjugacy class that generates a free factor of rank 1. Thus graph $\Delta$ admits a legal candidate. See [@francaviglia-martino]. 0.5cm [Folding at speed 1.]{}\[morphisms\]\[p:folding\] Now assume that $S,T$ represent points of Unprojectivized Outer space $\hX$ (i.e. they are universal covers of marked metric graphs), and $\tilde\phi:S\to T$ is an equivariant morphism. Equivariantly subdivide $S$ so that $\tilde\phi$ embeds each edge and the inverse image under $\tilde\phi$ of a vertex of $T$ is a vertex of $S$. Choose some $\epsilon>0$ smaller than half of the length of any edge in $S$. Then for $t\in [0,\epsilon]$ define the tree $S_t$ as the quotient of $S$ by the equivalence relation: $u\sim_t v$ if and only if there is a vertex $x$ with $d(x,u)=d(x,v)\leq t$ and $\tilde\phi(u)=\tilde\phi(v)$. Then $S_t$ represents a point in $\hX$ and $\tilde\phi$ factors as $S\overset{\tilde\phi_{0t}}\to S_t\overset{\tilde\phi_{t\infty}}\to T$ for some equivariant morphism $\tilde\phi_{t\infty}$ and the quotient map $\tilde\phi_{0t}:S\to S_t$, which is also an equivariant morphism. The trees $S_t$, $t\in [0,\epsilon]$ form a path in $\hX$, and $S_0=S$. We say that $S_t$ is [the path obtained from $S$ by folding all illegal turns at speed 1 with respect to $\tilde\phi$]{}. If $\tilde\phi$ induces only one gate at some vertex $v\in S$, then $S_t$ will have a valence 1 vertex for $t>0$. In that case we always pass to the minimal subtree of $S_t$. When $\tilde\phi$ induces a train track structure on $S$, $S_t$ is automatically minimal (if $S$ is). For simplicity we state Proposition \[slope 1 folding\] in the train track situation only. \[slope 1 folding\] Let $\tilde\phi:S\to T$ be an equivariant morphism between two trees in $\hX$ inducing a train track structure on $S$. There is a (continuous) path $S_t$ in $\hX$, $t\in [0,\infty)$, and there are equivariant morphisms $\tilde\phi_{st}:S_s\to S_t$ for $s\leq t$ so that the following holds: (1) $S_0=S$, $S_t=T$ for $t$ large; (2) $\tilde\phi_{tt}=Id$, $\tilde\phi_{su}=\tilde\phi_{tu}\tilde\phi_{st}$ for $s\leq t\leq u$; (3) $\tilde\phi_{0t}=\tilde\phi$ and $\tilde\phi_{tt'}=Id$ for large $t<t'$; (4) each $\tilde\phi_{st}$ isometrically embeds edges and induces at least two gates at every vertex of $S_s$; (5) for $s<t,t'$ the illegal turns at vertices of $S_s$ with respect to $\tilde\phi_{st}$ coincide with those with respect to $\tilde\phi_{st'}$, so $S_s$ has a well-defined train track structure; and (6) for every $s<t$ there is $\epsilon>0$ so that $S_{s+\tau}$, $\tau\in [0,\epsilon]$ is obtained from $S_s$ by folding all illegal turns at speed 1 with respect to $\tilde\phi_{st}$. Moreover, this path is unique. Uniqueness is clear from the definition of folding at speed 1. There can be no last time $s$ so that two paths satisfying the above conditions agree (including the maps $\tilde\phi_{tt'}$) up to $S_s$ but no further, by item (6). There are three methods to establish existence, and they will be only sketched. [\[slope 1 folding\].A. Stallings’ Method.]{} This works when $S$ and $T$ can be subdivided so that $\tilde\phi$ is simplicial and all edge lengths are rational (or fixed multiples of rational numbers). In our applications we can arrange that this assumption holds. Then we may subdivide further so that all edge lengths are equal. The path $S_t$ is then obtained exactly as in the Stallings’ beautiful paper [@stallings], by inductively identifying any pair of edges with a common vertex that map to the same edge in $T$. This operation of [ elementary folding]{} can be performed continuously to yield a 1-parameter family of trees, i.e. a path, between the original tree and the folded tree. Putting these paths together gives the path $S_t$. [\[slope 1 folding\].B. Via the vertical thickening of the graph of $\tilde\phi$.]{} This method is due to Skora [@skora:deformations], who built on the ideas of Steiner. Skora’s preprint was never published; the interested reader may find the details in [@matt] and [@2topologies]. Consider the graph of $\tilde\phi$ as a subset of $S\times T$ and define the “vertical $t$-thickening” of it as $$W_t=\{(x,y)\in S\times T\mid d(\tilde\phi(x),y)\leq t\}$$ Next, consider the decomposition $\mathcal D_t$ of $W_t$ into the path components of the sets $W_t\cap S\times\{y\}$, $y\in T$. Let $S_t=W_t/\mathcal D_t$ be the decomposition space with the metric defined as follows. A path in $W_t$ is [linear]{} if its projection to both $S$ and $T$ has constant speed (possibly speed 0). A piecewise linear path $\gamma$ in $W_t$ is [taut]{} if the preimages $\gamma^{-1}(\ell)$ of leaves in $\mathcal D_t$ are connected. Then define the distance in $S_t$ as the length of the projection to $T$ of any piecewise linear taut path connecting the corresponding leaves in $W_t$. In this way $S_t$ becomes a metric tree. The morphisms $\tilde\phi_{st}$ are induced by inclusion $W_s\hookrightarrow W_t$. [\[slope 1 folding\].C. Via integrating the speed 1 folding direction.]{} Starting at $S$ consider the path $S_t, t\in [0,\epsilon],$ obtained by folding all illegal turns at speed 1. Now extend this path by folding all illegal turns of $S_\epsilon$ at speed 1. Continue in this way inductively, and show that either $T$ is reached in finitely many steps, or there is a well defined limiting tree, from which folding can proceed. This is the approach taken in [@francaviglia-martino], to which the reader is referred for further discussion. One possible approach is as follows. Say $S_t$ is defined for $t\in [0,t_0)$ with $S_0=S$. To define the limiting tree $S_{t_0}$, note that for each conjugacy class, the length along the path is nonincreasing and thus converges. The limiting length function defines a tree, representing a point in compactified Outer space. The lengths of conjugacy classes are bounded below by their values in $T$, so the limiting tree $S_{t_0}$ is free simplicial and thus represents a point in Outer space. We may view the tree $S_{t_0}$ as the equivariant Gromov-Hausdorff limit of the path $S_t$. The maps $S\to S_t$, viewed as subsets of $S\times S_t$ via their graphs, subconverge to a morphism $S\to S_{t_0}$, and similarly by a diagonal argument one constructs morphisms $S_t\to S_{t_0}$ that compose correctly. To show uniqueness of such morphisms, one uses Gromov-Hausdorff limits and the fact that the only (equivariant) morphism $S_{t_0}\to S_{t_0}$ is the identity. The path $S_t, t\in [0,\infty)$ from Proposition \[slope 1 folding\] is [the path induced by $\tilde\phi$]{}. The following three lemmas are stated for clarity; their proofs are left as exercises. (The first is immediate from \[slope 1 folding\].B and implies the other two.) \[l:folding segments\] Let $S_t$, $t\in [0,\infty)$, be the path in $\hX$ induced by $\tilde \phi:S\to T$ and $[s_1,s_2]$ be a path in $S_0$. Suppose $\tilde\phi(s_1)=\tilde\phi(s_2)$ and set $h$ equal to the [ outradius of $\tilde\phi([s_1,s_2])$ with respect to $\tilde\phi(s_1)$]{}, i.e. $$h=\max_{s\in [s_1,s_2]}d_T(\tilde\phi(s_1),\tilde\phi(s))$$ Then $s_1$ and $s_2$ are identified by time $h$ but not before, i.e. $\tilde\phi_{0t}(s_1)=\tilde\phi_{0t}(s_2)$ iff $t\ge h$. \[exercise\] Let $S_t$ be the path in $\hX$ induced by $S\to T$ and $R\to S$ be an equivariant morphism with $R\in\hX$. The path $R_t$ induced by the composition $R\to S\to T$ and the path induced by $R\to S_{t_0}$ agree for $t\in [0,t_0]$. If $A$ is a nontrivial finitely generated subgroup of $\FF$ and $T\in\hX$, denote by $A\|T$ the minimal $A$-invariant subtree of $T$. \[exercise2\] Let $R, S, T\in\hX$, $R\to S$ and $S\to T$ be equivariant morphisms, and $A$ be a finitely generated subgroup of $\FF$. Suppose $S_t$ is induced by $S\to T$ and $R_t$ is induced by the composition $R\to S\to T$. If $A\|R\to A\|S$ is an isomorphism, then so is $A\|R_t\to A\|S_t$. If $S_t$ is a path in $\hX$ induced by the morphism $\tilde\phi:S\to T$, then $A\|S_t$ is a path in the Unprojectivized Outer space $\hX(A)$ of $A$. It is locally obtained by folding all illegal turns of the induced morphism $A\|S_t\to T$ and passing to minimal subtrees. For $t<t'$, there is a [generalized morphism]{} $A\|S_t\to A\|S_{t'}$, i.e. every segment in the domain has a partition into finitely many subsegments with the property that on each subsegment the map is an isometric embedding or degenerate. We will not need generalized morphisms in this paper. 0.5cm [Folding paths.]{} Suppose $S\to T$ is an equivariant morphism between trees in $\hX$ that induces a train track structure on $S$. The induced path $S_t$, $t\in [0,\infty),$ is a [folding path in $\hX$]{}. The projection of $S_t$ to $\X$ is a [folding path in $\X$]{}. A folding path $\gG_t$ in $\X$ can be parametrized by arclength, so that $d_\X(\gG_{t'},\gG_t)=t-t'$ for $t'\leq t$. \[n:folding path\] From now on, we switch to thinking of points of Outer space as finite graphs when discussing folding paths. We have several meanings for [folding path]{}. To avoid confusion, we will use the following notation: (1) \[i:unprojectivized path\] $\hG_t$, $t\in [0,\infty),$ denotes a folding path in $\hX$, i.e. a path in $\hX$ induced by an equivariant morphism $S\to T$ giving a train track structure on $S$. If $\o\in [0,\infty)$ is minimal such that $\hG_\o=T$, we also refer to $\hG_t$, $t\in [0,\o],$ as a folding path. We call $t$ the [ natural parameter]{}. (2) \[i:projectivized path\] $\G_t$, $t\in [0,\infty),$ (or $\G_t$, $t\in [0,\o]$) denotes a folding path in $\X$ obtained by projecting a path $\hG_t$ as in (\[i:unprojectivized path\]). So, $\G_t=\hG_t/vol(\hG_t)$. (3) \[i:arclength path\] $\bG_t$, $t\in [0,L],$ denotes a path as in (\[i:projectivized path\]), but reparametrized in terms of arclength with respect to $d_\X$. So, $\bG_{t(s)}=\G_s$ where $$t(s)=d_\X(\G_0,\G_s)=\log\bigg(\frac{vol(\hG_0)}{vol(\hG_s)}\bigg)$$ For all $t\le t'\in [0,L]$ there is a morphism $e^{t'-t}\bG_t\to \bG_{t'}$. Unless otherwise noted, a folding path $\bG_t$ without further adjectives or decorations will mean a folding path as in (\[i:arclength path\]). \[rescaling\] Let $\gG,\Sigma\in\X$. There is a geodesic in $\X$ from $\gG$ to $\Sigma$ which is the concatenation of two paths, the first is a (reparametrized) linear path in a single simplex of $\X$, and the second is a folding path in $\X$ parametrized by arclength. Fix an optimal map $\phi:\gG\to\Sigma$ and let $\Delta=\Delta(\phi)\subset \gG$ be its tension graph with maximal slope $\lambda=\lambda(\phi)$. If $\Delta=\gG$ then the rescaled map $\lambda\phi:\lambda\gG\to\Sigma$ is a morphism and satisfies the requirement that it induces a train track structure on $\gG$. Proposition \[slope 1 folding\] gives a folding path $\hG_t$ in $\hX$ from $\lambda\gG$ to $\Sigma$ and morphisms $\hat\phi_{st}:\hG_s\to\hG_t$. To see that $\hG_t$ projects to a geodesic $\G_t$ in $\X$, note that $\hat\phi_{su}=\hat\phi_{tu}\hat\phi_{st}$ for $s<t<u$ implies $d_\X(\G_s,\G_u)=d_\X(\G_s,\G_t)+d_\X(\G_t,\G_u)$. Now suppose $\Delta\neq\gG$. Denote by $e_1,\cdots,e_k$ the (topological) edges of $\gG$ outside of $\Delta$. For each tuple $x=(x_1,x_2,\cdots,x_k)$ of lengths in the cube $[0,\ell_1]\times [0,\ell_2]\times\cdots\times[0,\ell_k]$, where $\ell_i$ is the length of $e_i$ in $\gG$, denote by $\mu(x)$ the smallest maximal slope among maps $\gG\to\Sigma$ that are homotopic to $\phi$ rel $\Delta$, where $\gG$ is given the metric $x$ outside $\Delta$ (so $\mu(x)=\infty$ if some loop is assigned length 0). Among all $x$ in the cube with $\mu(x)=\lambda$ choose one with the smallest sum of the coordinates, say $x_0$. Denote by $\hG'\in\hX$ the graph $\gG$ with the metric $x_0$ outside of $\Delta$. (Some edges of $\hG'$ may get length 0 in which case its projection $\gG'$ to $\X$ is on the boundary of the original simplex.) Let $\phi_0:\hG'\to\Sigma$ be a map homotopic to $\phi$ rel $\Delta$, linear on edges, and with the maximal slope $\lambda(\phi_0)=\lambda$. We claim that $\phi_0$ is optimal with $\Delta(\phi_0)=\hG'$. Indeed, $\Delta(\phi_0)=\hG'$ (otherwise some edge length can be reduced contradicting the choice of $x_0$) and $\phi_0$ induces at least two gates at every vertex (otherwise $\phi_0$ may be perturbed so that the tension graph becomes a proper subgraph, see the proof of Proposition \[greengraph\] e.g. in [@b:bers]). Since we have maps $\gG\to\hG'$ and $\hG'\to\Sigma$ with slopes 1 and $\lambda$ respectively, we also have $d_\X(\gG,\Sigma)=d_\X(\gG,\gG')+d_\X(\gG',\Sigma)$. Detecting boundedness in the free factor complex ${\F}$ {#s:whitehead} ======================================================= In this section we define a coarse projection $\pi:\X\to\F$ and prove an analog of the inequality $d_{\mathcal C}(\alpha,\beta)\leq 1+i(\alpha,\beta)$ (see Lemma \[bounded crossing\]). An immediate consequence is that $\pi$ is coarsely Lipschitz (see Corollary \[proj X->F continuous\]). Recall that a nontrivial conjugacy class $x$ in $\FF$ is [simple]{} if any (equivalently some) representative is contained in a proper free factor. If $x$ is a simple class, denote by $\ff x$ the conjugacy class of a smallest free factor containing a representative of $x$. A proper connected subgraph $P$ of a marked graph $\gG$ that contains a circle defines a vertex $\ff P$ of $\F$. \[subgraph distance\] Let $\gG$ be a marked graph. If $P,Q\subset\gG$ are proper connected subgraphs defining free factors $\ff P,\ff Q$ then $d_{\F}(\ff P,\ff Q)\leq 4$. If $\rank(\FF)=2$ then $d_{\F}(\ff P,\ff Q)\leq 1$ (using the modified definition of $\F$). Now assume $\rank(\FF)\geq 3$. Enlarge $P,Q$ to connected graphs $P',Q'$ that contain all but one edge of $\gG$. Thus their intersection contains a circle $R$, so we have a path in $\F$ given by subgraphs $P,P',R,Q',Q$ and we see $d_\F(\ff P,\ff Q)\leq 4$. If $\gG$ is a marked graph, define $$\pi(\gG):=\{\ff P\mid P\mbox{ is a proper, connected, noncontractible subgraph of } \gG\}$$ The induced multi-valued, $Out(\FF)$-equivariant map $\X\to\F$ , still called $\pi$, given by $\bG\mapsto \pi(\bG)$ is coarsely defined in that, by Lemma \[subgraph distance\], the diameter of each $\pi(\gG)$ is bounded by 4. We refer to $\pi$ as the [coarse projection from $\X$ to $\F$]{}. \[bounded crossing\] Let $\gG$ be a marked graph and $x$ a simple class in $\FF$. If $x\|\gG$ crosses an edge $e$ $k$ times, then the distance in $\F$ between $\ff x$ and some free factor represented by a subgraph of $\gG$ is $\leq 6k+9$. This proof will use the fact, due to Reiner Martin [@reiner], that if the Whitehead graph of a simple class $x$ is connected then it has a cut point. The classical fact, due to Whitehead [@wh], is the analogous statement for the special case that $x$ is primitive[^3]. Stallings’ paper [@js:whitehead] is a good modern reference for Whitehead’s result and the reader is directed there for the definitions of Whitehead graph and Whitehead automorphism. First assume that $e$ is nonseparating. By collapsing a maximal tree in $\gG$ that does not contain $e$ we may assume that $\gG$ is a rose. Let $a_1,a_2,\cdots,a_m,c$ be the associated basis with $c$ corresponding to $e$ and set $A=\langle a_1,\cdots,a_m\rangle$. Thus $c^{\pm 1}$ appears in the cyclic word for $x$ $k$ times. If the Whitehead graph of $x$ is disconnected, consider a 1-edge blowup $\tilde \gG$ of $\gG$ so that $x$ realized in $\tilde \gG$ is contained in a proper subgraph. In this case $d_\F(\ff x,A)\leq 5$ by Lemma \[subgraph distance\] (4 for the distance between $A$ and the free factor determined by the image of $x$, and 1 more to get to $\ff x$). If the Whitehead graph is connected then it has a cut point [@wh; @reiner]. Let $\phi$ be the associated Whitehead automorphism. If the special letter is some $a_i^{\pm 1}$ then the free factor $A$ is $\phi$-invariant. If the special letter is $c^{\pm 1}$ then $d_\F(A,\phi(A))\leq 6$ ($d_\F(A,\langle c\rangle)\leq 3$ and $\langle c\rangle$ is fixed by $\phi$). But there are at most $k$ automorphisms of the latter kind in the process of reducing $x$ until its Whitehead graph is disconnected. Thus $d_\F(\ff x,A)\leq 6k+5$. Now assume $e$ is separating. By collapsing a maximal tree on each side of $e$ we may assume that $\gG$ is the disjoint union of two roses $R_A$ and $R_B$ connected by $e$. Let $a_1,\cdots,a_n$ and $b_1,\cdots,b_m$ be the bases determined by $R_A$ and $R_B$ respectively. Notice that the assumption about $e$ means that there are $k$ times when the cyclic word for $x$ changes from the $a_i$’s to the $b_j$’s or vice versa ($k$ is necessarily even here). If the Whitehead graph of $x$ with respect to $a_1,\cdots,a_n,b_1,\cdots,b_m$ is disconnected, we see as above that $d_\F(\ff x,A)\leq 5$ where $A=\langle a_1,\cdots,a_n\rangle$. Otherwise there is a cut point and let $\phi$ be the associated Whitehead automorphism. If the special letter is some $a_i^{\pm 1}$ then $A$ is $\phi$-invariant. Likewise, $\phi(A)$ is conjugate to $A$ if the special letter is $b_j^{\pm 1}$ and all the $a_i^{\pm 1}$ are on one side of the cut. If they are not on one side of the cut, then the subgraph spanned by the $a_i^{\pm 1}$’s is disconnected and we may consider the associated 1-edge blowup $\tilde R_A$ of $R_A$. Let $\tilde\gG$ be the 1-edge blowup of $\gG$ obtained by attaching $e\cup R_B$ to $\tilde R_A$ along either of the two vertices. The blowup edge $e'$ can be crossed by $x$ only if it is immediately followed or preceded by $e$ (but not both). Thus $x$ crosses $e'$ at most $k$ times. If $e'$ is nonseparating then by the first paragraph $d_\F(\ff x,\ff P)\leq 6k+5$ for some subgraph $P\subset\tilde\gG$ and so $d_\F(\ff x,\langle b_1,\cdots,b_m\rangle)\leq 6k+9$. If $e'$ is separating replace $A$ by a smaller free factor $A'$ and continue. We introduce the following notation. Suppose $\gG,\gG'$ are marked graphs, $A$ is a proper free factor of $\FF$, and $x$ is a simple class in $\FF$. Then: - $d_\F(\gG,\gG'):=\sup \{d_\F(A,A')\mid A\in\pi(\gG), A'\in\pi(\gG')\}$ - $d_\F(A,\gG'):=\sup\{d_\F(A,A')\mid A'\in\pi(\gG')\}$ - $d_\F(\gG,x):=d_\F(\gG,\ff x)$ - $d_\F(A, x):=d_\F(A, \ff x)$ For example, Lemmas \[subgraph distance\] and \[bounded crossing\] combine to give the following: \[bounded crossing 2\] Let $x$ be a simple class in $\FF$ and $\gG$ a marked graph so that $x\|\gG$ crosses some edge $\leq k$ times. Then $d_\F(\gG,x)\leq 6k+13$. \[r:bounded crossing 2\] Let $z$ be a conjugacy class in $\FF$ and $\bG\in\X$ a marked graph. Since $vol(\bG)=1$, there is an edge of $\bG$ that is crossed at most $[\ell(z\|\bG)]$ times by $z\|\bG$. \[proj X->F continuous\] If $d_\X(\gG,\gG')\leq \log K$ then $d_\F(\gG,\gG')\leq 12K+32$. Let $z$ be a candidate that realizes $d_\X(\gG,\gG')$. Thus $\ell(z\|\gG)<2$, $z\|\gG$ crosses some edge once and $\ell(z\|\gG')<2K$, so $z\|\gG'$ crosses some edge $<2K$ times. Therefore $$d_\F(\gG,\gG')\leq d_\F(\gG,z)+d_\F(z,\gG')\leq 19+(12K+13)=12K+32$$ In most of the paper we will be concerned with showing that distances in $\F$ are bounded above. We will use the obvious terminology: In Corollary \[proj X->F continuous\] we showed that the distance in $\F$ between projections of graphs from $\X$ are bounded above as a function of the distance in $\X$. When we say a distance in $\F$ is bounded without any variables, we mean bounded above by a universal constant that depends only on the rank $n$ of $\FF$. Note that Corollary \[proj X->F continuous\] says that $\pi:\X\to\F$ is [coarsely Lipschitz]{}: If $d_\X(\gG,\gG')\leq N$ with $N$ an integer, then $d_\F(\gG,\gG')\leq CN$ for a universal $C>0$. Indeed, choose a geodesic from $\gG$ to $\gG'$ and apply Corollary \[proj X->F continuous\] $N$ times to pairs of points at distance $\leq 1$. By the [injectivity radius]{} $injrad(\gG)$ of a metric graph $\gG$ we mean the length of a shortest embedded loop in $\gG$. If $A$ is a finitely generated subgroup of $\FF$ and $\hG\in\hX$ we denote by $A\|\hG$ the core of the covering space of $\hG$ corresponding to $A$. Thus there is a canonical immersion $A\|\hG\to \hG$. We adopt the following convention: Unless otherwise specified, the metric and illegal turn structures on $A\|\hG$ are the ones obtained by pulling back via $A\|\hG\to\hG$. \[lucas\] Suppose $A$ is a proper free factor of $\FF$ and $\bG\in\X$. If $injrad(A\|\bG)<k+1$ then $d_\F(A,\bG)\leq 6k+14$. This follows immediately from Remark \[bounded crossing\] and Lemma \[bounded crossing 2\]. Coarse paths in $\F$ obtained from folding paths by projecting will play a crucial role. \[c:coarse paths\] The collection of projections to $\F$ of folding paths in $\X$ is a coarsely transitive family: for any two proper free factors $A,B$ of $\FF$ there is a folding path $\bG_t$, $t\in [0,L]$, such that $A\in\pi(\bG_0)$ and $B\in \pi(\bG_L)$. The same is true for the subcollection consisting of folding paths induced by morphisms that satisfy the rationality condition from the Stallings method of folding in \[slope 1 folding\].A. Let $A,B$ be two free factors. Choose $\gG,\Sigma\in\X$ so that some subgraph of $\gG$ represents $A$ and some subgraph of $\Sigma$ represents $B$, and so that $\gG$ is a rose, and apply Proposition \[rescaling\] to obtain a geodesic from $\gG$ to $\Sigma$ that is the concatenation of two paths, the first a linear path from $\gG$ to $\gG'$ in a single simplex, and the second a folding path from $\gG'$ to $\gG$. The initial linear path keeps the underlying graph a rose and its coarse projection is constant. Thus, the desired folding path is the second path from $\gG'$ to $\Sigma$. To achieve rationality, choose $\Sigma$ to be a rose with rational edge lengths. Let $\phi:\gG\to \Sigma$ be an optimal map after adjusting the metric so that $\Delta(\phi)=\gG$. If the vertex of $\gG$ maps to the vertex of $\Sigma$, rationality is automatic. Otherwise the vertex of $\gG$ maps to a point in the interior of some edge. Perturb $\phi$ so that this point is rational, and adjust the edge lengths in $\gG$ so that the perturbed map is optimal, with the same train track structure. The new map satisfies rationality. More on folding paths {#s:folding} ===================== We now discuss folding in more detail. Let $\hG_t$, $t\in [0,\o]$, be a folding path in $\hX$ (from now on we replace trees by quotient graphs) with the natural parametrization. (See Notation \[n:folding path\] to recall our conventions.) So for $s<t$ we have maps $\hat\phi_{st}:\hG_s\to \hG_t$ that have slope 1 on each edge, immerse each edge, and induce train track structures. 0.5cm [Unfolding.]{} Traversing a folding path in reverse is [unfolding]{}. The main result of this subsection is Theorem \[t:gadget\] giving local and global pictures of both folding and unfolding. This result will only be used for the proofs of technical Propositions \[gafa\](surviving illegal turns) and \[left-right\](legal and illegal) which should be skipped in a first reading. In fact, Theorem \[t:gadget\] is obvious in the case of Stallings’ rational paths (see \[slope 1 folding\].A) and so could be avoided altogether (see Corollary \[c:coarse paths\]). An [(abstract) widget $W$ of radius $\epsilon$]{} is a metric graph that is a cone on finitely many, but at least 2, points all the same distance $\epsilon$ from the cone point. A widget has a canonical morphism to $W\to [0,\epsilon]$ that sends the cone point to $\epsilon$ and the other vertices to $0$. There is also a canonical path, parametrized by $[0,\epsilon]$, of finite trees from $W$ to $[0,\epsilon]$ that locally folds all legal turns of the morphism with speed 1. See Figure \[f:abstract widget\]. An [(abstract) gadget of radius $\epsilon$]{} is a union of finitely many widgets of radius $\epsilon$. The union is required to be disjoint except that widgets are allowed to meet in vertices and is also required to be a forest. There is a canonical path, parametrized by $[0,\epsilon]$, of forests obtained by folding each widget. ![An abstract widget of radius $\epsilon$.[]{data-label="f:abstract widget"}](widget.eps) Let $\hG\in\hX$. A [widget (resp. gadget) of radius $\epsilon$ in $\hG$]{}, is an embedding in $\hG$ of an abstract widget (resp.abstract gadget) of radius $\epsilon$. There is a natural path in $\hX$ starting with $\hG$ and parametrized by $[0,\epsilon]$ given by folding the gadget. See Figure \[f:widgets\]. An [(abstract) widget]{}, resp. [gadget]{} is a widget, resp. gadget, of some radius. \[t:gadget\] Let $\hG_t$, $t\in [0,\o]$ be a folding path in $\hX$ with its natural parametrization. There is a partition $0=t_0<t_1<\cdots<t_N=\o$ of $[0,\o]$ such that the restriction of $\hG_t$ to each $[t_i,t_{i+1}]$ is given by folding a gadget in $\hG_{t_i}$. In the context of Theorem \[t:gadget\], we say that, as we traverse $[t_i,t_{i+1}]$ in reverse, we are [unfolding a gadget]{}. The analogous result holds for the induced path $\bG_t$, $t\in [0,L]$, in $\X$ parametrized by arclength and we use the same terminology. For example, we say $[0,L]$ has a finite partition into subintervals such the restriction of $\bG_t$ to each subinterval is given by folding (unfolding) a gadget. Proposition \[p:tame\] is an immediate consequence of Theorem \[t:gadget\]. \[p:tame\] Let $\bG_t$, $t\in [0,L]$, be a folding path in $\X$. There is a partition of $[0,L]$ into finitely many subintervals so that the restriction of $\bG_t$ to each subinterval is a (reparametrized) linear path in a simplex of $\X$. In particular, a folding path in $\X$ changes an open simplex only at discrete times. Outside these times, illegal turns all belong to vertices with 2 gates, and one gate is a single direction. The restriction of a folding path to a simplex of $\X$ need not be linear. This can happen, for example, if an illegal turn becomes legal. The rest of this subsection is devoted to proving Theorem \[t:gadget\]. It is clear from our description of folding in \[slope 1 folding\].C that, for each $t_\in [0,\o)$, there is $\epsilon>0$ such that the restriction of $\hG_t$ to $[t_,t_+\epsilon]$ is given by folding a gadget in $\hG_{t_}$. To complete the proof of Theorem \[t:gadget\], we will show that, for each $t_\in (0,\omega]$, there is $\epsilon>0$ such that the restriction of $\hG_t$ to $[t_-\epsilon,t_]$ is given by folding a gadget in $\hG_{t_-\epsilon}$. Let $N$ be the closed $\epsilon$-neighborhood of a vertex $v$ in $\hG_{t_}$ of valence $\ge 3$. We will describe the preimage $N_\epsilon$ of $N$ in $\hG_{t_-\epsilon}$ for small enough $\epsilon>0$. As long as $\epsilon$ is small enough and $N_\epsilon\to N$ is not injective, we will find a connected gadget of radius $\epsilon$ in $N_\epsilon$ so that $N_\epsilon\to N$ folds this gadget. The gadget needed to complete the proof will be the disjoint union of these connected gadgets of radius $\epsilon$, one for each such vertex of $\bG_{t_}$. We will also equip $N$ and $N_\epsilon$ with height functions. For convenience, set $\hat\phi_\epsilon:=\hat\phi_{t_-\epsilon,t_}$. First we describe the height functions. Assume that $\epsilon$ is small enough so that $N$ is a cone on a finite set with cone point $v$. The height function on $N$ is the morphism $N\to [-\epsilon,\epsilon]$ given as follows. If the length in $N$ of $[v,w]$ is $\epsilon$ and the direction at $v$ determined by $[v,w]$ has more than one preimage in $N_\epsilon$, then map $[v,w]$ isometrically to $[0,\epsilon]$; otherwise map $[v,w]$ isometrically to $[0,-\epsilon]$. The height function $h$ on $N_\epsilon$ is the composition $N_\epsilon\to N\to [-\epsilon,\epsilon]$. Now we describe $N_\epsilon$ for small enough $\epsilon$, and justify this description immediately after that. In $N_\epsilon$, the preimage of $v$ is the set of points of height 0. The set $h^{-1}([0,\epsilon])$ is a gadget with widgets the closures of the components of $h^{-1}((0,\epsilon])$. Each of the height $\epsilon$ vertices has a unique direction in $\hG_{t_-\epsilon}$ not in the gadget, and we draw this direction upwards. Height 0 vertices may have additional directions not contained in the gadget, and we draw those downwards. See Figure \[f:widgets\]. ![An example of $N_\epsilon$ with 3 widgets, 3 vertices at height $\epsilon$ and 5 vertices at height 0. The union of the 3 widgets is a gadget.[]{data-label="f:widgets"}](widgets.v3.eps) All illegal turns in $N_\epsilon$ appear at vertices of height $\epsilon$ and these vertices have two gates in $\hG_{t_-\epsilon}$ (all downward directions form one gate and the single upward direction is the other gate). In particular, all turns at the height 0 vertices are legal. After the widgets are folded, in $\hG_{t_}$, the height 0 vertices get identified to $v$, each widget contributes an upward direction at $v$, and the downward directions at $v$ come from downward directions in $N_\epsilon$ based at height 0 vertices. Some pairs of directions may be illegal in $\hG_{t_}$, but in that case they have to come from directions in $N_\epsilon$ that don’t form a turn (i.e. that are based at different vertices). Now we justify our description of $N_\epsilon$. First choose $\epsilon_0$ small enough so that: (1) the closed $\epsilon_0$-neighborhood of $v$ is a cone on a finite set with cone point $v$ and (2) the cardinality of the preimage of $v$ in $N_{\epsilon}$ and the number of directions based at points in the preimage of $v$ is independent of $0<\epsilon\le\epsilon_0$; Note that it is possible to choose $\epsilon_0$ satisfying (2) because, since all $\hat\phi_{st}$ are surjective, the cardinality of $\hat\phi_\epsilon^{-1}(v)$ is non-increasing as $\epsilon$ decreases, and similarly the number of directions based at $\hat\phi_\epsilon^{-1}(v)$ is non-increasing. Choose $0<\epsilon_1<\epsilon_0$ so that the closed $\epsilon_1$-neighborhood $N'$ in $N_{\epsilon_0}$ of the preimage of $v$ contains no valence $\ge 3$ vertices of $\hG_{t_0-\epsilon_0}$ other than the preimages of $v$. We claim that, for $0<\epsilon\le\epsilon_1$, $N_\epsilon$ has the description given above. It is enough to consider the case $\epsilon=\epsilon_1$ because our description of $N_\epsilon$ is stable under decreasing $\epsilon$. By definition, $N_{\epsilon_1}$ is the preimage in $\hG_{t_-\epsilon_1}$ of $N$. It is equal to the closed $\epsilon_1$-neighborhood of the preimage of $v$. (Indeed, it is clear that the neighborhood is in $N_{\epsilon_1}$. If $w\in N_{\epsilon_1}$ does not map to $v$, then since $\hG_{t_-\epsilon_1}$ has a train track structure there is a direction at $w$ whose image in $N$ points toward $v$ and there is a legal path of length $\|h(w)\|$ starting at this direction. The endpoint of the path then maps to $v$.) Similarly, $N'$ is the preimage in $\hG_{t_-\epsilon_0}$ of $N$. In particular, $N'\to N_{\epsilon_1}$ is surjective. $N'$ is a disjoint union of cones on points of height $\pm\epsilon_1$ with cone points the preimages of $v$. Notice that all turns in $N'$ are legal (or else (2) fails). $N'\to N_{\epsilon_1}$ is an embedding off the points of height $\pm\epsilon_1$ in $N'$ (otherwise there would be a path in $N_{\epsilon_1}$ between distinct directions at preimages of $v$ whose image has outradius with respect to $v$ that is less than $\epsilon_1$, by Lemma \[l:folding segments\] these distinct directions would then be identified before time $t_$, and we contradict (2)). In fact, $N'\to N_{\epsilon_1}$ is an embedding off points of height $\epsilon$ (otherwise there would be a path $\sigma$ in $N_{\epsilon_1}$ that is cone on a pair of distinct points in the preimage of $v$ with cone point of height $-\epsilon_1$, contradicting the definition of $h$ since the directions at the ends of $\sigma$ are identified in $\hG_{t_}$). We see that $N_{\epsilon_1}$ is the obtained from $N'$ by identifying some pairs of points of height $\epsilon$. The picture is completed with a few observations. - [$N_{\epsilon_1}$ is connected:]{} By Lemma \[l:folding segments\], any two points in the the preimage of $v$ in $N_{\epsilon_1}$ are connected by a path in $N_{\epsilon_1}$ (or else they aren’t identified in $\hG_{t_}$). So, the preimage of $v$ is contained in a single component and from the picture we’ve developed so far we see that $N_{\epsilon_1}$ is connected. - [$N_{\epsilon_1}$ is a tree:]{} A loop $\sigma$ in $N_{\epsilon_1}$ has homotopically trivial image in $N$. Since $\hat\phi_{\epsilon_1}$ is a homotopy equivalence, $\sigma$ is homotopically trivial. - [Every point $w$ in $N_{\epsilon_1}$ of height $\epsilon_1$ has at least two downward directions:]{} Otherwise, there is a point $v'$ in the preimage of $v$ and a path $[v',w]$ of increasing height with the property that the induced direction at $v'$ is not identified by $\hat\phi_{\epsilon_1}$ with any other direction. This contradicts the definition of positive height. This completes the proof of Theorem \[t:gadget\]. Fix $\epsilon\le\epsilon_1$. We introduce a little more terminology for later use. By construction, every point in $\hG_{t_}$ at distance $\epsilon$ from $v$ has a unique direction pointing away from $v$. If $d$ is a direction at $v$, there is then a unique corresponding direction $d'$ pointing away from $v$ and based at a point at distance $\epsilon$ from $v$ determined by “$d$ points to $d'$”. If the height of $d'$ is $\epsilon$ then we denote $d'$ by $d^{\epsilon}$; if the height of $d'$ is $-\epsilon$ then we denote $d'$ by $d^{-\epsilon}$. We say that $d^{\epsilon}$ [points up]{} and $d^{-\epsilon}$ [points down]{}. The direction $d^{\pm\epsilon}$ has a unique lift $\tilde d^{\pm\epsilon}$ to $\hG_{t_-\epsilon}$. We say $\tilde d^{\epsilon}$ [points up* ]{} and $\tilde d^{-\epsilon}$ [ points down ]{}. See Figure \[f:notation\]. 0.5cm [Unfolding a path.]{} Given an immersed path $\gamma$ in $\hG_\omega$, one may try to “lift” it along the folding path, i.e. to find immersed paths $\gamma_t$ in $\hG_t$ that map to $\gamma$ (up to homotopy rel endpoints). This is always possible, since it is clearly possible locally. At discrete times new illegal turns may appear inside the path. Note that at discrete times the lifts are not unique, when an endpoint of the path coincides with the vertex of an illegal turn. Figure \[f:unfolding\] illustrates the nonuniqueness of lifts. ![The figure illustrates the ambiguity in lifting paths under unfolding which we see is parametrized by the preimage of $v$.[]{data-label="f:unfolding"}](unique.v3.eps) To get uniqueness, we can remove the end of the path that lifts nonuniquely. Thus we may have to remove segments[^4] at the ends whose size grows at speed 1. Now suppose there are illegal turns in the path. As we unfold, each illegal turn makes the length of the path grow with speed 2, and the illegal turn closest to an end moves away from the end at speed 1. We deduce that [lifting is unique between the first and last illegal turns along the path $\gamma$, including the germs of directions beyond these turns.]{} We call this the [unfolding principle]{}. In particular, this applies to illegal turns themselves: if a loop $z\|\hG_\omega$ has two occurrences of the same illegal turn, pulling back these turns produces two occurrences of the same path (most of the time a neighborhood of a single illegal turn, but see Figure \[f:unfold.v2\]). ![A path unfolds to a path with two illegal turns.[]{data-label="f:unfold.v2"}](unfold.v3.eps) But note that distinct illegal turns might pull back to the same illegal turn, see Figure \[f:fold\]. ![In this folding path the number of illegal turns grows from 1 to 2.[]{data-label="f:fold"}](fold.v3.eps) 0.5cm [.]{} \[p:illegality\] If a graph $\gG$ is equipped with a track structure, by the [* of $\gG$]{} we mean $$\m(\gG)=\sum_{v}\sum_{\Omega_v}(\|\Omega_v\|-1)$$ where the sum is over all vertices $v$ of $\gG$ and all gates $\Omega_v$ at $v$. Thus a gate that contains $k\geq 1$ directions contributes $k-1$ to the count. The right derivative of the function $$t\mapsto vol(\hG_t)$$ at $t=t_0$ is $-\m(\hG_0)$, the negative of the of $\hG_{t_0}$. If $\gG$ is marked and $z$ is a conjugacy class in $\FF$ then $\k(z\|\gG)$ is by definition the number of illegal turns in $z\|\gG$, i.e. the number of illegal turns in the illegal turn structure on $z\|\gG$ induced by $z\|\gG\to\gG$. [For the rest of this section,]{} $\bG_t$, $t\in [0,L]$, is a folding path in $\X$ parametrized by arclength. We will sometimes abbreviate $\m(\bG_t)$ by simply $\m_t$, $z\|\bG_t$ by $z_t$ and $\k(z_t)$ by $\k_t$. \[derivative\] (1) \[i:loop\] Let $z$ be a conjugacy class in $\FF$. The the right derivative of the length function $t\mapsto\ell(z_t)$ at $t=0$ is $$\frac d{dt}\ell(z_t)\|_{t=0^+}=\ell(z_{0})-2\frac{\k_0}{\m_0}$$ (2) \[i:segment\] Let $\sigma_0$ be a nondegenerate immersed path in $\bG_0$ whose initial and terminal directions are in illegal turns of $\bG_0$. Then, for small $t$, there is a corresponding path $\sigma_t$ whose initial and terminal directions are in illegal turns of $\bG_t$. The right derivative at 0 of the length $L_t$ of $\sigma_t$ is $L_0-2\frac{\k_0}{\m_0}$, where now $\k_0-1$ is the number of illegal turns in the interior of $\sigma_0$. If the initial and terminal directions of $\sigma_0$ are in the same gate, then the same is true for $\sigma_t$. (1): Let $\hG_t$ be the naturally parametrized by path in $\hX$ so that $\bG_{s(t)}=\G_t=\hG_t/vol(\G_t)$ and $vol(\hG_0)=1$ (see Notation \[n:folding path\]). For small $t\ge 0$, $vol(\hG_t)=1-\m_0t$, $\ell(z\|\hG_t)=1-2\k_0t$, and $s(t)=-\log(vol(\hG_t))$. The proof of the derivative formula in (2) is identical. We will sometimes abuse notation and write $\ell'(z_{t_0})$ for $\frac d{dt}\ell(z\|\bG_t)\|_{t=t_0^+}$. \[c:length\] Let $z$ and $w$ be conjugacy classes in $\FF$ and suppose $\m_t$ and $\k_t=\k(z_t)$ are constant for $t\in [0,\epsilon)$. (1) \[i:local\] The length $\ell(z_t)=a e^t+b$ on $[0,\epsilon)$ where $a=\ell(z_0)-\frac{2\k_0}{\m_0}$ and $b=\frac{2\k_0}{\m_0}$. (2) \[i:compare\] If $\ell'(z_0)\geq \ell'(w_0)$ then $\ell(z_t)-\ell(w_t)$ is nondecreasing on $[0,\epsilon)$. (3) \[i:average\] If the average length $A$ of a maximal legal segment in the loop $z_0$ is $>2/\m_0$ the loop grows in length on $[0,\epsilon)$, and if it is $<2/\m_0$ it shrinks. (\[i:local\]) and (\[i:compare\]) follow easily from Lemma \[derivative\]. For (\[i:average\]), $\k_0A=\ell(z_0)$ and so, by (\[i:local\]), $a=\ell(z_0)(1-2/\m_0A)$. \[c:global\] Let $z$ be a conjugacy class in $\FF$. The length $\ell(z_t)$ is piecewise exponential on $[0,L]$, i.e. $[0,L]$ has a finite partition such that the restriction of $\ell(z_t)$ to each subinterval is as in Corollary \[c:length\](\[i:local\]). Since we may subdivide $[0,L]$ as $0=s_0<s_1<\cdots<s_k=L$ so that, on each $[s_i,s_{i+1})$, $\m_t$ and $\k_t$ are constant, the corollary follows directly from Corollary \[c:length\](\[i:local\]) and Proposition \[p:tame\]. We will say that a segment [has endpoints illegal turns]{} if the path has been infinitesimally extended at each end, i.e. directions have been specified at the initial and terminal endpoints, and these directions together with the segment determine illegal turns. If the segment is degenerate then this means that the added directions form an illegal turn. \[c:segments\] Let $\sigma_L$ be an immersed path in $\bG_L$ with endpoints illegal turns and, for $t\in [0,L]$, let $\sigma_t$ be the immersed path in $\bG_t$ with endpoints illegal turns obtained by applying the Unfolding Principle to $\sigma_L$. Corollaries \[c:length\] and \[c:global\] hold if $\ell(z_t)$ is replaced by the length of $\sigma_t$. Since the derivative formula in Lemma \[derivative\](\[i:segment\]) is obtained from that in Lemma \[derivative\](\[i:loop\]) by such a replacement, proofs of corresponding statements are identical. \[3\] A legal segment $\sigma_0$ of length $L_0\geq 2$ inside $z_0$ gives rise to a legal segment $\sigma_t$ of length $L_t\geq 2+(L_0-2)e^t$ inside $z_t$ for $t\in [0,L]$. In particular, a legal segment of length $L_0\geq 3$ grows exponentially. $(L_t-2)'\ge L_t-2$ using Lemma \[derivative\](\[i:segment\]). Similar considerations control the lengths of topological edges $e$ of $\bG_t$ that are not [involved]{} in any illegal turn, i.e. the directions determined by the ends of $e$ aren’t in any illegal turns. \[l:good edges\] Suppose that $e_0$ is an edge of $\bG_\0$ that is not involved in an illegal turn. Then for small $t$, there is a corresponding edge $e_t$ of $\bG_t$ not involved in an illegal turn. Further, the length $L_t$ of $e_t$ satisfies $L_t=L_0e^t$. For small $t$, the morphism $e^t\bG_0\to\bG_t$ restricted to $e_0$ is an isometric embedding with image an edge of $\bG_t$. \[l:bounded everywhere\] Suppose $z$ is a conjugacy class such that $k(z_0)$ and $\ell(z_L)$ are bounded. Then $\ell(z_t)$ is bounded for all $t\in [0,L]$. By Corollary \[c:length\](\[i:average\]) and Corollary \[c:global\], if $\ell(z_t)/\k(z_t)\ge\ell(z_t)/\k(z_0)\ge 2\ge 2/m_t$, then $\ell(z_t)$ grows. Therefore, $\ell(z_t)\le\max\{2k(z_0), \ell(z_L)\}$ for all $t\in [0,L]$. By a [surface relation]{} we mean a conjugacy class that, with respect to some rose, crosses every edge twice and has a circle as its Whitehead graph (equivalently, attaching a 2-cell to the rose via the curve results in a surface). \[monogon\] Suppose $\m_t\ge m$, for all $t\in [0,L]$, and let $w$ be a conjugacy class in $\FF$. Assume $\k(w_0)=m$. If $\ell(w_L)\leq K$ then either (i) there is a simple class $u$ such that: - $\k(u_0)$ is bounded; and - $\ell(u_t)$ is bounded by a function of $K$ for all $t\in [0,L]$ (in particular $d_\F(\bG_0,\bG_L)$ is bounded by a function of $K$); or (ii) $w$ is a surface relation. Moreover, if $\m_t>m$ for all $t$ then (i) holds. Arguing as in Lemma \[l:bounded everywhere\], we have $\ell(w_t)\leq\max\{2,K\}$ (loops of length $>2$ with no more illegal turns than the grow under folding). So we may take $u=w$ provided $w$ is simple. See Lemma \[bounded crossing 2\] and Remark \[r:bounded crossing 2\] for the parenthetical remark. We now consider four cases. [Case 1.]{} $\ell(w_0)<2$. Then $w$ is simple as $w_0$ crosses some edge at most once. [Case 2.]{} $\ell(w_0)=2$. Then either $w$ is simple or $w_0$ crosses every edge exactly twice. In the latter case, collapse a maximal tree in $\bG_0$ – with respect to the resulting rose the Whitehead graph of $w$ is either a circle (and then $w$ is a surface relation) or the disjoint union of at least two circles (and then $w$ is simple). [Case 3.]{} $2<\ell(w_0)<2+injrad(\bG_0)$. Then either $w$ is simple or $w_0$ crosses every edge at least twice. Assume the latter. Under our assumption the edges crossed more than twice form a forest. Collapse a maximal tree that contains this forest and argue as in Case 2. [Case 4.]{} $\ell(w_0)\geq 2+injrad(\bG_0)$. Choose a conjugacy class $v$ with $\ell(v_0)=injrad(\bG_0)$. We now claim that $\ell(v_t)\leq\ell(w_t)-2$ for all $t$. This is clearly true at $t=0$. In fact, this condition [persists]{} in that there is no last time $t_0<L$ where it is true. Indeed, Lemma \[derivative\](\[i:loop\]) shows that $$\ell'(v_{t_0})\leq \ell(v_{t_0}) \leq \ell(w_{t_0})-2\leq \ell'(w_{t_0})$$ and so the inequality continues to hold for $t>t_0$ (see Corollary \[c:length\](\[i:compare\])). Thus $v$ is a simple class with both $\ell(v_t)$ bounded. We may take $u=v$. For the moreover part, we have $\ell'(w_t)\geq \ell(w_t)-2\frac m{m+1}$ for all $t$. Thus if $\ell(w_0)<2$ then $w$ is simple and the statement follows with $u=w$. If $\ell(w_0)\geq 2$, then we claim $d_\X(\bG_0,\bG_L)$ is bounded (see Corollary \[proj X->F continuous\]). Indeed, by Corollary \[c:length\] we have $\ell(w_t)\ge ae^t+b$ where $a=\ell(w_0)-2\frac{m}{m+1}\ge \frac{2}{m+1}$ and $b\ge 2$. In particular, $K\ge \frac{2}{m+1}e^L+2$. We may take $u=v$ such that $\ell(v_0)=injrad(\bG_0)$. We also have the following variant. \[monogon2\] Suppose in addition to the hypotheses of Lemma \[monogon\] that, for some illegal turn in $w\|\bG_0$, one of the two edges $e$ forming the turn is nonseparating and has length a definite fraction $p>0$ of $injrad(\bG_0)$. Then either: 1. there is a simple class $u$ such that: - $\k(u_0)$ is bounded; and - $\ell(u_t)$ is bounded as a function of $K$ and $p$ for all $t\in [0,L]$ (in particular $d_\F(\bG_0,\bG_L)$ is bounded in terms of $K$ and $p$); or 2. $w$ is a surface relation and any class $z$ such that $z_0$ contains a segment $S=e\cdots e$ that closes up (by identifying the two copies of $e$) to $w_0$ fails to be simple. Referring to the proof of Lemma \[monogon\], in Cases 1 and 4, ([i$'$]{}) holds; so assume we are in Cases 2 or 3. In fact we are free to assume $\ell(w_0) < 2+p~ injrad(\bG_0)$, for otherwise the argument of Case 4 shows that for $v$ with $\ell(v_0)=injrad(\bG_0)$ we have $\ell(v_t)\leq \frac{\ell(w_t)-2}p$ for any $t$ and ([ i$'$]{}) follows (and this time the bound also depends on $p$). Now the forest consisting of the edges crossed by $w_0$ more than twice (assuming all edges are crossed at least twice) does not include $e$, and we may collapse a maximal tree that contains this forest but does not contain $e$. Now $z_0$ can be thought of as $e\cdots e\cdots=eAeB$ with the subpath $S=eAe$ giving $w$. Since $e$ is not collapsed, the Whitehead graph of $z$ in the rose contains the Whitehead graph of $w$, which is a 1-manifold. So if $w$ is not simple, neither is $z$. Loops with long illegal segments {#s:gafa} ================================ The key result of this section is Proposition \[gafa\](surviving illegal turns) which is a generalization of [@gafa Lemma 2.10]. Before stating Proposition \[gafa\], we need a bit of terminology and some preliminary lemmas to be used in the proof. In this section, $\bG_t$, $t\in [0,L]$, is a folding path in $\X$ parametrized by arclength. Consider a conjugacy class $z$ in $\FF$ and the induced path of loops $z_t:=z\|\bG_t$. The illegal turns along $z_t$ are folding as $t$ increases, but at discrete times an illegal turn may become legal, or several illegal turns may collide and become one (e.g. see Figure \[f:unfold.v2\]). We say that a consecutive collection of illegal turns along $z$ [survives to $\bG_L$]{} if none of them become legal nor do they collide with a neighboring illegal turn in the collection, for any $t\in [0,L]$. In particular, each illegal turn in the collection in $z_t$ unfolds to a single illegal turn in the collection in $z_{t'}$ for $t'\le t$. We will call the portion[^5] of $z_t$ spanned by our collection the [good portion of $z_t$]{}. The turns in the collection in $z_t$ are, in order, $\tau_{t,1}, \tau_{t,2},\dots$ and have vertices $p_{t,1}, p_{t,2},\dots$. In particular, if $t'<t$ then $\tau_{t,i}$ unfolds to $\tau_{t',i}$. The image of $\tau_{t,i}$ in $\bG_t$ is $\bar\tau_{t,i}$. In this context, the Unfolding Principle gives the implication $\bar\tau_{t,i}=\bar\tau_{t,j}\implies\bar\tau_{t',i}=\bar\tau_{t',j}$. We also say that $\bar\tau_{t,i}$ [unfolds to]{} $\bar\tau_{t',i}$. Suppose that a consecutive collection of illegal turns along $z$ survives to $\bG_L$. For each $t\in [0,L]$ denote by $\mathcal T_t$ the set of turns that occur in the given consecutive collection, i.e. $\mathcal T_t=\{\bar\tau_{t,1}, \bar\tau_{t,2},\dots\}$. Let $D_t$ denote the set of directions in $\bG_t$ that occur in a turn in $\mathcal T_t$. Of course, $D_t$ is partitioned into equivalence classes with respect to the relation “being in the same gate”, but we consider a finer equivalence relation generated by $d\sim d'$ if $\{d,d'\}\in\mathcal T_t$. We will call the equivalence classes [subgates]{}. Each subgate, say at the vertex $v$, gives rise to a “Whitehead graph”: the vertices of the graph are the directions in the subgate, and an edge is drawn between $d$ and $d'$ if $\{d,d'\}\in\mathcal T_t$. \[sublemma\] In the situation above, let $d_1,d_2,\cdots,d_k$ be the vertices along an embedded closed curve in the Whitehead graph of a subgate (that is, $d_i\neq d_j$ for $i\neq j$) at $\bG_t$. Then for any $t'<t$ there is an induced embedded closed curve in the Whitehead graph of a subgate of $\bG_{t'}$. Specifically, each turn $\{d_i,d_{i+1}\}$ (taken $\mbox{mod } k$, so including $\{d_k,d_1\}$) unfolds to a turn in $\mathcal T_{t'} $; Lemma \[sublemma\] says that these turns also form an embedded closed curve in a subgate (in particular, all are based at the same vertex). Let $\hG_{s(t)}$ be the path in $\hX$ giving rise to $\bG_t$ (see Notation \[n:folding path\]). As combinatorial graphs, we identify $\bG_t$ and $\hG_{s(t)}$ (they differ only by homothety). In particular, a direction in one is naturally identified with a direction in the other and we will use the same names for two such directions. If we set $s:=s(t)$ and $s':=s(t')$, it suffices to argue in the case $s'=s-\epsilon$ for small $\epsilon>0$; for the conclusion clearly holds in the limit. To that end, we use the description of $N_\epsilon$ developed in Section \[s:folding\]. So, let $v$ be the vertex in $\hG_s$ that is the base of the directions $d_i$ in our subgate, $N$ be the $\epsilon$-neighborhood in $\hG_s$ of $v$, and $N_\epsilon$ be the preimage of $N$ in $\hG_{s'}$. Set $d_i^=d_i^\epsilon$ or $d_i^{-\epsilon}$ depending on whether or not $d_i$ points up or down and similarly for $\tilde d_i^$. For $i\not= j$, let $[d^_i,d^_j]$ denote the path in $\hG_s$ from the base of $d^_i$ to the base of $d^_j$ extended infinitesimally by the outgoing (germs of) directions $d^_i$ and $d^_j$. Similarly, $[\tilde d^_i,\tilde d^_j]$ denotes the unique immersed path lifting $[d^_i,d^_j]$. Note that if $z\|\hG_s$ contains the illegal turn $\{d_i,d_j\}$ then it also contains the (infinitesimally extended) path $[d^_i,d^_j]$ and $z\|\hG_{s'}$ contains $[\tilde d^_i,\tilde d^_j]$. If $\tilde d_i^$ points down then it is [supported]{} by a widget $W$ if $\tilde d_i^$ is in the downward direction from some $x\in W$ of height 0. In this case, we also say that $\tilde d_i^$ is [supported]{} by $x$. If $\tilde d_i^$ points up, then it is [ supported]{} by the unique widget at which it is based. See Figure \[f:notation\]. ![An upward and a downward direction supported by the leftmost widget.[]{data-label="f:notation"}](notation.eps) By hypothesis, $[\tilde d^_i,\tilde d^_{i+1}]$ contains a unique illegal turn that we abusingly denote $\{\tilde d^_i,\tilde d^_{i+1}\}$. Recall that all illegal turns in $\bG_{s'}$ have height $\epsilon$ and are in a widget. Note that if $[\tilde d^_i,\tilde d^_{i+1}]$ (or indeed any path between height $\pm\epsilon$ vertices) contains its unique illegal turn in $W$ then it falls into one of the following three cases: - $\tilde d^_i$ and $\tilde d^_{i+1}$ point up in distinct widgets adjacent to $W$ - $\tilde d^_i$ and $\tilde d^_{i+1}$ point down and are supported by $W$ - one of $\tilde d^_i$ or $\tilde d^_{i+1}$ points down and is supported by $W$ and the other points up in a widget adjacent to $W$ We now make the following observations. (1) If $\tilde d^_i$ points up and is supported by the widget $W$, then all directions pointing upward at height 0 in $W$ are mapped to $d_i$. If $\tilde d^_i$ points down, then it is in a unique downward direction from a height 0 point and this direction maps to $d_i$. (2) The directions $\tilde d^_i$ are all distinct. Indeed, by hypothesis the $d_i$ (hence the $d^_i$) are distinct, and $\tilde d^_i$ is the unique lift of $d^_i$. (3) It is not possible for a single widget to support both an upward $\tilde d^_i$ and a downward $\tilde d^_j$. Indeed, this would force $\{d_i,d_j\}$ to be illegal and then $[\tilde d^_i,\tilde d^_j]$ would have a height 0 illegal turn. However, all illegal turns occur at height $\epsilon$. (4) It is not possible for adjacent widgets to both support upward $\tilde d^_i$ and $\tilde d^_j$. This is because this would force an illegal turn at the common height 0 vertex formed by directions that map to $d_i$ and $d_j$. (5) It is not possible for downward $\tilde d^_i$ and $\tilde d^_j$ to be supported by the same height 0 vertex. Indeed, this would then force an illegal turn based at this height 0 vertex. Recall that we want to prove $\{\tilde d^_1,\tilde d^_{2}\}, \{\tilde d^_2,\tilde d^_{3}\},\dots,\{\tilde d^_k,\tilde d^_{1}\}$ gives rise to an embedded closed curve in the Whitehead graph of a subgate at $\bG_{s'}$. To do this we prove two things. - (there is a loop) The turns $\{\tilde d^_i,\tilde d^_{i+1}\}$ are all based at the same vertex $w$, i.e. the base of the illegal turn crossed by $[\tilde d^_i,\tilde d^_{i+1}]$ is independent of $i$. - (the loop is embedded) For $i\not=j$, $\tilde d^_i$ and $\tilde d^_{j}$ are not in the same direction from $w$. To see there is a loop, suppose we have three consecutive directions $\tilde d^_{i-1},\tilde d^_i,\tilde d^_{i+1}$ that determine two illegal turns in $N_\epsilon$ not based at the same vertex. There are two cases. First suppose $\tilde d^_i$ points down and is supported by some height 0 vertex $x$. Paths from $\tilde d_i^$ to $\tilde d^_{i\pm 1}$ lead through two distinct widgets each containing $x$. Since our directions $\tilde d^_1,\dots,\tilde d^_k$ are cyclically ordered and $N_\epsilon$ is a tree, there must be some $j$ with $j\neq i\neq j+1$ so that $[\tilde d^_j,\tilde d^_{j+1}]$ passes through $x$. (Indeed, otherwise all $\tilde d^_j$, for $j\not=i$, lie in the same component of $N_\epsilon\setminus\{w\}$. Since $\tilde d^_{i-1}$ and $\tilde d^_{i+1}$ lie in distinct components, this is a contradiction.) The path $[\tilde d^_j,\tilde d^_{j+1}]$ cannot terminate at $\tilde d^_i$ by (2), and by (5) it cannot terminate in any downward direction supported by $x$, but must continue to another (adjacent) widget. By (3) the widgets containing $x$ do not support upward $\tilde d^_j$ and $\tilde d^_{j+1}$, so the path crosses two illegal turns, contradiction. The other case is that $\tilde d^_i$ is upward, say based at a height $\epsilon$ vertex $x$ inside a widget $W$. Then there are distinct widgets $W_+$ and $W_-$ adjacent to $W$ so that $[\tilde d^_i,\tilde d^_{i\pm 1}]$ crosses an illegal turn in $W_\pm$. Again there must be some $j$ so that $[\tilde d^_j,\tilde d^_{j+1}]$ either crosses $x$ (if $W_+\cap W_-=\emptyset$) or crosses the intersection point $W_+\cap W_-$ (if there is one, and then this point is in $W$ as well). In the latter case the path does not terminate at any direction supported by this point by (3) nor at an upward direction supported by $W_\pm$ by (4). Thus this path has two illegal turns, contradiction. In the former case, $[\tilde d^_j, \tilde d^_{j+1}]$ must have at least 3 illegal turns: one at $x$ and one on each side of $x$, by (3) and (4). We have established the first item. To see that the loop is embedded, suppose that there are $i\not= j$ such that $\tilde d^_i$ and $\tilde d^_j$ are in the same direction from $w$. Each of these directions is contained in a path between height $\pm \epsilon$ vertices with one illegal turn, that illegal turn being based at $w$. Considering the three cases listed above for such paths, $\tilde d^_i$ and $\tilde d^_j$ have to either be both downward and supported by the same height 0 vertex of the widget $W$ containing $w$, contradicting (5), or one is downward and the other upward supported in the same widget adjacent to $W$, contradicting (3), or they are upward and supported by widgets adjacent to $W$ and to each other, contradicting (4). Given a finite, simple[^6], connected graph $\gG$, we may consider the components $C_i$ of $\gG$ cut open along its set $CV$ of cut vertices, i.e. $C_i$ is the closure in $\gG$ of a component of $\gG\setminus CV$. Let $\sim$ be the equivalence relation on the edges of $\gG$ generated by $e\sim e'$ if there is an embedded loop containing $e$ and $e'$. \[l:graph\] Suppose $\gG$ is a finite, simple, connected graph. Let $\{C_i\}$ be the components of $\gG$ cut open along its cut vertices. (1) \[i:component\] $e\sim e'$ if and only if they are in the same $C_i$. (2) \[i:count\] The cardinality of $\{C_i\}$ is at most the number of edges of $\gG$ with equality iff $\gG$ is a tree. (1): Since a circle has no cut vertex, if $e$ and $e'$ are contained in an embedded loop, then they are in the same $C_i$. If $e$ and $e'$ are in the same $C_i$ then there is an edge path $e_1,\dots, e_p$ in $C_i$ from $e$ to $e'$. The common vertex between $e_i$ and $e_{i+1}$ is not a cut vertex of $C_i$ (indeed, $C_i$ has no cut vertices). It follows that $e_i$ and $e_{i+1}$ are contained in an embedded loop in $C_i$. Hence $e=e_1\sim e_2\sim\dots\sim e_p=e'$. (2): If $\gG$ is a tree then $\{C_i\}$ is the set of closed edges of $\gG$. In particular, we have equality. In general, consider the bipartite tree whose vertex set is the disjoint union of $\{ C_i\}$ and the set $\{v\}$ of cut points of $\gG$ and whose edges are given by the relation $v\in C_i$. For the equality, note that, since $\gG$ is simple, if $\gG$ is not tree then some $C_i$ contains more than one edge. We will consider the equivalence relation on $\mathcal T_t$ generated by $\bar\tau_{t,i}\sim\bar\tau_{t,j}$ if there is an embedded loop in the Whitehead graph $WG_t$ of a subgate containing $\bar\tau_{t,i}$ and $\bar\tau_{t,j}$. By Lemma \[l:graph\](\[i:component\]), equivalent turns are represented by edges in the same component of $WG_t$ cut open along its cut vertices. According to the Lemma \[sublemma\], unfolding equivalent turns produces equivalent turns, and in particular the turns have the same vertex. Recall that $\m_t=\m(\bG_t)$ is the of $\bG_t$, see Page . \[l:\|classes\|\] (1) \[i:subgates\] $\|\,\mathcal T_t\,/\!\!\sim\!\!\|\le\m_t$ with equality iff subgates coincide with gates and the Whitehead graph of subgates are trees. (2) $\|\,\mathcal T_t\,/\!\!\sim\!\!\|\le \|\,\mathcal T_t\,\|$ with equality iff Whitehead graphs of subgates are trees. The first item follows from definitions and the second is a consequence of Lemma \[l:graph\](\[i:count\]). We are now ready for the main result of this section. Let $\hm$ denote the maximal possible number of illegal turns for any train track structure on any element of $\X$. \[gafa\] Let $z$ be a simple class and $\bG_t$, $t\in [0,L],$ a folding path in $\X$ parametrized by arclength. Assume that $M=2\hm+1$ consecutive illegal turns of $z\|\bG_0$ survive to $\bG_L$ and that the legal segments between them in $z\|\bG_L$ have bounded size. Then $d_\F(\bG_0,\bG_L)$ is bounded. We saw above that two illegal turns in our consecutive collection in $z\|\bG_t$ that give the same element of $\mathcal T_t$ also give the same element of $\mathcal T_{t'}$ for $t'\le t$, i.e. we saw $\bar\tau_{t,i}=\bar\tau_{t,j}$ implies $\bar\tau_{t',i}=\bar\tau_{t',j}$. However, distinct illegal turns might unfold to the same illegal turn, i.e.$\bar\tau_{t',i}=\bar\tau_{t',j}$ does not imply $\bar\tau_{t,i}=\bar\tau_{t,j}$. So $\|\mathcal T_{t'}\|\le\|\mathcal T_t\|$. By partitioning $[0,L]$ into a bounded number of subintervals and renaming, we may assume in proving Proposition \[gafa\] that $\|\mathcal T_t\|$ is constant on $[0,L]$. Likewise, we may assume that $\|\,\mathcal T_t\,/\!\!\sim\!\!\|$ is constant (in general $\|\,\mathcal T_t\,/\!\!\sim\!\!\|$ may decrease under unfolding when a new circle is formed). The proof of Proposition \[gafa\] breaks into two cases: (1) Some subgate contains a circle in its Whitehead graph at time $L$. (2) There are no circles in Whitehead graphs of subgates at time $L$. In Lemma \[l:case 1\], resp. Lemma \[l:case 2\], we show that Proposition \[gafa\] holds in Case 1, resp. Case 2. So once we have proved these next two lemmas, we will also have proved Proposition \[gafa\]. In Lemma \[l:case 1\], we prove a little more. \[l:case 1\] Suppose that, in addition to the hypotheses of Proposition \[gafa\], some subgate contains a circle in its Whitehead graph at time $L$. Then, there is a partition $0=t_0<t_1<\dots<t_N=L$ of $[0,L]$ into boundedly many subintervals and simple classes $u_1, u_2, \dots, u_N$ such that - $\k(u_{i}\|\bG_{t_{i-1}})$ is bounded (recall that $\k(u_i\|\bG_{t_{i-1}})$ is the number of illegal turns in $u_i\|\bG_{t_{i-1}}$); and - $\ell(u_i\|\bG_t)$ is bounded for all $t\in [t_{i-1},t_i]$. In particular $d_\F(\bG_0,\bG_L)$ is bounded. In light of Lemma \[l:bounded everywhere\], we only need to prove there are simple $u_i$ such that: - $\k(u_{i}\|\bG_{t_{i-1}})$; and - $\ell(u_i\|\bG_{t_i})$ is bounded. Since some subgate contains a circle in its Whitehead graph at time $L$, by Lemma \[sublemma\] the same is true at every $t$. Choose distinct illegal turns $\tau_{L,i}, \tau_{L,j}$ in $z_L$ that are equivalent in $\mathcal T_L$ so that the number of illegal turns between in the good portion of $z_L$ is smaller than the number of equivalence classes. (This is possible because circles in Whitehead graphs have more than one edge.) Let $[p_{t,i},p_{t,j}]$ denote the resulting segment in the good portion of $z_t$ and let $\sigma_t$ be the loop obtained by closing up our segment, i.e. by identifying $p_{t,i}$ and $p_{t,j}$. We refer to $\sigma_t$ as a [monogon]{} because it is immersed in $\bG_t$ except possibly at the point $p_{t,i}=p_{t,j}$. In particular, the conjugacy class $w(t)$ in $\FF$ represented by $\sigma_t$ is nontrivial. Of course, $w(t)$ is also represented by the immersed circle $w(t)\|\bG_t$ which is obtained by tightening $\sigma_t$. By construction and Lemma \[l:\|classes\|\](\[i:subgates\]), $\k(w(t)\|\bG_t)\le\|\,\mathcal T_t\,/\!\!\sim\!\!\|<\m_t$. In particular, $w(L)\|\bG_L$ has bounded length. We claim that $w(0)=w(L)$. Once this claim is established, the last sentence of Lemma \[monogon\] (with $m$ equal to the constant $\|\,\mathcal T_t\,/\!\!\sim\!\!\|$) completes Case 1. To prove our claim, we must show that $\sigma_{0}$ and $\sigma_L$ determine the same conjugacy class. In folding $z_{0}$ to $z_L$, maximal arcs in the directions of $\tau_{0,k}$ are identified in $\bG_L$, i.e. they have the same image which is an immersed arc $\alpha_{k}$ in $\bG_L$. If we tighten the image of $[p_{0,i},p_{0,j}]$ in $\bG_L$, the result is the image of $[p_{L,i},p_{L,j}]$ extended at its ends by $\alpha_{i}$ and $\alpha_{j}$. The claim follows since $\alpha_{i}=\alpha_{j}$. Indeed, $\tau_{0,i}$ and $\tau_{0,j}$ are equivalent, and so in the same subgate, and so in the same gate. We see $\alpha_i=\alpha_j$ at least if $L$ is small enough and that this condition persists. \[l:case 2\] Suppose that, in addition to the hypotheses of Proposition \[gafa\], no subgate contains a circle in its Whitehead graph at time $L$. Then $d_\F(\bG_0,\bG_L)$ is bounded. Since there are no circles in Whitehead graphs of subgates at time $L$, $s:=\|\,\mathcal T_L\,/\!\!\sim\!\!\|=\|\mathcal T_L\|\le \m_L$. By our assumptions that $\|\,\mathcal T_t\,/\!\!\sim\!\!\|$ and $\|\mathcal T_t\|$ are constant on $[0,L]$, there are no circles for any $t$. In particular, $s\le \m_t$ for all $t\in [0,L]$. See Lemma \[l:\|classes\|\]. First assume that there are two occurrences of the same illegal turn in the consecutive collection at time $L$ that are separated by $<s-1$ illegal turns, i.e. there are $\tau_{L,i}$ and $\tau_{L,j}$ with $\bar\tau_{L,i}=\bar\tau_{L,j}$ and $0<j-i<s-1$. Closing up gives a curve with $<s$ illegal turns, so again the conclusion follows by arguing as in Case 1. So from now on we assume that this does not happen, i.e. all $s$ illegal turns occur repeatedly in a cyclic order in the consecutive collection along $z_t$. If it so happens that one of these illegal turns at $t=0$ involves an edge $e$ which is nonseparating and has length a definite fraction $p$ of $injrad(\bG_0)$, we argue using Lemma \[monogon2\] as follows. Consider the loop in $\bG_0$ obtained by closing up the segment that starts with $e$ and ends at the next occurrence of the same illegal turn. (Such a segment exists since there are $M>\hm$ illegal turns in our collection.) If the edge following this segment is $e$, we can appeal to Lemma \[monogon2\] to deduce the conclusion of the lemma (because $z$ is simple). If the edge following the segment is not $e$, then the last edge $\bar e$ of the segment is $e$ with the opposite orientation and closing up forces cancellation. If the tightened loop has length $<2$, it is simple and its image in $\bG_L$ is bounded, so the conclusion follows (cf. the first sentence of the proof of Lemma \[monogon\]). If the tightened loop has length $\geq 2$, then the original segment from $e$ to $\bar e$ has length $\geq 2+2p~injrad(\bG_0)$ and the same argument as in Case 4 of Lemma \[monogon\] (using Lemma \[derivative\](\[i:segment\]) instead of Lemma \[derivative\](\[i:loop\]); see also proof of Lemma \[monogon2\]) shows that for $\ell(v\|\bG_0)=injrad(\bG_0)$ we must have $\ell(v\|\bG_t)$ bounded. To summarize, the conclusion of the lemma holds whenever the following condition is satisfied at $t=0$: - There is a nonseparating edge $e$ in $\bG_t$ such that: (i) $e$ is in the good portion of $z_t$; (ii) $\ell_{\bG_t}(e)\ge p~injrad(\bG_t)$; and (iii) $e$ is involved in some turn in our collection $\mathcal T_t$. So now all that remains is to reduce to the case where $(\star)$ is satisfied at $t=0$. As a warmup, first consider the case where $\m(\bG_t)=s$ for all $t$. In particular, every edge that is involved in an illegal turn of $\bG_t$ is involved in an illegal turn of our collection. Let $\beta\in [0,L]$ be the first time that a nonseparating edge of length $\geq \frac 1{3n-3}injrad(\bG_\beta)$ is involved in an illegal turn of $\bG_\beta$. If there is no such $\beta$ then we claim that the conclusion of the lemma holds. Indeed, if $v$ is a conjugacy class with $\ell(v_0)=injrad(\bG_0)$ where $v_0:=v\|\bG_0$ then there is a nonseparating edge $e_0$ of length $\geq \frac 1{3n-3}injrad(\bG_0)$ in $v_0$. By Lemma \[l:good edges\], there is a corresponding edge $e_t$ for small $t$. Note that $$\ell(e_t)=\ell(e_0)e^t\ge\frac{\ell(v_0)e^t}{3n-3}\ge\frac{\ell(v_t)}{3n-3}\ge\frac 1{3n-3}injrad(\bG_t)$$ and so $e_t$ is not involved in any illegal turns. In fact, we see that there can be no first time where $\ell(e_t)<\frac 1{3n-3}injrad(\bG_t)$. We have $\ell(v_t)\le e^t\ell(v_0)\le (3n-3){e^t}\ell(e_0)=(3n-3)\ell(e_t)<3n-3$ is bounded. If there is such a $\beta$, then for the same reason the conclusion holds for $\bG_t$, $t\in [0,\beta]$. Also, the conclusion holds for $\bG_t$, $t\in [\beta,L]$ since ($\star$) holds at $t=\beta$. It remains to consider the case when $\m_t$ is perhaps sometimes $>s$. Let $\bG_t$, $t\in [0,L']$ be the path in $\X$ starting at $\bG_0$ obtained as in \[slope 1 folding\].C by folding only the $s$ illegal turns in our collection $\mathcal T_t$ (and then normalizing and reparametrizing by arclength). Note that by the Unfolding Principle each illegal turn $\tau_{0,i}$ our collection at time 0 comes with a pair of legal paths that get identified in the folding process. Folding only these turns amounts to identifying these paths. There are induced morphisms $\bG_0\to e^{-L'}\bG'_{L'}\to e^{-L}\bG_L$. (In particular, $\bG'_t$ is a folding path.) We can scale the second of these morphisms to obtain $\phi:\bG'_{L'}\to \bG_L$. By the special case $\m(\bG'_t)=s$, there are simple classes $v$ and $u$ and a partition of $[0,L']$ into two subintervals such that $\ell(v\|\bG'_t)$ is bounded on the first and $\ell(u\|\bG'_t)$ is bounded on the second. Note that $\phi$ is an immersion on the segment spanned by our collection of turns, i.e. this segment is now legal in the train track structure induced by $\phi$. Let $x\|\bG'_{L'}$ be obtained by identifying consecutive occurrences of the same oriented element of our collection (possible since $M>2\hm$). So, $\ell(x\|\bG'_{L'})$ is legal with respect to $\bG'_{L'}\to\bG_L$. Also $\k(x\|\bG_0)$ and $\ell(x\|\bG_L)$ are bounded. If $\ell(x\|\bG_L)<2$, then $x$ is simple and the conclusions of the lemma hold. If $\ell(x\|\bG'_{L'})\ge 2$, then $d_\X(\bG'_{L'},\bG_L)=L-L'$ is bounded. Indeed, since $$\ell(w\|\bG_L)=\ell(w\|\bG'_{L'})e^{L-L'}\ge 2e^{L-L'}$$ is bounded, so is $L-L'$. In particular, $\ell(u\|\bG_L)$ is bounded. We have completed the proof of Proposition \[gafa\]. \[d:illegal loop\] An immersed path or a loop in a metric graph $\gG$ equipped with a train track structure is [illegal]{} if it does not contain a legal segment of length 3 (in the metric on $\gG$). \[unfolding\] Let $z$ be a simple class and $\bG_t$, $t\in [0,L],$ a folding path in $\X$ parametrized by arclength. Assume $z_t$ is illegal for all $t$. Then either $\ell(z_L)<\ell(z_0)/2$ or $d_\F(\bG_0,\bG_L)$ is bounded. There are two cases. First suppose that the average distance between consecutive illegal turns in $z_0$ is $\geq 1/\hm$. Then, by Proposition \[gafa\](surviving illegal turns), after a bounded progress in $\F$ the loop $z$ must lose at least $1/M$ of its illegal turns. Repeating this a bounded number of times, we see that, after bounded progress in $\F$, $\k_t\le \k_0/6\hm$. Thus either the length of $z_t$ is less than $1/2$ of the length of $z_0$ or the average distance between illegal turns at time $t$ is $\geq 3$, so there is a legal segment. Now suppose the average distance between illegal turns in $z_0$ is $<1/\hm$. By Lemma \[derivative\], $$\ell'(z_0)=\ell(z_0)-2\frac{\k_0}{\m_0}$$ We are assuming that $\k_0/\m_0>\ell(z_0)$ so the above derivative is $<-\ell(z_0)$. Thus in this case the length of $z$ decreases exponentially, until either half the length is lost after a bounded distance in $\X$, or the average distance between illegal turns becomes $\geq 1/\hm$, when the above argument finishes the proof. One source of asymmetry between legality and illegality is that a long legal segment gets predictably longer under folding, while a long illegal segment may not get longer under unfolding. For example, take a surface relation inside a subgraph where folding amounts to an axis of a surface automorphism. But the lemma above implies that an illegal segment inside a loop representing a simple class will get predictably longer under unfolding after definite progress in $\F$. Projection to a folding path {#s:projection} ============================ We thank Michael Handel for pointing out the technique for proving the following lemma (see [@michael-lee Proposition 8.1]). \[michael2\] Let $\bG\in\X$ be a metric graph with a train track structure and $A<\FF$ a free factor. Suppose $A\|\bG$ satisfies: - there is an illegal loop $a$ in $A\|\bG$ (see Definition \[d:illegal loop\]), - there is an immersed legal segment in $A\|\bG$ of length $3(2n-1)$, $n=\rank(\FF)$. Then $d_\F(A,\bG)$ is bounded. If the injectivity radius of $A\|\bG$ is $\leq 3(2n-1)$ the conclusion follows from Corollary \[lucas\]. Choose a complementary free factor $B$ to $A$ and add a wedge of circles to $A\|\bG$ representing $B$ to get a graph $H$. Extend $A\|\bG\to \bG$ to a homotopy equivalence (difference of markings) $H\to\bG$, which is an immersion on each 1-cell. Pull back the metric to $H$ and consider the folding path in $\hX$ induced by $H\to\bG$. Let $H'$ be the first graph on this folding path with injectivity radius $3(2n-1)$. Now give $H'$ the pullback illegal turn structure via $H'\to\bG$. Since $A\|\bG\to H'$ is an immersion, the interior of the legal segment in $A\|\gG$ embeds in $H'$. Since $H'$ has at most $2n-2$ topological vertices, the interior of the legal segment meets at most $2n-2$ topological vertices. So there is a legal segment of length $3$ inside one of the topological edges of $H'$. Thus $a\|H'$ does not cross this topological edge and hence $d_\F(a,H')\leq 1+4$ (1 bounds the distance between $\ff a$ and the free factor given by the subgraph of $H'$ traversed by $a$ and 4 coming from Lemma \[subgraph distance\]). Since $d_\F(a,A)\leq 1$ and $d_\F(H',\gG)$ is bounded by Corollary \[lucas\] applied to a shortest loop in $H'$, the statement follows. The following is a slight generalization. \[michael3\] Let $\gG\in\X$ be a metric graph with a train track structure and $A<\FF$ a free factor. Suppose $A\|\gG$ satisfies: - there is a loop $a$ in $A\|\gG$ with the maximal number of pairwise disjoint legal segments of length 3 bounded by $N$, - there is an immersed legal segment in $A\|\gG$ of length $3(2n-1)$, $n=\rank(\FF)$. Then $d_\F(A,\gG)$ is bounded as a function of $N$. The proof is similar. With $H'$ defined as in the proof of Lemma \[michael2\], there is an edge in $H'$ which is crossed by $a\|H'$ at most $N$ times. Hence, by Lemma \[bounded crossing 2\], $d_\F(H',a)\le 6N+13.$ For the rest of this section, let $\bG_t$, $t\in [0,L],$ be a folding path in $\X$ parametrized by arclength and let $A$ be the conjugacy class of a proper free factor. A folding path has a natural orientation given by the parametrization. We will think of this orientation as going left to right. \[p:I\] Set $$I:=(18\hm(3n-3)+6)(2n-1)$$ where recall that $\hm$ is the maximal possible number of illegal turns in any $\gG\in\X$ (so $\hm$ is some linear function of the rank). The number $I$ comes from Proposition \[left-right\](legal and illegal) which will say that if $A\|\bG_0$ has a long (i.e. of length $>3$) legal segment and $A\|\bG_L$ has a long (i.e. of length $>I$) illegal segment then $d_\F(\bG_0,\bG_L)$ is bounded. Recall that a segment is illegal if it does not contain a legal subsegment of length 3. This motivates the following definitions. Denote by $\operatorname{left}_{\bG_t}(A)$ (or just $\operatorname{left}(A)$ if the path $\bG_t$ is understood) the number $$\inf \{t\in [0,L]\mid A\|\bG_t \mbox{ has an immersed legal segment of length } 3\}$$ The [left projection $\operatorname{Left}_{\bG_t}(A)$ of $A$ to the path $\bG_t$]{} is $\bG_{\operatorname{left}(A)}$. Denote by $\operatorname{right}_{\bG_t}(A)$ (or just $\operatorname{right}(A)$ if $\bG_t$ is understood) the number $$\sup \{t\in [0,L]\mid A\|\bG_t \mbox{ has an immersed illegal segment of length } I\}$$ The [right projection $\operatorname{Right}_{\bG_t}(A)$ of $A$ to the path]{} is $\bG_{\operatorname{right}(A)}$. If the above sets are empty, we interpret $\inf$ as $L$ and $\sup$ as $0$. We make analogous definitions for any simple class $a$, so e.g.$\operatorname{left}_{\bG_t}(a)$ is the first time $a\|\bG_t$ contains a legal segment of length 3 (here wrapping around is allowed, i.e. $a\|\bG_t$ may be legal and of length $<3$). In all cases, we may suppress subscripts if the path is understood. Note that the first set displayed above is closed under the operation of increasing $t$. Clearly, $\operatorname{left}(A)\leq\operatorname{right}(A)$. We also generalize these definitions from free factors to marked graphs in the obvious way. If $H\in\X$, $\operatorname{left}(H):=\min\operatorname{left}(\pi(H))$, $\operatorname{Left}(H):=\bG_{\operatorname{left}(H)}$, $\operatorname{right}(H):=\max\operatorname{right}(\pi(H))$, and $\operatorname{Right}(H):=\bG_{\operatorname{right}(H)}$. \[L2\] Suppose $B<A$ are free factors. Then: - $\operatorname{left}(A)\leq \operatorname{left}(B)$, $\operatorname{right}(A)\geq \operatorname{right}(B)$; and - either $d_\X({\operatorname{Left}(A)},{\operatorname{Left}(B)})=e^{\operatorname{left}(B)-\operatorname{left}(A)}$ is bounded or the distance in $\F$ between $A$ and $\{\bG_t\mid t\in [\operatorname{left}(A),\operatorname{left}(B)]\}$ is bounded. The first bullet is clear since we are taking $\inf$ and $\sup$ over smaller sets. Denote by $\operatorname{left}'(A)$ the first time along the folding path that $A\|\bG_t$ has a legal segment of length $3(2n-1)$. Then $\operatorname{left}(A)\leq\operatorname{left}'(A)$, and $\operatorname{left}'(A)-\operatorname{left}(A)$ is bounded by Corollary \[3\]. If $\operatorname{left}(B)\leq \operatorname{left}'(A)$ we are done, so suppose $\operatorname{left}'(A)<\operatorname{left}(B)$. It follows from Lemma \[michael2\] that the set of $\bG_t$’s for $t\in [\operatorname{left}'(A),\operatorname{left}(B)]$ has a bounded projection in $\F$, and the projection is close to $A$. Given a constant $K>0$, we will say that a coarse path $\gamma:[\alpha,\omega]\to\F$ is a [reparametrized quasi-geodesic]{} if there is a subdivision $\alpha=t_0<t_1<\cdots<t_m=\omega$ such that $diam_\F(\gamma([t_i,t_{i+1}]))\leq K$, $m\leq d_\F(\gamma(\alpha),\gamma(\omega))$, and $\|i-j\|\leq d_\F(\gamma(t_i),\gamma(t_j))+2$ for all $i,j$. In particular, a map $[0,m]\to \F$ given by mapping $x\in [0,m]$ to an element of $\gamma(t_{[x]})$ is a quasi-geodesic with constants depending only on $K$. A collection $\{\gamma_i\}_{i\in I}$ of reparametrized quasi-geodesics is [uniform]{} if the $K$ appearing in the definition of $\gamma_i$ is independent of $i$ and is, in fact, a function of the rank $n$ of $\FF$ alone. A [coarse Lipschitz]{} function $f:X\to Y$ between metric spaces is one that satisfies $d_Y(f(x_1),f(x_2))\leq K~ d_X(x_1,x_2)+K$ for all $x_1,x_2\in X$. A function $f:X\to A\subseteq X$ is a [coarse retraction]{} if $d(a,f(a))\leq K$ for all $a\in A$. In all these cases, $f$ is allowed to be multivalued with the bound of $K$ on the diameter of a point image. \[coarse Lipschitz\] For any folding path $\bG_t$ the projection $$\F\to\pi(\bG_t)$$ $$A\mapsto \pi({\operatorname{Left}(A)})$$ is a coarse Lipschitz retraction with constants depending only on $\rank(\FF)$. Consequently, the collection of paths $\{\pi(\bG_t)\}$ is a uniform collection of reparametrized quasi-geodesics in $\F$. That the map is coarsely Lipschitz follows from Proposition \[L2\]. To prove that it is a coarse retraction, we need to argue that $\pi(\operatorname{Left}(\bG_{t_0}))$ is bounded distance from $\pi(\bG_{t_0})$ for $t_0\in [0,L]$. Let $a$ be the conjugacy class of a legal candidate in $\bG_{t_0}$, so $\operatorname{left}(a)\leq t_0$ and $d_\F(a,\bG_{t_0})$ is bounded. By Proposition \[L2\] again, it is enough to argue that $d_\F({\operatorname{Left}(a)},\bG_{t_0})$ is bounded. Let $t'$ be the smallest parameter such that $a_{t'}$ is legal. Then $\ell(a_{t'})\le 2$ so $d_\F(a,\bG_{t'})$ is bounded. Now note that $\operatorname{left}(a)=t'$ since for $t<t'$ any legal segment of length 3 in $a_t$ would force $\ell(a_{t'})>3$. The argument for the second part is from [@bo:hyp]. Let $\bG_t$ be a folding path so that $\pi(\bG_t)$ is a coarse path joining free factors $A$ and $B$. Choose a geodesic $C_i$, $i=0,\cdots,m$ of free factors joining $C_0=A$ and $C_m=B$ in $\F$. Consider the coarse projection $D_i$ of $C_i$ to $\pi(\bG_t)$. By Proposition \[L2\] the diameter of the segment bounded by $D_i$ and $D_{i+1}$ is uniformly bounded. Now the $D_i$’s may not occur monotonically along $\pi(\bG_t)$. To fix this, let $i_1<i_2<\cdots<i_k$ be the sequence defined inductively by $i_1=0$ and $i_{j+1}$ is the smallest index such that $D_{i_{j+1}}$ occurs after $D_{i_j}$ in the order on $\bG_t$ given by $t$. Then by construction the interval between $D_{i_j}$ and $D_{i_{j+1}}$ has uniformly bounded diameter and the number $k$ is bounded by $m=d_\F(A,B)$. Call a subdivision satisfying these properties a [admissible]{}. To ensure the last property $\|i-j\|\leq d_\F(\gamma(t_i),\gamma(t_j))+2$ take an admissible subdivision with minimal $k$. \[d:projection to Gt\] Given $A\in\F$, $H\in\X$, and a folding path $\bG_t$, $\pi(\operatorname{Left}(A))$ is [the projection of $A$ to $\pi(\bG_t)$]{} and $\pi(\operatorname{Left}(H))$ is [the projection of $H$ to $\pi(\bG_t)$]{}. For $\kappa>0, C>0$ we say a folding path $\bG_t$ makes [ $(\kappa,C)$-definite progress in $\F$]{} if for any $D>0$ and $s<t$, $d_\X(\bG_s,\bG_t)>D\kappa+C$ implies $d_\F(\bG_s,\bG_t)>D$. For any folding path $\bG_t$ the projection $$\F-R(\bG_t)\to \{\bG_t\}$$ $$A\mapsto {\operatorname{Left}(A)}$$ where $R(\bG_t)$ is the set of free factors at a certain bounded distance from $\bG_t$ measured in $\F$, is coarsely Lipschitz (with respect to the path metric in $\F-R(\bG_t)$). Moreover, the projection is coarsely defined and coarsely Lipschitz on all of $\F$ provided $\bG_t$ makes $(\kappa,C)$-definite progress in $\F$ (with constants depending on $\kappa,C$). \[forward immersion\] Let $\bG_t$, $t\in [0,L],$ be a folding path and $A$ a free factor. The length of any illegal segment contained in a topological edge of $A\|\bG_L$ is less than $$\frac 32 m\cdot edgelength(A\|\bG_0)+6$$ where $m=\max\{m_t\mid t\in [0,L]\}$ and $edgelength(A\|\bG_0)$ is the maximal length of a topological edge in $A\|\bG_0$. Fix an illegal segment of length $\ell_L$ in the interior of a topological edge of $A\|\bG_L$. We will assume that the endpoints are illegal turns and argue $$\ell_L\leq \frac 32 m\cdot edgelength(A\|\bG_0) \tag{$\diamond$}$$ After adding $<3$ on each end we recover any illegal path. By the Unfolding Principle our path lifts to an illegal segment bounded by illegal turns inside some topological edge of $A\|\bG_t$, whose length will be denoted $\ell_t$. In particular, $\ell_0\leq edgelength(A\|\bG_0)$. In order to obtain a contradiction, assume ($\diamond$) fails. Let $t_0$ be the first time the right derivative of $\ell_t$ is nonnegative (if such $t_0$ does not exist then $\ell_L\leq \ell_0$ and ($\diamond$) holds, contradiction). Thus $\ell_{t_0}\leq \ell_0$ and the average length of a maximal legal segment inside the path is $\geq 2/\m_{t_0}\geq 2/m$ by Corollaries \[c:length\](\[i:average\]) and \[c:segments\]. Since $\ell_L\ge \frac 32m\ell_{t_0}$, the average length of a legal segment of our path is guaranteed to be $\ge 3$, contradicting the hypothesis that our segment is illegal. Thus ($\diamond$) holds and the lemma follows. Recall that the number $I$ used in the next proposition was defined on Page . \[left-right\] Let $\bG_t$, $t\in [0,L],$ be a folding path and $A$ a free factor. Assume that $A\|\bG_0$ has a legal segment of length $3$, and that $A\|\bG_L$ has an illegal segment of length $I$. Then $d_\F(\bG_0,\bG_L)$ is bounded. Since a legal segment of length 3 grows to a legal segment of length $>12(3n-3)(2n-1)$ in bounded time (Corollary \[3\]), by replacing $\bG_0$ with $\bG_{t}$ for a bounded $t$, we may assume that $A\|\bG_0$ has a legal segment of length $12(3n-3)(2n-1)$. In order to obtain a contradiction, assume the distance $d_\F(\bG_0,\bG_L)$ is large. Let $\tau\in [0,L]$ then be chosen so that $d_\F (\bG_0, \bG_\tau)$, $d_\F (\bG_\tau , \bG_L )$ and $d_\F (\bG_\tau , A)$ are all large. Wedge $A\|\bG_\tau$ onto a rose representing a complementary free factor to $A$ in order to obtain a graph $H'\in\hX$ and a difference of markings morphism $H'\to\bG_\tau$ extending $A\|\bG_\tau\to\bG_\tau$ and which is an isometric immersion on every edge. In particular, $H'\to\bG_\tau$ induces a train track structure on $H'$. If $H'$ has bounded injectivity radius then $d_\F (A, \bG_\tau )$ is bounded, contradicting the choice of $\bG_\tau$. So suppose the injectivity radius of $H'$ is large and fold $H'\to \bG_\tau$ until a graph $H''$ is reached which is the last time there is an edge $E''$ of length $4$. Cf. [@michael-lee Proposition 8.1]. In particular, $vol(H'')\le 4(3n-3)$. We continue by folding with speed 1 the subset of those illegal turns of $H''\to\bG_\tau$ that don’t involve $E''$. Since $H''\to\bG_\tau$ induces a train track structure on $H''$, so does our subset. For small $t$, we obtain a graph $H''_t$ where the image $E''_t$ of $E''$ (perhaps no longer topological) still has length 4 and a morphism $H''_t\to\bG_\tau$ inducing a train track structure. We continue folding all illegal turns not involving $E''_t$ (as in \[slope 1 folding\].C) until we obtain morphism $H\to\bG_\tau$ inducing a train track structure and isometrically immersing both the image $E$ in $H$ of $E''$ and its complement. The only illegal turns of $H\to\bG_\tau$ involve $E$. In fact, since our illegal turns give a train track structure, the only illegal turns involve the topological edge containing $E$. We will now use $E$ for the name of this topological edge. After folding from $H''$ to $H$, some edge lengths may now be $>4$. But since $vol(H)\le vol(H'')$, edge lengths in $H$ are at most $4(3n-3)$ (and $E$ still has length at least 4). We may assume that the complement of $E$ does not have a valence 1 vertex. Indeed, assuming otherwise, with respect to the train track structure induced by $H\to\bG_\tau$ there are two possibilities for the illegal turns (which recall must involve $E$), see Figure \[f:lollipop\]. In the left picture, the length of $E$ stays constant under folding. We continue folding until the separating edge folds in with $E$, and this is our new $H$. The right picture is impossible: $E$ is a “monogon” (of length at least 4) and the folding towards $\bG_\tau$ stops before the loop degenerates. But this means that $\bG_\tau$ has volume $>2$, a contradiction. ![Two possibilities when $E$ is a loop attached to a separating edge. The square represents the remainder of the graph.[]{data-label="f:lollipop"}](lollipop.eps) We will also assume for concreteness that the complement of $E$ is connected, and denote by $B$ the free factor determined by it. When the complement is disconnected, there are two free factors determined by the components. The changes are straightforward and left to the reader. We have morphisms $H\to\bG_\tau\to \bG_L$ and now also bring in the pullback illegal turn structure via $H\to\bG_\tau$. To distinguish between the two structures, terms like $p$-legal and $p$-illegal will refer to this pullback, i.e. the one induced by $H\to\bG_L$. Terms like $i$-legal and $i$-illegal will refer to the structure induced by $H\to\bG_\tau$. The same terminology will be applied to turns in $\hat H_t$ (resp. $H_t$ and $K_t$) with respect to $\hat\psi_s:\hat H_s\to\hG_s$ (resp. $H_t\to \bG_t$ and $K_t\to\bG_t$) constructed below. Note that $H$ may have $p$-illegal turns in the interior of topological edges. (Consider that perhaps $H=H'$.) By construction, $i$-illegal turns must involve $E$. There are two cases. [Case 1.]{} $E$ contains a $p$-legal segment of length 3. As $G_\tau$ folds toward $\bG_L$, we will use the technique of \[slope 1 folding\].C to fold $H$ and produce a new path (though usually not a folding path) in $\X$. To describe this path, it is convenient to view the folding path $\bG_t$ as in Proposition \[slope 1 folding\], i.e. without rescaling and folding with speed 1. So, let $\hG_{s(t)}$, $t\in [\tau,L],$ be the folding path $\hX$ induced by the morphism $\bG_\tau\to e^{\tau-L}\bG_L$ with natural parameter $s$. We claim that, for $s\in [s(\tau),s(L)]$, there is a path $\hat H_s$ in $\hX$ that starts at $H$ and satisfies: (1) $\hat H_s=B\|\hG_s\cup \hat E_s$, where $\hat E_s$ is a topological edge containing a $p$-legal segment of length at least $3\, vol(\hG_s)$, (2) the immersion $B\|\hG_s\to\hG_s$ extends to a morphism (difference of markings) $\hat\psi_s:\hat H_s\to \hG_s$ inducing a train track structure on $\hat H_s$. In particular, $\hat \psi_s$ is an isometric immersion on $\hat E_s$, (3) $A\|\hG_s\to \hG_s$ factors through $\hat\psi_s$. In particular, $A\|\hG_s\to\hat H_s$ is an isometric immersion. By construction (1–3) hold for $\hat H_{s(\tau)}:=H$ and $\hat E_{s(\tau)}:=E$. Following \[slope 1 folding\].C, assume $\hat H_s$ has been defined on a subinterval $J$ of $[s(\tau),s(L)]$ containing $s(\tau)$. If $J=[s(\tau),s_0]$ with $s_0\not=s(L)$, then we can, for small time $\epsilon>0$, fold all $p$-illegal turns of $\hat H_{s_0}$ at speed 1 (see Page ) thereby extending the path to $[s(\tau), s_0+\epsilon]$. We see that (1–3) hold for $s\in [s_0,s_0+\epsilon]$. Indeed, $\hat H_{s_0}\to\hG_{s}$ factors through $\hat H_{s_0}\to\hat H_{s}$ and so $\hat H_s\to\hG_s$ is a morphism. By Lemma \[3\], $\hat H_s$ has a topological edge $\hat E_s$ containing a $p$-legal segment of length at least $3\,vol(\hG_s)$ and whose complement has core representing $B$. Since $B\|\hat H_{s_0}$, $A\|\hat H_{s_0}$, and the interior of $\hat E_{s_0}$ contain no $i$-illegal turn, the same is true at $s$. There must be an $i$-illegal turn of $\hat H_{s_0}$ involving both $\hat E_{s_0}$ and an edge in $B\|\hat H_{s_0}$ (or else $\hat H_s$ has a monogon as above which has been ruled out). We move to the case $J=[s(\tau),s_0)$. As in \[slope 1 folding\].C, we may define a limit tree $\hat H_{s_0}\in \hX$. By Lemma \[3\], $\hat H_{s_0}$ has a topological edge containing a $p$-legal segment of length at least $3\,vol(\bG_{s_0})$ and whose complement has core representing $B$. The limit of these morphisms is a morphism, so (2) and (3) also hold. Finally, $\hat E_{s_0}$ can’t be a loop connected to $B\|\hat H_{s_0}$ by a separating edge (or else the same would have been true at smaller $s$). Set $H_s:=\hat H_s/vol(\hG_s)$ and define the image of $\hat E_s$ in $H_s$ to be $E_s$. Reverting to our original parametrization, we now have our original path $\bG_t$, $t\in [\tau,L],$ in $\X$ and a new path $H_t$, $t\in [\tau, L],$ in $\hX$ such that, for each $t$, $H_t=B\|H_t\cup E_t$, there is a morphism $\psi_t:H_t\to \bG_t$, $A\|\bG_t$ isometrically immerses in $H_t$, $B\|\bG_t$ isometrically embeds in $H_t$, and $E_t$ contains a $p$-legal segment of length at least 3. We need one more modification to control the length of $E_t$. Define $K_t\in\hX$ as follows. If the length of $E_t$ is $\le 4$, $K_t:=H_t$. If the length of $E_t$ in $H_t$ is $>4$, define $K_t$ to be the graph obtained by folding $i$-illegal turns of $H_t\to\bG_t$ until the length of $E_t$ is 4. Since $A\|\bG_t$ and $B\|\bG_t$ are immersed in $H_t$, the only effect is to fold pieces of the end of $E_t$ into $B\|\bG_t$. In particular, the analogues of (1–3) hold for $K_t$, except it is possible that $E_t$ no longer has a $p$-legal segment of length 3. By keeping in mind that the length of $E_t$ in $K_t$ is at most 4 and applying Lemma \[forward immersion\] to $\bG_t$, $t\in [\tau, L],$ and $B$, the length of any $p$-illegal segment contained in a topological edge of $K_t$ is bounded by $$\frac 32\hm\cdot edgelength(B\|\bG_\tau)+6\le \frac 32\hm\cdot 3\, edgelength(K_\tau)+6\le 18\hm(3n-3)+6$$ Since the number of topological vertices of $K_t$ is $\le 2n-2$, a $p$-illegal segment in $K_t$ of length $I=(18\hm(3n-3)+6)(2n-1)$ meets some topological vertex of $K_t$ twice. We see that, for $t\in [\tau,L]$, either the $injrad(K_t)$ is bounded by $I$ (in which case $d_\F(\bG_t, B)$, and hence $d_\F(\bG_t, G_\tau)$, is bounded), or there are no $p$-illegal segments in $K_t$ of length $I$ and hence the same holds for $A\|\bG_t$. Applying this to $t=L$ we see that $d_\F(\bG_\tau,\bG_\L)$ is bounded, contradicting the choice of $\bG_\tau$. [Case 2.]{} $E$ doesn’t contain a $p$-legal segment of length 3. In particular, the interior of $E$ crosses a $p$-illegal turn. Let $\hG_{s(t)}$, $t\in [0,\tau],$ be the folding path in $\hX$ giving rise to $\bG_t$ and ending at $\bG_\tau$. We will produce a path $\hat H_{s(t)}$, $t\in [0,\tau]$ (usually not a folding path) in $\hX$ ending at $H$ and, for each $s\in [s(0),s(\tau)]$, satisfying: 1. $\hat H_s=B\|\hG_s\cup \hat E_s$, where $\hat E_s$ is a single edge, 2. the immersion $B\|\hG_s\to\hG_s$ extends to a morphism (difference of markings) $\hat\psi_s:\hat H_s\to\hG_s$ which is an isometric immersion on $\hat E_s$, 3. $A\|\hG_s\to\hG_s$ factors through $\hat\psi_s$. In particular, $A\|\hG_s\to\hat H_s$ is an isometric immersion. Note that (1–3) hold for $\hat H_{s(\tau)}=H$. Let $0=s_0<s_1<\dots<s_N=s(\tau)$ be a partition of $[0,s(\tau)]$ so that the restriction of $\hG_t$ to each $[s_i,s_{i+1}]$ is given by folding a gadget. Assume $\hat H_s$ has been defined for $s\in [s_{i},s_N]$ satisfying (1–3). We now work to extend $\hat H_s$ over $[s_{i-1},s_N]$ still satisfying (1–3). We first define $\hat H_s$, $s\in (s_{i-1},s_i]$, via the following local operations. $\hat H_s$ is defined as $B\|\hG_s$ with an edge $\hat E_s$ attached, and we specify the attaching points. Consider first the case that a direction $e$ of an end of $\hat E_{s_i}$ forms a $i$-illegal turn with a direction $b$ in $B\|\hG_{s_i}$ (such a direction is then unique). Intuitively, as $s$ decreases, $B\|\hG_s$ unfolds and we choose to fold $b$ and $e$ with speed 1. A more elaborate description follows. Let $\hat\phi=\hat\phi_{ss_i}:\hG_{s}\to \hG_{s_i}$ be the folding morphism. It induces a morphism $\hat\phi_B:B\|\hG_{s}\to B\|\hG_{s_i}$. Let $\epsilon=s_i-s$ and let $\tilde N$ be the $\epsilon$-neighborhood in $B\|\hat H_{s_i}=B\|\hG_{s_i}$ of the vertex $v$ of $e$. $N(B)$ is a subset of the $\epsilon$-neighborhood $N$ of $v$ in $\hG_{s_i}$. $N_{\epsilon}$ denotes the preimage in $\hG_{s}$ of $N$ and $\tilde N_{\epsilon}$ is the preimage of $\tilde N$ in $B\|\hG_{s}$. Using the language of widgets, we attach the end of $\hat E_{s}$ corresponding to $e$ to the base of $\tilde b^$ in $\tilde N_\epsilon$. (To recall notation, see Figure \[f:notation\].) To define $\hat E_{s}$ delete a length $\epsilon$ segment from the end of $\hat E_{s_i}$. $B\|\hG_{s}\to\hG_{s}$ now extends to a morphism $\hat\psi_{s}:\hat H_{s}\to\hG_{s}$. Figure \[f:case2.1\] illustrates the diagram $$\begindc{\commdiag}[40] \obj(1,2)[12]{$\tilde N_{\epsilon}$} \obj(2,1)[21]{$N$} \obj(2,2)[22]{$\tilde N$} \obj(1,1)[11]{$N_{\epsilon}$} \mor{11}{21}{} \mor{12}{22}{}[\atleft,\solidarrow] \mor{22}{21}{} \mor{12}{11}{}[\atright,\solidarrow] \enddc$$ with ends of $\hat E_{s}$ and $\hat E_{s_i}$ attached to (resp.) $\tilde N_{\epsilon}$ and $\tilde N$. ![The thickened segments represent $\tilde b^$ and $b$. The curved segments represent ends of $\tilde E_{s}$ and $\tilde E_{s_i}$.[]{data-label="f:case2.1"}](case.2.1.v3.eps) Now suppose that the direction $e$ does not form an $i$-illegal turn with any direction in $B\|\hG_{s_i}$. Then there is a natural way to construct the attaching point in $\hG_s$ by watching $\hG_{s_i}$ unfold to $\hG_s$. In terms of widgets, we attach the end of $\tilde E_{s}$ corresponding to $e$ to the point in $B\|\hG_{s}$ closest to $\tilde e^$. Figure \[f:case2.2\] illustrates the diagram $$\begindc{\commdiag}[40] \obj(1,2)[12]{$\tilde N_\epsilon$} \obj(2,1)[21]{$N$} \obj(2,2)[22]{$\tilde N$} \obj(1,1)[11]{$N_{\epsilon}$} \mor{11}{21}{} \mor{12}{22}{}[\atleft,\solidarrow] \mor{22}{21}{} \mor{12}{11}{}[\atright,\solidarrow] \enddc$$ with $\hat E_{s}$ and $\hat E_{s_i}$ attached. ![To decide where to attach $\hat E_s$ mimic what happens in $\hG_s$. The thickened lines represent $\hat E_{s}$ and $\hat E_{s_i}$.[]{data-label="f:case2.2"}](case.2.2.v3.eps) There is a unique homotopy class of paths in $\hG_{s}$ connecting the images of attaching points. The map $\hat\psi_{s}$ is defined so that it isometrically immerses $\hat E_{s}$ to the immersed path the above homotopy class. Now suppose $\hat H_s$ is defined for $s\in (s_{i-1},s(\tau)]$ and we want to define $\hat H_{s_{i-1}}$. Let $\sigma$ be a conjugacy class in $\FF$. By construction, for $s\in (s_{i-1},s_i)$, $\ell(\sigma\|\bG_s)=\ell(\sigma\|\bG_{s_{i}})+(2x-y)(s_i-s)$ where $x$ is the number of $p$-illegal turns crossed by $\sigma\|\bG_s$ that are $i$-legal and $y$ is the number of $i$-illegal turns crossed. Note that $x$ and $y$ are constant on $(s_{i-1},s_i)$. In particular, $\lim_{s\to s_{i-1}^+}\ell(\sigma\|\hat H_s)$ exists, thereby defining $\hat H_{s_{i-1}}$. That $\hat H_{s_{i-1}}$ is in $\hX$ follows from the existence of the limiting morphism $\hat H_{s_{i-1}}\to\hG_{s_{i-1}}$. Finally, note that $\hat E_s$ doesn’t degenerate to a point in this limit. Indeed, $\hat E_{s(\tau)}$ crosses a $p$-illegal turn and this property persists by the Unfolding Principle (as $s$ decreases, $p$-illegal turns in $\hat E_s$ move away from the endpoints of $\hat E_s$ which balances any loss at the ends of $\hat E_s$ due to $i$-illegal turns). Set $H_s=\hat H_s/vol(\hG_s)$ and revert to our original parametrization. We now have a path $H_t$, $t\in [0,\tau]$. Define $K_t$ exactly as before, i.e. if $\ell(E_t)>4$, then fold $i$-illegal turns of $H_t\to\bG_t$ until $E_t$ has length 4. A $p$-legal segment of length $>3\cdot 4(3n-3)$ interior to an edge of $K_s$ would produce an edge in $B\|\bG_s$, hence also in $B\|\bG_\tau$, of length $>3\cdot 4(3n-3)$. We would then have an edge of length $>4(3n-3)$ in $K_\tau$, contradiction. A $p$-legal segment in $K_s$ of length $>12(3n-3)(2n-1)$ is then forced meet least $2n-1$ topological vertices which implies $injrad(K_s)\le 12(3n-3)(2n-1)$. By assumption, $A\|\bG_0$, hence also $K_0$, has a $p$-legal segment of length $>12(3n-3)(2n-1)$. Arguing as at the end of Case 1, we get the contradiction that $d_\X(\bG_0,\bG_\tau)$ is bounded. \[left-right2\] The image in $\F$ of the interval $[\operatorname{Left}_{\bG_t}(A),\operatorname{Right}_{\bG_t}(A)]$ has bounded diameter. The endpoints have bounded $\F$-distance by Proposition \[left-right\](legal and illegal), and therefore the whole interval projects to a bounded set by Corollary \[coarse Lipschitz\]. Corollary \[left-right2\] says that the projection of $A\in\F$ to $\pi(\bG_t)$ is bounded distance from $\pi([\operatorname{Left}_{\bG_t}(A)),\operatorname{Right}_{\bG_t}(A)])$. (Recall Definition \[d:projection to Gt\].) We will, in Lemma \[criterion\], see a way to estimate where the this projection lies. To prove Lemma \[criterion\], we first need a simple lemma about cancelling paths in a graph and then a general fact that in a different form appears in [@yael Proposition 5.10 and claim on p. 2218]. If $X$ is an edge-path in a graph, then $[X]$ denotes the path obtained from $X$ by [tightening]{}, i.e. $[X]$ is the immersed edge-path homotopic rel endpoints to $X$. If the endpoints of $X$ coincide and the resulting closed path is not null-homotopic, then $[[X]]$ denotes the loop obtained from $X$ by [tightening]{}, i.e.$[[X]]$ is the immersed circle freely homotopic to $X$. \[l:simple\] Let $V$ be an immersed edge path in a graph $G$. Suppose $V$ represents an immersed circle, i.e. $V$ begins and ends at a vertex $P$ and has distinct initial and terminal directions. Let $W$ be a nontrivial initial edge subpath of $V$ ending at a vertex $Q$ and let $V'=[W^{-1}VW]$ (so that $V'$ also represents an immersed circle). Also, let $X$ and $Y$ be immersed edge paths in $G$ starting at $P$ and $Q$ respectively. Suppose that $WY$ is immersed in $G$. Then: (1) \[i:3 segments\] the maximal common initial subpath of $X$, $WY$, and $WV'Y$ has the form $V^NW'$ for some $N\ge 0$ and some initial subpath $W'$ of $V$. (2) \[i:2 segments\] The maximal initial subpath of $X$ and $VX$ has the form $V^NW'$ for some $N\ge 0$ and some initial subpath $W'$ of $V$. The proof of Lemma \[l:simple\] is left to the reader. Note that (2) follows from (1) applied to $X$, $VX$, and $V^2X$. If $a$ and $b$ are conjugacy classes in $\FF$ and $\bG\in\X$ then by [$a\|\bG$ and $b\|\bG$ share a segment of length $K$]{} we mean that there is an isometric immersion $[0,K]\to \bG$ that lifts to both $a\|\bG$ and $b\|\bG$. \[general\] There is a constant $C_n>0$ so that the following holds. Suppose $G,H\in\X$, $z$ is any class (not necessarily simple), and $K>0$. If $\ell(z\|G)\geq C_nK\ell(z\|H)$ then there is a class $u$ in $\FF$ such that: - $\ell(u\|H)<2$; - $d_\F(H,u)$ is bounded; and - $u\|G$ and $z\|G$ share a segment of length $K$. In particular, $u$ is simple. If $\ell(z\|H)<2$ then $u=z$ [works]{}, i.e. $z$ satisfies the conclusions of the lemma. So, in the proof of Proposition \[general\] we assume $\ell(z\|H)\ge 2$. Fix a map $\phi:H\to G$ so that each edge is immersed (or collapsed) and each vertex has at least two gates (e.g. first change the metric on $H$ as in Proposition \[rescaling\] so that the tension graph is all of $H$, but in the rest of the proof we use the original metric). In the proof we will not keep track of exact constants, but will talk about “long segments in common with $z\|G$”. For example, suppose an edge path $A$ in $H$ is the concatenation $A=BC$ of two sub-edge paths, and we look at $[\phi(A)]$, which is the tightening of the composition $[\phi(B)][\phi(C)]$. If $[\phi(A)]$ contains a long segment in common with $z\|G$, then so does $[\phi(B)]$ or $[\phi(C)]$ (or both), but “long” in the conclusion means about a half of “long” in the assumption. The number of times this argument takes place will be bounded, and at the end the length can be taken as large as we want by choosing the original length (i.e. $C_n$) large. Represent $z\|H$ as a composition of $\sim\ell(z\|H)$ paths $z_i$ where each $z_i$ is either an edge, or a combinatorially long (but of length $\leq 1$) path contained in the thin subgraph (union of immersed loops of small length). Thus the loop $z\|G$ is obtained by tightening the composition of the paths $[\phi(z_i)]$. In the process of tightening, everything must cancel except for a (possibly degenerate) segment $\sigma_i\subset [\phi(z_i)]$ in each path, and at least one $\sigma_i$ must have length $\geq\sim C_nK$. So we conclude that, for some $z_i$, $[\phi(z_i)]$ contains a long segment in common with $z\|G$. There are now two cases, depending on whether $z_i$ is contained in the thin part or is an edge. Lemmas \[l:closing1\] and \[l:closing2\] will prove in turn that in each case the conclusions of the proposition hold. After proving these lemmas, we will have completed the proof of Proposition \[general\]. \[l:closing1\] Assume that, in addition to the hypotheses of Proposition \[general\], there is an edge $e$ so that $\phi(e)$ contains a segment of length $\sim C_nK$ in common with $z\|G$. Then the conclusions of Proposition \[general\] hold. Start extending $e$ to a legal edge path until an edge is repeated. There are several possibilities. [Type 0.]{} The first repetition is $e$ itself, i.e. we have $e..e$. Then identifying the $e$’s gives a legal loop $u$ that crosses each edge at most once, and $u$ works. [Type 1.]{} The first repetition is either $e^{-1}$ or another edge with reversed orientation, i.e. $e..e^{-1}$ or $e..a..a^{-1}$. Schematically we picture this as a monogon. Note that there are two ways to traverse the monogon starting with $e$ and ending with $e^{-1}$, both legal. [Type 2.]{} The first repetition is an edge $a$ different from $e$ and with the same orientation, i.e. $e..a..a$. We picture this as a spiral. A monogon or spiral has its [tail]{} and its [loop]{}. In $e..e^{-1}$ the tail is $e$ and the loop is represented by the edge path between $e$ and $e^{-1}$; in $e..a..a^{-1}$, the tail is $e..a$ and the loop is represented by the edge path between $a$ and $a^{-1}$; and in $e..a..a$ the tail is $e..a$ and the loop is represented by the edge path between the $a$’s. We can also extend $e$ in the opposite direction until an edge repeats, and so we have three subcases. [Subcase 1.]{} Type 1-1, i.e. we have Type 1 on both sides. Here we have a morphism to $H$ from a graph $Y$ as in Figure \[f:type.1.1\] whose induced illegal turns form a subset of those indicated. ![Type 1-1.[]{data-label="f:type.1.1"}](type.1.1.eps) If there is an edge $b$ (different from $e$) crossed by both monogons then by switching from one copy of $b$ to the other we may form a legal loop $u$ that crosses $e$ once and all other edges at most twice. Indeed, if at least one copy of $b$ is in a loop of a monogon, then there is a legal segment in $Y$ of the form $b..e..b$ that crosses $e$, $b$, and $b^{-1}$ only as indicated. If both copies of $b$ are in tails, then there is either $b..e..b$ as above or $b..e..b^{-1}..b$. In the latter case, choose the first $b$ as close as possible to $e$ (to guarantee all edges in our legal loop are crossed at most twice), and this loop works. If there is no such $b$, the loop $u$ that traverses both monogons once is legal, crosses each edge at most twice, and crosses some edge once, so $u$ works. [Subcase 2.]{} Type 1-2, i.e. we have a monogon on one side and a spiral on the other. ![Type 1-2.[]{data-label="f:type.1.2"}](type.1.2.eps) If some edge $b$ different from $e$ is crossed by both the spiral and the monogon, we can form a legal loop $u$ that crosses $e$ once as in Subcase 1, and we are done. Otherwise, let $u$ be the loop that crosses both the spiral and the monogon, so it has (potentially) one illegal turn. We claim that either the loop $v$ of the spiral or $u$ works. Indeed, $u$ crosses some edge once and all edges at most twice. Write $u$ as $Ev'E^{-1}v^{-1}$, where $E$ is the edge-path formed by the two tails and $v'$ is the loop of the monogon. Schematically, $u$ can be drawn as in Figure \[f:monogon\]. Now consider the image $\phi(u)$. To see how much cancellation occurs, let $Z$ be the maximal common initial segment of $[\phi(E)]$ and $[\phi(vE)]$. By Lemma \[l:simple\](\[i:2 segments\]), $Z$ has the form $[\phi(v)]^NW'$ for some initial segment $W'$ of $[\phi(v)]$. If $Z$ shares a long segment with $z\|G$ then $v$ works. Otherwise, $u\|G$ shares a long segment with $z\|G$, and so works. [Subcase 3.]{} Type 2-2, i.e. we have two spirals. ![Type 2-2.[]{data-label="f:type.2.2"}](type.2.2.eps) The first case is that the two spirals do not contain any edges in common except for $e$. Then the loop $u$ that crosses both spirals can be written in the form $u=Ev'E^{-1}v^{-1}$, pictured as a bigon (see Figure \[f:bigon\]). The argument is now similar to Subcase 2. Consider the maximal common initial segment $V$ (resp. $V'$) of $\phi(E)$ and $[\phi(vE)]$ (resp. $\phi(E^{-1})$ and $[\phi(v'E^{-1})]$). If either $V$ or $V'$ shares a long segment with $z\|G$ then $v$ or $v'$ works. Otherwise, $u$ works. The second case is that some edge $b$, other than $e$, occurs on both spirals and we can construct a loop that crosses $e$ only once by jumping from one $b$ to the other. If this loop is legal, we can take it for $u$. Otherwise, it has one illegal turn, and there are two possibilities. Suppose first that $b$ in the tail of a spiral whose loop is $v$. There is then a segment of the form $b..e..b^{-1}..v..b$ where the only illegal turn is the initial point of $v$. Let $u$ be the loop obtained by identifying the first and last $b$’s. Exactly as in Subcase 2, either the cancelling segments of $[\phi(u)]$ share a long segment with $z\|G$ (in which case $v$ works) or else $u\|G$ shares a long segment with $z\|G$. Secondly, suppose that $b$ appears in the loop $v$ of a spiral. Here there is a segment of the form $b..e..w$ where $w$ is an initial segment of $v$ ending with $b$ and the only illegal turn is the initial vertex of $w$. Let $u$ be the loop obtained by identifying the $b$’s in $b..e..w$. Let $u'$ be the loop obtained by identifying the first and last $b$’s of $b..e..vw$. By Lemma \[l:simple\](\[i:3 segments\]), either the cancelling segments in one of $\phi(u)$ and $\phi(u')$ shares a long segment with $z\|G$ (in which case $v$ works) or else one of $u\|G$ and $u'\|G$ shares a long segment with $z\|G$. Finally, by choosing the first $b$ of our segment as close as possible to $e$, we guarantee that the loop we produce crosses each edge at most twice and some edge once and so works. \[l:closing2\] Suppose that, in addition to the hypotheses of Proposition \[general\], there is no edge as in Case 1, but there is a path $w$ in the thin part of $H$ of length $\leq 1$ such that $[\phi(w)]$ contains a segment of length $\sim C_nK$ in common with $z\|G$. Then the conclusions of Proposition \[general\] hold. First, if necessary, concatenate $w$ with a combinatorially bounded path in the thin part so that its endpoints coincide. After this operation $[\phi(w)]$ still has a long piece in common with $z$ since $\phi$-images of edges do not. If taking $u=w$ does not work, then the path $[\phi(w)]$ has the form $VUV^{-1}$ with $V$ having a long piece in common with $z\|G$. Choose a combinatorially short loop $a$ in the thin part, based at the endpoints of $w$. We aim to show that either $a$ or $aw$ works. Write $[\phi(a)]=ABA^{-1}$ so that $B=[[B]]$. The loop $[[\phi(aw)]]$ is obtained by tightening the loop $ABA^{-1}UVU^{-1}$ which has at most two illegal turns (the two occurrences of $\{A,U\}$). See Figure \[f:aw\]. ![$\phi(aw)$[]{data-label="f:aw"}](aw.eps) There are two cases. If the maximal common initial segment of $A$ and $U$ is a proper segment of $A$ then our loop becomes immersed after cancelling the copies of this this common initial segment at the two illegal turns. Here $aw$ works. If $U=AU'$, then after cancelling the common $A$’s, we are left with $BU'VU'^{-1}$. Since the initial and terminal directions of $B$ are distinct, one of the turns $\{B, U'\}$ and $\{B^{-1},U'\}$ is legal. Using Lemma \[l:simple\](\[i:2 segments\]) again, either a power of $B$ has a long segment in common with $U$ (in which case $a$ works) or not (in which case $aw$ works). We have completed the proof of Proposition \[general\]. \[criterion\] Let $\bG_t$, $t\in[0,L],$ be a folding path in $\X$ parametrized by arclength, $H\in\X$, and $z$ a class in $\FF$. For some $\tau\in [0,L]$, assume that $\ell(z\|\bG_\tau)\geq\ell(z\|H)$. (i) If $z\|\bG_\tau$ is legal then either $\operatorname{left}(H)\le \tau$ or $d_\F(\operatorname{Left}(H),\bG_\tau)$ is bounded. (Recall Definition \[d:projection to Gt\].) (ii) If $z\|\bG_\tau$ has no immersed illegal segment of length $I$ then either $\operatorname{left}(H)\le \tau$ or $d_\F(\operatorname{Left}(H),\bG_\tau)$ is bounded. (iii) If $z$ is simple and $z\|\bG_\tau$ is illegal then either $d_\F(\operatorname{Right}(H),\bG_\tau)$ is bounded or $\operatorname{right}(H)\ge\tau$. We first prove (i). Let $C_n$ be the constant from Proposition \[general\](closing up to a simple class). Since $z\|\bG_\tau$ is legal, by replacing $\tau$ by $\tau+e^{3C_n}$ we may assume that $\ell(z\|\bG_\tau)\geq 3C_n\ell(z\|H)$. Proposition \[general\] then provides a simple class $u$ with $\ell(u\|H)<2$ and $d_\F(H,u)$ bounded so that $u\|\bG_\tau$ has a segment of length 3 in common with $z\|\bG_\tau$; in particular $u\|\bG_\tau$ contains a legal segment of length 3 and so $\operatorname{left}_{\bG_t}(u)\le\tau$. We are done by Lemma \[bounded crossing 2\] and Corollary \[coarse Lipschitz\]. The proof of (i$'$) is similar. Because $z\|\bG_\tau$ has no illegal segment of length $I$, its length grows at an exponential rate under folding. Also, the property of not having an illegal segment of length $I$ is stable under folding. Therefore we may assume that $\ell(z\|\bG_\tau)\ge C_nI\ell(z\|H)$ and conclude that $u\|\bG_\tau$ produced by Proposition \[general\] shares a length $I$ segment with $z\|\bG_\tau$ and so has a legal segment of length 3. The proof of (ii) is also analogous. Using Lemma \[unfolding\], by moving left from $\bG_\tau$ a bounded amount in $\F$ we may assume $\ell(z\|\bG_\tau)\geq C_nI\ell(z\|H)$. Then Proposition \[general\] provides a simple class $u$ with $\ell(u\|H)<2$ and $d_\F(H,u)$ bounded so that $u\|\bG_\tau$ has a segment of length $I$ in common with $z\|\bG_\tau$. In particular, this segment is illegal. We summarize the conclusions in Lemma \[criterion\](i) and (i$'$) \[resp. (ii)\] by saying that [the projection of $H$ to $\bG_t$ is coarsely to the left \[resp. right\] of $\bG_\tau$, measured in $\F$.]{} (Recall Corollary \[left-right2\].) Hyperbolicity {#s:hyperbolicity} ============= The following proposition provides a blueprint for proving hyperbolicity of $\F$. In the case of the curve complex the same blueprint was used by Bowditch [@bo:hyp]. \[hyperbolicity\] $\F$ is hyperbolic if and only if the following holds for projections of folding paths. There is $C>0$ so that: (i) (Fellow Travel) Any two projections $\pi(G_t)$ and $\pi(H_t)$ of folding paths that start and end “at distance 1” (coarsely interpreted) are in each other’s Hausdorff $C$-neighborhood. (ii) (Symmetry) If $\pi(G_t)$ goes from $A$ to $B$ and $\pi(H_t)$ from $B$ to $A$ then the two projections are in each other’s Hausdorff $C$-neighborhood. (iii) (Thin Triangles) Any triangle formed by projections of three folding paths is $C$-thin. More precisely, in (i), if $G$ and $H$ are the initial points of the two paths, the hypothesis means that there exist adjacent free factors $A$ and $B$ such that $A\in\pi(G)$ and $B\in\pi(H)$, and similarly for terminal points. It is clear that (i)-(iii) are necessary for hyperbolicity, since projections of folding paths form a uniform collection of reparametrized quasi-geodesics and in hyperbolic spaces quasi-geodesics stay in bounded neighborhoods of geodesics. The converse is due to Bowditch [@bo:hyp] (a variant was used earlier by Masur-Minsky [@MM]). Here is a sketch. We will show that any loop $\alpha$ in $\F$ of length $L$ bounds a disk of area $\sim L\log L$. (Think of bounded length loops as bounding disks of area 1.) Subdivide $\alpha$ into $3\times 2^N$ segments of size $\sim 1$ and think of it as a polygon. Subdivide it into triangles in a standard way: a big triangle in the middle with vertices $2^N$ segments apart, then iteratively bisect remaining polygonal paths. Represent each diagonal by the image of a folding path – up to bounded Hausdorff distance the choices are irrelevant. Using Thin Triangles, each triangle of diameter $D$ can be filled with a disk of area $\sim D$. Adding the areas of all the triangles gives $\sim N\times 2^N\sim L\log L$. Proposition \[contraction\] generalizes Algom-Kfir’s result [@yael]. \[contraction\] Let $H,H'\in\X$ with $d_\X(H,H')\leq M$ and let $\bG_t$ be a folding path such that $d_\X(H,\bG_t)\geq M$ for all $t$. Then the distance between the projections of $H$ and $H'$ to $\bG_t$ is uniformly bounded in $\F$. Denote by $\bG_1$ the leftmost of $\operatorname{left}_{\bG_t}(z)$ and by $\bG_2$ the rightmost of $\operatorname{right}_{\bG_t}(z)$ as $z\|H$ varies over candidates in $H$. Then the interval $[\bG_1,\bG_2]$ is bounded in $\F$ by Lemma \[bounded crossing 2\], Proposition \[L2\], and Corollary \[left-right2\]. Let $z_1\|H$ be a candidate that realizes the distance to $\bG_1$, so $\ell(z_1\|\bG_1)\geq e^M\ell(z_1\|H)$ and $\ell(z_1\|H')\leq {e^M}\ell(z_1\|H)$. Combining these inequalities gives $\ell(z_1\|\bG_1)\geq\ell(z_1\|H')$, so Lemma \[criterion\](ii) shows $\operatorname{Right}_{\bG_t}(H')$ is coarsely to the right of $\bG_1$, measured in $\F$. In the same way one argues that $\operatorname{Left}_{G_t}(H')$ is coarsely to the left of $\bG_2$, measured in $\F$. The claim follows. \[c:contraction\] A folding line that makes $(\kappa,C)$-definite progress in $\F$ is strongly contracting in $\X$ (with the constants depending on $\kappa$ and $C$). This simply means that, in the situation of Proposition \[contraction\], projections of $H$ and $H'$ to $\bG_t$ are at a uniformly bounded distance in $\X$ (depending on $\kappa$ and $C$), measured from left to right. Note that a folding line that makes definite progress in $\F$ is necessarily in a thick part of $\X$ (i.e. the injectivity radius of $\bG_t$ is bounded below by a positive constant). The converse does not hold (but recall that it does hold in Teichmüller space, and Corollary \[c:contraction\] is the direct analog of Minsky’s theorem [@minsky] that Teichmüller geodesics in the thick part are strongly contracting). One can avoid the use of the technical Proposition \[gafa\](surviving illegal turns) in the proof of Proposition \[contraction\]. Fellow Travelers and Symmetry {#s:FT} ============================= We will fix constants $C_1$, $C_2$, and $D$ from Proposition \[L2\] and Proposition \[contraction\], so that: - If $B<A$ are free factors at $\F$-distance $\geq C_1$ from a folding path $\bG_t$ then the $\X$-distance along $\bG_t$ between $\operatorname{left}(A)$ and $\operatorname{left}(B)$ is bounded by $D$, - The $\F$-diameter of the projection of a path of length $M$ to any folding path at distance $\geq M$ is always $\leq C_2$, - $C_1>C_2$. \[FT0\] Fix $C$ sufficiently large. Suppose $\bG_t$ and $H_\tau$ are two folding paths, $d_\F(\bG_t,H_\tau)\geq C$ for all $t,\tau$, but the initial points and the terminal points are at $\F$-distance $\leq 10C$. Then the projections of the two paths to $\F$ are uniformly bounded in diameter. The same holds if the initial point of $\bG_t$ is $10C$-close to the terminal point of $H_\tau$ and the terminal point of $\bG_t$ is $10C$-close to the initial point of $H_\tau$. Subdivide $H_\tau$ into a minimal number of segments whose $\F$-diameter is bounded by $C_1$. Say the subdivision points are $s_0<s_1<s_2<\cdots<s_m$. Let $\bG_{t_i}=\operatorname{left}_{\bG_t}(H_{s_i})$. When $C>2C_1$ we have that the distance, measured from left to right, between $\bG_{t_i}$ and $\bG_{t_{i+1}}$ along $\bG_t$ is $\leq C_1D$ (here $C>2C_1$ is needed so that interpolating free factors are also far from $\bG_t$). The $\F$-distance between $\bG_{t_0}$ and the initial point of $\bG_t$, and also between $\bG_{t_m}$ and the terminal point of $\bG_t$ is bounded (recall Corollary \[coarse Lipschitz\]). Further, $d_\X(\bG_{t_0},\bG_{t_m})\leq mC_1D$ as long as $\bG_{t_0}$ is to the left of $\bG_{t_m}$ (if it is to the right, the whole path $\bG_t$ is $\F$-bounded). So the projection of $[\bG_{t_0},\bG_{t_m}]$ to $H_\tau$ is bounded by $mC_2$, as long as the $\X$-distance between $\bG_t$ and $H_\tau$ is bounded below by $C_1D$ (if not then the $\F$-distance between $\bG_t$ and $H_\tau$ is bounded, contradiction when $C$ is sufficiently large, see Corollary \[proj X->F continuous\]). So, $$mC_2+2(\mbox{const})\geq (m-1)C_1$$ and since $C_1>C_2$ this implies that $m$ is bounded above. The claim follows. The proof is analogous in the “anti-parallel” case. Fellow Traveler and Symmetry properties are now immediate. \[FT\] Let $\bG_t$ and $H_\tau$ be folding paths whose initial points are at $\F$-distance $\leq R$ and the same holds for terminal points. Then $\pi(\bG_t)\subset\F$ and $\pi(H_\tau)\subset\F$ are in each other’s bounded Hausdorff neighborhoods, the bound depending only on $R$. The same holds when the initial point of $\pi(\bG_t)$ is $R$-close to the terminal point of $\pi(H_\tau)$ and the terminal point of $\bG_t$ is $R$-close to the initial point of $H_\tau$. Let $C>R$ be a sufficiently large constant as in Proposition \[FT0\]. If $\pi(\bG_t)$ is not contained in the Hausdorff $C$-neighborhood of $\pi(H_\tau)$ there is a subpath $[\bG_{t_1},\bG_{t_2}]$ such that no point of it is $C$-close to $\pi(H_\tau)$, but the endpoints $\bG_{t_1},\bG_{t_2}$ are within $10C$. Then there is a subpath $[H_{\tau_1},H_{\tau_2}]$ of $H_\tau$ whose endpoints are within $10C$ of the endpoints of $[\bG_{t_1},\bG_{t_2}]$ (but notice that we don’t know in advance if the orientations are parallel or anti-parallel). Now by Proposition \[FT0\] the set $\pi([\bG_{t_1},\bG_{t_2}])$ is in a bounded Hausdorff neighborhood of $\pi(H_\tau)$. By the same argument $\pi(H_\tau)$ is contained in a bounded Hausdorff neighborhood of $\pi(\bG_t)$. \[FT2\] Note that, in the situation of Proposition \[FT\], any $\bG_{t_0}$ is bounded $\F$-distance from its projection to $H_\tau$. Indeed, if $H_{\tau_0}$ is bounded $\F$-distance from $\bG_{t_0}$ then from Corollary \[coarse Lipschitz\] it follows that the projection of $\bG_{t_0}$ is bounded $\F$-distance from $H_{\tau_0}$. \[FT for projections\] Let $\bG_t$ and $H_\tau$ be folding paths whose initial points are at $\F$-distance $\leq R$ and the same holds for terminal points. There is a uniform bound, depending only on $R$, to $d_\F(\operatorname{Left}_{\bG_t}(A),\operatorname{Left}_{H_\tau}(A))$ for any free factor $A$. The same holds in the anti-parallel case. For notational simplicity, we may assume that $A=\langle a\rangle$ is cyclic. First suppose $\bG_t$ and $H_\tau$ are parallel. Modulo interchanging the two paths, we can assume that the projection of $\operatorname{Left}_{\bG_t}(a)$ to $H_\tau$ is “ahead” of $\operatorname{Left}_{H_\tau}(a)$ (i.e., when $\operatorname{Left}_{\bG_t}(a)$ is projected to $H_\tau$, it is coarsely right of $\operatorname{Left}_{H_\tau}(a)$ measured in $\F$) and the projection of $\operatorname{Left}_{H_\tau}(a)$ to $\bG_t$ is “behind” (defined analogously) $\operatorname{Left}_{\bG_t}(a)$. By Lemma \[criterion\] if $\ell(a\|\operatorname{Left}_{\bG_t}(a))\leq \ell(a\|\operatorname{Right}_{H_\tau}(a))$ then the projection of $\operatorname{Left}_{\bG_t}(a)$ to $H_\tau$ is behind $\operatorname{Right}_{H_\tau}(a)$, and the claim is proved. If $\ell(a\|\operatorname{Left}_{\bG_t}(a))\geq \ell(a\|\operatorname{Right}_{H_\tau}(a))$ then the projection of $\operatorname{Right}_{H_\tau}(a)$ to $\bG_t$ is ahead of $\operatorname{Left}_{\bG_t}(a)$, and we are done again. Now suppose $\bG_t$ and $H_\tau$ are anti-parallel. There are two subcases to consider, see Figure \[f:ahead\]. In the first case $\operatorname{Left}_{\bG_t}(A)$ is ahead the projection to $\bG_t$ of $\operatorname{Left}_{H_\tau}(A)$. This is a symmetric condition with respect to interchanging $\bG_t$ and $H_\tau$ (by Remark \[FT2\]). Say $\ell(a\|\operatorname{Left}_{\bG_t}(a))\leq \ell(a\|\operatorname{Left}_{H_\tau}(a))$. Then the projection of $\operatorname{Left}_{\bG_t}(a)$ to $H_\tau$ is ahead of $\operatorname{Left}_{H_\tau}(a)$ by Lemma \[criterion\], and the claim follows. The “behind” case is similar, but we consider right projections. Say $\ell(a\|\operatorname{Right}_{\bG_t}(a))\leq \ell(a\|\operatorname{Right}_{H_\tau}(a))$. Then the projection of $\operatorname{Right}_{\bG_t}(a)$ to $H_\tau$ is behind $\operatorname{Right}_{H_\tau}(a)$, and the claim again follows. Thin Triangles {#s:thin} ============== \[thin\] Triangles in $\F$ made of images of folding paths are uniformly thin. More precisely, if $A,B,C$ are three free factors coarsely joined by images of folding paths $AB$, $AC$, $BC$ and $\hat C$ is the projection of $C$ to $AB$, then $A\hat C$ is in a bounded Hausdorff neighborhood of $AC$ and $B\hat C$ is in a bounded Hausdorff neighborhood of $BC$. We will consider points $U,V,W\in\X$ and folding paths $H_t$ from $V$ to $W$ and $\bG_t$ from $U$ to $W$. Denote by $P$ the rightmost of $\operatorname{Right}_{\bG_t}(z)$ as $z$ ranges over candidates in $V$. By Lemma \[bounded crossing 2\], $P$ has bounded $\F$-distance from the projection of $V$ to $\bG_t$. See Figure \[f:thin\]. We will prove that $\pi(VP)\cup\pi(PW)$ and $\pi(VW)$ are contained in uniform Hausdorff neighborhoods of each other (by $VP$ we mean a folding path from $V$ to $P$ etc). The basic idea is that $VP\cup PW$ behaves like a folding path and the claim is an instance of the Fellow Traveler Property. [Claim 1.]{} $d_\X(V,W)\geq d_\X(V,P)+d_\X(P,W)-C$ for a universal constant $C$. To prove the claim, let $v\|V$ be a candidate for $d_\X(V,P)$. By definition of $P$, $v\|P$ has only bounded length illegal subsegments. It follows that after removing the 1-neighborhood of each illegal turn, a definite percentage of the length of $v\|P$ remains, and hence $$\ell(v\|W)\geq e^{d_\X(P,W)} \epsilon\ell(v\|P)=e^{d_\X(V,P)+d_\X(P,W)} \epsilon\ell(v\|V)$$ for a fixed $\epsilon>0$. Thus $$d_\X(V,W)\geq\log\frac{\ell(v\|W)}{\ell(v\|V)}\geq d_\X(V,P)+d_\X(P,W) +\log \epsilon$$ By $Q$ denote the point on $VW$ such that $d_\X(V,Q)=d_\X(V,P)$. From Claim 1 we see that $d_\X(Q,W)\leq d_\X(P,W)\leq d_\X(Q,W)+C$. [ Claim 2.]{} $d_\F(P,Q)$ is bounded. This time let $v\|V$ be a candidate for $d_\X(V,W)$. Thus $$\ell(v\|Q)=e^{d_\X(V,Q)}\ell(v\|V)=e^{d_\X(V,P)}\ell(v\|V)\geq \ell(v\|P)$$ Let $Q'$ be the point along $QW$ with $d_\X(Q,Q')=\log(3(2n-1)C_n)$ (if such a point does not exist, both $P$ and $Q$ are uniformly close to $W$ and we are done). Then $\ell(v\|Q')=3(2n-1)C_n\ell(v\|Q)\geq 3(2n-1)C_n\ell(v\|P)$, so Proposition \[general\](closing up to a simple class) implies that there is a simple class $p$ such that $p\|P$ is of length $<2$ and $p\|Q'$ has a legal segment of length $3(2n-1)$. Now $p\|Q'$ cannot contain many disjoint legal segments of length 3, for otherwise $p$ would grow to a much longer length along $Q'W$ than along $PW$. Now Claim 2 follows from Lemma \[michael3\]. By Proposition \[FT\] we are free to replace a folding path by another whose endpoints project close to the endpoints of the original, and we are allowed to reverse orientations. By Proposition \[FT for projections\] these replacements affect the projections by a bounded amount, so that the projection $\hat C$ of $C$ to $AB$ is coarsely well-defined, independently of the choice of a folding path whose projection coarsely connects $A$ and $B$ with either orientation. In particular, we may assume that $U,V,W$ project near $A,C,B$ respectively and we may consider folding paths $UW$ and $VW$ that end at the same graph $W$. The above discussion then shows that $B\hat C$ is contained in a bounded Hausdorff neighborhood of $BC$. Making analogous choices of the folding paths, the claim about $A\hat C$ follows similarly. The following fact shows that for the purposes of this paper the collection of folding paths can be replaced by the larger collection of geodesic paths. \[p:any geodesic\] Let $V,P,W\in\X$ so that $d_\X(V,P)+d_\X(P,W)=d_\X(V,W)$ and let $V'\in\X$ be such that $d_\X(V,V')+d_\X(V',W)=d_\X(V,W)$, $d_\F(V,V')$ is bounded, and there is a folding path $\bG_t$ from $V'$ to $W$ (see Proposition \[rescaling\]). Then the $\F$-distance between $P$ and some $\bG_t$ is uniformly bounded. Moreover, if $H_\tau$ is a geodesic in $\X$ from $V$ to $P$, then the induced correspondence $\tau\mapsto t$ can be taken to be monotonic with respect to $\tau$. Consequently, the set of projections to $\F$ of geodesics in $\X$ is a uniform collection of reparametrized quasi-geodesics in $\F$. The proof is a variant of the discussion above. Let $P'$ be the point on $\bG_t$ with $d_\X(V,P')=d_\X(V,P)$ (if such a point does not exist we assign $V'$ to $P$). We need to argue that $d_\F(P,P')$ is bounded. Let $v\|V$ be a candidate realizing $d_\F(V,W)$. Thus $v\|\bG_t$ is legal for all $t$ and $\ell(v\|P)=\ell(v\|P')$. Let $Q'$ be a point on $\bG_t$ with $d_\X(P',Q')=\log (3(2n-1)C_n)$ (if such a point does not exist, both $P$ and $P'$ are close to $W$ and we are done). Then $\ell(v\|Q')=3(2n-1)C_n\ell(v\|P)$, so Proposition \[general\](closing up to a simple class) implies that there is a simple class $p$ such that $\ell(p\|P)<2$ and with $p\|Q'$ containing a legal segment of length $3(2n-1)$. Now $p\|Q'$ cannot contain many disjoint legal segments of length 3, for otherwise $p$ would grow to a much longer length along $Q'W$ than along $PW$. Lemma \[michael3\] implies that $d_\F(P,Q')$, and therefore $d_\F(P,P')$, is bounded. $\F$ is $\delta$-hyperbolic. Images of geodesic paths in $\X$ are in uniform Hausdorff neighborhoods of geodesics with the same endpoints. Furthermore, an element of $Out(\FF)$ has positive translation length in $\F$ if and only if it is fully irreducible. The action of $Out(\FF)$ on $\F$ satisfies Weak Proper Discontinuity (see [@BF])). Since we have checked Conditions (i), (ii), and (iii) in Proposition \[hyperbolicity\], $\F$ is hyperbolic. The second statement is a consequence of Proposition \[p:any geodesic\] and last the two statements follow from: - There are coarsely well-defined, Lipschitz maps from $\F$ to the hyperbolic complexes $\mathcal X$ constructed in [@BF2 Sections 4.4.1 and 4.4.2]. - Given a fully irreducible element $f$ of $Out(F_n)$, there is an $\mathcal X$ on which $f$ has positive translation length ([@BF2 Main Theorem]). - Further, for every $x\in\mathcal X$ and every $C > 0$ there is $N>0$ such that $\{g\in Out(\FF)\mid d_{\mathcal X}(x, xg)\le C, d_\mathcal X(xf^N, xf^Ng)\le C\}$ is finite ([@BF2 Section 4.5]). [^1]: Both authors gratefully acknowledge the support by the National Science Foundation. [^2]: The standard usage of the term [train track map]{} is to self-maps of a graph, but this natural extension of the terminology should not cause any confusion. [^3]: an element of some basis for $\FF$. [^4]: We use “segment” synonymously with “immersed path”, but usually to connote a smaller piece of something. [^5]: usually a segment, but possibly all of $z_t$ [^6]: no multiple edges, no edges that are loops
--- abstract: 'Over the past few years, Generative Adversarial Networks (GANs) have garnered increased interest among researchers in Computer Vision, with applications including, but not limited to, image generation, translation, imputation, and super-resolution. Nevertheless, no GAN-based method has been proposed in the literature that can successfully represent, generate or translate 3D facial shapes (meshes). This can be primarily attributed to two facts, namely that (a) publicly available 3D face databases are scarce as well as limited in terms of sample size and variability (e.g., few subjects, little diversity in race and gender), and (b) mesh convolutions for deep networks present several challenges that are not entirely tackled in the literature, leading to operator approximations and model instability, often failing to preserve high-frequency components of the distribution. As a result, linear methods such as Principal Component Analysis (PCA) have been mainly utilized towards 3D shape analysis, despite being unable to capture non-linearities and high frequency details of the 3D face - such as eyelid and lip variations. In this work, we present 3DFaceGAN, the first GAN tailored towards modeling the distribution of 3D facial surfaces, while retaining the high frequency details of 3D face shapes. We conduct an extensive series of both qualitative and quantitative experiments, where the merits of 3DFaceGAN are clearly demonstrated against other, state-of-the-art methods in tasks such as 3D shape representation, generation, and translation.' author: - 'Stylianos Moschoglou^\^' - 'Stylianos Ploumpis^\^' - Mihalis Nicolaou - Athanasios Papaioannou - Stefanos Zafeiriou bibliography: - 'egbib.bib' date: 'Received: date / Accepted: date' title: '3DFaceGAN: Adversarial Nets for 3D Face Representation, Generation, and Translation' --- ![image](uv_diagram.pdf){width="0.9\linewidth"} Introduction ============ GANs are a promising unsupervised machine learning methodology implemented by a system of two deep neural networks competing against each other in a zero-sum game framework . GANs became immediately very popular due to their unprecedented capability in terms of implicitly modeling the distribution of visual data, thus being able to generate and synthesize novel yet realistic images and videos, by preserving high-frequency details of the data distribution and hence appearing authentic to human observers. Many different GAN architectures have been proposed over the past few years, such as the Deep Convolutional GAN (DCGAN) and the Progressive GAN (PGAN) , which was the first to show impressive results in generation of high-resolution images. A type of GANs which has also been extensively studied in the literature is the so-called Conditional GAN (CGAN) , where the inputs of the generator as well as the discriminator are conditioned on the class labels. Applications of CGANs include domain transfer , image completion , image super-resolution and image translation . Despite the great success GANs have had in 2D image/video generation, representation, and translation, no GAN method tailored towards tackling the aforementioned tasks in 3D shapes has been introduced in the literature. This is primarily attributed to the lack of appropriate decoder networks for meshes that are able to retain the high frequency details . ![image](low_high.pdf){width="100.00000%"} ![image](rep_inter.pdf){width="90.00000%"} \[fig:rep\_inter\] In this paper, we study the task of representation, generation, and translation of 3D facial surfaces using GANs. Examples of the applications of 3DFaceGAN in the tasks of 3D face translation as well as 3D face representation and generation are presented in Fig. \[fig:low\_high\] and Fig. \[fig:rep\_inter\], respectively. Due to the fact that (a) the use of volumetric representation leads to very low-quality representation of faces , and (b) the current geometric deep learning approaches , and especially spectral convolution, preserve only the low-frequency details of the 3D faces, we study approaches that use 2D convolutions in a UV unwrapping of the 3D face. The process of unwrapping a 3D face in the UV domain is shown in Fig. \[fig:\_preprocessing\_uvs\]. Overall, the contributions of this work can be summarized as follows. - We introduce a novel autoencoder-like network architecture for GANs, which achieves state-of-the-art results in tasks such as 3D face representation, generation, and translation. - We introduce a novel training framework for GANs, especially tailored for 3D facial data. - We introduce a novel process for generating realistic 3D facial data, retaining the high frequency details of the 3D face. The rest of the paper is structured as follows. In Section \[sec:3D\_rep\], we succinctly present the various methodologies that can be utilized in order to feed 3D facial data into a deep network and argue why the UV unwrapping of the 3D face was the method of choice. In Section \[sec:3DFaceGAN\], we present all the details with respect to 3DFaceGAN training process, losses, and model architectures. Finally, in Section \[sec:experiments\], we provide information about the database we collected, the preprocessing we carried out in the databases we utilized for the experiments and lastly we present extensive quantitative and qualitative experiments of 3DFaceGAN against other state-of-the-art deep networks. 3D face representations for deep nets {#sec:3D_rep} ===================================== The most natural representation of a 3D face is through a 3D mesh. Adopting a 3D mesh representation requires application of mesh convolutions defined on non-Euclidean domains (i.e., geometric deep learning methodologies[^1]). Over the past few years, the field of geometric deep learning has received significant attention . Methods relevant to this paper are auto-encoder structures such as [@ranjan2018generating; @litany2017deformable]. Nevertheless, such auto-encoders, due to the type of convolutions applied, mainly preserve low-frequency details of the meshes. Furthermore, architectures that could potentially preserve high-frequency details, such as skip connections, have not yet been attempted in geometric deep learning. Therefore, geometric deep learning methods are not yet suitable for the problem we study in this paper. ![image](net_diagram.pdf){width="0.9\linewidth"} Another way to work with 3D meshes is to concatenate the coordinates of the 3D points in an 1D vector and utilize fully connected layers to decode correctly the structure of the point cloud . Nevertheless, in this way the triangulation and spatial adjacent information is lost and the number of the parameters describing this formulation is extremely large which makes the network hard to train. Recently, many approaches aim at regressing directly on the latent parameters of a learned model space, e.g., PCA, rather than the 3D coordinates of points . This formulation limits the geometrical details of the 3D representations and is restricted to their latent model space. In contrast, a 3D volumetric space is introduced in [@jackson2017large] as a representation of a 3D structure and exploits a Volumetric Regression Network which outputs a discretized version of the 3D structure. Due to discretization, the predicted 3D shape has low quality and corresponds to non-surface points that are difficult to handle. Lastly, in [@feng2018joint], a UV spatial map framework is utilized where the 3D coordinates of the points are stored in a UV space instead of the texture values of the mesh. This formulation exhibits a very good representation for 3D meshes where there are no overlapping regions and the mesh is optimally unwrapped. Since the 3D mesh is transferred in a 2D UV domain, we are then able to use 2D convolutions, with the whole range of capabilities they offer. As a result, this is our preferred methodology for preprocessing the 3D face scans, as further explained in Section \[sec:data\_preprocessing\]. 3DFaceGAN {#sec:3DFaceGAN} ========= In this Section we describe the training process, network architectures, and loss functions we utilized for 3DFaceGAN. Moreover, we discuss the framework we utilized for 3D face generation as well as present an extension of 3DFaceGAN which is able to handle data annotated with multiple labels. Objective function ------------------ The main objective of the generator $G$ is to retrieve a facial UV map $x$ as input and generate a fake one, $G\left(x\right)$, which in turn should be as close as possible to the real target facial UV map $y$. For example, in the case of 3D face translation, the input can be a neutral face and the output a certain expression (e.g., happiness) or in the case of 3D face reconstruction the input can be a 3D facial UV map and the output a reconstruction of the particular 3D facial UV map. The goal of the discriminator $D$ is to distinguish between the real ($y$) and fake ($G\left(x\right)$) facial UV maps. Throughout the training process, $D$ and $G$ compete against each other until they reach an equilibrium, i.e., until $D$ can no longer differentiate between the fake and the real facial UV maps. [0cm]{}[0cm]{} Adversarial loss. To achieve the 3DFaceGAN objective, we propose to utilize the following loss for the adversarial part. That is, $$\begin{aligned} \begin{array}{ll} \mathcal{L}_D = \mathbb{E}_{y}\left[\mathcal{L}\left(y\right)\right] - \lambda_{adv}\cdot\mathbb{E}_{x}\left[\mathcal{L}\left(G(x)\right)\right],\label{eq:adv_loss}\\ \mathcal{L}_G = \mathbb{E}_{x}\left[\mathcal{L}\left(G(x)\right)\right], \end{array}\end{aligned}$$ where $D\left(\cdot\right)$ refers to the output of the discriminator $D$, $\mathcal{L}\left(x\right)\doteq {\left\lVertx - D(x)\right\rVert}_1$, and $\lambda_{adv}$ is the hyper-parameter which controls how much weight should be put on $\mathcal{L}\left(G(x)\right)$. The higher the $\lambda_{adv}$, the more emphasis $D$ puts on the task of differentiating between the real and fake data. The lower the $\lambda_{adv}$, the more emphasis $D$ puts on reconstructing the actual real data. There is a fine line between which task $D$ should primarily focus on by adjusting $\lambda_{adv}$. In our experiments we deduced that for relatively low values of $\lambda_{adv}$ we retrieve optimal performance as then $D$ is able to influence the updates of $G$ in such a way that the generated facial UV maps are more realistic. During the adversarial training, $D$ tries to minimize $\mathcal{L}_{D}$ whereas $G$ tries to minimize $\mathcal{L}_G$. Similar to recent works such as [@zhao2016energy; @berthelot2017began], the discriminator $D$ has the structure of an autoencoder. Nevertheless, the main differences are that (a) we do not make use of the margin $m$ as in [@zhao2016energy] or the equilibrium constraint as in [@berthelot2017began], and (b) we use the autoencoder structure of the discriminator and pre-train it with the real UV targets prior to the adversarial training. Further details about the training procedure are presented in Section \[sec:training\]. Reconstruction loss. With the utilization of the adversarial loss , the generator $G$ is trying to “fool” the discriminator $D$. Nevertheless, this does not guarantee that the fake facial UV will be close to the corresponding real, target one. To impose this, we use an $L1$ loss between the fake sample $G\left(x\right)$ and the corresponding real one, $y$, so that they are as similar as possible, as in [@isola2017image]. Namely, the reconstruction loss is the following. $$\begin{aligned} \mathcal{L}_{rec} &= \mathbb{E}_{x}{\left\lVertG\left(x\right) - y\right\rVert}_1.\label{eq:rec_loss} \end{aligned}$$ Full objective. In sum, taking into account and , the full objective becomes $$\label{eq:full_loss} \begin{array}{ll} \mathcal{L}_{D} = \mathbb{E}_{y}\left[\mathcal{L}\left(y\right)\right] - \lambda_{adv}\cdot\mathbb{E}_{x}\left[\mathcal{L}\left(G(x)\right)\right], \\ \mathcal{L}_{G} = \mathbb{E}_{x}\left[\mathcal{L}\left(G(x)\right)\right] + \lambda_{rec}\cdot \mathcal{L}_{rec}, \end{array}$$ where $\lambda_{rec}$ is the hyper-parameter that controls how much emphasis should be put on the reconstruction loss. Overall, the discriminator $D$ tries to minimize $\mathcal{L}_D$ while the generator $G$ tries to minimize $\mathcal{L}_{G}$. Training procedure {#sec:training} ------------------ In this Section, we first describe how we pre-train the discriminator (autoencoder) $D$ and then provide details with respect to the adversarial training of 3DFaceGAN. [0cm]{}[0cm]{} Pre-training the discriminator. The majority of GANs in the literature utilize discriminator architectures with logit outputs that correspond to a prediction on whether the input fed into the discriminator is [real]{} or [fake]{}. Recently proposed GAN variations have nevertheless taken a different approach, namely by utilizing autoencoder structures as discriminators . Using an autoencoder structure in the discriminator $D$ is of paramount importance in the proposed 3DFaceGAN. The benefit is twofold: (a) we can pre-train the autoencoder $D$ acting as discriminator prior to the adversarial training, which leads to better quantitative as well as more compelling visual results [^2], and (b) we are able to compute the actual UV space dense loss, as compared to simply deciding on whether the input is real or fake. As we empirically show in our experiments and ablation studies, this approach encourages the generator to produce more realistic results than other, state-of-the-art methodologies. Adversarial training. Before starting the adversarial training, we initialize the weights and biases[^3] for both the generator $G$ and the discriminator $D$ utilizing the learned parameters estimated after the pre-training of $D$ (the architecture of $G$ is identical to the architecture of $D$). During the training phase of 3DFaceGAN, we freeze the parameter updates in the decoder parts for both the generator $G$ and the discriminator $D$. Furthermore, we utilize a low learning rate on the encoder and bottleneck parts of $G$ and $D$ so that overall the parameter updates are relatively close to the ones found during the pre-training of $D$. Network architectures. The network architectures for both the discriminator $D$ and the generator $G$ are the same. In particular, each network is consisted of 2D convolutional blocks with kernel size of three, stride and padding size of one. Down-sampling is achieved by average 2D pooling with kernel and stride size of two. The convolution filters grow linearly in each down-sampling step. Up-sampling is implemented by nearest-neighbor with scale factor of two. The activation function that is primarily used is ELU , apart from the last layer of both $D$ and $G$ where Tanh is utilized instead. At the bottleneck we utilize fully connected layers and thus project the tensors to a latent vector $b\in\mathbb{R}^{N_b}$. To generate more compelling visual results, we utilized skip connections in the first layers of the decoder part of both the generator and the discriminator. Further details about the network architectures are provided in Table \[table:architecture\]. 3D face generation {#sec:generation} ------------------ Variational autoencoders (VAEs) are widely used for generating new data using autoencoder-like structures. In this setting, VAEs add a constraint on the latent embeddings of the autoencoders that forces them to roughly follow a normal distribution. We can then generate new data by sampling a latent embedding from the normal distribution and pass it to the decoder. Nevertheless, it was empirically shown that enforcing the embeddings in the training process to follow a normal distribution leads to generators that are unable to capture high frequency details . To alleviate this, we propose to generate data using Algorithm \[algo:generate\_data\], which better retains the generated data fidelity, as shown in Section \[sec:experiments\]. 3DFaceGAN for multi-label 3D data {#sec:multi_label} --------------------------------- Over the last few years, databases annotated with regards to multiple labels are becoming available in the scientific community. For instance, 4DFAB is a publicly available 3D facial database containing data annotated with respect to multiple expressions. We can extend 3DFaceGAN to handle data annotated with regards to multiple labels as follows. Without any loss of generality, suppose there are three labels in the database (e.g., expressions neutral, happiness and surprise). We adopt the so-called one-hot representation and thus denote the existence of a particular label in a datum by $1$ and the absence by $0$. For example, a 3D face datum annotated with the label happiness will have the following label representation: $l=[0, 1, 0]$, where the first entry corresponds to the label neutral, the second to the label happiness and the third to the label surprise. We then choose the desired $l$ we want to generate (e.g., if we want to translate a neutral face to a surprised one, we would choose $l=[0, 0, 1]$) and then spatially replicate it and concatenate it in the input that is then fed to the generator. The real target is the actual expression (in this case surprise) with the corresponding $l$ spatially replicated and concatenated. Apart from this change, the rest of the training process is exactly the same as the one described in Section \[sec:training\]. Finally, to generate 3D facial data with respect to a particular label, we follow the same process as the one presented in Algorithm \[algo:generate\_data\], with the only difference being that we extract different pairs of ($\boldsymbol{\mu}_Z$, $\boldsymbol{\Sigma}_Z$) for every subset of the data, each corresponding to a particular label in the database. We then choose the pair ($\boldsymbol{\mu}_Z$, $\boldsymbol{\Sigma}_Z$) corresponding to the desired label and sample from this multi-variate Gaussian distribution. Step 1: Train 3DFaceGAN utilizing . Step 2: Extract the trained $G$, and for all $N$ training facial UV maps: Step 3: Concatenate column-wise all of the bottlenecks, i.e., $\mathbf{Z} = \left[\mathbf{z}_1, \mathbf{z}_2, \ldots, \mathbf{z}_N\right]$. Step 4: Extract the mean $\boldsymbol{\mu}_Z$ of $\mathbf{Z}$ and the covariance $\boldsymbol{\Sigma}_Z$ of the zero-mean $\mathbf{Z}$. Step 5: To generate new data, retain only the trained Bottleneck$_2$ and the Decoder part of $G$ (see Table \[table:architecture\] for the network structures) and sample a new $\mathbf{z}_i$ (i.e., Bottleneck$_2$ input) from the multivariate Gaussian $\mathcal{N}\left(\boldsymbol{\mu}_Z,\boldsymbol{\Sigma}_Z\right)$. Experiments {#sec:experiments} =========== In this Section we (a) describe the databases which we used to carry out the experiments utilizing 3DFaceGAN, (b) provide information with respect to the data preprocessing we conducted prior to feeding the 3D data into the network, (c) succinctly describe the baseline state-of-the-art algorithms we employed for comparisons and (d) provide quantitative as well as qualitative results on a series of experiments that demonstrate the superiority of 3DFaceGAN. Databases --------- ### The Hi-Lo database [Hi-Lo]{} database contains approximately $6,000$ 3D facial scans captured during a special exhibition in the Science Museum, London. It is divided into the high quality data (Hi) recorded with a 3dMD face capturing system and the low quality (Lo) data captured with a V1 Kinect sensor. All the subjects were recorded in neutral expression. The overlapping subjects that were recorded in both frameworks were approximately $3,000$. The 3dMD apparatus utilizes a 4 camera structured light stereo system which can create 3D triangular surface meshes composed of approximately $60,000$ vertices joined into approximately $120,000$ triangles. Moreover, the low quality database was captured with a KinectFusion framework . In contrast to the 3dMD system, multiple frames are required to build a single 3D representation of the subject’s face. The fused meshes were built by employing a $6,083$ voxel grid. In order to accurately reconstruct the entire surface of the faces, a circular motion scanning pattern was carried out. Each subject was instructed to stay still in a fixed pose during the entire scanning process with a neutral facial expression. The frame rate for every subject was constant at $8$ frames per second. Furthermore, all $3,000$ subjects provided metadata about themselves, including their gender, age, and ethnicity. The database covers a wide variety of age, gender ($48\%$ male, $52\%$ female), and ethnicity ($82\%$ White, $9\%$ Asian, $5\%$ Mixed Heritage, $3\%$ Black and $1\%$ other). [Hi-Lo]{} database was utilized for the experiments of 3D face representation and generation, where we utilized the high quality data to train 3DFaceGAN. Moreover, [Hi-Lo]{} database was used for demonstrating the capabilities of 3DFaceGAN in a 3D face translation setting, where the low quality data are translated into high quality ones. In all of the training tasks, $85\%$ of the data were used for training and the rest were used for testing. ### 4DFAB database 4DFAB database contains 3D facial data from $180$ subjects ($60$ females, $120$ males), aged from $5$ to $75$ years old. The subjects vary in their ethnicity background, coming from more than $30$ different ethnic groups. For the capturing process, the DI4D dynamic capturing system [^4] was used. 4DFAB contains data varying in expressions, such as [neutral]{}, [happiness]{}, and [surprise]{}. As a result, we utilized it to showcase 3DFaceGAN’s capability in successfully handling data annotated with multiple labels in the task of 3D face translation as well as generation. In all of the training tasks, $85\%$ of the data were used for training and the rest were used for testing. Data preprocessing {#sec:data_preprocessing} ------------------ In order to feed the 3D data into a deep network several steps need to be carried out. Since we employ various databases, the representation of the facial topology is not consistent in terms of vertex number and triangulation. To this end, we need to find a suitable template $T$ that can easily retain the information of all raw scans across all databases and describe them with the same triangulation/topology. We utilized the mean face mesh of the LSFM model proposed by [@booth20163d], which consists of approximately $54,000$ vertices that are sufficient to capture high frequency facial details. We then bring the raw scans in dense correspondence by morphing non-rigidly the template mesh to each one of them. For this task, we utilize an optimal-step Non-rigid Iterative Closest Point algorithm in combination with a per vertex weighting scheme. We weight the vertices according to the Euclidean distance measured from the tip of the nose. The greater the distance from the nose tip, the bigger the weight that is assigned to that vertex, i.e., less flexible to deform. In that way we are able to avoid the noisy information recorded by the scanners on the outer regions of the raw scans. Following the analysis of the various methods of feeding 3D meshes in deep networks in Section \[sec:3D\_rep\], we chose to describe the 3D shapes in the UV domain. UV maps are usually utilized to store texture information. In our case, we store the spatial location of each vertex as an RGB value in the UV space. In order to acquire the UV pixel coordinates for each vertex, we start by unwrapping our mesh template $T$ into a 2D flat space by utilizing an optimal cylindrical unwrapping technique proposed by [@booth2014optimal]. Before storing the 3D coordinates into the UV space, all meshes are aligned in the 3D spaces by performing the General Procrustes Analysis and are normalized to be in the scale of $[1,-1]$. Afterwards, we place each 3D vertex in the image plane given the respective UV pixel coordinate. Finally, after storing the original vertex coordinates, we perform a 2D nearest point interpolation in the UV domain to fill out the missing areas in order to produce a dense representation of the originally sparse UV map. Since the number of vertices in $S_T$ is more than $50K$, we choose a $256\times 256\times 3$ tensor as the UV map size, which assists in retrieving a high precision point cloud with negligible re-sampling errors. A graphical representation of the preprocessing pipeline can be seen in Figure \[fig:\_preprocessing\_uvs\]. ![image](joined_gen_graphs.pdf){width="0.9\linewidth"} Method Mean std AUC FR (%) --------------- -------- -------- ----------- ------------- 3DFaceGAN 0.0031 0.0028 0.741 1.42e-7 CoMA 0.0038 0.0037 0.716 3.66e-7 PCA 0.0040 0.0040 0.711 0.91e-6 PGAN 0.0041 0.0041 0.705 1.22e-6 : Generalization metric for the meshes of the test set for the 3D face representation task. The table reports the mean error (Mean), the standard deviation (std), the Area Under the Curve (AUC), and the Failure Rate (FR) of the Cumulative Error Distributions of Fig. \[fig:error\_plot\]a.[]{data-label="tab:error_gen"} Method AUC FR (%) --------------- ----------- ------------- 3DFaceGAN 0.741 1.42e-7 3DFaceGAN\_V3 0.736 2.62e-7 3DFaceGAN\_V2 0.704 3.15e-6 Baseline (AE) 0.697 4.24-6 : Ablation study generalization results for the 3D face representation task. The table reports the Area Under the Curve (AUC) and Failure Rate (FR) of the Cumulative Error Distributions of Fig. \[fig:error\_plot\]b.[]{data-label="tab:error_ablation"} ![image](qual_rep_coma.pdf){width="0.9\linewidth"} ![Generated faces utilizing 3DFaceGAN.[]{data-label="fig:generations"}](generations.pdf){width="0.8\linewidth"} ![image](quality_1.pdf){width="0.82\linewidth"} Training -------- We trained all 3DFaceGAN models utilizing Adam with $\beta_1=0.5$ and $\beta_2=0.999$. The batch size we used for the pre-training of the discrminator was $32$ for a total of $300$ epochs. The batch size we used for 3DFaceGAN was $16$ for a total of $300$ epochs. For our model we used $n=128$ convolution filters and a bottleneck of size $b=128$. The total number of trainable parameters was $38.5\times 10^6$. The learning rates that we used for both the pre-training and training of the discriminator was $5e-5$ and the same was for the training of the generator. We linearly decayed the learning rate by $5\%$ every $30$ epochs during training. For the rest of the parameters, we used $\lambda_{adv}=1e-3$, $\lambda_{rec}=1$. Overall training time on a GV100 NVIDIA GPU was about $5$ days. 3D Face Representation {#sec:representation} ---------------------- In the 3D face representation (reconstruction) experiments, we utilize the high quality 3D face data from the [Hi-Lo]{} database to train the algorithms. In particular, we feed the high quality 3D data as inputs to the models and use the same data as target outputs. Before providing the qualitative as well as quantitative results, we briefly describe the baseline models we compared against as well as provide information about the error metric we used for the quantitative assessment. ### Baseline models In this Section we briefly describe the state-of-the-art models we utilized to compare 3DFaceGAN against. ### Vanilla Autoencoder (AE) {#vanilla-autoencoder-ae .unnumbered} Vanilla Autoencoder follows exactly the same structure of the discriminator we used in 3DFaceGAN. We used the same values for the hyper-parameters and the same optimization process. This is the main baseline we compared against and the results are provided in the ablation study in Section \[sec:ablation\_rep\]. ### Convolutional Mesh Autoencoder (CoMA) {#convolutional-mesh-autoencoder-coma .unnumbered} In order to train CoMA , we use the authors’ publicly available implementation and utilize the default parameter values, the only difference being that the bottleneck size is $128$, to make a fair comparison against 3DFaceGAN, where we also used a bottleneck size of $128$. ### Principal Component Analysis (PCA) {#principal-component-analysis-pca .unnumbered} We employ and train a standard PCA model based on the meshes of our database we used for training. We aimed at retaining the $98\%$ of variance of our available training data which corresponds to the first $50$ principal components. ### Progressive GAN (PGAN) {#progressive-gan-pgan .unnumbered} In order to train PGAN , we used the authors’ publicly available implementation with the default parameter values. After the training is complete, in order to represent a test 3D datum, we invert the generator $G$ as in [@lucic2017gans] and [@mahendran2015understanding], i.e., we solve $z^{} = \operatorname{\arg\!\min}{\left\lVertx - G(z)\right\rVert}$ by applying gradient descent on $z$ while retaining $G$ fixed . ![image](joined_translation_graph.pdf){width="0.9\linewidth"} Method AUC Failure Rate (%) ---------------- ----------- -------------------- 3DFaceGAN 0.827 5.49e-6 pix2pixHD 0.760 5.18e-5 pix2pix 0.757 1.81e-5 Denoising CoMA 0.742 2.41e-4 : High quality 3DRMSE results for the 3D face translation task. The table reports the Area Under the Curve (AUC) and Failure Rate of the Cumulative Error Distributions of Fig. \[fig:error\_plot\_translation\]a. \[tab:dense\_fit\_error\_1\] Method AUC Failure Rate (%) ------------------------- ----------- -------------------- 3DFaceGAN 0.827 5.49e-6 3DFaceGAN\_V3 0.819 8.70e-6 3DFaceGAN\_V2 0.794 1.38e-5 Baseline (Denoising AE) 0.758 1.95e-5 : Ablation study 3DRMSE results for the 3D face translation task. The table reports the Area Under the Curve (AUC) and Failure Rate of the Cumulative Error Distributions of Fig. \[fig:error\_plot\_translation\]b. \[tab:ablation\_fit\_error\_1\] ### Error metric A common practice when it comes to evaluating statistical shape models is to estimate the intrinsic characteristics, such as the [generalization]{} of the model . The [generalization]{} metric captures the ability of a model to represent [unseen]{} 3D face shapes during the testing phase. Table \[tab:error\_gen\] presents the generalization metric for 3DFaceGAN compared against the baseline models. In order to compute the generalization error for a given model, we compute the per-vertex Euclidean distance between every sample of the test set and its corresponding reconstruction. We observe that the model which holds the best error results and thus demonstrates greater generalization capabilities is the proposed 3DFaceGAN with mean error $0.0031$ and standard deviation $0.0028$. Additionally, as shown in Fig. \[fig:error\_plot\]a, which depicts the cumulative error distribution of the normalized dense vertex erors, 3DFaceGAN outperforms all of the baseline models. ### Ablation study {#sec:ablation_rep} In this ablation study we investigate the importance of pre-training the discriminator $D$ prior to the adversarial training of 3DFaceGAN as well as the freezing of the weights in the decoder parts of both $D$ and $G$. More specifically, we compare 3DFaceGAN against the Vanilla Autoencoder (AE) and another two 3DFaceGAN possible variations, namely (a) the simplest case, where the discriminator and generator structures are retained as is, but no pre-training takes place prior to the adversarial training (we refer to this methodology as 3DFaceGAN\_V2), (b) the case where (i) the discriminator and generator structures are retained as is, (ii) we pre-train the discriminator and initialize both the generator and the discriminator with the learned weights with no parameters frozen during the adversarial training (we refer to this methodology as 3DFaceGAN\_V3). As shown in Fig. \[fig:error\_plot\]b and Table \[tab:ablation\_fit\_error\_1\], 3DFaceGAN outperforms Vanilla AE and 3DFaceGAN\_V2 by a large margin. Moreover, 3DFaceGAN also outperforms 3DFaceGAN\_V3. As a result, not only does 3DFaceGAN have the best performance among the compared 3DFaceGAN variants, but it also requires less training time compared to 3DFaceGAN\_V3, as the parameters in the decoder parts of both the generator and the discriminator are not updated during the training phase and thus need not be computed. 3D Face Translation {#sec:face_translation} ------------------- In the 3D face translation experiments, we utilize the low and high quality 3D face data from the [Hi-Lo]{} database to train the algorithms. In particular, we feed the low quality 3D data as inputs to the models and use the high quality data as target outputs. Before providing the qualitative as well as quantitative results, we briefly describe the baseline models we compared against as well as provide information about the error metric we used for the quantitative assessment. ### Baseline models {#sec:translation} In this Section we briefly describe the state-of-the-art deep models we utilized to compare 3DFaceGAN against. ### Denoising Vanilla Autoencoder (Denoising AE) {#denoising-vanilla-autoencoder-denoising-ae .unnumbered} Denoising Vanilla Autoencoder follows exactly the same structure as the Vanilla AE in Section \[sec:representation\], the only difference being the inputs fed to the network. This is the main baseline we compared against and the results are provided in the ablation study in Section \[sec:ablation\_trans\]. ### Denoising Convolutional Mesh Autoencoder (Denoising CoMA) {#denoising-convolutional-mesh-autoencoder-denoising-coma .unnumbered} Denoising CoMA , follows exactly the same structure as the Vanilla AE in Section \[sec:representation\], the only difference being again the inputs fed to the network. ### pix2pix {#pix2pix .unnumbered} pix2pix is amongst the most widely utilized GANs for image to image translation applications. We used the official implementation and hyper-parameter initializations provided by the authors in . ![Reconstruction quality of our proposed GAN network along with pix2pixHD and pix2pix in the 3D face translation task. As it can be seen, the mean error of 3DFaceGAN is considerably less than the other two approaches.[]{data-label="fig:heat_maps"}](heat_maps2.pdf){width="1\linewidth"} ### pix2pixHD {#pix2pixhd .unnumbered} More recently pix2pixHD was proposed, which can be considered as an extension of pix2pix and which is able to better handle data of higher resolution. We used the official implementation and hyper-parameter initializations provided by the authors in [@wang2018high]. As evinced in Fig. \[fig:qualitative\], Fig. \[fig:error\_plot\_translation\], and Fig. \[fig:heat\_maps\], pix2pixHD outperforms pix2pix , and this is expected since pix2pixHD uses more intricate structures for both the generator and discriminator networks. ### Error metric For each low quality test mesh we aim to estimate the high quality representation based on the 3dMD ground truth data. The error metric between the estimated and the real high quality mesh is a standard 3D Root Mean Square Error (3DRMSE) where the Euclidean distances are computed between the two meshes and normalized based on the inter-ocular distance of the test mesh. Before computing the metric error we perform dense alignment between each test mesh and its corresponding ground truth by implementing an iterative closest point (ICP) algorithm . In order to avoid any inconsistencies in the alignment we compute a point-to-plain rather than a point-to-point error. Finally, the measurements are performed in the inner part of the face, where we crop each test mesh at a radius of $150$mm around the tip of the nose. As can be clearly seen in Fig. \[fig:error\_plot\_translation\]a as well as in Table \[tab:dense\_fit\_error\_1\], 3DFaceGAN outperforms all of the compared state-of-the-art methods. ### Ablation study {#sec:ablation_trans} For the ablation study in this set of experiments, we use exactly the same 3DFaceGAN variants as the ones we utilized in Section \[sec:ablation\_rep\]. Moreover, instead of the vanilla AE in this experiment we utilize the denoising AE. As evinced in Fig. \[fig:error\_plot\_translation\]b and Table \[tab:ablation\_fit\_error\_1\], 3DFaceGAN clearly outperforms all of the compared models. ![image](expressions.pdf){width="1\linewidth"} Multi-label 3D Face Translation {#sec:multi_label_translation} ------------------------------- In this experiment we utilize 4DFAB for the multi-label transfer of expressions. In particular, we feed the neutral faces to the models and receive as outputs either the ones bearing the label happiness or surprise. It should be noted here that whereas 3DFaceGAN requires only a single model to be trained under the multi-label expression translation scenario, the rest of the compared models require different trained models for each label, i.e., a model for expression happiness and a model for expression surprise. As baseline models for comparisons, we use exactly the same as the ones in Section \[sec:face\_translation\], the only difference being the inputs fed to network as well as the corresponding targets. Qualitative comparisons against the compared methods are presented in Fig. \[fig:expression\_qual\]. 3D Face Generation {#d-face-generation} ------------------ In the 3D face generation experiment, we utilized the high quality data of the Hi-Lo database to train the algorithms. In particular, we feed the high quality 3D data as inputs to the models and use the same data as target outputs. ### Baseline models {#baseline-models-1} The baseline models we used in this set of experiments are the same as the ones presented in Section \[sec:representation\]. Method Mean std --------------- ---------- ----------- 3DFaceGAN 1.28 0.183 CoMA 1.40 0.205 PCA 1.43 0.232 PGAN 1.79 0.189 : Specificity metric on the test set for the 3D face generation task. We generate $10,000$ random faces from each model. The table reports the mean error (Mean) and the standard deviation (std).[]{data-label="tab:specificity"} ![Generated faces with expression utilizing 3DFaceGAN multi-label approach.[]{data-label="fig:gen_expressions"}](gen_expressions.pdf){width="0.8\linewidth"} ### Error metric The metric of choice to quantitatively assess the performance of the models in this set of experiments is [specificity]{} . For a randomly generated 3D face, [specificity]{} metric measures the distance of this 3D face to its nearest real 3D face belonging in the test, in terms of minimum per vertex distance over all samples of the test set. To evaluate this metric, we randomly generate $n=10,000$ face meshes from each model. Table \[tab:specificity\] reports the specificity metric for 3DFaceGAN compared against the baseline models. In order to generate random meshes utilizing 3DFaceGAN, we sample from a multivariate Gaussian distribution, as explained in Section \[sec:generation\]. To generate random meshes utilizing PGAN , we sample new latent embeddings from the multivariate normal distribution and feed them to the generator $G$. To generate random faces utilizing CoMA , we utilize the proposed variational convolutional mesh autoencoder structure, as described in . For the PCA model , we generate meshes directly from the latent eigenspace by drawing random samples from a Gaussian distribution defined by the principal eigenvalues. As shown in Table \[tab:specificity\], 3DFaceGAN achieves the best specificity error, outperforming all compared methods by a large margin. In Fig. \[fig:generations\], we present various visualizations of realistic 3D faces generated by 3DFaceGAN. As can be clearly seen, 3DFaceGAN is able to generate data varying in ethnicity, age, etc., thus capturing the whole population spectrum. Multi-label 3D Face Generation ------------------------------ In this set of experiments, we utilized the 4DFAB data to generate random subjects of various expressions such as happiness and surprise, as seen in Fig. \[fig:gen\_expressions\]. The 3D faces were generated utilizing the methodology detailed in Section \[sec:multi\_label\]. As evinced, 3DFaceGAN is able to generate expressions of subjects varying in age and ethnicity, while retaining the high-frequency details of the 3D face. Conclusion ========== In this paper we presented the first GAN tailored for the tasks of 3D face representation, generation, and translation. Leveraging the strengths of autoencoder-based discriminators in an adversarial framework, we propose 3DFaceGAN, a novel technique for training on large-scale 3D facial scans. As shown in an extensive series of quantitative as well as qualitative experiments against other state-of-the-art deep networks, 3DFaceGAN improves upon state-of-the-art algorithms for the tasks at-hand by a significant margin. Stylianos Moschoglou is supported by an EPSRC DTA studentship from Imperial College London, Stylianos Ploumpis by the EPSRC Project EP/N007743/1 (FACER2VM), and Stefanos Zafeiriou by the EPSRC Project EP/S010203/1 (DEFORM). [^1]: A thorough overview describing the first attempts towards geometric deep learning can be found in [@bronstein2017geometric]. [^2]: note that pre-training $D$ is not possible when the outputs are logits since there are no fake data to compare against prior to the adversarial training. [^3]: for brevity in the text, we will use the term parameters to refer to the weights and the biases from this point onwards. [^4]: http://www.di4d.com
--- abstract: 'We show that the strongly correlated $4f$-orbital symmetry of the ground state is revealed by linear dichroism in core-level photoemission spectra as we have discovered for YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$. Theoretical analysis tells us that the linear dichroism reflects the anisotropic charge distributions resulting from crystalline electric field. We have successfully determined the ground-state $4f$ symmetry for both compounds from the polarization-dependent [angle-resolved]{} core-level spectra at a low temperature well below the first excitation energy. The excited-state symmetry is also probed by temperature dependence of the linear dichroism where the high measuring temperatures are of the order of the crystal-field-splitting energies.' author: - Takeo Mori - Satoshi Kitayama - Yuina Kanai - Sho Naimen - Hidenori Fujiwara - Atsushi Higashiya - Kenji Tamasaku - Arata Tanaka - Kensei Terashima - Shin Imada - Akira Yasui - Yuji Saitoh - Kohei Yamagami - Kohei Yano - Taiki Matsumoto - Takayuki Kiss - Makina Yabashi - Tetsuya Ishikawa - Shigemasa Suga - 'Yoshichika $\bar{\rm O}$nuki' - Takao Ebihara - Akira Sekiyama title: \| Probing Strongly Correlated 4$f$-Orbital Symmetry of the Ground State\ in Yb Compounds by Linear Dichroism in Core-Level Photoemission --- Intrroduction ============= Strongly correlated electron systems show a variety of intriguing phenomena like unconventional and/or high-temperature superconductivity, spin and charge ordering, formation of heavy fermions, non-trivial (Kondo) semiconducting behavior, and quantum criticality. Among them, such Yb-based single-crystalline compounds as YbRh$_2$Si$_2$ [@YRS00] and $\beta$-YbAlB$_4$ [@YbAlB4SN08] showing the quantum criticality in ambient pressure, a Kondo semiconductor YbB$_{12}$ [@Iga98; @Susaki99], valence-fluctuating YbAl$_3$ [@YbAl3C02; @YbAl3E03] and YbCu$_2$Si$_2$ [@RC2S2; @YCS09], and a very heavy fermionic YbCo$_2$Zn$_{20}$ [@PNASCo; @SaigaCo; @OnukiCo] have been synthesized with excellent quality and thus intensively studied within a couple of decades. Since the strong Coulomb repulsion (effective value of 6-10 eV) works between the 4$f$ electrons in the Yb sites, an ionic picture is a good starting point to discuss and reveal their electronic structure as well as the origins of various phenomena in the crystalline solids. The majority of the Yb ions in the above-mentioned materials is in trivalent 4$f^{13}$ (one 4$f$ hole) configurations although there are to some extent Yb$^{2+}$ (4$f^{14}$) components due to the hybridization between the 4f orbitals and other valence-band states crossing the Fermi level. The Yb$^{3+}$ $4f$ levels are split by the spin-orbit coupling ($>1$ eV) and further split by the crystalline electric field (CEF, from a few to several tens meV) in solids as shown in Fig. \[bothLD\](a). Ground-state $f$-orbital symmetry determined by the CEF splitting is very fundamental information of the realistic strongly correlated electron systems. In contrast to the case of transition-metal oxides in which the electron correlations work among the $d$-orbital electrons, the ground-state symmetry is not straightforward revealed since it is unclear which sites act as effective [“ligands”]{} for the $f$ sites. A standard experimental technique for determining the 4$f$ levels with their symmetry is to analyze the inelastic neutron scattering spectra and anisotropy in the magnetic susceptibility of single crystals. However, the magnetic 4$f$-4$f$ excitations are often hampered by the phonon excitations with the same energy scale. Moreover, it is difficult to uniquely determine the symmetry of all $f$ levels by the analysis of the magnetic anisotropy since there are too many free parameters for a unique description of the CEF potential. Actually, the ground-state 4$f$-orbital symmetry is not clear for here reported YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$ although a couple of possible solutions have been proposed. Polarized neutron scattering [@WillersCePt3Si] for large single crystals is principally powerful to determine the $f$-orbital symmetry, but time-consuming. Recently linear dichroism (LD) in the 3$d$-to-4$f$ soft X-ray absorption spectroscopy (XAS) [@WillersCePt3Si; @Hansmann08LD; @WillersCeTIn5; @WillersCe122] for single crystals has been reported for the heavy fermion systems with nearly Ce$^{3+}$ ($4f^1$) configurations as a powerful tool to determine the 4$f$ ground state owing to the dipole selection rules. However, it is difficult to apply this technique to probe the Yb$^{3+}$ states since there is only single-peak structure ($3d^94f^{14}$ final state) at the $M_5$ absorption edge. On the other hand, the selection rules work also in the photoemission process while the excited electron energy is much higher than that in the absorption process. We have discovered that the atomic-like multiplet-split structure in the core-level photoemission spectra shows easily detectable LD reflecting the outer 4$f$ spatial distribution probed by the created core hole. In this letter, we demonstrate that the 4$f$-orbital symmetry of the ground state for YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$ in tetragonal symmetry is determined by the LD in the Yb 3$d_{5/2}$ core-level photoemission, and further that the excited-state $4f$ symmetry is also probed by the temperature-dependent data. YbRh$_2$Si$_2$ is known as the first discovered Yb system showing quantum criticality in ambient pressure as mentioned above. YbCu$_2$Si$_2$ is a counterpart of the first discovered heavy fermion superconductor CeCu$_2$Si$_2$ [@CeCuSi2] with respect to an electron-hole symmetry. However, it does not show superconductivity, being recognized as one of the valence-fluctuating systems with anisotropic magnetic susceptibility [@RC2S2; @YCS09; @YCSPES] which implies the anisotropic 4$f$-hole spatial distribution. ![\[bothLD\](Color online) (a) Schematically drawn Yb$^{3+} 4f$ levels split by the spin-orbit coupling (SOC) and further split by the crystalline electric field (CEF) in tetragonal symmetry. Filled (open) circle denotes an occupied 4$f$ electron (hole). (b) Geometry for the LD in HAXPES measurements, where $\theta$ is the angle of photoelectron detection direction to the \[001\] direction (normal direction to the cleaved surfaces). (c) Polarization-dependent Yb 3$d_{5/2}$ core-level HAXPES spectra for YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$ at $\theta =0^\circ$ (along the \[001\] direction). The spectra are normalized by the overall $3d_{5/2}$ spectral weight displayed in this graph. LD for YbRh$_2$Si$_2$ (YbCu$_2$Si$_2$) is also shown by the dashed (solid) line in the lower panel. (d) Simulated polarization-dependent 3$d_{5/2}$ photoemission spectra for the Yb$^{3+}$ ions assuming the pure $J_z$ ground state, together with the corresponding 4$f$-hole spatial distributions.](Fig1R.eps){width="8cm"} Experimental ============ Since the binding energy of the Yb 3$d_{5/2}$ core-level is higher than 1.5 keV, hard X-ray photoemission spectroscopy (HAXPES) at least $h\nu > 2.5$ keV is preferable to avoid the surface contributions deviated from the bulk ones [@SugaYbAl3; @KitayamaTokimeki]. We have performed LD in HAXPES [@SekiyamaAuAg; @ASHAXPES2013] at BL19LXU of SPring-8 [@YabashiPRL01] by using a MBS A1-HE hemispherical photoelectron spectrometer. A Si(111) double-crystal monochromator selected 7.9 keV radiation with linear polarization along the horizontal direction (the so-called degree of linear polarization $P_L > +0.98$), which was further monochromatized by a Si(620) channel-cut crystal. In order to switch the linear polarization of the excitation light from the horizontal to vertical directions, two single-crystalline (100) diamonds were used as a phase retarder with the (220) reflection placed downstream of the channel-cut crystal. $P_L$ of the polarization-switched x-ray after the phase retarder was estimated as $-0.96$, corresponding to the vertically linear polarization components of 98%. Since the detection direction of photoelectrons was set in the horizontal plane with the angle to the incident photons of 60$^{\circ}$ as shown in Fig. \[bothLD\](b), the experimental configuration at the horizontally (vertically) polarized light excitation corresponds to the p-polarization (s-polarization). The excitation light was focused onto the samples with the spot size of $\sim$25 $\mu$m $\times$ 25 $\mu$m by using an ellipsoidal Kirkpatrik-Baez mirror. The single crystals of YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$ synthesized by a flux method were cleaved along the (001) plane [in situ]{}, where the base pressure was $\sim$1$\times10^{-7}$ Pa. The sample and surface quality was checked by the absence of any core-level spectral weight caused by a possible impurity including oxygen and carbon. The energy resolution was set to 250 meV. Results and Discussions ======================= The polarization-dependent Yb 3$d_{5/2}$ core-level HAXPES spectra of YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$ at $\theta = 0^{\circ}$ (photoelectron detection is along the \[001\] direction) are shown in Fig.\[bothLD\](c). There are a single peak at the binding energy of $\sim$1520 eV and a multiple-peak structure ranging from 1525 to 1535 eV in all spectra. Since the 4$f$ subshell is fully occupied in the Yb$^{2+}$ sites with spherically symmetric 4$f$ distribution, the former single peak is ascribed to the Yb$^{2+}$ states. The $3d^94f^{13}$ (one $4f$ hole with one $3d$ core hole) final states for the Yb$^{3+}$ components show the atomic-like multiplet-split peak structure in the 1525-1535 eV range. Clear LD defined by the difference of the spectral weight between the s- and p-polarization configurations is seen in the Yb$^{3+}$ $3d_{5/2}$ spectral weight depending on material, where it is relatively weaker for YbRh$_2$Si$_2$. For instance, one of the Yb$^{3+}$ $3d_{5/2}$ peak at 1527 eV is stronger in the s-polarization configuration (s-pol.) than in the p-polarization configuration (p-pol.) whereas a structure with the 1529.5-eV peak and 1530.5-eV shoulder is stronger in the p-pol. for both compounds. Possible photoelectron diffraction effects are ruled out of the origin of LD based on the fact that the degree of LD is mutually different between both compounds with the same ThCr$_2$Si$_2$ crystal structure. To clarify the origin of LD in the Yb$^{3+}$ $3d_{5/2}$ core-level HAXPES spectra, we have performed ionic calculations including the full multiplet theory [@Thole85] and the local CEF splitting using the XTLS 9.0 program [@XTLS]. All atomic parameters such as the 4$f$-4$f$ and 3$d$-4$f$ Coulomb and exchange interactions (Slater integrals) and the 3$d$ and 4$f$ spin-orbit couplings have been obtained by Cowan’s code [@Cowan] based on the Hartree-Fock method. The Slater integrals (spin-orbit couplings) are reduced down to 88% (98%) to fit the core-level photoemission spectra [@YbB12PRB2009]. We show the polarization-dependent core-level spectra at $\theta=0^{\circ}$ for pure $\|J_z\rangle$ states of the Yb$^{3+}$ ions with $J = 7/2$ in Fig.\[bothLD\](d). LD depends strongly on $\|J_z\|$, reflecting the Coulomb interactions between the 3$d$ and 4$f$ holes with anisotropic spatial distributions in the final state. The result of the simulations tells us that the observed LD in the core-level HAXPES spectra originates from the anisotropic 4$f$ hole distribution under CEF in the initial state. In the case of Yb$^{3+}$ ions in tetragonal symmetry, the eightfold degenerate $J=7/2$ state splits into four doublets as $$\begin{aligned} &&\|\Gamma_7^1\rangle = c\|\pm5/2\rangle+\sqrt{1-c^2}\|\mp3/2\rangle,\label{G71} \\ &&\|\Gamma_7^2\rangle =- \sqrt{1-c^2}\|\pm5/2\rangle+c\|\mp3/2\rangle,\label{G72} \\ &&\|\Gamma_6^1\rangle = b\|\pm1/2\rangle+\sqrt{1-b^2}\|\mp7/2\rangle,\label{G61}\\ &&\|\Gamma_6^2\rangle =\sqrt{1-b^2}\|\pm1/2\rangle-b\|\mp7/2\rangle,\label{G62}\end{aligned}$$ where the coefficients $0\leq b\leq 1, 0\leq c\leq 1$ defining the actual charge distributions, and CEF splitting energies depend on the CEF parameters $B_2^0, B_4^0, B_4^4, B_6^0$, and $B_6^4$ in Stevens formalism [@Stevens]. Since all CEF splitting energies are highly expected to be much larger ($\gtrsim100$ K) than the measured temperature of 14 K for YbRh$_2$Si$_2$ [@YRSINS1] and YbCu$_2$Si$_2$ [@INS82; @JETP98], it is justified to assume that only the lowest state is populated. As shown in Fig.\[bothLD\](d), LD is qualitatively different between $\|\Gamma_6^{1,2}\rangle$ and $\|\Gamma_7^{1,2}\rangle$, where LDs for $\|J_z\|=5/2$ and 3/2 are qualitatively consistent with those for the experimental data. Therefore, the $\|\Gamma_6^{1,2}\rangle$ ground state formed by the $\|J_z\|$ = 7/2 and 1/2 components is ruled out for both compounds. ![(Color online) (a) Polarization-dependent Yb$^{3+}$ $3d_{5/2}$ core-level HAXPES spectra of YbRh$_2$Si$_2$ at $\theta = 0^\circ$ and 60$^\circ$ and their LDs, where the Shirley-type background has been subtracted from the raw spectra. The spectra are normalized by the Yb$^{3+}$ $3d_{5/2}$ spectral weight. (b) Simulated polarization-dependent core-level photoemission spectra and their LDs \[dashed (solid) line for $\theta=0^\circ$ (60$^\circ$)\] for the Yb$^{3+}$ ion with the $\|J_z\|=3/2$ ($\Gamma_7$) ground state at the same geometrical configurations as those for the experiments. The inset shows the corresponding 4$f$ hole spatial distribution in the initial state. (c) Crystal structure of YbRh$_2$Si$_2$ and YbCu$_2$Si$_2$.[]{data-label="YRSLD"}](Fig2R.eps){width="7.5cm"} To more accurately determine the ground-state orbital symmetry, we have also performed the polarization-dependent core-level HAXPES for YbRh$_2$Si$_2$ at different $\theta$ of 60$^\circ$ corresponding to the incident photon direction parallel to the \[001\] direction as shown in Fig.\[YRSLD\](a). Compared to LD at $\theta=0^\circ$, the sign of LD at $\theta=60^\circ$ is flipped as recognized at the bottom of the figure. We have found that our data are best described by the pure $\|J_z\| = 3/2$ ground state as shown in Fig.\[YRSLD\](b). So far, it has been unclear whether $\|\Gamma_6^1 \rangle$ with dominant $\|J_z\|=1/2$ or $\|\Gamma_7^1 \rangle$ with dominant $\|J_z\|=3/2$ forms the ground state [@YRSCEF1; @YRSCEF2] while a comparison of the slab calculations for subsurface YbRh$_2$Si$_2$ to the low-energy angle-resolved photoemission data has suggested the $\|\Gamma_7^1 \rangle$ ground state [@YRSARPES10]. Here the $\|J_z\|=3/2$ ($\Gamma_7$) ground state is unambiguously revealed for YbRh$_2$Si$_2$ from our LD in HAXPES at $\theta=0^\circ$ and 60$^\circ$. Since there is no anisotropy within the $ab$ plane for the 4$f$ hole distribution with the $\|J_z\|=3/2$ ground state as depicted in the inset of Fig. \[YRSLD\], it is naturally concluded that the Yb 4$f$ holes are hybridized with the partially filled neighbor Rh 4$d$ and Si 3$sp$ states where the $4f$ hole distribution spreads over both sites in Fig.\[YRSLD\](c). ![(Color online) (a) Polarization-dependent Yb$^{3+}$ $3d_{5/2}$ core-level HAXPES spectra of YbCu$_2$Si$_2$ at $\theta = 0^\circ$ and 60$^\circ$ and their LDs, where the Shirley-type background has been subtracted from the raw spectra. (b) Simulated polarization-dependent core-level photoemission spectra and their LDs at the same geometrical configurations as those for the experiments for the Yb$^{3+}$ ion with the $\|\Gamma_7^1\rangle$ and $\|\Gamma_7^2\rangle$ ground states with the $\|J_z\|=3/2$ (5/2) component of 87% (13%). LD at $\theta=0^\circ$ is represented by a dashed line whereas that at $\theta=60^\circ$ for the $\|\Gamma_7^1\rangle$ ($\|\Gamma_7^2\rangle$) ground state is shown by a thin (thick) solid line. (c) 4$f$ hole spatial distribution for the initial $\|\Gamma_7^1\rangle$ and $\|\Gamma_7^2\rangle$ states.[]{data-label="YCSLD"}](Fig3R.eps){width="7cm"} The polarization-dependent Yb$^{3+}$ $3d_{5/2}$ HAXPES spectra of YbCu$_2$Si$_2$ at $\theta=0^\circ$ and 60$^\circ$ are shown in Fig.\[YCSLD\](a). In contrast to the data for YbRh$_2$Si$_2$, the sign of LD is not flipped between two angles of $\theta$ whereas LD is reduced at $\theta=60^\circ$, being inconsistent with the simulations for the pure $\|J_z\| = 3/2$ state in Fig.\[YRSLD\](b). Our detailed analysis indicates that the data set of polarization-dependent HAXPES is best described by the ground state of $$\|\Gamma_7^2\rangle = -0.36\|\pm5/2\rangle+0.93\|\mp3/2\rangle \label{YCSGS}$$ as shown in Fig.\[YCSLD\](b), where the precision of $c^2$ in Eq.(\[G72\]) is $\pm$0.05. The state of $\|\Gamma_7^1\rangle = 0.36\|\pm5/2\rangle+0.93\|\mp3/2\rangle$ with the same amount of the $\|J_z\|=3/2$ components as in Eq.(\[YCSGS\]), of which the spatial 4$f$ hole distribution shows the same shape as for $\|\Gamma_7^2\rangle$ with rotation within the $ab$ plane by 45$^\circ$ \[see Fig.\[YCSLD\](c), hereafter called as in-plane rotation\], gives the same spectra and LD as those for $\|\Gamma_7^2\rangle$ at $\theta=0^\circ$. Therefore, the data at $\theta=60^\circ$ enable us to discriminate the in-plane rotation of the $4f$ charge distributions and unambiguously determine the ground state. The 4$f$ hole distribution for $\|\Gamma_7^2\rangle$ is elongated along the Si sites not the Cu sites with fully occupied 3$d$ levels [@YCS09] as shown in Figs.\[YRSLD\](c) and \[YCSLD\](c), leading to the conclusion that the 4$f$ holes are primarily hybridized with the Si 3$sp$ states. The polarization-dependent Yb$3d_{5/2}$ core-level HAXPES spectra shown here are well reproduced apart from the Yb$^{2+}$ contributions by the simulations for the atomic-like models as seen in the analysis of many LD data in XAS for Ce compounds [@WillersCePt3Si; @Hansmann08LD; @WillersCeTIn5; @WillersCe122], where the hybridization effects are not explicitly taken into account. Such a successful analysis is owing to the highly localized nature of the Yb$^{3+}$ sites in the $3d$ core-level photoemission final states due to the core hole-$4f$ Coulomb interactions ($\sim$10 eV) giving a sufficient energy splitting between the Yb$^{3+}$ and Yb$^{2+}$ final states, and the configuration dependence of the hybridization strengths [@Cdep] leading to the reduced hybridization in the final states. The analysis needs to be extended by using the Anderson impurity model for strongly hybridized systems showing the core-level spectral line shape highly deviated from the atomic-like multplet-split structure, which is not the case for the data displayed here. ![(Color online) (a) Temperature dependence of the polarization-dependent Yb$^{3+}$ $3d_{5/2}$ HAXPES spectra at $\theta=0^\circ$ and their LDs for YbCu$_2$Si$_2$. (b) Simulated temperature dependence of the polarization-dependent core-level photoemission spectra and their LDs at $\theta=0^\circ$ for the Yb$^{3+}$ ion with the 4$f$ levels in (c), where the coefficient $b^2$ in Eqs. (\[G61\]) and (\[G62\]) for the excited $\|\Gamma_6^{1,2}\rangle$ states of 0.4 with the lower state with predominant $\|J_z\|=7/2$ components. (c) Schematically drawn Yb$^{3+} 4f$ $J=7/2$ levels with symmetry split by CEF determined for YbCu$_2$Si$_2$ together with the fraction of the predominant component for each doublet. []{data-label="LDTdep"}](Fig4Rev.eps){width="8.5cm"} We have also performed the temperature- and polarization-dependent HAXPES at $\theta=0^\circ$ for YbCu$_2$Si$_2$ as shown in Fig.\[LDTdep\](a), verifying that LD is reduced at high temperatures without a flip of its sign. The temperature dependence originates from a partial occupation of the excited state $i$ split by $\Delta_i$ from the ground state at high temperatures $T$ with a fraction of $\exp[-\Delta_i/(k_BT)]/\{1+\sum_i\exp[-\Delta_i/(k_BT)]\}$, which leads to the isotropic spectra at a sufficiently high temperature. Taking the fact that the magnetic excitations are clearly seen at $\sim$210 and $\sim$360K for YbCu$_2$Si$_2$ [@INS82] into account, we can determine to some extent the $4f$-orbital symmetry of the excited states based on our data and simulations, where $\|\Gamma_7^1\rangle$ with dominant $\|J_z\|=5/2$ gives larger LD than the experimental one and $\|\Gamma_6\rangle$ formed by the the $\|J_z\|=7/2$ and 1/2 states shows a sign-flipped LD compared to the data as suggested by the simulations in Fig. \[bothLD\](d). Figure \[LDTdep\](b) shows one of the best simulated temperature-dependent HAXPES spectra and LDs for the $4f$-level scheme with symmetry shown in Fig. \[LDTdep\](c), which has been optimized with some ambiguities in determining the parameters. The other $4f$-level scheme with the first excitation energy of $\sim$100 K similar to a previously proposed one [@JETP98] is inconsistent with the experimental result since the simulations with this scheme give larger temperature dependence of LD involving a sign-flipping or an enhancement of LD at 100 K. Here optimized CEF splitting energies for the $4f$-level scheme in Fig.\[LDTdep\](c) is similar to another previously proposed one in Ref. based on the magnetic excitations in Ref. . Conclusions =========== In conclusion, we have shown that the 4$f$-orbital symmetry of the ground states as well as that of the excited states in the Yb compounds is probed by the LD in the core-level HAXPES and its $\theta$ dependence. The ground-state symmetry is unambiguously determined by the LD under the sole assumption that only the lowest state is populated. LD in the core-level HAXPES has the advantage over LD in XAS in discriminating the same shape of the 4$f$ charge distributions with in-plane rotation as shown for YbCu$_2$Si$_2$ owing to the experimental parameter $\theta$ in addition to the polarization. The discrimination of the in-plane rotation of charge distribution is also feasible by the polarization-dependent non-resonant inelastic X-ray scattering [@NIXSCeCu2Si2], but the throughput and energy resolution are much better for LD in the core-level HAXPES. Therefore, the experimental technique demonstrated here will be very promising to reveal the strongly correlated orbital symmetry of the ground and excited states in the atomic-like partially filled subshell in solids, complementing the neutron scattering. We thank T. Kadono, F. Honda, Y. Nakata, Y. Nakatani, T. Yamaguchi, H. Fuchimoto, T. Yagi, S. 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--- abstract: 'We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.' address: - 'Département de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France' - \| Mathematisches Institut der Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2\ 91054 Erlangen, Germany author: - Nabile Boussaid - Sylvain Golénia date: Version of title: Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies --- Introduction ============ We study properties of relativistic massive charged particles with spin-$1/2$ (e.g., electron, positron, (anti-)muon, (anti-)tauon,$\ldots$). We follow the Dirac formalism, see [@Dirac]. Because of the spin, the configuration space of the particle is vector valued. To simplify, we consider finite dimensional and trivial fiber. Let $\nu\geq 2$ be an integer. The movement of the free particle is given by the Dirac equation, $$i \hbar \frac{\partial \varphi}{\partial t }= D_m \varphi, \mbox{ in } L^2({\mathbb{R}}^3; {\mathbb{C}}^{2\nu}),$$ where $m>0$ is the mass, $c$ the speed of light, $\hbar$ the reduced Planck constant, and $$\label{e:op} D_m:= c \hbar\, \alpha\cdot P+mc^2\beta= -\rmi c\hbar\sum_{k=1}^3\alpha_k\partial_k + mc^2\beta.$$ Here we set ${\alpha}:=\left(\alpha_1, \alpha_2, \alpha_3\right)$ and $\beta:=\alpha_4$. The $\alpha_i$, for $i\in\{1,2,3, 4\}$, are linearly independent self-adjoint linear applications, acting in ${\mathbb{C}}^{2\nu}$, satisfying the anti-commutation relations: $$\label{e:Diracrep0} \alpha_i\alpha_j+\alpha_j\alpha_i=2\delta_{ij}\bone_{{\mathbb{C}}^{2\nu}}, \mbox{ where } i,j\in\{1,2,3,4\}.$$ For instance, when $\nu=2$, one may choose the Pauli-Dirac representation: $$\begin{gathered} \label{e:Diracrep} \alpha_i=\left( \begin{array}{cc} 0&\sigma_i\\ \sigma_i&0 \end{array} \right)\quad\mbox{ and }\quad \beta=\left(\begin{array}{cc} {{\rm{Id}} }_{{\mathbb{C}}^\nu}&0\\ 0&-{{\rm{Id}} }_{{\mathbb{C}}^\nu} \end{array} \right) \\ \nonumber \mbox{ where } \sigma_1=\left( \begin{array}{cc} 0&\; 1\\ 1&\;0 \end{array} \right),\quad \quad\sigma_2=\left( \begin{array}{cc} 0&-\rmi\\ \rmi&0 \end{array} \right)\quad\mbox{ and } \quad\sigma_3=\left( \begin{array}{cc} 1&0\\ 0&-1 \end{array} \right),\end{gathered}$$ for $i=1,2,3$. We refer to [@Thaller]\[Appendix 1.A\] for various equivalent representations. In this paper we do not choose any specific basis and work intrinsically with . We refer to [@LawsonMichelsohn] for a discussion of the representations of the Clifford algebra generated by . We also renormalize and consider $\hbar=c=1$. The operator $D_m$ is essentially self-adjoint on ${\mathcal{C}}^\infty_c({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$ and the domain of its closure is ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$, the Sobolev space of order $1$ with values in ${\mathbb{C}}^{2\nu}$. We denote the closure with the same symbol. Easily, using Fourier transformation and some symmetries, one deduces the spectrum of $D_m$ is purely absolutely continuous and given by $(-\infty, -m]\cup[m, \infty)$. In this introduction, we focus on the dynamical and spectral properties of the Hamiltonian describing the movement of the particle interacting with $n$ fixed, charged particles. We model them by fixed points $\{a_i\}_{i=1,\ldots, n}\in {\mathbb{R}}^{3n}$ with respective charges $\{z_i\}_{i=1,\ldots, n}\in {\mathbb{R}}^{n}$. Doing so, we tacitly suppose that the particles $\{a_i\}$ are far enough from one another, so as to neglect their interaction. Note we make no hypothesis on the sign of the charges. The new Hamiltonian is given by $$\begin{aligned} \label{e:H} H_\gamma:=D_m + \gamma V_c(Q), \mbox{ where } V_c:=v_c \otimes{{\rm{Id}} }_{{\mathbb{C}}^{2\nu}} \mbox{ and } v_c(x):=\sum_{k=1,\ldots ,n} \frac{z_i}{\|x-a_i\|},\end{aligned}$$ acting on $\mathcal{C}_c^\infty({\mathbb{R}}^3\setminus\{a_i\}_{i=1, \ldots, n}; {\mathbb{C}}^{2\nu})$, with $a_i\neq a_j$ for $i\neq j$. The $\gamma\in {\mathbb{R}}$ is the coupling constant. The index $c$ stands for coulombic multi-center. The notation $V(Q)$ indicates the operator of multiplication by $V$. Here, we identify $L^2({\mathbb{R}}^3;{\mathbb{C}}^{2\nu})\simeq L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^{2\nu}$, canonically. Remark the perturbation $V_c$ is not relatively compact with respect to $D_m$, then one needs to be careful to define a self-adjoint extension for $D_m$. Assuming $$\begin{aligned} \label{e:atomic} Z:=\|\gamma\| \max_{i=1, \ldots, n}(\|z_i\|)< \sqrt{3}/2,\end{aligned}$$ the theorem of Levitan-Otelbaev ensures that $H_\gamma$ is essentially self-adjoint and its domain is the Sobolev space ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$, see [@AraiYamada; @Kalf; @Klaus; @LandgrenRejto; @LandgrenRejtoKlaus; @LevitanOtelbaev] for various generalizations. This condition corresponds to the nuclear charge $\alpha_{\rm at}^{-1} Z\leq 118 $, where $\alpha_{\rm at}^{-1}= 137.035999710(96)$. Note that using the Hardy-inequality, the Kato-Rellich theorem will apply till $Z<1/2$ and is optimal in the matrix-valued case, see [@Thaller]\[Section 4.3\] for instance. For $Z<1$, one shows there exists only one self-adjoint extension so that its domain is included in ${\mathscr{H}}^{1/2}({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$, see [@Nenciu]. This covers the nuclear charges up to $Z=137$. When $n=1$ and $Z=1$, this property still holds true, see [@EstebanLoss]. Surprisingly enough, when $n=1$ and $Z>1$, there is no self-adjoint extension with domain included in ${\mathscr{H}}^{1/2}({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$, see [@Xia]\[Theorem 6.3\]. We mention also the work of [@VoronovGitmanTyutin] for $Z>1$. In [@Nenciu] and $Z<1$, one shows the essential spectrum is given by $(-\infty, -m]\cap [m, \infty)$ for all self-adjoint extension. For all $Z$, one refers to [@GeorgescuMantoiu]\[Proposition 4.8.\], which relies on [@Xia]. In [@GoleniaGeorgescu] one gives some criteria of stability of the essential spectrum for some very singular cases. In [@BerthierGeorgescu], one proves there is no embedded eigenvalues for a more general model and till the coupling constant $Z<1$. For all energies being in a compact set included in $(-\infty, -m)\cap (m, \infty)$, [@GeorgescuMantoiu] obtains some estimates of the resolvent. This implies some propagation estimates and that the spectrum of $H_\gamma$ is purely absolutely continuous. Similar results have been obtained for magnetic potential of constant direction, see [@Yokoyama] and more recently [@RichardTiedra]. In this paper we are interested in uniform estimates of the resolvent at threshold energies. The energy $m$ is called the electronic threshold and $-m$ the positronic threshold. In Theorem , we obtain a uniform estimation of the resolvent over $[-m-\delta, -m]\cup [m, m+\delta]$, see and deduce some propagation properties, see . One difficulty is that in the case $n=1$ and $z_i<0$, it is well known there are infinitely many eigenvalues in the gap $(-m, m)$ converging to the $m$ as soon as $\gamma \neq 0$ (see for instance [@Thaller]\[Section 7.4\] and references therein). This is a difficult problem and, to our knowledge, this result is new for the multi-center case. There is a larger literature for non-relativist models, e.g., $-\Delta +V$ in $L^2({\mathbb{R}}^n;{\mathbb{C}})$. The question is intimately linked with the presence of resonances at threshold energy, [@JensenNenciu; @FournaisSkibsted; @Nakamura; @Richard; @Yafaev82]. We mention also [@BurqPlanchonStalkerTahvildarZadeh] for applications to Strichartz estimates and [@DerezinskiSkibsted] for applications to scattering theory. Before giving the main result, we shall discuss some commutator methods. The first stone was set C.R. Putnam a self-adjoint operator $H$ acting in a Hilbert space ${\mathscr{H}}$, see [@Putnam] and for instance [@ReedSimon]\[Theorem XIII.28\]. One supposes there is a bounded operator $A$ so that $$\label{e:KatoPutnam} C:=[H,iA]_\circ>0,$$ where $>$ means non-negative and injective. The commutator has to be understood in the form sense. When it extends into a bounded operator between some spaces, we denote this extension with the symbol $\circ$ in subscript, see Appendix \[s:dev-commut\]. The operator $A$ is said to be conjugate to $H$. One deduces some estimation on the imaginary part of the resolvent, i.e., one finds some weight $B$, a closed injective operator with dense domain, so that $$\begin{aligned} \sup_{\Re(z)\in {\mathbb{R}}, \Im(z)>0} \Im \langle f, (H- z )^{-1} f \rangle \leq \\|B f\\|^2.\end{aligned}$$ This estimation is equivalent to the global propagation estimate, c.f. [@Kato] and [@ReedSimon]\[Theorem XIII.25\]: $$\begin{aligned} \int_{\mathbb{R}}\\|B^{-1} e^{itH} f\\|^2 dt \leq 2 \\|f\\|^2\end{aligned}$$ One infers that the spectrum of $H$ is purely absolutely continuous with respect to the Lebesgue measure. In particular, $H$ has no eigenvalue. To deal with the presence of eigenvalues, the fact that $A$ is bounded and with the $3$-body-problem, Eric Mourre has the idea to localized in energy the estimates and to allow a compact perturbation, see [@Mourre]. With further hypothesis, one shows an estimate of the resolvent (and not only on the imaginary part). The applications of this theory are numerous. The theory was immediately adapted to treat the $N$-body problem, see [@PerrySigalSimon]. The theory was finally improved in many directions and optimized in many ways, see [@AmreinBoutetdeMonvelGeorgescu] for a more thorough discussion of these matters. We mention also [@GeorgescuGerardMoller; @GoleniaJecko; @Gerard] for recent developments. As we are concern about thresholds, Mourre’s method does not seem enough, as the estimate of the resolvent is given on an interval which is strictly smaller than the one used in the commutator estimate. In [@BoutetMantoiu] one generalizes the result of Kato-Putnam approach. Under some conditions, one allows $A$ to be unbounded. They obtain a global estimate of the resolvent. Note this implies the absence of eigenvalue. In [@FournaisSkibsted], in the non-relativistic context, by asking some positivity on the Virial of the potential, see below, one is able to conciliate the estimation of the resolvent above the threshold energy and the accumulation of eigenvalue under it. In [@Richard], one presents an abstract version of the method of [@FournaisSkibsted]. To give an idea, we shall compare the theories on a non-optimal example. Take $H:=-\Delta+V$ in $L^2({\mathbb{R}}^3)$, with $V$ being in the Schwartz space. Consider the generator of dilation $A:= (P\cdot Q + Q\cdot P)/2$, one looks at the quantity $$\begin{aligned} [H, iA]_\circ - cH = -(2-c) \Delta - W_V(Q), \mbox{ where } W_V(Q):= Q \cdot \nabla V(Q) + c V(Q), \end{aligned}$$ with $c\in (0,2)$ and seeks some positivity. The expression $W_V$ is called the Virial of $V$. In [@FournaisSkibsted], one uses extensively that $W_V(x)\leq -c\langle x\rangle^{-\alpha}$ for some $\alpha,c>0$ and $\|x\|$ big enough. In [@Richard], one notices that it suffices to suppose that $W_V(x)\leq 0$ and to take advantage of the positivity of the Laplacian. We take the opportunity to mention that it is enough to suppose that $W_V(x)\leq c'\|x\|^{-2}$, for some small positive constant $c'$, see Theorem \[t:nonrelamain\]. Observe also that these methods give different weights. For instance, [@FournaisSkibsted] obtains better weights in the scale of $\langle Q\rangle^\alpha$ and [@Richard] can obtain singular weights like $\|Q\|$, see Appendix \[s:WeakMourreTheory\]. Finally, [@FournaisSkibsted] deals only with low energy estimates and [@Richard] works globally on $[0, \infty)$. We also point out [@Herbst] which relies on commutator techniques and deals with smooth homogeneous potentials. In this article, we revisit the approach of [@Richard] and make several improvements, see Appendix \[s:WeakMourreTheory\]. Our aim is twofold: to treat dispersive non self-adjoint operator and to obtain estimates of the resolvent uniformly in a parameter. At first sight, these improvements are pointless from the standpoint of the Coulomb-Dirac problem we treat. In reality, they are the key-stone of our approach. As a direct by-product of the method, we obtain some new results for dispersive Schrödinger operators. The following $V_2$ term corresponds to the absorption coefficient of the laser energy by material medium absorption term in the Helmholtz model, see [@Jackson] for instance. \[t:nonrela\] Let $n\geq 3$. Suppose that $V_1, V_2\in L^\infty({\mathbb{R}}^n;{\mathbb{R}})$ satisfy: 1. $\nabla V_i$, $ Q\cdot \nabla V_i(Q)$, $\langle Q\rangle (Q\cdot \nabla V_i)^2(Q)$ are bounded, for $i\in\{1,2\}$. 2. There are $c_1\in [0,2)$ and $\displaystyle c_1'\in \big[0, 4(2-c_1)/(n-2)^2\big)$ such that $$\begin{aligned} W_{1}(x):= x\cdot (\nabla V_1)(x) + c_{1} V_1(x) \leq \frac{c_{1}'}{\|x\|^2}, \mbox{ for all } x\in {\mathbb{R}}^n. \end{aligned}$$ and $$\begin{aligned} V_2(x)\geq 0 \mbox{ and } -x\cdot (\nabla V_2)(x)\geq 0, \mbox{ for all } x\in {\mathbb{R}}^n.\end{aligned}$$ On ${\mathcal{C}}^\infty_c({\mathbb{R}}^n)$, we define $H:= -\Delta + V(Q), \mbox{ where } V:=V_1 + i V_2$. The closure of $H$ defines a dispersive closed operator with domain ${\mathscr{H}}^2({\mathbb{R}}^n)$. We keep denoting it with $H$. Its spectrum included in the upper half-plane. The operator $H$ has no eigenvalue in $[0,\infty)$. Moreover, $$\begin{aligned} \label{e:nonrela} \sup_{\lambda\in [0,\infty), \, \mu>0} \big\\|\, \|Q\|^{-1}(H -\lambda + \rmi \mu )^{-1} \|Q\|^{-1}\big\\| <\infty. \end{aligned}$$ Note we require nor smoothness on the potentials neither that they are relatively compact with respect to the Laplacian. We refer to Appendix \[s:nonrela\] for further comments, the case $c_1=0$ and a stronger result. We come back to the main application, namely the operator $H_\gamma$ defined by . As the Dirac operator is vector-valued, coulombic interaction are singular and as we are interested in both thresholds, we were not able to use directly the ideas of [@FournaisSkibsted; @Richard]. Indeed, it is unclear for us if one can actually deal with thresholds energy and keep the “positivity” of the something close to the quantity $[H_\gamma, iA]- c H_\gamma$, for some self-adjoint operator $A$. We hedge this fundamental problem. First of all we cut-off the singularities of the potential $V_c$ and consider the operator $H^{\rm bd}_\gamma = D_m+\gamma V$ in Section \[s:red\]. We recover the singularities of the operator by perturbation in Proposition \[p:collage\]. In Section , we explicit the resolvent of $H^{\rm bd}_\gamma-z$ relatively to a spin-down/up decomposition. This transfers the analysis to the one of an elliptic operator of second order, $\Delta_{m,v,z}$, see Section \[s:reduction\]. The drawback is that this operator is dispersive and also depends on the spectral parameter $z$. We elude the latter by studying the family $\{\Delta_{m,v,\xi}\}_{\xi\in{\mathcal{E}}}$ uniformly in ${\mathcal{E}}$. In Section \[s:LAPano\], we explain how to deduce the estimation of the resolvent of $H^{\rm bd}_\gamma$ having the one of $\Delta_{m,v,z}$. In the Section \[s:positive\], we establish some positive commutator estimates for $\Delta_{m,v,z}$ and derive the sought estimates of the resolvent, see Theorem . For the last step, we rely on the theory developed in Appendix \[s:WeakMourreTheory\]. The main result of this introduction is the following one. \[t:main\] There are $\kappa, \delta, C>0$ such that $$\begin{aligned} \label{e:main} \sup_{\|\lambda\|\in [m, m+\delta], \, \varepsilon>0, \|\gamma\|\leq \kappa }\\|\langle Q\rangle^{-1}(H_\gamma -\lambda - \rmi \varepsilon )^{-1} \langle Q\rangle^{-1}\\|\leq C. \end{aligned}$$ In particular, $H_\gamma$ has no eigenvalue in $\pm m$. Moreover, there is $C'$ so that $$\begin{aligned} \label{e:Kato} \sup_{\|\gamma\|\leq \kappa}\int_{\mathbb{R}}\\| \langle Q \rangle^{-1} e^{-it H_\gamma } E_{\mathcal{I}}(H_\gamma) f \\|^2 dt \leq C'\\|f\\|^2,\end{aligned}$$ where ${\mathcal{I}}=[-m-\delta, -m]\cup [m, m+\delta]$ and where $E_{\mathcal{I}}(H_\gamma)$ denotes the spectral measure of $H_\gamma$. A more general result is given in Theorem \[t:vraimain\]. In Theorem \[t:vraimain2\] we discuss the weights $\langle P\rangle^{1/2} \|Q\|$ and in Remark \[r:noLAP\] the weights $\|Q\|$. If one is not interested in the uniformity in the coupling constant, using [@GeorgescuMantoiu], one can consider all $\delta>0$ and deduce . The propagation estimate refers as Kato smoothness and it is a well-known consequence of , see [@Kato]. Using some kernel estimates, one can obtain directly for the free Dirac operator, i.e., $\gamma=0$, see for instance [@Thaller]\[Section 1.E\] and [@KatoYajima]. One may find an alternative proof of this fact in [@IftimoviciMantoiu] which relies on some positive commutator techniques. In this study, we are mainly interested by long range perturbations of Dirac operators. Concerning limiting absorption principle for short range perturbations of Dirac operators there are some interesting works such as [@DAnconaFanelli] for small perturbations without discrete spectrum or [@Boussaid] for potentials producing discrete spectrum. These authors were mainly interested by time decay estimates similar to . In the short range case, the limiting absorption principle is a key ingredient to establish Strichartz estimates for perturbed Dirac type equations see [@Boussaid2; @DAnconaFanelli2]. For free Dirac equations there are some direct proofs, see [@EscobedoVega; @MachiharaNakamuraOzawa; @MachiharaNakamuraNakanishiOzawa]. Time decay estimates such as or Strichartz are crucial tool to establish well posedness results [@EscobedoVega; @MachiharaNakamuraOzawa; @MachiharaNakamuraNakanishiOzawa] and stability results [@Boussaid; @Boussaid2] for nonlinear Dirac equations. The paper is organized as follows. In the second section we reduced the analysis of the resolvent of the Dirac operator perturbed with a bounded potential to the one of family of non self-adjoint operators. In the third part, we analyze these operators and obtain some estimates of the resolvent. In the fourth part, we state the main results of the paper. For the convenience of the reader, we expose some commutator expansions in the Appendix \[s:dev-commut\]. In the Appendix \[s:WeakMourreTheory\], we develop the abstract positive commutator theory. At last in Appendix \[s:nonrela\], we give a direct application to the theory in the context of the Helmholtz equation. [Notation:]{} In the following $\Re$ and $\Im$ denote the real and imaginary part, respectively. The smooth function with compact support are denoted by ${\mathcal{C}}^\infty_c$. Given a complex-valued function $F$, we denote by $F(Q)$ the operator of multiplication by $F$. We mention also the notation $P=-\rmi \nabla$. We use the standard $\langle \cdot \rangle:= (1+ \|\cdot\|^2)^{1/2}$. [Acknowledgments: ]{} We would like to thank Lyonel Boulton, Bertfried Fauser, Vladimir Georgescu, Thierry Jecko, Hubert Kalf, Andreas Knauf, Michael Levitin, François Nicoleau, Heinz Siedentop and Xue Ping Wang for useful discussions. The first author was partially supported by ESPRC grant EP/D054621. Reduction of the problem {#s:red} ======================== In this section, we study the resolvent of the perturbed Dirac operator $$\begin{aligned} \label{e:Hbd} H^{\rm bd}_\gamma=D_m+\gamma V, \mbox{ where } V:=v \otimes {{\rm{Id}} }_{{\mathbb{C}}^{2\nu}} \mbox{ and $v$ \underline{bounded}}. \end{aligned}$$ In Section \[s:mainresult\], we explain how to cover some singularities. Due to the method, we will consider only small coupling constants. We will show the limiting absorption principle $$\begin{aligned} \label{e:mainlow} \sup_{\|\lambda\|\in [m, m+\delta], \, \varepsilon>0, \, \|\gamma\|\leq \kappa} \\|\langle Q\rangle^{-1}(H_\gamma^{\rm bd} -\lambda - \rmi \varepsilon )^{-1} \langle Q\rangle^{-1}\\|\leq C,\end{aligned}$$ for some $\kappa>0$. We notice this is equivalent to $$\begin{aligned} \label{e:mainlow2} \sup_{\lambda\in [m, m+\delta], \, \varepsilon>0, \, \|\gamma\|\leq \kappa} \\|\langle Q\rangle^{-1}(H_\gamma^{\rm bd} -\lambda - \rmi \varepsilon )^{-1} \langle Q\rangle^{-1}\\|\leq C,\end{aligned}$$ Indeed, by setting $\alpha_5:=\alpha_1\alpha_2\alpha_3\alpha_4$ and using the anti-commutation relation , we infer $$\begin{aligned} \alpha_5\left(D_m+\gamma V\right)\alpha_5^{-1}= -D_m+\gamma V.\end{aligned}$$ Note that $\alpha_5$ is unitary and stabilizes ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2n})$. This gives $$\begin{aligned} \label{e:gauge} \alpha_5 \varphi( D_m +\gamma V) \alpha_5^{-1}= \varphi\big( -(D_m -\gamma V)\big), \mbox{ for all } \varphi\in {\mathcal{C}}({\mathbb{R}}; {\mathbb{C}}).\end{aligned}$$ The non self-adjoint operator {#s:reduction} ----------------------------- Here, we relate the resolvent of in a point $z\in {\mathbb{C}}\setminus {\mathbb{R}}$ with the one of some non self-adjoint Laplacian type operator $\Delta_{m,v,z}$, chosen in . We fix a compact set ${\mathcal{I}}$ being the area of energy we are concentrating on. In the next section, we explain how to recover a limiting absorption principle for $H^{\rm bd}_\gamma$ over ${\mathcal{I}}$ given the one of $\Delta_{m,\gamma v,z}$. We consider a potential $v\in L^\infty({\mathbb{R}}^3;{\mathbb{R}})$, not necessarily smooth, satisfying $$\label{a:BoundV} \\|v\\|_\infty \leq m/2\mbox{ and }\nabla v\in L^\infty({\mathbb{R}}^3;{\mathbb{R}}^3).$$ It particular, $(v(Q)-m-z)^{-1}$ stabilizes ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$ for all $z$ in ${\mathbb{C}}\setminus {\mathbb{R}}$. Since $\beta=\alpha_4$ satisfies , we deduct that $\beta$ has the eigenvalues $\pm 1$ and the eigenspaces have the same dimension. Let $P^+$ be the orthogonal projection on the spin-up part of the space, i.e., on $\ker(\beta -1)$. Let $P^-:= 1-P^+$. Since $\alpha_j$ satisfies , for $j\in \{1,2, 3\}$, we get $P^{\pm}\alpha_jP^{\pm}=0$. We set: $$\begin{aligned} \alpha^+_j:= P^+ \alpha_j P^- \mbox{ and } \alpha^-_j:= P^- \alpha_j P^+, \mbox{ for } j\in\{1, 2, 3\}.\end{aligned}$$ They are partial isometries: $$\begin{aligned} \big(\alpha^+_j\big)^= \alpha^-_j, \quad \alpha^+_j \alpha^-_j = P^+ \mbox{ and } \alpha^-_j \alpha^+_j = P^-, \mbox{ for } j\in\{1,2, 3\}.\end{aligned}$$ The relation of anti-commutation gives: $$\begin{aligned} \label{e:Diracrep1} \alpha_i^-\alpha_j^+ + \alpha_j^-\alpha_i^+= 2\delta_{ij} \mbox{ and } \alpha_i^+\alpha_j^- + \alpha_j^+\alpha_i^-= 2\delta_{ij}, \mbox{ for } i,j\in\{1,2, 3\}.\end{aligned}$$ We set ${\mathbb{C}}^\nu_\pm:= P^\pm{\mathbb{C}}^{2\nu}$. In the direct sum ${\mathbb{C}}^{\nu}_-\oplus {\mathbb{C}}^{\nu}_+$, with a slight abuse of notation, one can write $$\begin{aligned} \beta = \left(\begin{array}{cc} {{\rm{Id}} }_{{\mathbb{C}}^\nu} & 0 \\ 0 & -{{\rm{Id}} }_{{\mathbb{C}}^\nu} \end{array}\right) \mbox{ and } \alpha_j= \left(\begin{array}{cc} 0 & \alpha_j^+ \\ \alpha_j^- & 0 \end{array}\right), \mbox{ for } j\in\{1,2, 3\}.\end{aligned}$$ We now split the Hilbert space ${\mathscr{H}}=L^2({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$ into the spin-up and down part: $$\begin{aligned} \label{e:split} {\mathscr{H}}={\mathscr{H}}^+\oplus {\mathscr{H}}^-, \mbox{ where } {\mathscr{H}}^\pm := L^2({\mathbb{R}}^3; {\mathbb{C}}^{\nu}_\pm)\simeq L^2({\mathbb{R}}^3; {\mathbb{C}}^{\nu}) . \end{aligned}$$ We define the operator: $$\label{e:Delta} \Delta_{m,v,z}:=\alpha^+\!\cdot P\frac{1}{m -v(Q) +z}\alpha^-\!\cdot P + v(Q)$$ on ${\mathcal{C}}^\infty_c({\mathbb{R}}^3; {\mathbb{C}}^{\nu}_+)$. It is well defined by (\[a:BoundV\]). It is closable as its adjoint has a dense domain. We consider the minimal extension, its closure. We denote its domain by ${\mathcal{D}}_{\rm min}(\Delta_{m,v,z})$ and keep the same symbol for the operator. It is well known that even for symmetric operators one needs to be careful with domains as the domain of the adjoint could be much bigger than the one of the closure. In the next Proposition, we care about this problem in our non-symmetric setting. \[p:domain\] Let $z\in {\mathbb{C}}\setminus {\mathbb{R}}$ such that $\Re(z)\geq 0$. Under the hypotheses , we have that $$\begin{aligned} {\mathcal{D}}(\Delta_{m,v,z})= {\mathcal{D}}(\Delta_{m,v,\overline{z}}^ )= {\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+) \mbox{ and } \Delta_{m,v,z}=\Delta_{m,v,\overline{z}}^.\end{aligned}$$ We mimic the Kato-Rellich approach and compare $\Delta_{m,v,z}$ with the more convenient operator $\tilde \Delta_z:=\big(1/(m+z)\big) \Delta_{1,0,0}$. Its domain is ${\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$ and its spectrum is $\{(m+\overline{z})t \mid t\in [0,\infty)\}$. We now show there is $a\in [0,1)$ and $b\geq 0$ such that $$\begin{aligned} \label{e:KR} \\|B f\\|^2\leq a\left\\|\tilde\Delta_z f\right\\|^2 +b\\|f\\|^2, \end{aligned}$$ holds true for all $f\in {\mathcal{C}}^\infty_c({\mathbb{R}}^3, {\mathbb{C}}^\nu_+)$, where $$\begin{aligned} B:= \frac{v}{(m-v+z)} \tilde \Delta_z -i \frac{(\alpha^+ \cdot \nabla v)(Q)}{(m-v(Q)+z)^2}\, \alpha^- \cdot P+ v(Q).\end{aligned}$$ Since $\\|v\\|_\infty \leq m/2$, $\Re(z)\geq 0$ and $\Im(z)>0$, we infer $a_0:=\\|v/(m-v+z)\\|_\infty<1$. Set $M:=\\|(\alpha^+\cdot \nabla v)(\cdot)/(m-v(\cdot)+z)^2\\|_\infty$. Take $\varepsilon, \varepsilon'\in (0,1)$ $$\begin{aligned} \\|Bf\\|^2 &\leq (1+\varepsilon)a_0^2\left\\|\tilde \Delta_z f\right\\|^2 + \left(1+ \frac{1}{\varepsilon}\right) \left\\|\frac{(\alpha^+ \cdot \nabla v)(Q)}{(m-v(Q)+z)^2}\, \alpha^- \cdot P f + v(Q )f\right\\|^2, \\ &\leq (1+ \varepsilon) a_0^2\left\\|\tilde \Delta_z f\right\\|^2 +\frac{4M^2}{\varepsilon} \left\\|\alpha^- \cdot P f\right\\|^2 + \frac{4 \\|v\\|_\infty}{ \varepsilon^2 }\\|f\\|^2, \\ &\leq \left((1+ \varepsilon) a_0^2 + \varepsilon'\right) \left\\|\tilde \Delta_z f\right\\|^2 + \left( \frac{4 \\|v\\|_\infty}{ \varepsilon^2 }+ \frac{2\|m+z\|^2M^2}{\varepsilon \varepsilon' } \right) \left\\|f\right\\|^2.\end{aligned}$$ By choosing $\varepsilon$ and $\varepsilon' $ so that the first constant is smaller than $1$, is fulfilled. Now, observe that $\\|B(\tilde \Delta_z+\mu)^{-1}\\|^2\leq a +b\mu^{-2}$ for $\mu>0$. Fix $\mu_0>0$ such that $\\|B(\tilde \Delta_z+\mu_0)^{-1}\\|<1$. Then $(1+B(\tilde \Delta_z+\mu_0)^{-1})$ is bijective. Noticing that $$\begin{aligned} \big({{\rm{Id}} }+ B (\tilde \Delta_z+\mu_0)^{-1}\big)(\tilde \Delta_z+\mu_0) = \Delta_{m,v,z}+ \mu_0, \end{aligned}$$ we infer that $\Delta_{m,v,z}+ \mu_0$ is bijective from ${\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$ onto $L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$. In particular ${\mathcal{D}}(\Delta_{m,v,z})= {\mathscr{H}}^2({\mathbb{R}}^3;{\mathbb{C}}^\nu_+)$. Directly, one has ${\mathcal{D}}(\Delta_{m,v,z})\subset {\mathcal{D}}(\Delta_{m,v,\overline{z}}^)$ and $\Delta_{m,v,z}\subset \Delta^_{m,v,\overline{z}}$ (inclusion of graphs). Take now $f\in {\mathcal{D}}(\Delta_{m,v,\overline{z}}^ )$. Since $\Delta_{m,v,z}+ \mu_0$ is surjective, there is $g\in {\mathcal{D}}(\Delta_{m,v,z})$ so that $$\begin{aligned} (\Delta_{m,v,z}+ \mu_0) g = (\Delta_{m,v,\overline{z}}^* + \mu_0) f.\end{aligned}$$ In particular, $(\Delta_{m,v,\overline{z}}^* + \mu_0) (f-g)=0$. As $\Delta_{m,v,z}+ \mu_0$ is surjective, we derive that $\ker (\Delta_{m,v,\overline{z}}^+\mu_0)=\{0\}$. In particular $f=g$, ${\mathcal{D}}(\Delta_{m,v,z})= {\mathcal{D}}(\Delta_{m,v,\overline{z}}^)$ and $\Delta_{m,v,z}=\Delta^_{m,v,\overline{z}}$. As a corollary, we derive: The spectrum of $\Delta_{m,v,z}$ is contained in the lower/upper half-plane which does not contain $z$. In particular, $c+z$ is always in the resolvent set of $\Delta_{m,v,z}$ for any $c\in{\mathbb{R}}$. Take now $f\in {\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$. Since $$\begin{aligned} \label{e:imdelta} \Im \langle f, \Delta_{m,v,z}\, f\rangle = \langle \alpha^-\!\cdot P f, \frac{-\Im(z)}{\big(m-v(Q)+\Re(z)\big)^2+\Im(z)^2}\, \alpha^-\!\cdot P f\rangle, \end{aligned}$$ is of the sign of $-\Im(z)$. Since $\Delta_{m,v,z}$ is a closed operator having the same domain of its adjoint, the spectrum of $\Delta_{m,v,z}$ is contained in the closure of its numerical range, see Lemma \[l:NRT\]. We give a kind of Schur’s lemma, so as to compute the inverse of the Dirac operator. \[l:inv\] Suppose (\[a:BoundV\]). Take $z\in {\mathbb{C}}\setminus {\mathbb{R}}$ such that $\Re(z)\geq 0$. In the spin-up/down decomposition ${\mathscr{H}}={\mathscr{H}}^+\oplus {\mathscr{H}}^-$, we have $(H^{\rm bd}_1-z)^{-1}=$ $$\begin{aligned} \left(\begin{array}{c} (\Delta_{m,v,z}+m-z)^{-1} \\ \displaystyle \frac{1}{m -v(Q) +z}\alpha^-\!\cdot P (\Delta_{m,v,z}+m-z)^{-1} \end{array}\right. & \\ &\hspace{-4cm} \left.\begin{array}{c} \displaystyle (\Delta_{m,v,z}+m-z)^{-1}\alpha^+\!\cdot P\frac{1}{m -v(Q) +z} \\ \displaystyle \frac{1}{m -v(Q) +z}\alpha^-\!\cdot P (\Delta_{m,v,z}+m-z)^{-1} \alpha^+\!\cdot P\frac{1}{m -v(Q) +z} -\frac{1}{m -v(Q) +z} \end{array}\right) \end{aligned}$$ The operator $(H^{\rm bd}_1-z)^{-1}$ is bounded from $L^2({\mathbb{R}}^3;{\mathbb{C}}^{2\nu})$ into ${\mathscr{H}}^1({\mathbb{R}}^3;{\mathbb{C}}^{2\nu})$. However, this improvement in the Sobolev scale does not hold if one looks at the matricial terms separately. There is a real compensation coming from the off-diagonal terms. First note that $\alpha^- \cdot P(\Delta_{m,v,z}+m-z)^{-1} \alpha^+\cdot P$ is a bounded operator in $L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_-)$ and a priori not into ${\mathscr{H}}^s({\mathbb{R}}^3; {\mathbb{C}}^\nu_-)$, with $s>0$. Indeed, $\alpha^+\cdot P$ sends $L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_-)$ into ${\mathscr{H}}^{-1}({\mathbb{R}}^3; {\mathbb{C}}^\nu_+) $, then $(\Delta_{m,v,z}+m-z)^{-1}$ to ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$ and the left $\alpha^-\cdot P$ sends again into $L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_-)$. On the other hand, the term $(\Delta_{m,v,z}+m-z)^{-1}$ is bounded from $L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$ into ${\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$, which is much better than expected. Let $f\in L^2({\mathbb{R}}^3;{\mathbb{C}}^{2\nu})$. By self-adjointness of $H^{\rm bd}_1$, there is a unique $\psi\in{\mathscr{H}}^1({\mathbb{R}}^3;{\mathbb{C}}^{2\nu})$ such that $(H^{\rm bd}_1-z)\psi=f$. We separate the upper and lower spin components and denote $f=(f_+, f_-)$ and $\psi=(\psi_+, \psi_-)$ in ${\mathscr{H}}={\mathscr{H}}^+\oplus {\mathscr{H}}^-$. We rewrite the equation $(D_m +V(Q)-z)\psi =f$ to get: $$\label{e:system} \begin{cases} \alpha^+\!\cdot P \psi_- + m \psi_+ + v(Q)\psi_+ -z\psi_+=f_+,\\ \alpha^-\!\cdot P \psi_+ - m \psi_- +v(Q)\psi_- -z\psi_-=f_-. \end{cases}$$ From the second line, we get $(v(Q)-m-z)\psi_-=f_- -\alpha^-\!\cdot P \psi_+$. Since $z$ is not real, we can take the inverse and infer $\psi_-=(v(Q)-m-z)^{-1}(f_- -\alpha^-\!\cdot P \psi_+)$. Since $\psi_- \in {\mathscr{H}}^{1}$, we can apply it $\alpha^+\!\cdot P$ and obtain a vector of $L^2({\mathbb{R}}^3;{\mathbb{C}}^\nu_+)$. Now, since $f_-$ is in $L^2({\mathbb{R}}^3;{\mathbb{C}}^\nu_-)$ and since $(v(Q)-m-z)^{-1}$ is bounded, we have $\alpha^+\!\cdot P (v(Q)-m-z)^{-1}f_-\in {\mathscr{H}}^{-1}({\mathbb{R}}^3;{\mathbb{C}}^\nu_+)$ and since $(v(Q)-m-z)^{-1}\alpha^-\cdot P \psi_+$ is in $L^2({\mathbb{R}}^3;{\mathbb{C}}^\nu_-)$, we rewrite the system: $$\begin{cases} \displaystyle \left(\alpha^+\!\cdot P\frac{1}{m -v(Q) +z}\,\alpha^-\!\cdot P +v(Q) + m-z\right)\psi_+= &\displaystyle f_+ +\alpha^+\!\cdot P\frac{1}{m -v(Q) +z}f_-,\\ \hfill\psi_-=&\displaystyle \frac{1}{m -v(Q) +z}\left(\alpha^-\!\cdot P \psi_+ -f_-. \right) \end{cases}$$ To conclude it remains to show that $\Delta_{m,v,z}+m-z$ is invertible in ${\mathcal{B}}({\mathscr{H}}^{1}, {\mathscr{H}}^{-1})$, so as to invert it in the system. Using , we have $\|\Im\langle u, (\Delta_{m,v,z}-z) u \rangle\| \geq c \\|u\\|_{{\mathscr{H}}^1}^2$. Then $\\|(\Delta_{m,v,z}+m-z) u\\|_{{\mathscr{H}}^{-1}}\geq c\\|u\\|_{{\mathscr{H}}^1} $ and $\\|(\Delta_{m,v,z}+m-z)^* u\\|_{{\mathscr{H}}^{-1}}\geq c\\|u\\|_{{\mathscr{H}}^1}$ hold. Thus, $\Delta_{m,v,z}-z$ is bijective from ${\mathscr{H}}^1$ onto ${\mathscr{H}}^{-1}$. From one limiting absorption principle to another {#s:LAPano} ------------------------------------------------- The main motivation for the operator $\Delta_{m,v,z}$ is to deduce a limiting absorption principle for $H^{\rm bd}_\gamma$ starting with one for $\Delta_{m,\gamma v,z}$. Consider the upper right term in Lemma \[l:inv\], the basic idea would be to put by force the weight $\langle Q\rangle^{-1}$ and to say that every terms are bounded. However, we have that $$\begin{aligned} \underbrace{\langle Q\rangle^{-1} (\Delta_{m,v,z}+m-z)^{-1} \langle Q \rangle^{-1}}_{{\rm bounded\, from\, LAP\, for\, } \Delta_{m,v,z}}\, \underbrace{\langle Q\rangle \alpha^+\cdot P \langle Q\rangle^{-1}}_{\rm unbounded}\, \frac{1}{m -v(Q) +z}. \end{aligned}$$ One needs to take advantage that one seeks an estimate on a bounded interval of the spectrum. Therefore, we start with a lemma of localization in the momentum space and elicit a solution in Lemma \[l:LAPchange\]. Note also that one may consider $\Im z <0$ by taking the adjoints in the two next lemmata. We shall also use estimates which are uniform in the coupling constant, due to Proposition \[p:collage\]. \[l:steptoLAP\] Set ${\mathcal{I}}\subset {\mathbb{R}}$ a compact interval. Let $V$ be a bounded potential and $\kappa>0$. There is an even function $\varphi\in{\mathcal{C}}^\infty_c({\mathbb{R}};{\mathbb{R}})$ such that the following estimations of the resolvent are equivalent: $$\label{e:LAPchange1} \sup_{\Re z\in {\mathcal{I}}, \Im z>0, \|\gamma\|\leq \kappa} \left\\|\langle Q\rangle^{-1} \varphi(\alpha\cdot P) (D_m+\gamma V(Q)-z)^{-1} \varphi(\alpha\cdot P) \langle Q\rangle^{-1}\right\\|<\infty,$$ $$\label{e:LAPchange1''} \sup_{\Re z\in {\mathcal{I}}, \Im z>0, \|\gamma\|\leq \kappa} \left\\|\langle Q\rangle^{-1}(D_m+\gamma V(Q)-z)^{-1}\varphi(\alpha\cdot P)\langle Q\rangle^{-1}\right\\|<\infty,$$ $$\label{e:LAPchange1'} \sup_{\Re z\in {\mathcal{I}}, \Im z>0, \|\gamma\|\leq \kappa} \left\\|\langle Q\rangle^{-1}(D_m+\gamma V(Q)-z)^{-1}\langle Q\rangle^{-1}\right\\|<\infty.$$ It is enough to consider $\Im z \in (0,1]$. Set ${\mathcal{J}}:={\mathcal{I}}\times(0,1]\times [-\kappa,\kappa]$, $H_\circ:=\alpha\cdot P$ and $H_\gamma :=D_m+\gamma V$. We choose $\varphi_1\in{\mathcal{C}}^\infty_c({\mathbb{R}})$ with value in $[0,1]$, being even and equal to $1$ in a neighborhood of $0$. We define $\varphi_R(\cdot):=\varphi_1(\cdot/R)$ and $\widetilde \varphi_R:=1- \varphi_R$. We first notice that $\langle Q\rangle \in {\mathcal{C}}^1(H_\circ)$, see Appendix \[s:dev-commut\]. There is a constant $C>0$ so that $$\begin{aligned} \label{e:kommu} \left\|\langle \langle Q\rangle f, \alpha\cdot P f \rangle - \langle \alpha \cdot P f, \langle Q\rangle f\rangle\right\| = \left\|\langle f, (\alpha\cdot \nabla \langle \cdot \rangle)(Q) f\rangle\right\|\leq C\\|f\\|^2 \end{aligned}$$ holds true for all $f \in {\mathcal{C}}^\infty_c ({\mathbb{R}}^3 ;{\mathbb{C}}^{2\nu})$. This is usually not enough to deduce the ${\mathcal{C}}^1$ property, see [@GeorgescuGerard]. We use [@GoleniaMoroianu]\[Lemma A.2\] with the notations $A:=H_\circ$, $H:=\langle Q\rangle$, $\cchi_n(x):=\varphi(x/n)$ and with ${\mathscr{D}}:={\mathcal{C}}^\infty_c ({\mathbb{R}}^3;{\mathbb{C}}^{2\nu})$. The hypotheses are fulfilled and we deduce that $\langle Q\rangle \in {\mathcal{C}}^1(H_\circ)$. By the resolvent equality, we have: $$\begin{aligned} \nonumber (H_\gamma-z)^{-1}\tilde \varphi_R(H_\circ)\big({{\rm{Id}} }+ W( H_\circ -z)^{-1}\tilde \varphi_R(H_\circ)\big)=&\, ( H_\circ-z)^{-1} \tilde \varphi_R(H_\circ) \\ \label{e:LAPchange3} &-(H_\gamma-z)^{-1} \varphi_R( H_\circ) W( H_\circ-z)^{-1} \tilde \varphi_R( H_\circ), \end{aligned}$$ where $W:=\gamma V+m\beta$. Note that the support of $\tilde \varphi_R$ vanishes as $R$ goes to infinity. We have $$\begin{aligned} \\|\langle Q\rangle ( H_\circ -z)^{-1}\tilde \varphi_R( H_\circ) \langle Q \rangle^{-1}\\|\leq {\mathcal{O}}(1/R), \mbox{ uniformly in } (z, \gamma)\in {\mathcal{J}}.\end{aligned}$$ Indeed, if we commute with $\langle Q\rangle$, the part in $( H_\circ -z)^{-1}\tilde \varphi_R( H_\circ)$ is a ${\mathcal{O}}(1/R)$ by functional calculus. For the other part, Lemma \[l:est3\] gives $$\begin{aligned} \\|[\langle Q\rangle, ( H_\circ -z)^{-1}\tilde \varphi_R( H_\circ)] \langle Q\rangle^{-1}\\|\leq {\mathcal{O}}(1/R^2), \mbox{ uniformly in } (z, \gamma)\in {\mathcal{J}}.\end{aligned}$$ Remembering $V$ is bounded and choosing $R$ big enough, we infer there is a constant $c\in (0,1)$, so that $$\begin{aligned} \label{e:LAPchangeagain} \\|W \langle Q\rangle(H_\circ-z)^{-1}\tilde\varphi_R( H_\circ)\langle Q\rangle^{-1}\\|\leq c, \mbox{ uniformly in } (z, \gamma)\in {\mathcal{J}}.\end{aligned}$$ We fix $R$ and choose $\varphi:=\varphi_R$. We now prove the equivalence. Observe that $\langle Q\rangle^{-1}\varphi(H_\circ)\langle Q\rangle$ is bounded, since $\langle Q\rangle \in {\mathcal{C}}^1(H_\circ)$. One infers directly that $\Rightarrow$ $\Rightarrow$ . It remains to prove $\Rightarrow$ . Thanks to , we deduce from that: $$\begin{aligned} \nonumber \langle Q\rangle^{-1}(H_\gamma-z)^{-1}\tilde \varphi(H_\circ)\langle Q\rangle^{-1}=&\, \Big(\langle Q\rangle^{-1}( H_\circ-z)^{-1} \tilde \varphi( H_\circ)\langle Q\rangle^{-1} \\ \label{e:LAPchange2} & -\langle Q\rangle^{-1}(H_\gamma-z)^{-1} \varphi( H_\circ) \langle Q\rangle^{-1} \quad W \langle Q \rangle ( H_\circ-z)^{-1} \tilde \varphi( H_\circ)\langle Q\rangle^{-1}\Big) \\ \nonumber &\times \big({{\rm{Id}} }+ W\langle Q\rangle(H_\circ -z)^{-1}\tilde \varphi(H_\circ) \langle Q\rangle^{-1}\big)^{-1}. \end{aligned}$$ Note that the last line and the right part of the second line of the r.h.s. are uniformly bounded in $(z, \gamma)\in {\mathcal{J}}$ by . We multiply on the left by the bounded operator $\langle Q\rangle^{-1}\varphi(H_\circ)\langle Q\rangle$. The first term of the r.h.s. is bounded uniformly by functional calculus. For the second one, we use . We infer: $$\sup_{(z, \gamma)\in {\mathcal{J}}} \left\\|\langle Q\rangle^{-1} \varphi (H_\circ) (H_\gamma-z)^{-1} \tilde \varphi (H_\circ) \langle Q\rangle^{-1}\right\\|<\infty.$$ Doing like in , on the left hand side, we get $$\label{e:LAPchange4} \sup_{(z, \gamma)\in {\mathcal{J}}} \left\\|\langle Q\rangle^{-1} \tilde \varphi(H_\circ) (H_\gamma-z)^{-1} \varphi(H_\circ) \langle Q\rangle^{-1}\right\\|<\infty.$$ Finally, to control $\langle Q\rangle^{-1} \tilde\varphi (H_\circ) (H_\gamma-z)^{-1} \tilde\varphi (H_\circ) \langle Q\rangle^{-1}$, we multiply on the left by the bounded operator $\langle Q\rangle^{-1}\tilde\varphi (H_\circ) \langle Q\rangle$ and deduce the boundedness using . \[l:LAPchange\] Take $\kappa\in (0,1]$ and a compact interval ${\mathcal{I}}\subset [0,\infty)$. Suppose (\[a:BoundV\]) and that $$\label{e:LAPchange0} \sup_{\Re z\in {\mathcal{I}},\Im z\in (0,1], \|\gamma\|\leq \kappa} \left\\|\langle Q\rangle^{-1}(\Delta_{m,\gamma v,z}+m-z)^{-1}\langle Q\rangle^{-1}\right\\|<\infty$$ hold true. Then, we have $$\label{e:LAPchange} \sup_{\Re z\in {\mathcal{I}}, \Im z>0, \|\gamma\|\leq \kappa} \left\\|\langle Q\rangle^{-1}(D_m+ \gamma v(Q)\otimes {{\rm{Id}} }_{{\mathbb{C}}^{2\nu}} -z)^{-1}\langle Q\rangle^{-1}\right\\|<\infty.$$ Set $H_\circ:=\alpha\cdot P$ and $ {\mathcal{J}}:={\mathcal{I}}\times (0,1]\times [-\kappa, \kappa]$. By Lemma \[l:steptoLAP\], it is enough to show for a chosen $\varphi$. Since $\varphi$ is even and constant in a neighborhood of $0$, by setting $\psi(\cdot):=\varphi(\sqrt{\|\cdot\|})$, we have $\psi\in{\mathcal{C}}^\infty_c({\mathbb{R}})$ and that $\varphi(H_\circ)= \psi\big((\alpha\cdot P)^2 \big)$. In particular, we obtain $\varphi(H_\circ)$ stabilizes ${\mathscr{H}}^{\pm}$ and have the right to let it appear in spin decomposition of the resolvent of $H$ of Lemma \[l:inv\]. We treat only the upper right corner of the expression as the others are managed in the same way. We need to bound the term: $$\begin{aligned} \langle Q\rangle^{-1} \varphi(H_\circ) (\Delta_{m,\gamma v,z}+m-z)^{-1}\alpha^+ \cdot P\frac{1}{m -\gamma v(Q) +z} \varphi(H_\circ)\langle Q\rangle^{-1} &= \\ &\hspace{-9cm}\langle Q\rangle^{-1} \varphi(H_\circ)\langle Q\rangle\quad \langle Q\rangle^{-1}(\Delta_{m,\gamma v,z}+m-z)^{-1}\langle Q\rangle^{-1} \quad \langle Q\rangle \alpha^+ \cdot P\frac{1}{m -\gamma v(Q) +z} \varphi(H_\circ)\langle Q\rangle^{-1}. \end{aligned}$$ The middle term is controlled by the hypothesis. Thanks to , one has that $[\varphi(H_\circ),\langle Q\rangle]$ is bounded; the first term is bounded. For the last one, we commute: $$\begin{aligned} \langle Q\rangle \alpha^+ \cdot P\frac{1}{m -\gamma v(Q) +z} \varphi(H_\circ)\langle Q\rangle^{-1} =& \langle Q\rangle \left[\alpha^+ \cdot P,\frac{1}{m -\gamma v(Q) +z}\right] \langle Q\rangle^{-1}\, \quad \langle Q\rangle \varphi(H_\circ)\langle Q\rangle^{-1} \\ &+ \langle Q\rangle\frac{1}{m -\gamma v(Q) +z} \langle Q\rangle^{-1}\, \quad \langle Q\rangle\alpha^+ \cdot P\varphi(H_\circ)\langle Q\rangle^{-1}. \end{aligned}$$ We estimate uniformly in $(z, \gamma)\in {\mathcal{J}}$. By , we get $\\|\langle Q\rangle (m-\gamma v(Q)+z)^{-1} \langle Q\rangle^{-1}\\|$ is bounded as $\langle Q\rangle$ commute with $v$. By , we also obtain that $\\|\langle Q\rangle [\alpha^+ \cdot P, (m-\gamma v+z)^{-1}]\langle Q\rangle^{-1}\\|$ is also controlled. At last, it is enough to consider $\langle Q\rangle \partial_j \varphi(H_\circ)\langle Q\rangle^{-1}$, which is easily bounded by Lemma \[l:est3\] for instance. We come to other types of weights. Motivated by the non-relativistic case, see Theorem \[t:nonrelamain\], we are interested in singular weights like $\|Q\|$. But, as noticed in Remark \[r:noLAP\], the operator $\|Q\|^{-1}(H^{\rm bd}-z)^{-1} \|Q\|^{-1}$ is even not bounded. Therefore, we enlarge the space in momentum and try the first reasonable weight, namely $\langle P\rangle^{1/2}\|Q\|$. Given $z\in {\mathbb{C}}\setminus {\mathbb{R}}$ and using the Hardy inequality, one reaches $$\begin{aligned} \\|\langle P\rangle^{-1}\|Q\|^{-1} ( H^{\rm bd}_\gamma -v -z)^{-1}\|Q\|^{-1}\\|&\leq \\| \langle P\rangle^{-1} \|P\|\, \\| \cdot \\|\, \|P\|^{-1} \|Q\|^{-1}\\|^2 \cdot \\|( H^{\rm bd}_\gamma -v -z)^{-1} \|P\|\,\\| \\ &\leq \, C(\kappa) \langle z \rangle / \|\Im(z)\|.\end{aligned}$$ By interpolation, one infers $$\begin{aligned} \\|\langle P\rangle^{-1/2}\|Q\|^{-1} ( H^{\rm bd}_\gamma -v -z)^{-1}\|Q\|^{-1}\langle P\rangle^{-1/2}\\| \leq C(\kappa) \langle z \rangle / \|\Im(z)\|<\infty.\end{aligned}$$ The upper bound seems relatively sharp in $z$. However, under the same hypotheses as before, we obtain: \[l:LAPchange2\] Take $\kappa\in (0,1]$ and a compact interval ${\mathcal{I}}\subset [0,\infty)$. Suppose (\[a:BoundV\]) and that hold true. Then, there is $C>0$ so that $$\label{e:LAPchange'} \sup_{\Re z\in {\mathcal{I}}, \Im z >0, \|\gamma\|\leq \kappa} \|\langle f, (D_m+ \gamma v(Q)\otimes {{\rm{Id}} }_{{\mathbb{C}}^{2\nu}} -z)^{-1} f \rangle\|\leq C \\| \langle P\rangle^{1/2}\|Q\| f\\|^2.$$ It is enough to consider $\Im z \in (0,1]$. Set $H_\gamma :=D_m+\gamma v(Q)\otimes {{\rm{Id}} }_{{\mathbb{C}}^{2\nu}}$ and ${\mathcal{J}}:={\mathcal{I}}\times(0,1]\times [-\kappa,\kappa]$. Let $f=(f_+,f_-)$, with $f_{\pm}\in {\mathcal{C}}^\infty_c({\mathbb{R}}^3\setminus\{0\}; {\mathbb{C}}^{\nu}_{\pm})$. Lemma \[l:inv\] and give a constant $C>0$, uniform in $(z, \gamma)\in {\mathcal{J}}$, so that: $$\begin{aligned} \left\|\left\langle f, (H_\gamma -z)^{-1}f\right\rangle\right\|\leq& \,4/m^2 \\|f_-\\|^2 + 2 \left\\|\|Q\|^{-1}(\Delta_{m,\gamma v,z}+m-z)^{-1}\| Q\|^{-1}\right\\| \times \\ &\hspace{-2cm}\left(\big\\|\|Q\|f_+\big\\|^2+\big\\|\|Q\|\alpha^+\cdot P (m-v(Q)+z)^{-1}f_-\big\\|^2+\big\\|\|Q\|\alpha^+\cdot P(m-v(Q)+\overline{z})^{-1}f_-\big\\|^2\right) \\ \leq& \,C \left(\big\\|\|Q\|f_+\big\\|^2 + \big\\|\|Q\|\alpha^+\cdot P f_-\big\\|^2 + \\|f_-\\|^2 \right). \end{aligned}$$ Note that the Hardy inequality gives that $\\|f_-\\|\leq 2 \big\\| \|Q\| \alpha^+\cdot P f_- \big\\|$. Then, by commuting $\|Q\|$ with $\alpha^+\cdot P$ over ${\mathcal{C}}^\infty_c({\mathbb{R}}^3\setminus\{0\}; {\mathbb{C}}^\nu)$, we find $C'>0$, so that: $$\begin{aligned} \sup_{(z, \gamma)\in {\mathcal{J}}}\left\|\left\langle f, \|Q\|^{-1}(H_\gamma -z)^{-1} \|Q\|^{-1}f\right\rangle\right\|\leq& C' \left(\big\\|{{\rm{Id}} }\otimes P_+\, f\big\\|^2 + \big\\|\langle P\rangle \otimes P_-\, f\big\\|^2\right), \end{aligned}$$ for all $f\in {\mathcal{C}}^\infty_c({\mathbb{R}}^3\setminus\{0\}; {\mathbb{C}}^{2\nu})$. Here we identify, $L^2({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})\simeq L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^{2\nu}$. We now exchange the role of $P_+$ and $P_-$. Considering the operator $$\alpha^-\!\cdot P\frac{1}{m +v(Q) +z}\alpha^+\!\cdot P -v(Q) \mbox{ in } L^2({\mathbb{R}}^3; {\mathbb{C}}^{\nu}_-),$$ which leads to the same arguments as for $\Delta_{m,-v,z}$ if one identifies ${\mathbb{C}}^{\nu}_-\simeq {\mathbb{C}}^{\nu}_+$, one obtains also that $$\begin{aligned} \left\|\left\langle f, \|Q\|^{-1}(H_\gamma -z)^{-1}\|Q\|^{-1} f\right\rangle\right\|\leq& C' \left(\big\\|{{\rm{Id}} }\otimes P_-\, f\big\\|^2 + \big\\|\langle P\rangle \otimes P_+\, f\big\\|^2\right), \end{aligned}$$ for all $f\in {\mathcal{C}}^\infty_c({\mathbb{R}}^3\setminus\{0\}; {\mathbb{C}}^{2\nu})$. By interpolation, e.g., [@BerghLofstrom]\[Theorem 4.4.1 and Theorem 6.4.5.(7)\], we infer: $$\begin{aligned} \left\|\left\langle f, \|Q\|^{-1}(H_\gamma -z)^{-1} \|Q\|^{-1}f\right\rangle\right\|\leq& C'' \big\\|\langle P\rangle^{1/2} f\big\\|^2, \end{aligned}$$ for all $f\in {\mathcal{C}}^\infty_c({\mathbb{R}}^3\setminus\{0\}; {\mathbb{C}}^{2\nu})$. The latter is a core in ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$. Positive commutator estimates. {#s:positive} ============================== In the previous section, we saw how to deduce some estimate of the resolvent for $D_m+V(Q)$ starting with some of $\Delta_{m,v,z}$, namely . First, one technical problem is that operator depends on the spectral parameter; hence we will study a family of operators uniformly in the spectral parameter. Secondly, we are concerned about the interval $[m, m+\delta]$ and we know that there is no such estimate above $(m- \varepsilon , m)$ as eigenvalues usually accumulates to $m$ from below. Since the theory developed in Appendix \[s:WeakMourreTheory\] gives some estimates for $\Re z\in [0, \infty)$, we will perform a shift. Therefore, we study the operator $$\begin{aligned} \Delta_{2m,\gamma v,\xi}, \mbox{ uniformly in } (\gamma, \xi)\in {\mathcal{E}}= {\mathcal{E}}(\kappa, \delta):= [-\kappa,\kappa]\times [0,\delta]\times(0,1].\end{aligned}$$ Here we use a slight abuse of notation identifying ${\mathbb{C}}\simeq {\mathbb{R}}^2$. One should read $\Re\xi\in [0, \delta], \Im\xi\in(0, 1]$ and $\|\gamma\|\leq \kappa$. Note the uniformity in the coupling constant is used in Proposition \[p:collage\]. To show , and therefore with the help of Lemma \[l:steptoLAP\], it is enough to prove the following fact. Note we strengthen the hypothesis . \[t:mourreD\] Suppose that $v\in L^\infty({\mathbb{R}}^3; {\mathbb{R}})$ satisfies the hypothesis (H1) and (H2) from Theorem \[t:vraimain\]. Then there are $\delta, \kappa, C_{\rm LAP} >0$ such that $$\begin{aligned} \label{e:mourreD} \sup_{\Re z\geq 0,\Im z>0, (\gamma, \xi)\in {\mathcal{E}}} \|\langle f, (\Delta_{2m,\gamma v,\xi}-z)^{-1} f\rangle \| \leq C_{\rm LAP} \big\\|\, \|Q\| f \big\\|^2.\end{aligned}$$ We will show the theorem in the end of the section. We proceed by checking the hypothesis of Appendix \[s:WeakMourreTheory\]. We recall and fix some notation: $$\begin{aligned} S:= \Delta_{1,0,0}= \alpha^+ \cdot P\, \alpha^- \cdot P = -\Delta_{{\mathbb{R}}^3} \otimes {{\rm{Id}} }_{{\mathbb{C}}^\nu_+} \mbox{ in } {\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+) \simeq {\mathscr{H}}^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^\nu_+\end{aligned}$$ and set ${\mathscr{S}}:=\dot{\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$, the homogeneous Sobolev space of order $1$, i.e., the completion of ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$ under the norm $\\|f\\|_{\mathscr{S}}:= \\| S^{1/2} f\\|^2$. Consider the strongly continuous one-parameter unitary group $\{W_t\}_{t\in {\mathbb{R}}}$ acting by: $$\begin{aligned} (W_t f)(x)= e^{3t/2} f(e^tx), \mbox{ for all } f\in L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+).\end{aligned}$$ This is the $C_0$-group of dilatation. Easily, by interpolation and duality, one gets $$\begin{aligned} W_t {\mathscr{S}}\subset {\mathscr{S}}\mbox{ and } W_t {\mathscr{H}}^s({\mathbb{R}}^3; {\mathbb{C}}^\nu_+) \subset {\mathscr{H}}^s({\mathbb{R}}^3; {\mathbb{C}}^\nu_+), \mbox{ for all } s\in {\mathbb{R}}.\end{aligned}$$ Consider now its generator $A$ in $L^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$. By the Nelson lemma, it is essentially self-adjoint on ${\mathcal{C}}^\infty_c({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$. It acts as follows: $$\begin{aligned} \displaystyle A=\frac{1}{2}(P\cdot Q+ Q\cdot P)\otimes {{\rm{Id}} }_{{\mathbb{C}}^\nu_+} \mbox{ on } {\mathcal{C}}^\infty_c({\mathbb{R}}^3; {\mathbb{C}}^\nu_+) \simeq {\mathcal{C}}^\infty_c({\mathbb{R}}^3)\otimes {\mathbb{C}}^\nu_+. \end{aligned}$$ In the next Proposition, we will choose the upper bound $\kappa$ of the coupling constant and state the commutator estimates. \[p:commu1\] Let $\delta\in (0, 2m)$. Suppose that the hypotheses (H1) and (H2) are fulfilled. Then there are $c_1, \kappa>0$ such that $$\begin{aligned} \label{e:commu10} {\mathcal{D}}(\Delta_{2m,\gamma v,\xi}) = {\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^\nu_+), \quad (\Delta_{2m,\gamma v,\xi})^* = \Delta_{2m,\gamma v, \overline{\xi}}, \\ \label{e:commu11} [\Re(\Delta_{2m,\gamma v,\xi}),\rmi A]_\circ- c_v\Re(\Delta_{2m, \gamma v,\xi}) \geq c_1 S>0, \\ \mp[\Im (\Delta_{2m, \gamma v, \Re(\xi) \pm\rmi \Im(\xi)}) , \rmi A]_\circ\geq 0,\end{aligned}$$ hold true in the sense of forms on ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$, for all $(\gamma, \xi)\in {\mathcal{E}}$. We start with a first restriction on $\kappa$. We set $\kappa\leq (2m-\delta) / \\|v\\|_\infty$. Hence, $$\begin{aligned} \label{e:bd} \delta \leq 2m - \gamma v(\cdot) + \Re(\xi) \leq 4m, \mbox{ for all } (\gamma, \xi)\in {\mathcal{E}}.\end{aligned}$$ In particular, $0$ is not in the essential image of $2m- \gamma v +\Re(\xi)$; Proposition \[p:domain\] gives . We turn to the commutator estimates. It is enough to compute the commutators in the sense of form on ${\mathcal{C}}^\infty_c({\mathbb{R}}^3; {\mathbb{C}}^\nu)$, since it is a core for $\Delta_{2m,v,\xi}$ and $A$. $$\begin{aligned} \nonumber \big[\Delta_{2m,\gamma v,\xi},\rmi A\big] =& \left[\alpha^+ \cdot P\, \frac{1}{2m -\gamma v +\xi}\alpha^- \cdot P,\rmi A\right] +\gamma [v,\rmi A]\\ \label{e:firstcommu} =&\, 2\,\alpha^+\cdot P\, \frac{1}{2m -\gamma v +\xi}\alpha^- \cdot P -\gamma\, \alpha^+\cdot P\, \frac{Q\cdot \nabla v(Q)}{(2m - \gamma v +\xi)^2}\alpha^-\cdot P - \gamma\, Q\cdot\nabla v(Q).\end{aligned}$$ Then, we have $[\Re(\Delta_{2m,\gamma v,\xi}), \rmi A]- c_v\Re(\Delta_{2m, \gamma v,\xi})=$ $$\begin{aligned} =&\,(2-c_v)\, \alpha^+\cdot P\frac{2m -\gamma v+\Re(\xi)}{\big(2m -\gamma v+\Re(\xi)\big)^2+\Im(\xi)^2}\alpha^-\cdot P \\ & -\gamma\,\alpha^+\cdot P\left(\frac{Q\cdot \nabla v(Q)\big(\big(2m -\gamma v+\Re(\xi)\big)^2-\Im(\xi)^2\big)}{\big(\big(2m - \gamma v+\Re(\xi)\big)^2+\Im(\xi)^2\big)^2 }\right) \alpha^-\cdot P - \gamma\, Q\cdot\nabla v(Q) - c_v \gamma v(Q). \\ \geq & (2-c_v)\frac{\delta}{16 m^2+1} S - \kappa\, \\|Q\cdot \nabla v(Q)\\|\, \frac{ 16 m^2 +1}{ \delta^4} S -\kappa \frac{c_v'}{\|Q\|^2} \geq c_1 S,\end{aligned}$$ where $\displaystyle c_1:= \frac{\delta (2-c_v) }{32 m^2+2}$ and by assuming that $\displaystyle \kappa \leq \frac{c_1}{2 (4 c_v'+ \\|Q\cdot \nabla v(Q)\\| ( 16 m^2 +1)/\delta^4)}$. Note the “$4$” comes from the Hardy inequality. This gives . At last, we have: $$\begin{aligned} [\Im\Delta_{2m, \gamma v,\xi}, \rmi A] &= -2 \Im(\xi)\, \alpha^+\cdot P \frac{\big(2m-\gamma v+\Re(\xi)\big)^2+\Im(\xi)^2 - \gamma Q\cdot \nabla v(Q)\big(2m-\gamma v+\Re(\xi)\big) } {\big(\big(2m-\gamma v+\Re(\xi)\big)^2+\Im(\xi)^2\big)^2} \alpha^- \cdot P.\end{aligned}$$ This is of the sign of $-\Im(\xi)$, when we further impose $\kappa \leq \delta^2/(8 m \\|Q\cdot \nabla v(Q)\\|)$. We now bound some commutators. \[p:commu2\] Let $\delta\in (0, 2m)$. Suppose that the hypotheses (H1) and (H2) are fulfilled. Consider the $c_1, \kappa>0$ from Proposition \[p:commu1\]. There is $c$ and $C$ depending on $c_v, \delta, \kappa$ and $v$, such that $$\begin{aligned} \label{e:mourrestrict0chk} \|\langle \Delta_{2m,\gamma v,\overline{\xi}}\, f, Ag \rangle - \langle Af, \Delta_{2m,\gamma v,\xi}\, g \rangle \|\leq c\\|f\\|\cdot\\|(\Delta_{2m,\gamma v,\xi} \pm i) g\\|, \end{aligned}$$ holds true, for all $f,g \in {\mathscr{H}}^2({\mathbb{R}}^3; {\mathbb{C}}^+_\nu)\cap {\mathcal{D}}(A)$ and $$\begin{aligned} \label{e:commu21} \|\langle f, [[\Delta_{2m,\gamma v,\xi},\rmi A]_\circ, \rmi A]_\circ f\rangle\| \leq C \langle f, S f\rangle. \end{aligned}$$ holds true for all $f\in{\mathscr{H}}^{1}({\mathbb{R}}^3; {\mathbb{C}}^\nu_+)$. We take $\kappa$ as in the proof of Proposition \[p:commu1\]. We first find $c>0$, uniform in $(\gamma, \xi)\in {\mathcal{E}}$, so that $$\begin{aligned} \label{e:commuS} \|\langle f, [ \Delta_{2m, \gamma v, \xi }, iA ]_\circ g\rangle \| \leq c \big(\\| f \\|\cdot\\|g\\| + \\|f\\|\cdot \\|Sg\\|\big), \mbox{ for all } f,g \in {\mathcal{C}}^\infty_c({\mathbb{R}}^3; {\mathbb{C}}^\nu_+).\end{aligned}$$ We recall that the latter space is a core for $A$, $S$ and $\Delta_{2m, \gamma v, \xi }$. Taking in account , observe that $$\begin{aligned} \left\|\frac{2}{2m -\gamma v +\xi} -\gamma \frac{Q\cdot \nabla v(Q)}{(2m - \gamma v +\xi)^2}\right\|\leq \frac{2}{\delta} + \kappa \frac{\\|Q\cdot \nabla v(Q)\\|}{\delta^2}. \end{aligned}$$ It remains to find $a,b>0$, which are uniform in $(\gamma, \xi)\in {\mathcal{E}}$, such that the following estimation holds: $$\begin{aligned} \\|\Delta_{2m, \gamma v, \xi } f\\|\geq a \\|S f\\| - b\\|f\\|, \mbox{ for all } f\in {\mathcal{C}}^\infty_c({\mathbb{R}}^3, {\mathbb{C}}^\nu_+).\end{aligned}$$ This follows from $\\|\Delta_{2m, \gamma v, \xi } f\\|^2\geq a^2 \\|S f\\|^2 - b^2\\|f\\|^2$. Take $\varepsilon, \varepsilon' \in (0,1)$. $$\begin{aligned} \\|\Delta_{2m,v,z}f\\|^2 &\geq (1-\varepsilon)\left\\|\frac{1}{2m-\gamma v(Q)+\xi} S f\right\\|^2 + \left(1- \frac{1}{\varepsilon}\right) \left\\|\frac{\gamma (\alpha^+\cdot \nabla v)(Q)}{(2m-\gamma v(Q)+\xi)^2} \alpha^- \cdot P f\right\\|^2. \\ &\geq (1- \varepsilon )\frac{1}{1+ 16m^2} \left\\|S f\right\\|^2 + \left(1- \frac{1}{\varepsilon}\right) \frac{ \kappa \\|\alpha^+\cdot \nabla v(Q) \\| }{\delta^4} \left\\|\alpha^- \cdot P f\right\\|^2, \\ &\hspace{-2cm}\geq \left((1- \varepsilon )\frac{1}{1+ 16m^2} + \varepsilon' (\varepsilon-1) \frac{ \kappa \\|\alpha^+\cdot \nabla v(Q) \\| }{2 \varepsilon \delta^4}\right) \left\\|S f\right\\|^2 + (\varepsilon- 1) \frac{ \kappa \\|\alpha^+\cdot \nabla v(Q) \\| }{2 \varepsilon\varepsilon' \delta^4} \left\\|f\right\\|^2. \end{aligned}$$ Choosing $ \varepsilon'$ small enough, we infer . We turn to . Again, it is enough to compute in the form sense on ${\mathcal{C}}^\infty_c({\mathbb{R}}^3;{\mathbb{C}}^\nu)$. $$\begin{aligned} [[\Delta_{2m,v,z},\rmi A],\rmi A] =&\, 4\, \alpha^+\cdot P\frac{1}{2m -v +z}\alpha^-\cdot P +4\, \alpha^+\cdot P\frac{Q\cdot \nabla v(Q)}{(2m -v +z)^2}\alpha^-\cdot P\\ &- \alpha^+\cdot P\frac{Q\cdot \nabla\big(Q\cdot \nabla v(Q)\big)}{(2m -v +z)^2}\alpha^-\cdot P+ 2\, \alpha^+\cdot P\frac{(Q\cdot \nabla v(Q))^2}{(2m -v +z)^3}\alpha^-\cdot P + (Q\cdot \nabla)^2 v(Q).\end{aligned}$$ Note that (H1) ensures that $\\|(Q\cdot \nabla)^2 v(Q) f\\|^2\leq 4 \\|\, \|Q\|(Q\cdot \nabla)^2 v(Q)\\|^2 \\|S f\\|^2$ is controlled by $S$. Relying again on , the bound follows. We finally turn to the proof of the main result of this section. Since we have $e^{itA}{\mathscr{H}}^2\subset {\mathscr{H}}^2$ and , we obtain that $\Delta_{2m,\gamma v,\xi}\in{\mathcal{C}}^1(A, {\mathscr{H}}^2, {\mathscr{H}})$, for all $(\gamma, \xi)\in {\mathcal{E}}$. By interpolation, we deduce that $\Delta_{2m,\gamma v,\xi}\in{\mathcal{C}}^1(A, {\mathscr{H}}^1, {\mathscr{H}}^{-1})$. Now taking in account , we infer $\Delta_{2m,\gamma v,\xi}\in{\mathcal{C}}^2(A, {\mathscr{H}}^1, {\mathscr{H}}^{-1})$, for all $(\gamma, \xi)\in {\mathcal{E}}$. Using Propositions \[p:commu1\] and \[p:commu2\], we can apply Theorem \[t:mourrestrict\]. We derive there is a finite $C'$ so that $$\begin{aligned} \sup_{\Re z\geq 0,\Im z>0, (\gamma, \xi)\in {\mathcal{E}}} \|\langle f, (\Delta_{2m,\gamma v,\xi}-z)^{-1} f\rangle \| \leq C' \left(\\|S^{-1/2} f\\|^2 + \\|S^{-1/2} A f\\|^2\right).\end{aligned}$$ The Hardy inequality concludes. Main result {#s:mainresult} =========== In this section, we will prove the main result of this paper and deduce Theorem \[t:main\]. \[t:vraimain\] Let $\gamma\in {\mathbb{R}}$. Suppose that $v\in L^\infty({\mathbb{R}}^3; {\mathbb{R}})$ satisfies the hypothesis: 1. $\\|v\\|_\infty\leq m/2$ and $\nabla v$, $ Q\cdot \nabla v(Q)$, $\langle Q\rangle (Q\cdot \nabla v)^2(Q)$ are bounded. 2. There are $c_v\in [0,2)$ and $c_v'\geq 0$ such that $$\begin{aligned} x\cdot (\nabla v)(x) + c_v v(x) \leq \frac{c_v'}{\|x\|^2}, \mbox{ for all } x\in {\mathbb{R}}^3\setminus\{0\}.\end{aligned}$$ Set $V_1(Q):= v(Q)\otimes {{\rm{Id}} }_{{\mathbb{C}}^{2n}}$, where $L^2({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})\simeq L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^{2\nu}$. 1. Consider $V_2\in L^1_{\rm loc}({\mathbb{R}}^3; {\mathbb{R}}^{2\nu})$ satisfying: $$\begin{aligned} \langle Q\rangle^2 V_2(Q) \in {\mathcal{B}}\big({\mathscr{H}}^{1}({\mathbb{R}}^3; {\mathbb{C}}^{2\nu}), L^2 ({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})\big). \end{aligned}$$ Then, there are $\kappa, \delta, C>0$, such that $H_\gamma:= D_m + \gamma V(Q)$, where $V:=V_1 + V_2$, is self-adjoint with domain ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$. Moreover, $$\begin{aligned} \label{e:vraimain} \sup_{\|\lambda\|\in [m, m+\delta], \, \varepsilon>0, \|\gamma\|\leq \kappa }\\|\langle Q\rangle^{-1}(H_\gamma -\lambda - \rmi \varepsilon )^{-1} \langle Q\rangle^{-1}\\|\leq C. \end{aligned}$$ In particular, $H_\gamma$ has no eigenvalue in $\pm m$. Moreover, there is $C'$ so that $$\begin{aligned} \label{e:vraiKato} \sup_{\|\gamma\|\leq \kappa}\int_{\mathbb{R}}\\| \langle Q \rangle^{-1} e^{-it H_\gamma } E_{\mathcal{I}}(H_\gamma) f \\|^2 dt \leq C'\\|f\\|^2,\end{aligned}$$ where ${\mathcal{I}}=[-m-\delta, -m]\cup [m, m+\delta]$ and where $E_{\mathcal{I}}(H_\gamma)$ denotes the spectral measure of $H_\gamma$. In [@FournaisSkibsted] and in [@Richard], one takes advantage that the Virial of the potential is negative, in order to prove the limiting absorption principle for some self-adjoint Schrödinger operators, see Remark \[r:virial\]. Here, we cannot allow this hypothesis as we are also interested in positronic threshold, i.e., we seek a result for $v$ and $-v$, see . We recover the positivity using some Hardy inequality and small coupling constants. First note that is a consequence of , see [@Kato]. Consider the case $V_2=0$. Note that, in Section \[s:red\], the operator $H_\gamma$ is denoted by $H^{\rm}_\gamma$. The self-adjointness is clear. We first apply Theorem \[t:mourreD\] and obtain . By choosing $\xi=z$ and as $\\| \|Q\| f\\| \leq \\| \langle Q\rangle f\\|$, we infer . In turn, it implies . Finally, using the unitary transformation $\alpha_5$, follows from . For a general $V_2$, we use Proposition \[p:collage\]. It remains to explains how to add the singular part $V_2$ of the potential by perturbing the limiting absorption principal. This is somehow standard. Note that unlike [@JensenNenciu], for instance, we do not distinguish the nature of the singularity at the threshold energy, as we work with small coupling constants. \[p:collage\] Assume that Theorem \[t:vraimain\] holds true for $V_2=0$. Take now $V_2$ satisfying (H3). Then there is $\kappa' \in (0, \kappa]$, so that $$\begin{aligned} H_\gamma:= D_m + \gamma (V + V_2)(Q)\end{aligned}$$ is self-adjoint with domain ${\mathscr{H}}^1({\mathbb{R}}^3; {\mathbb{C}}^{2\nu})$, for all $\|\gamma\|\leq \kappa$. Moreover, $$\begin{aligned} \sup_{\Re z\in [m, m+\delta], \Im z>0, \|\gamma\|\leq \kappa'} \left\\|\langle Q\rangle^{-1}(H_\gamma -z)^{-1}\langle Q\rangle^{-1}\right\\|<\infty. \end{aligned}$$ Up to a smaller $\kappa$, Kato-Rellich ensures the self-adjointness. We turn to the estimate of the resolvent. Easily, one reduces to the case $\|\Im(z)\|\leq 1$. From the resolvent identity, we have: $$\begin{aligned} \nonumber \langle Q\rangle^{-1} (H_\gamma -z)^{-1}\langle Q\rangle^{-1} \quad \langle Q\rangle \left\{ I+\gamma V_2(H_\gamma^{\rm bd} -z)^{-1}\right\}\langle Q\rangle^{-1} =&\, \langle Q\rangle^{-1} (H_\gamma^{\rm bd} -z)^{-1} \langle Q\rangle^{-1}\end{aligned}$$ Considering Lemma \[l:LAPchange\] and Theorem \[t:mourreD\], the result follows if we can invert the second term of the l.h.s. uniformly in the parameters. Therefore, we show there is $\kappa'\in (0,\kappa]$ so that $$\begin{aligned} \sup_{\Re(z)\in [m, m+\delta], \Im(z)\in (0,1], \|\gamma\|\leq \kappa'} \\|\langle Q\rangle \gamma V_2(H_\gamma^{\rm bd} -z)^{-1} \langle Q\rangle^{-1}\\| <1. \end{aligned}$$ Using the identity of the resolvent, we get $$\begin{aligned} \langle Q\rangle V_2(H_\gamma^{\rm bd} -z)^{-1}\langle Q\rangle^{-1} =& \langle Q\rangle V_2 (H_0^{\rm bd} -i)^{-1} \quad \langle Q \rangle^{-1} \\ &- \langle Q\rangle V_2 (H_0^{\rm bd} -i )^{-1}\langle Q\rangle \quad (\gamma V -z+i) \quad \langle Q\rangle^{-1} (H_\gamma^{\rm bd} -z )^{-1} \langle Q\rangle^{-1}. \end{aligned}$$ The first term of the r.h.s. is bounded by using (H3). To control the last term, remember that $z$ is bounded and use again Lemma \[l:LAPchange\] and Theorem \[t:mourreD\]. It remains to notice that $$\langle Q\rangle V_2 (H_0^{\rm bd} -i )^{-1}\langle Q\rangle=\langle Q\rangle^2 V_2 (H_0^{\rm bd} -i )^{-1}-\langle Q\rangle V_2 (H_0^{\rm bd} -i )^{-1} \quad [H_0^{\rm bd},\langle Q\rangle]_\circ(H_0^{\rm bd} -i )^{-1}$$ is bounded. Indeed, the assumption (H3) controls the terms in $V_2$ and $\langle Q\rangle\in {\mathcal{C}}^1(H_0)$ and $[H_0^{\rm bd},\langle Q\rangle]_\circ$ is bounded, see proof of Lemma \[l:steptoLAP\]. At last, Theorem \[t:main\] is an immediate corollary of Theorem \[t:vraimain\]. Indeed, one has: \[ex:multi\] For $i=1, \ldots, n$, we choose $a_i\in {\mathbb{R}}^3$ the site of the poles and $Z_i\in {\mathbb{R}}$ its charge. We set: $$\begin{aligned} v_c := \sum_{i=1}^n \frac{z_i}{\|\cdot- a_i\|}\end{aligned}$$ Note that $$\begin{aligned} Q\cdot \nabla v_c (Q) + v_c:= \sum_{i=1}^n a_i \cdot \nabla v(Q).\end{aligned}$$ Choose now $\varphi\in{\mathcal{C}}^\infty_c({\mathbb{R}}^3)$ radial with values in $[0,1]$. Moreover, we ask that $\varphi$ restricted to the ball $B(0, \max(\|a_i\|))$ is $1$. Consider the support large enough, so that $\\|\tilde \varphi v\\|_\infty \leq m/2$, where $\tilde \varphi:\ 1-\varphi$. Set $v:= \tilde \varphi v_c$. Straightforwardly, the hypothesis (H1) and (H2) are satisfied. Note that (H3) follows from the Hardy inequality. In [@Herbst], one considers smooth potential independent of $\|x\|$ of the form $v(x):= \tilde v(\|x\|/x)$, with $v\in{\mathcal{C}}^\infty(S^{2})$, see also Remark \[r:Herbst\]. Here, by taking $c_v=0$ in Theorems \[t:vraimain\] and \[t:vraimain2\], one obtains a relativistic equivalent of this result. We point out that this perturbation is not relatively compact with respect to the Dirac operator. We now discuss singular weights in $\|Q\|$. \[r:noLAP\] It is important to note that unlike in the non-relativistic case, see Theorem \[t:nonrelamain\], one cannot replace the weights $\langle Q\rangle$ in by $\|Q\|$. Indeed, with the notation of Theorem \[t:vraimain\], $V_2=0$ and $z\in {\mathbb{C}}$, consider a function $f$ in ${\mathcal{C}}^\infty_c({\mathbb{R}}^3\setminus \{0\};{\mathbb{C}}^{2\nu})$ and notice the expression $R^{-3/2}\big\\|\,\|Q\|(H_\gamma-z) \|Q\| f(\cdot/R)\big\\|_2$ tends to $0$, as $R$ goes to $0$. Therefore, there is no $z\in{\mathbb{C}}$ such that the operator $\|Q\|(H_\gamma -z) \|Q\|$ has a bounded inverse. We finally give a second result with a weight allowing some singularity in $\|Q\|$. Using Lemma \[l:LAPchange2\] instead of Lemma \[l:LAPchange\] in the proof of Theorem \[t:vraimain\], we infer straightforwardly: \[t:vraimain2\] Let $\gamma\in {\mathbb{R}}$ and take $v\in L^\infty({\mathbb{R}}^3; {\mathbb{R}})$ satisfying (H1) and (H2). Then, there are $\kappa, \delta, C>0$, such that $H_\gamma:= D_m + \gamma v(Q)\otimes {{\rm{Id}} }_{{\mathbb{C}}^{2\nu}}$ satisfies $$\begin{aligned} \label{e:vraimain2} \sup_{\|\lambda\|\in [m, m+\delta], \, \varepsilon>0, \|\gamma\|\leq \kappa }\\| \langle P\rangle^{-1/2} \|Q\|^{-1}(H_\gamma -\lambda - \rmi \varepsilon )^{-1} \|Q\|^{-1}\langle P\rangle^{-1/2}\\|\leq C. \end{aligned}$$ Moreover, there is $C'$ so that $$\begin{aligned} \label{e:vraiKato2} \sup_{\|\gamma\|\leq \kappa}\int_{\mathbb{R}}\\| \langle P\rangle^{-1/2} \|Q\|^{-1} e^{-it H_\gamma } E_{\mathcal{I}}(H_\gamma) f \\|^2 dt \leq C'\\|f\\|^2,\end{aligned}$$ where ${\mathcal{I}}=[-m-\delta, -m]\cup [m, m+\delta]$ and where $E_{\mathcal{I}}(H_\gamma)$ denotes the spectral measure of $H_\gamma$. Keeping in mind Proposition \[p:collage\], one sees that one can only add trivial potentials in the perturbation theory of the limiting absorption principle. Thence, it is an open question whether one can cover the example \[ex:multi\] with the weights $\langle P\rangle^{1/2}\|Q\|$. Commutator expansions. {#s:dev-commut} ====================== This section is a small improvement of [@GoleniaJecko]\[Appendix B\], see also [@DerezinskiGerard; @HunzikerSigalSoffer]. We start with some generalities. Given a bounded operator $B$ and a self-adjoint operator $A$ acting in a Hilbert space ${\mathscr{H}}$, one says that $B\in {\mathcal{C}}^k(A)$ if $t\mapsto e^{-itA}B e^{itA}$ is strongly ${\mathcal{C}}^k$. Given a self-adjoint operator $B$, one says that $B\in {\mathcal{C}}^k(A)$ if for some (hence any) $z\notin \sigma(B)$, $t\mapsto e^{-itA}(B-z)^{-1} e^{itA}$ is strongly ${\mathcal{C}}^k$. The two definitions coincide in the case of a bounded self-adjoint operator. We recall a result following from Lemma 6.2.9 and Theorem 6.2.10 of [@AmreinBoutetdeMonvelGeorgescu]. \[th:abg\] Let $A$ and $B$ be two self-adjoint operators in the Hilbert space ${\mathscr{H}}$. For $z\notin \sigma(A)$, set $R(z):=(B-z)^{-1}$. The following points are equivalent to $B\in{\mathcal{C}}^1(A)$: 1. For one (then for all) $z\notin \sigma(B)$, there is a finite $c$ such that $$\begin{aligned} \label{e:C1a} \|\langle A f, R(z) f\rangle - \langle R(\overline{z}) f, Af\rangle\| \leq c \\|f\\|^2, \mbox{ for all $f\in{\mathcal{D}}(A)$}. \end{aligned}$$ 2. 1. There is a finite $c$ such that for all $f\in {\mathcal{D}}(A)\cap{\mathcal{D}}(B)$: $$\label{e:C1b} \|\langle Af, B f\rangle- \langle B f, Af\rangle\|\leq \, c\big(\\|B f\\|^2+\\|f\\|^2\big).$$ 2. For some (then for all) $z\notin \sigma(B)$, the set $\{f\in{\mathcal{D}}(A) \mid R(z)f\in{\mathcal{D}}(A)$ and $R(\overline{z})f\in{\mathcal{D}}(A) \}$ is a core for $A$. Note that the condition (3.b) could be uneasy to check, see [@GeorgescuGerard]. We mention [@GoleniaMoroianu]\[Lemma A.2\] and [@GerardLaba]\[Lemma 3.2.2\] to overcome this subtlety. As $(B+i)^{-1}$ is a homeomorphism between ${\mathscr{H}}$ onto ${\mathcal{D}}(B)$, $(B+i)^{-1} {\mathcal{D}}(A)$ is dense in ${\mathcal{D}}(B)$, endowed with the graph norm. Moreover, gives $(B+i)^{-1}{\mathcal{D}}(A)\subset {\mathcal{D}}(A)$. Therefore $(B+i)^{-1}{\mathcal{D}}(A)\subset {\mathcal{D}}(B)\cap {\mathcal{D}}(A)$ are dense in ${\mathcal{D}}(B)$ for the graph norm. Remark that ${\mathcal{D}}(B)\cap {\mathcal{D}}(A)$ is usually not dense in ${\mathcal{D}}(A)$, see [@GeorgescuGerardMoller]. Note that yields the commutator $[A, R(z)]$ extends to a bounded operator, in the form sense. We shall denote the extension by $[A, R(z)]_\circ$. In the same way, since ${\mathcal{D}}(B)\cap {\mathcal{D}}(A)$ is dense in ${\mathcal{D}}(B)$, ensures that the commutator $[B, A]$ extends to a unique element of ${\mathcal{B}}\big({\mathcal{D}}(B), {\mathcal{D}}(B)^\big)$ denoted by $[B, A]_\circ$. Moreover, when $B\in {\mathcal{C}}^1(A)$, one has: $$\begin{aligned} \big[A, (B-z)^{-1}\big]_\circ =\quad \underbrace{(B-z)^{-1}}_{{\mathscr{H}}\leftarrow {\mathcal{D}}(B)^}\quad \underbrace{[B, A]_\circ}_{{\mathcal{D}}(B)^\leftarrow {\mathcal{D}}(B)} \quad \underbrace{(B-z)^{-1}}_{{\mathcal{D}}(B)\leftarrow {\mathscr{H}}}.\end{aligned}$$ Here we use the Riesz lemma to identify ${\mathscr{H}}$ with its anti-dual ${\mathscr{H}}^$. We now recall some well known facts on symbolic calculus and almost analytic extensions. For $\rho\in{\mathbb{R}}$, let ${\mathcal{S}}^\rho$ be the class of function $\varphi\in{\mathcal{C}}^\infty({\mathbb{R}};{\mathbb{C}})$ such that $$\begin{aligned} \label{eq:regu} \forall k\in{\mathbb{N}}, \quad C_k(\varphi) :=\sup _{t\in{\mathbb{R}}}\, \langle t\rangle^{-\rho+k}\|\varphi^{(k)}(t)\|<\infty . \end{aligned}$$ Equipped with the semi-norms defined by (\[eq:regu\]), ${\mathcal{S}}^\rho$ is a Fréchet space. Leibniz’ formula implies the continuous embedding: ${\mathcal{S}}^\rho\cdot {\mathcal{S}}^{\rho'} \subset {\mathcal{S}}^{\rho+\rho'}$. We shall use the following result, e.g., [@DerezinskiGerard]. \[l:dg\] Let $\varphi\in{\mathcal{S}}^\rho$ with $\rho\in{\mathbb{R}}$. For all $l\in {\mathbb{N}}$, there is a smooth function $\varphi^{\mathbb{C}}:{\mathbb{C}}\rightarrow {\mathbb{C}}$, such that: $$\begin{aligned} \label{eq:dg1} \varphi^{\mathbb{C}}\|_{{\mathbb{R}}}=\varphi,\quad &&\left\|\frac{\partial \varphi^{\mathbb{C}}}{\partial \overline{z}}(z) \right\|\leq c_1 \langle \Re(z) \rangle^{\rho-1 -l} \|\Im(z)\|^l\\\label{eq:dg2} && {\mathrm{supp}}\varphi^{\mathbb{C}}\subset\{x+iy\mid \|y\|\leq c_2 \langle x\rangle\},\\\label{eq:dg3} && \varphi^{\mathbb{C}}(x+iy)= 0, \mbox{ if } x\not\in{\mathrm{supp}}\varphi . \end{aligned}$$ for some constants $c_1$, $c_2$ depending on the semi-norms of $\varphi$ in ${\mathcal{S}}^\rho$ and not on $\varphi$. One calls $\varphi^{\mathbb{C}}$ an almost analytic extension* of $\varphi$. Let $A$ be a self-adjoint operator, $\rho < 0$ and $\varphi\in {\mathcal{S}}^{\rho}$. By functional calculus, one has $\varphi(A)$ bounded. The Helffer-Sjöstrand’s formula, see [@HelfferSjostrand] and [@DerezinskiGerard] for instance, gives that for all almost analytic extension of $\varphi$, one has: $$\begin{aligned} \label{eq:int} \varphi(A) = \frac{i}{2\pi}\int_{\mathbb{C}}\frac{\partial \varphi^{\mathbb{C}}}{\partial \overline{z}}(z-A)^{-1}dz\wedge d\overline{z}. \end{aligned}$$ Note the integral exists in the norm topology, by with $l=1$. Next we come to a commutator expansion. Here $B$ is not necessarily bounded while in [@GoleniaJecko], one considers the case $B$ bounded. We denote by ${\mathrm{ad}}_A^j(B)$ the extension of the $j$-th commutator of $A$ with $B$ defined inductively by ${\mathrm{ad}}_A^p(B):=[{\mathrm{ad}}_A^{p-1}(B),A]_\circ$, when it exists. \[p:regu\] Let $k\in {\mathbb{N}}^\ast$ and $B\in{\mathcal{C}}^k(A)$ be self-adjoint. Suppose ${\mathrm{ad}}_A^j(B)$ are bounded operators, for $j=1,\ldots, k$. Let $\rho < k$ and $\varphi\in {\mathcal{S}}^{\rho}$. Suppose that ${\mathcal{D}}(B)\cap {\mathcal{D}}(\varphi(A))$ is dense in ${\mathcal{D}}(\varphi(A))$ for the graph norm. Then, the commutator $[\varphi(A), B]_\circ$ belongs to ${\mathcal{B}}\big({\mathcal{D}}(\varphi'(A)), {\mathscr{H}}\big)$ and satisfies $$\begin{aligned} \label{e:ega} [\varphi(A), B]_\circ = \sum_{j=1}^{k-1} \frac{1}{j!} \varphi^{(j)}(A){\mathrm{ad}}_A^j(B) + \frac{i}{2\pi}\int_{\mathbb{C}}\frac{\partial\varphi^{\mathbb{C}}}{\partial \overline{z}}(z-A)^{-k} {\mathrm{ad}}_A^k(B) (z-A)^{-1} dz\wedge d\overline{z}, \end{aligned}$$ where the integral exists for the topology of ${\mathcal{B}}({\mathscr{H}})$. We cannot use the directly with $\varphi$ as the integral does not seem to exist. We proceed as in [@GoleniaJecko]. Take $\cchi_1\in {\mathcal{C}}^\infty_c({\mathbb{R}};{\mathbb{R}})$ with values in $[0,1]$ and being $1$ on $[-1,1]$. Set $\cchi_R:=\cchi(\cdot/R)$. As $R$ goes to infinity, $\cchi_R$ converges pointwise to $1$. Moreover, $\{\cchi_R\}_{R\in [1,\infty]}$ is bounded in ${\mathcal{S}}^0$. We infer $\varphi_R:=\varphi \cchi_R$ tends pointwise to $\varphi$ and that $\{\varphi_R\}_{R\in [1,\infty]}$ is bounded in ${\mathcal{S}}^\rho$. Now, note that $$\begin{aligned} \label{e:ligne} \hspace{1cm} [\varphi_R(A), B] & =\sum_{j=1}^{k-1} \frac{i}{2\pi}\int_{\mathbb{C}}\frac{\partial\varphi^{\mathbb{C}}_R }{\partial\overline{z}} (z-A)^{-j-1}{\mathrm{ad}}_A^{j}(B) dz\wedge d\overline{z} \\ \nonumber &\, + \frac{i}{2\pi}\int_{\mathbb{C}}\frac{\partial\varphi^{\mathbb{C}}_R }{\partial \overline{z}}(z-A)^{-k} {\mathrm{ad}}_A^k(B) (z-A)^{-1} dz\wedge d\overline{z}. \end{aligned}$$ in the form sense on ${\mathcal{D}}(B)$. Using , the integral converges in norm. We write $[\varphi_R(A), B]_\circ$ on the l.h.s. The first term of the r.h.s. is $\sum_{j=1}^{k-1} \varphi^{(j)}_R(A){\mathrm{ad}}_A^j(B)/ j!$. Now we let $R$ goes to infinity. On the l.h.s. we use the Lebesgue converges. On the r.h.s. we expand the commutator in in the form sense on ${\mathcal{D}}\big(\varphi(A)\big)\cap {\mathcal{D}}(B)$, take the limit by functional calculus and finish by density in ${\mathcal{D}}\big(\varphi(A)\big)$. The hypothesis on the density of ${\mathcal{D}}(B)\cap {\mathcal{D}}(\varphi(A))$ in ${\mathcal{D}}(\varphi(A))$ could be delicate to check. It follows by the Nelson Lemma from the fact that the $C_0$-group $\{e^{it A^k}\}_{t\in {\mathbb{R}}}$ stabilizes ${\mathcal{D}}(B)$. We mention that for $k=1$, since $[B,iA]_\circ$ is bounded, [@GeorgescuGerard]\[Lemma 2\] ensures this invariance of the domain. The rest of the previous expansion is estimated as in [@GoleniaJecko]. We rely on the following important bound. Let $c>0$ and $s\in [0,1]$, there exists some $C>0$ so that, for all $z=x+iy\in\{a+ib\mid 0<\|b\|\leq c\langle a\rangle \}$: $$\begin{aligned} \label{eq:majoA} \big\\| \langle A\rangle^s (A-z)^{-1}\big\\|\leq C \langle x \rangle^{s}\cdot \|y\|^{-1}. \end{aligned}$$ \[l:est3\] Let $B\in{\mathcal{C}}^k(A)$ self-adjoint. Suppose ${\mathrm{ad}}_A^j(B)$ are bounded operators, for $j=1,\ldots, k$. Let $\varphi\in{\mathcal{S}}^\rho$, with $\rho< k$. Let $I_k(\varphi)$ the rest of the development of order $k$ of $[\varphi(A), B]$ in . Let $s, s' \in [0,1]$ such that $\rho+s+s'<k$. Then $\langle A \rangle^{s} I_k(\varphi)\langle A \rangle^{s'}$ is bounded and it is uniformly bounded when $\varphi$ stays in a bounded subset of ${\mathcal{S}}^\rho$. Let $R>0$. If $\varphi$ stays in a bounded subset of $\{\psi \in {\mathcal{S}}^\rho\mid [-R;R]\cap {\mathrm{supp}}(\varphi)=\emptyset\}$ then $\langle R\rangle^{k-\rho-s-s'} \\|\langle A \rangle^{s} I_k(\varphi)\langle A \rangle^{s'}\\|$ is uniformly bounded. We will follow ideas from [@DerezinskiGerard]\[Lemma C.3.1\]. In this proof, all the constants are denoted by $C$, independently of their value. Given a complex number $z$, $x$ and $y$ will denote its real and imaginary part, respectively. Since $B\in{\mathcal{C}}^k(A)$, ${\mathrm{ad}}^k_A(B)$ is bounded. We start with the second assertion. Let $\varphi\in {\mathcal{S}}^\rho$, $R>0$ such that $[-R;R]\cap {\mathrm{supp}}(\varphi)=\emptyset$. Notice that, by , $\varphi^{\mathbb{C}}(x+iy)= 0$ for $\|x\|\leq R$. By , $$\begin{aligned} \\|\langle A \rangle^{s}I_k(\varphi)\langle A \rangle^{s'}\\|\leq&\, \frac{1}{\pi} \int \big\|\frac{\partial\varphi^{\mathbb{C}}}{\partial \overline{z}}\big\|\cdot \frac{\langle x\rangle^{s}}{\|y\|^k} \cdot \\|{\mathrm{ad}}^k_A(B)\\|\cdot \frac{\langle x\rangle^{s'}}{\|y\|} dx\wedge dy\\ \leq&\, C(\varphi)\int_{\|x\|\geq R}\int_{\|y\|\leq c_2\langle x\rangle } \langle x \rangle^{\rho+s+s'-1-l}\|y\|^l \|y\|^{-k-1} dx\wedge dy, \end{aligned}$$ for any $l$, by . Recall that $dz\wedge d\overline{z}=-2i dx\wedge dy$. We choose $l=k+1$. We have, $$\begin{aligned} \\|\langle A\rangle^{s}I_k(\varphi)\langle A\rangle^{s'} \\|\leq C(\varphi)\int_{\|x\|\geq R} \langle x \rangle^{\rho+s+s'-k-1}dx \leq C(\varphi)\langle R\rangle^{\rho+s+s'-k}. \end{aligned}$$ Since $C(\varphi)$ is bounded when $\varphi$ stays in a bounded subset of ${\mathcal{S}}^\rho$, this yields the second assertion. For the first one, we can follow the same lines, replacing $R$ by $0$ in the integrals, and arrive at the result. A non-selfadjoint weak Mourre theory {#s:WeakMourreTheory} ==================================== In this section, we adapt ideas coming from [@FournaisSkibsted] and [@Richard] in order to obtain a limiting absorption principle for a family of closed operators $\{H^{\pm}(p)\}_{p\in {\mathcal{E}}}$. We ask that they have a common domain $$\begin{aligned} \label{e:dom0} {\mathscr{D}}:={\mathcal{D}}\big(H^+(p)\big)={\mathcal{D}}\big(H^-(p)\big), \mbox{ for all } p\in {\mathcal{E}}.\end{aligned}$$ We choose $p_0\in {\mathcal{E}}$ and endow ${\mathscr{D}}$ with the graph norm of $H^+(p_0)$. We also ask that $$\begin{aligned} \label{e:adj} \big(H^+(p)\big)^=H^-(p), \mbox{ for all } p\in {\mathcal{E}}.\end{aligned}$$ In particular, we have that ${\mathcal{D}}\big((H^{\pm}(p))^\big)= {\mathscr{D}}$. In the sequel, we forgo $p$, when no confusion can arises. Since $H^\pm$ are densely defined, share the same domain and are adjoint of the other, we have that $\Re (H^\pm)$ and $\Im (H^\pm)$ are closable operators on ${\mathscr{D}}$, indeed their adjoints are densely defined. We denote by $\Re (H^\pm)$ and by $\Im (H^\pm)$ the closure of these operators. It is possible that they are not self-adjoint, albeit there are symmetric. However, ${\mathscr{D}}$ is a core for them. Their domain is possibly bigger than ${\mathscr{D}}$. We suppose that $H^+$ is dissipative, i.e., $$\begin{aligned} \mbox{ $\langle f, \Im(H^+) f \rangle \geq 0$, for all $f\in {\mathscr{D}}$.} \end{aligned}$$ This gives also that $\Im (H^-)\leq 0$. By the numerical range theorem (see Lemma \[l:NRT\]), we infer that $\sigma(H^{\pm})$ is included in the half-plan containing $\pm i$. Take now a non-negative self-adjoint operator $S$, independent* of $p\in {\mathcal{E}}$, with form domain ${\mathscr{G}}:={\mathcal{D}}(S^{1/2})\supset{\mathscr{D}}$. We assume that $S$ is injective. We have $\langle f, S f\rangle> 0$ for all $f\in {\mathscr{G}}\setminus \{0\}$ and simply write $S> 0$. One defines ${\mathscr{S}}$ as the completion of ${\mathscr{G}}$ under the norm $\\|f\\|_{\mathscr{S}}^2:=\langle f, S f\rangle$. We obtain ${\mathscr{G}}\subset {\mathscr{S}}$ with dense and continuous embedding. Moreover, since ${\mathscr{G}}= \langle S^{1/2}\rangle^{-1}{\mathscr{H}}$, ${\mathscr{S}}$ is also the completion of ${\mathscr{H}}$ under the norm given by $\\|S^{1/2} \langle S^{1/2}\rangle^{-1}\cdot\\|$. We use the Riesz Lemma to identify ${\mathscr{H}}$ with ${\mathscr{H}}^$, its anti-dual. The adjoint space ${\mathscr{S}}^$ of ${\mathscr{S}}$ is exactly the domain of $\langle S^{1/2}\rangle S^{-1/2}$ in ${\mathscr{H}}\simeq{\mathscr{H}}^$. Note that $S^{-1}$ is an isomorphism between ${\mathscr{S}}$ and ${\mathscr{S}}^$. We get the following scale with continuous and dense embeddings: $$\begin{aligned} \label{e:scale} \begin{array}{cccccccccc} &&&&&& {\mathscr{S}}^&&& \\ &&&&&& \downarrow&\searrow && \\ {\mathscr{D}}&\longrightarrow &{\mathscr{G}}& \longrightarrow &{\mathscr{H}}&\simeq& {\mathscr{H}}^ &\longrightarrow &{\mathscr{G}}^& \longrightarrow{\mathscr{D}}^. \\ &&&\searrow &\downarrow &&&&& \\ &&&&{\mathscr{S}}&&&&& \end{array}\end{aligned}$$ To perform this analysis, we consider an external operator, the conjugate operator. Let $A$ be a self-adjoint operator in ${\mathscr{H}}$. We assume $S\in {\mathcal{C}}^1(A)$. Let $W_t:=e^{itA}$ be the $C_0$-group associated to $A$ in ${\mathscr{H}}$. We ask: $$\begin{aligned} \label{e:stab} W_t{\mathscr{G}}\subset {\mathscr{G}}\mbox{ and } W_t {\mathscr{S}}\subset {\mathscr{S}}, \mbox{ for all } t\in{\mathbb{R}}. \end{aligned}$$ By duality, we have $W_t$ stabilizes ${\mathscr{G}}^$ and also ${\mathscr{S}}^$ (but may be not ${\mathscr{D}}$ or ${\mathscr{D}}^$). The restricted group to these spaces is also a $C_0$-group. We denote the generator by $A$ with the subspace in subscript. Given ${\mathscr{H}}_i\subset{\mathscr{H}}_j$ be two of those spaces. One easily shows that $A\|_{{\mathscr{H}}_i}\subset A\|_{{\mathscr{H}}_j}$ and that $A\|_{{\mathscr{H}}_j}$ is the closure of $A\|_{{\mathscr{H}}_i}$ in ${\mathscr{H}}_j$. Moreover, one has $$\begin{aligned} \label{e:dom} {\mathcal{D}}(A\|_{{\mathscr{H}}_i})= \left\{f\in {\mathcal{D}}\big(A\|_{{\mathscr{H}}_j}\big)\cap{\mathscr{H}}_i \mbox{ such that } A\|_{{\mathscr{H}}_j} f\in {\mathscr{H}}_i\right\}. \end{aligned}$$ We now explain how to check the second hypothesis of , see also [@Richard]. \[r:inv\] The second invariance of the domains of follows from the first one and from $$\begin{aligned} \|\langle S f, A f\rangle- \langle A f, S f\rangle\| \leq c \\|S^{1/2} f\\|^2, \mbox{ for all } f\in{\mathcal{D}}(S)\cap{\mathcal{D}}(A). \end{aligned}$$ As $(S+i)^{-1}$ is a homeomorphism between ${\mathscr{H}}$ onto ${\mathcal{D}}(S)$, $(S+i)^{-1} {\mathcal{D}}(A)$ is dense in ${\mathcal{D}}(S)$, endowed with the graph norm. Moreover, since $S\in {\mathcal{C}}^1(A)$, one has $(S+i)^{-1}{\mathcal{D}}(A)\subset {\mathcal{D}}(A)$. Therefore $(S+i)^{-1}{\mathcal{D}}(A)\subset {\mathcal{D}}(S)\cap {\mathcal{D}}(A)$ are dense in ${\mathcal{D}}(S)$, hence in ${\mathscr{G}}$ and in ${\mathscr{S}}$. The commutator $[S,A]$ has a unique extension to an element of ${\mathcal{B}}({\mathscr{S}}, {\mathscr{S}}^)$, in the form sense. We denote it by $[S,A]_\circ$. Take now $f\in {\mathscr{G}}\cap{\mathcal{D}}(A)$, which is a dense set in ${\mathscr{G}}$. On one hand we have $\tau\mapsto \\|W_\tau f\\|_{\mathscr{S}}^2$ is bounded when $\tau$ is in a compact set (since ${\mathscr{G}}\, \hookrightarrow {\mathscr{S}}$. On the other hand, the Gronwall lemma concludes by noticing: $$\begin{aligned} \\|W_t f\\|_{\mathscr{S}}^2 = \langle f, S f\rangle + \int_0^t \langle W_\tau f, [S,iA]_\circ W_\tau f \rangle\, d\tau \leq \\|S^{1/2} f\\|^2 + c \int_0^{\|t\|} \\|W_\tau f\\|_{\mathscr{S}}^2\, d\tau. \end{aligned}$$ Let ${\mathscr{K}}\subset {\mathscr{H}}$ be a space which is stabilized by $W_t$. Consider $L\in {\mathcal{B}}({\mathscr{K}}, {\mathscr{K}}^)$. We say that $L\in {\mathcal{C}}^k(A; {\mathscr{K}}, {\mathscr{K}}^)$, when $t\rightarrow W_{-t}L W_t$ is strongly ${\mathcal{C}}^k$ from ${\mathscr{K}}$ into ${\mathscr{K}}^$. When ${\mathscr{K}}={\mathscr{H}}$, this class is the same as ${\mathcal{C}}^k(A)$, see [@AmreinBoutetdeMonvelGeorgescu]\[Theorem 6.3.4 a.\]. \[t:mourrestrict\] Let $H^{\pm}=H^{\pm}(p)$, with $p\in {\mathcal{E}}$ as above. Let $A$ be self-adjoint such that holds true. Suppose that $H^\pm\in{\mathcal{C}}^2(A; {\mathscr{G}}, {\mathscr{G}}^)$ and that there is a constant $c$, independent of $p$, such that $$\begin{aligned} \label{e:mourrestrict0} \|\langle H^{\mp}f, Ag \rangle - \langle Af, H^{\pm} g \rangle \|\leq c\\|f\\|\cdot\\|(H^{\pm} \pm i) g\\|, \mbox{ for all } f,g \in {\mathscr{D}}\cap {\mathcal{D}}(A).\end{aligned}$$ Take $c_1\geq 0$ independent of $p$ and assume that $$\begin{aligned} \label{e:mourrestrict1} [\Re (H^\pm), \rmi A]_\circ -c_1\Re (H^\pm) \geq S > 0, &\\ \label{e:mourrestrict2} \pm c_1[\Im (H^\pm),\rmi A]_\circ\geq 0, & \pm \Im (H^\pm)\geq 0,\end{aligned}$$ in the sense of forms on ${\mathscr{G}}$. Suppose also there exists $C>0$ independent of $p\in {\mathcal{E}}$ such that $$\label{Eq:SecondCommBound} \left \|\langle f, \big[\big[ H^\pm, A\big]_\circ,A\big]_\circ f\rangle\right\|\leq C \\|S^{1/2} f\\|^2, \mbox{ for all } f\in{\mathscr{G}}.$$ Then, there are $c$ and $\mu_0>0$, both independent of $p$, such that $$\begin{aligned} \label{e:LAP} \|\langle f, (H^\pm -\lambda \pm i \mu)^{-1} f\rangle \| \leq c \left(\\|S^{-1/2} f\\|^2 + \\|S^{-1/2} A f\\|^2\right)\leq c \\|f\\|_{{\mathcal{D}}(A\|_{{\mathscr{S}}^})}, \end{aligned}$$ for all $p\in {\mathcal{E}}$, $\mu\in (0, \mu_0)$ and $\lambda\geq 0$, in the case $c_1>0$ and $\lambda\in {\mathbb{R}}$ if $c_1=0$. In the self-adjoint setting, the case $c_1=0$ is treated in [@BoutetKazantsevaMantoiu; @BoutetMantoiu]. Comparing with [@Richard], who deal with the case of one self-adjoint operator and for $c_1>0$. We give some few improvements. First, we do not ask ${\mathscr{D}}$ to be the domain of $S$. Moreover, we drop the hypothesis that the first commutator $[H, \rmi A]_\circ$ is bounded from below. For the latter, we use more carefully the numerical range theorem in our proof. Finally, unlike [@Richard], we shall not go into interpolation theory so as to improve the norm in the limiting absorption principle. Indeed, in the context of the model we are considering here, we reach the weights we are interested in without it. We stick to an intermediate and explicit result, which is closer to [@IftimoviciMantoiu]. Therefore, for the sake of clarity, we present then the easiest proof possible and pay an important care about domains. We also mention that there exists other Mourre-like theory for non-self-adjoint operators, [@ABCF; @Royer]. We focus on the case $c_1>0$, as for the case $c_1=0$, one replaces “$\lambda>0$” by “$\lambda\in {\mathbb{R}}$”. We define $H_\varepsilon^\pm := H^\pm \pm\rmi \e[ H^\pm,\rmi A]_\circ$ with the common domain ${\mathscr{D}}$ for $\varepsilon\geq 0$. Since $H^\pm \pm \rmi$ is bijective, by writing $H_\varepsilon^\pm \pm \rmi= \big(1 \pm \rmi \e [ H^\pm,\rmi A]_\circ(H^\pm \pm \rmi)^{-1}\big)(H^\pm \pm \rmi)$ and using , we get there is $\varepsilon_0$ such that $H_\varepsilon^\pm(p) \pm \rmi$ is bijective and closed for all $\|\varepsilon\|\leq \varepsilon_0$ and all $p\in {\mathcal{E}}$. Therefore $(H_\varepsilon^\pm \pm \rmi)^$ is also bijective from ${\mathcal{D}}\big((H_{\varepsilon}^\pm)^\big)$ onto ${\mathscr{H}}$. Now since $(H_\varepsilon^\pm\pm\rmi)^$ is an extension of $H_{\varepsilon}^\mp \mp\rmi$ which is also bijective, we infer the equality of the domains and that $(H_\varepsilon^\pm)^= H_{\varepsilon}^\mp$ for $\varepsilon\leq \varepsilon_0$. Since $H^\pm\in {\mathcal{C}}^1(A; {\mathscr{G}},{\mathscr{G}}^)$, we obtain that $\Re(H^\pm)$ and $\Im(H^\pm)$ are in ${\mathcal{C}}^1(A; {\mathscr{G}},{\mathscr{G}}^)$. In this space we have $[ H^\pm, A]_\circ= [ \Re(H^\pm), A]_\circ + i [ \Im(H^\pm), A]_\circ$. Now, take $f\in {\mathscr{G}}$. Take $\varepsilon, \lambda, \mu \geq 0$. We get: $$\begin{aligned} \nonumber &\hspace{-1cm} -c_1 \varepsilon \left\langle f,\Re(H^\pm_\varepsilon-\lambda \pm \rmi \mu )f\right\rangle \mp \left\langle f,\Im (H^\pm_\varepsilon-\lambda \pm \rmi \mu ) f\right\rangle= \\ \nonumber &=-c_1 \varepsilon \left\langle f,\left(\Re(H^\pm) \pm \e[ \Im (H^\pm),\rmi A]_\circ -\lambda\right)f\right\rangle \mp \left\langle f, \left(\Im( H^\pm)\mp\mu\mp\e[\Re (H^\pm), \rmi A]_\circ\right)f\right\rangle \\ \nonumber &=\varepsilon \left\langle f,\big([\Re (H^\pm), \rmi A]_\circ -c_1\Re (H^\pm) \big)f\right\rangle +\left(c_1\lambda \varepsilon +\mu \right)\left\\|f\right\\|^2 \mp\left\langle f, \left(c_1\e^2 [ \Im (H^\pm),\rmi A]_\circ +\Im (H^\pm)\right)f \right \rangle \\ \label{e:numericalrange} &\geq (c_1\lambda \varepsilon + \mu)\left\\|f\right\\|^2 + \varepsilon \\| S^{1/2} f\\|^2. \end{aligned}$$ We start with a crude bound. For $\varepsilon, \mu>0$, we get: $$\begin{aligned} (c_1 \varepsilon +1)\, \\|(H_\varepsilon^\pm-\lambda \pm\rmi\mu)f \\|_{{\mathscr{G}}^}\geq {\min(c_1\lambda \varepsilon +\mu, \varepsilon ) } \\|f \\|_{\mathscr{G}}. \end{aligned}$$ Since $H_\varepsilon^\pm-\lambda \pm\rmi\mu\in{\mathcal{B}}({\mathscr{G}}, {\mathscr{G}}^)$ and since they are adjoint of the other, we infer the injectivity and that the ranges are closed. They are bijective and the inverse is bounded by the open mapping theorem. $$\begin{aligned} G^\pm_\e:=G^\pm_\e(\lambda, \mu)=(H_\varepsilon^\pm-\lambda \pm\rmi\mu)^{-1} \mbox{ exists in } {\mathcal{B}}({\mathscr{G}}^, {\mathscr{G}}), \mbox{ for } \lambda \geq 0 \mbox{ and } \varepsilon, \mu>0. \end{aligned}$$ Here we lighten the notation but keep in mind the dependency in $\lambda$ and $\mu$. Moreover, $$\begin{aligned} \label{e:G0} \\|G_\varepsilon^\pm \\|_{{\mathcal{B}}({\mathscr{G}}^, {\mathscr{G}})}\leq (c_1 \varepsilon +1)/\min(c_1\lambda \varepsilon +\mu, \varepsilon), \mbox{ for } \lambda \geq 0 \mbox{ and } \varepsilon, \mu>0. \end{aligned}$$ This bound seems not enough to lead the whole analysis. Then, we first restrict the domain of $G_\varepsilon^\pm$ to ${\mathscr{H}}$ and improve it. Since this inequality holds also true on the common domain of $H_\varepsilon^\pm$ (and of its adjoint), we can apply the numerical range theorem, Lemma \[l:NRT\]. Since $S\geq 0$, we get the spectrum of $H_\varepsilon^+ -\lambda +\rmi\mu$ is contained in the lower half-plane delimited by the equation $y\leq -c_1 \varepsilon x-\mu$. Hence, for $\varepsilon\in (0, \varepsilon_0]$ and $\mu>0$, $H_\varepsilon^\pm -\lambda \pm\rmi\mu$ is bijective and by taking $\varepsilon_0$ smaller, one has the distance from $0$ to the boundary of the cone bigger than $\mu/2$. Then, $$\begin{aligned} \label{e:G1} \\|G^\pm_\e \\|_{{\mathcal{B}}({\mathscr{H}})}\leq 2/ \mu, \mbox{ for } \mu>0 \mbox{ and } \varepsilon\in[0, \varepsilon_0]. \end{aligned}$$ Note also that $(G^\pm_\e)^= G^\mp_\e$. Take $\varepsilon, \mu>0$. We fix $f\in {\mathscr{H}}$ and set: $$\begin{aligned} F_\e^\pm:=\left\langle f, G^\pm_\e f\right\rangle. \end{aligned}$$ Since $G^\pm_\e {\mathscr{H}}\subset {\mathscr{D}}\subset {\mathscr{S}}$ and using , we infer $$\begin{aligned} \nonumber \left\\|S^{1/2}G^\pm_\e f\right\\|^2 &\leq c_1\left\|\Re \left\langle G^\pm_\e f,(H^\pm_\varepsilon -\lambda \pm \rmi \mu) G^\pm_\e f\right\rangle\right\|+ \frac{1}{\e}\left\|\Im \left\langle G^\pm_\e f,(H^\pm_\varepsilon -\lambda \pm \rmi \mu) G^\pm_\e f\right\rangle\right\| \\ \label{e:estG} &\leq \max\left(c_1,\frac{1}{\e}\right)\left\|F_\e^\pm\right\|.\end{aligned}$$ Hence up to a smaller $\e_0>0$, we obtain $\left\\|S^{1/2}G^\pm_\e f\right\\|^2\leq \left\|F_\e^\pm\right\|/ \e$ for all $\varepsilon \in(0, \varepsilon_0]$. Moreover, if $f\in{\mathcal{D}}(S^{-1/2})$, we obtain $$\left\|F_\e^\pm\right\|\leq \left\\|S^{-1/2} f\right\\| \left\\|S^{1/2}G^\pm_\e f\right\\|\leq \left\\|S^{-1/2} f\right\\| \frac{ \sqrt{\left\|F_\e^\pm\right\|}}{\sqrt{\e}}$$ and deduce $$\label{e:estFe} \left\|F_\e^\pm\right\|\leq \frac{1}{\e}\left\\|S^{-1/2} f\right\\|^2, \mbox{ for all } \varepsilon \in(0, \varepsilon_0] .$$ We now show that $G_\varepsilon^\pm\in {\mathcal{C}}^1(A)$. First note that $G_\varepsilon^\pm$ is a bijection from ${\mathscr{H}}$ onto ${\mathscr{D}}$. Then by taking the adjoint, it is also a bijection from ${\mathscr{D}}^$ onto ${\mathcal{H}}$. Remember now that $W_t$ stabilizes ${\mathscr{G}}$ and ${\mathscr{G}}^$. By the resolvent equality in ${\mathcal{B}}({\mathscr{H}})$, we have: $$\begin{aligned} [G_\varepsilon^\pm , W_t]= - \underbrace{ G_\varepsilon^\pm}_{{\mathscr{H}}\longleftarrow {\mathscr{G}}^}\quad \underbrace{\big[H^\pm \pm\rmi \e[ H^\pm,\rmi A], W_t\big]}_{{\mathscr{G}}^* \longleftarrow {\mathscr{G}}}\quad \underbrace{G_\varepsilon^\pm}_{{\mathscr{G}}\longleftarrow {\mathscr{H}}} \end{aligned}$$ Let now take the derivative in $0$. Since $H^{\pm}$ and $[ H^\pm,\rmi A]$ are in ${\mathcal{C}}^1(A; {\mathscr{G}}, {\mathscr{G}}^)$ (the former being in ${\mathcal{C}}^2(A; {\mathscr{G}}, {\mathscr{G}}^)$), the right hand side has a strong limit for all element in ${\mathscr{H}}$. Hence, $G_\varepsilon^\pm \in {\mathcal{C}}^1(A; {\mathscr{H}}, {\mathscr{H}})$ which is the same as $G_\varepsilon^\pm \in {\mathcal{C}}^1(A)$, see [@AmreinBoutetdeMonvelGeorgescu]\[Theorem 6.3.4 a.\]. Easily, it follows that $G_\varepsilon^\pm {\mathcal{D}}(A)\subset {\mathcal{D}}(A)\cap {\mathscr{D}}$ and one can safely expand the commutator in the next computation. Take $f\in{\mathcal{D}}(A)$. $$\begin{aligned} \frac{d}{d\e}F_\e^\pm&= \left\langle f, \frac{d}{d\e}G^\pm_\e f\right\rangle =\pm\rmi \left\langle G^\mp_\e f, [ H^\pm,\rmi A]_\circ G^\pm_\e f\right\rangle\\ &=\pm \left\langle G^\mp_\e f,Af\right\rangle \mp \left\langle Af, G^\mp_\e f\right\rangle - \varepsilon \left\langle G^\mp_\e f, \big[[H,iA]_\circ, iA\big] G^\pm_\varepsilon f \right\rangle. \end{aligned}$$ Here the last commutator in taken in the form sense. Now use three times and the bound , which is uniform in $p\in {\mathcal{E}}$, then integrate to obtain $$\label{e:inegdiff} \left\|F_\e^\pm - F_{\e'}^\pm\right\|\leq \int_\e^{\e'}\left\{2 \frac{ \sqrt{\|F_s^\pm\|}} {\sqrt{s}}\left\\|S^{-1/2}A f\right\\| + C \left\|F_s^\pm\right\|\right\}\;ds, \mbox{ for } 0<\varepsilon\leq\varepsilon'\leq \varepsilon_0$$ and for all $f\in{\mathcal{D}}(S^{-1/2}A)\cap{\mathcal{D}}(A)$. We give a first estimation. Using and the Gronwall lemma, see [@AmreinBoutetdeMonvelGeorgescu]\[Lemma 7.A.1\] with $\theta=1/2$ or [@Oguntuase]\[Lemma 2.6\] with $p=1/2$, we infer there are some constants $C, C', C'', C'''$, independent of $\e\in(0,\e_0]$, $\lambda\geq 0$, $\mu>0$ and of $p\in {\mathcal{E}}$, so that $$\begin{aligned} \nonumber \left\|F_\e^\pm\right\|&\leq e^{C(\e-\e_0)} \left(\left\|F_{\e_0}^\pm\right\|^{1/2} + \int_\e^{\e_0}\left\{\frac{ 1}{\sqrt{\eta}} e^{-\frac{1}{2}C(\eta-\e_0)}\right\}d\eta\,\, \left\\|S^{-1/2}A f\right\\|\right)^2 \\ \nonumber &\leq C''\left(\left\|F_{\e_0}^\pm\right\| +\left(\sqrt{\e}-\sqrt{\e_0}\right)^2 \left\\|S^{-1/2}A f\right\\|^2\right) \\ \label{e:last} &\leq C''\left(\frac{1}{\e_0} \left\\|S^{-1/2} f\right\\|^2 +\left(\sqrt{\e}-\sqrt{\e_0}\right)^2 \left\\|S^{-1/2}A f\right\\|^2\right) \leq C''' \\|f\\|^2_{\tilde {\mathscr{S}}^}\end{aligned}$$ for $f\in{\mathcal{D}}(S^{-1/2})\cap{\mathcal{D}}(S^{-1/2}A)\cap{\mathcal{D}}(A)$ and where $\tilde {\mathscr{S}}^$ is the completion of ${\mathcal{D}}(A\|_{{\mathscr{S}}^})$ under the norm $\\|f\\|^2_{\tilde {\mathscr{S}}^}:= \left\\|S^{-1/2} f\right\\|^2 + \left\\|S^{-1/2}A f\right\\|^2$. Here one notices that the norm is well defined for elements of ${\mathcal{D}}(A\|_{{\mathscr{S}}^})$ by taking in account . We now plug this back in . Since the inverse of the square root is integrable around $0$, we find $C''''$ with the same independence so that $$\begin{aligned} \left\|F_\e^\pm-F_{\e'}^\pm\right\| \leq \int_\e^{\e'}\left\{2 \frac{\sqrt{C'''}}{\sqrt{s}} + CC''' \right\}ds\,\, \\|f\\|_{\tilde {\mathscr{S}}^}^2= C'''' \big(\sqrt{\varepsilon'}-\sqrt{\varepsilon}\,\big) \\|f\\|_{\tilde {\mathscr{S}}^}^2.\end{aligned}$$ Then, $\{F_\e^\pm\}_{\e\in(0, \varepsilon_0]}$ is a Cauchy sequence. We denote by $F_{0^+}^\pm$ the limit, as $\varepsilon$ goes to $0$. It remains to notice that $F_{0^+}^\pm= F_0^{\pm}$. Indeed, using and , one has the stronger fact that $$\begin{aligned} \\|G_0^\pm - G_\e^\pm\\|_{{\mathcal{B}}({\mathscr{H}})}\leq \varepsilon \\|G^\pm_\e\\|_{{\mathcal{B}}({\mathscr{H}})}\cdot \\| [H^{\pm}(p),iA] (H^{\pm}(p)-\lambda\pm \rmi\mu)^{-1} \\|_{{\mathcal{B}}({\mathscr{H}})} \leq \frac{c\varepsilon}{\mu^2}. \end{aligned}$$ This gives us . For the convenience of the reader, we give a proof of the following well known fact: \[l:NRT\] Let $H$ be a closed operator. Suppose that ${\mathscr{D}}:={\mathcal{D}}(H)={\mathcal{D}}(H^)$. The numerical range of $H$ is defined by ${\mathcal{N}}:=\{ \langle f, Hf \rangle$ with $ f\in {\mathscr{D}}$ and $ \\|f\\|=1\}$. We have that $\sigma(H) \subset \overline{{\mathcal{N}}}$, the closure of ${\mathcal{N}}$. Moreover, if $\lambda \notin \sigma(H)$, then $\\|(H-\lambda)^{-1}\\|\leq 1/d(\lambda, {\mathcal{N}})$. Let $\lambda \notin \overline{{\mathcal{N}}}$. There is $c:=d(\lambda, {\mathcal{N}})>0$, such that $\|\langle f, Hf \rangle - \lambda\|\geq c$. Then, $$\begin{aligned} \\|(H-\lambda)f \\|\geq c \\|f\\|, \quad \\|(H^-\overline\lambda)f \\|\geq c \\|f\\|,\end{aligned}$$ for all $f\in {\mathscr{D}}$ and $\\|f\\|=1$. From the second part, we get the range of $(H-\lambda)$ is dense. Then, since $H$ is closed, the first part gives that the range of $(H-\lambda)$ is closed. Hence, using again the first inequality, $H-\lambda$ is bijective. The open mapping theorem concludes. Application to non-relativistic dispersive Hamiltonians {#s:nonrela} ======================================================= In this section, we give an immediate application to the theory exposed in Appendix \[s:WeakMourreTheory\]. We do not discuss the uniformity with respect to the external parameter. The latter would be used in the heart of our approach, see Section \[s:positive\]. We discuss shortly the Helmholtz equation, see [@BenamouCastella; @BenamouLafitte; @Wang2; @WangZhang]. In [@Royer], one studies the size of the resolvent of $$\begin{aligned} H_h := -h^2 \Delta +V_1(Q) -ih V_2(Q), \mbox{ as } h\rightarrow 0.\end{aligned}$$ This operator models accurately the propagation of the electromagnetic field of a laser in material medium. The important improvement between [@Royer] and the previous ones, is that he allows $V_2$ to be a smooth function tending to $0$ without any assumption on the size of $\\|V_2\\|_\infty$. Note he supposes the coefficients are smooth as some pseudo-differential calculus is used to applied the non self-adjoint Mourre theory he develops. Then, he discusses trapping conditions in the spirit of [@Wang2]. Here, we will stick to the quantum case and choose $h=-1$. To simplify the presentation and expose some key ideas of Section \[s:positive\], we focus on $L^2({\mathbb{R}}^n; {\mathbb{C}})$, with $n\geq 3$. For dimensions $1$ and $2$, one needs to adapt the first part of (H2) and the weights in . \[t:nonrelamain\] Suppose that $V_1, V_2\in L^1_{\rm loc}({\mathbb{R}}^n;{\mathbb{R}})$ satisfy: 1. $V_i$ are $\Delta$-operator bounded with a relative bound $a<1$, for $i\in\{1,2\}$. 2. $\nabla V_i$, $ Q\cdot \nabla V_i(Q)$ are in ${\mathcal{B}}({\mathscr{H}}^2({\mathbb{R}}^n);L^2({\mathbb{R}}^n))$ and $\langle Q\rangle (Q\cdot \nabla V_i)^2(Q)$ is bounded, for $i\in\{1,2\}$. 3. There are $c_1\in [0,2)$ and $\displaystyle c_1'\in \big[0, 4(2-c_1)/(n-2)^2\big)$ such that $$\begin{aligned} W_{V_1}(x):= x\cdot (\nabla V_1)(x) + c_{1} V_1(x) \leq \frac{c_{1}'}{\|x\|^2}, \mbox{ for all } x\in {\mathbb{R}}^n. \end{aligned}$$ and $$\begin{aligned} V_2(x)\geq 0 \mbox{ and } -c_1 x\cdot (\nabla V_2)(x)\geq 0, \mbox{ for all } x\in {\mathbb{R}}^n.\end{aligned}$$ On ${\mathcal{C}}^\infty_c({\mathbb{R}}^n)$, we define $H:= -\Delta + V(Q), \mbox{ where } V:=V_1 + i V_2$. The closure of $H$ defines a dispersive closed operator with domain ${\mathscr{H}}^2({\mathbb{R}}^n)$. We keep denoting it with $H$. Its spectrum included in the upper half-plane. Moreover, $H$ has no eigenvalue in $[0,\infty)$ and $$\begin{aligned} \label{e:nonrelamain} \sup_{\lambda\in [0,\infty), \, \mu>0} \big\\|\, \|Q\|^{-1}(H -\lambda + \rmi \mu )^{-1} \|Q\|^{-1}\big\\| <\infty. \end{aligned}$$ If $c_1=0$, $H$ has no eigenvalue in ${\mathbb{R}}$ and holds true for $\lambda \in {\mathbb{R}}$. The quantity $W_{V_1}$ is called the virial* of $V_1$. For $h$ fixed and for a compact ${\mathcal{I}}$ included in $(0,\infty)$, [@Royer] shows some estimates of the resolvent above ${\mathcal{I}}$. Here we deal with the threshold $0$ and with high energy estimates. On the other hand, as he avoids the threshold, he reaches some very sharp weights. As mentioned above, one can improve the weights $\|Q\|$ to some extend by the use of Besov spaces, see [@Richard]. In [@Royer] one makes an hypothesis on the sign of $V_2$ but not on the one of $x\cdot (\nabla V_2)(x)$. Note that if one supposes $c_1=0$, we are also in this situation. We take the opportunity to point out [@Wang], where one discusses the presence of possible eigenvalues in $0$ for non self-adjoint problems. \[r:virial\] Taking $V_2=0$, we can compare the results with [@FournaisSkibsted; @Richard]. In [@FournaisSkibsted], one uses in a crucial way that $W_{V_1}(x)\leq - c \langle x\rangle^\alpha$ in a neighborhood of infinity, for some $\alpha, c>0$. In [@Richard], one remarks that the condition $W_{V_1}(x)\leq 0$ is enough to obtain the estimate. Here we mention that the condition (H2) is sufficient. Note this example is not explicitly discussed in [@Richard] but is covered by his abstract approach. In [@BoutetKazantsevaMantoiu], for the special case $c_1=0$, one uses extensively the condition (H2). This implies for $\lambda\in {\mathbb{R}}$. \[r:Herbst\] Unlike in [@Royer], we stress that $V$ is not supposed to be a relatively compact perturbation of $H$ and that the essential spectrum of $H$ can be different of $[0, \infty)$. In [@Herbst], see also [@BoutetKazantsevaMantoiu], one studies $V_2=0$ and $V_1(x):= v(x/\|x\|)$, with $v\in {\mathcal{C}}^\infty(S^{n-1})$. We improve the weights of [@Herbst]\[Theorem 3.2\] from $\langle Q\rangle $ to $\|Q\|$. We can also give a non-self-adjoint version. Consider $V_1$ satisfying (H1) and being relatively compact with respect to $\Delta$ and $V_2(x):= v(x/\|x\|)$, where $v\in {\mathcal{C}}^0(S^{n-1})$, non-negative. If $v^{-1}(0)$ is non-empty, one shows $[0,\infty)$ is included in the essential spectrum of $H$ by using some Weyl sequences. Using (H0) and adapting the proof of Kato-Rellich, e.g., [@ReedSimon]\[Theorem X.12\], one obtains easily ${\mathcal{D}}(H)={\mathcal{D}}(H^)={\mathscr{H}}^2({\mathbb{R}}^n)$. Let $S:=c_s(-\Delta)^{1/2}$, with $c_s:=2-c_{1}- (n-2)^2 c_{1}'/4>0$. Set ${\mathscr{S}}:=\dot{\mathscr{H}}^1({\mathbb{R}}^n)$, the homogeneous Sobolev space of order $1$, i.e., the completion of ${\mathscr{H}}^1({\mathbb{R}}^n)$ under the norm $\\|f\\|_{\mathscr{S}}:= \\| S^{1/2} f\\|^2$. Consider the strongly continuous one-parameter unitary group $\{W_t\}_{t\in {\mathbb{R}}}$ acting by: $(W_t f)(x)= e^{nt/2} f(e^tx)$, for all $f\in L^2({\mathbb{R}}^3)$. This is the $C_0$-group of dilatation. By interpolation and duality, one derives $W_t {\mathscr{S}}\subset {\mathscr{S}}\mbox{ and } W_t {\mathscr{H}}^s({\mathbb{R}}^3) \subset {\mathscr{H}}^s({\mathbb{R}}^n)$, for all $s\in {\mathbb{R}}$. Consider now its generator $A$ in $L^2({\mathbb{R}}^n)$. By the Nelson lemma, it is essentially self-adjoint on ${\mathcal{C}}^\infty_c({\mathbb{R}}^n)$ and acts as follows: $A=(P\cdot Q+ Q\cdot P)/2$ on ${\mathcal{C}}^\infty_c({\mathbb{R}}^n)$. By computing on ${\mathcal{C}}^\infty_c({\mathbb{R}}^n)$ in the form sense, we obtain that $$\begin{aligned} \label{e:nonrelamain1} [\Re(H), iA] - c_{1} \Re(H) = -(2-c_{1}) \Delta - W_{V_1}\geq S,\end{aligned}$$ here we used the Hardy inequality for the last step. Furthermore, $\Im(H)=V_2(Q)\geq 0$, $$\begin{aligned} \label{e:nonrelamain2} [\Im(H), iA]= -Q\cdot\nabla(V_2)(Q),\end{aligned}$$ and also $$\begin{aligned} \label{e:nonrelamain3} [[H, iA], iA] = -4 \Delta + (Q\cdot \nabla V)^2(Q).\end{aligned}$$ Since $W_t$ stabilizes ${\mathscr{G}}:={\mathscr{H}}^1$ and as , and extend to bounded operators from ${\mathscr{H}}^1$ into ${\mathscr{H}}^{-1}$, we infer that $H$ and $H^$ are in ${\mathcal{C}}^2(A; {\mathscr{H}}^1, {\mathscr{H}}^{-1})$ and also and . Now since ${\mathcal{C}}^\infty_c({\mathbb{R}}^3)$ is a core for $H$, $H^$ and $A$, and give , with notation $H^+=H$ and $H^-=H^$. 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--- abstract: \| High electric-field transport parameters are calculated using an analytical Fokker-Planck approach (FPA), where transport is modeled as a drift-diffusion process in energy space. We have applied the theory to the case of Si, taking into account the six intervalley phonons, aiming to test the FPA. The obtained results show a quite reasonable agreement with experimental data and Monte Carlo simulations confirming in this case that the FPA works very well for high enough electric fields. PACS numbers: 72.10.Di; 72.20.Ht. address: \| Departmento de Física, Universidade Federal de São Carlos,\ 13565-905, São Carlos, São Paulo, Brazil author: - 'F. Comas\* and Nelson Studart' title: 'High-field transport properties of bulk Si: A test for the Fokker-Planck approach' --- More than three decades ago the Fokker-Planck approach (FPA) was proposed as an alternative for the Boltzmann transport equation (BTE) in the calculation of semiconductor transport properties [@b1; @b2]. Recently the theory has been revisited in a series of works [@b3; @b4; @b5; @b6; @b7] where a relatively detailed analysis of both mathematical and physical aspects of this formalism was developed. In these papers the same model system was considered in calculations using the FPA and the BTE by means of the Monte Carlo method. The results of both approaches showed good agreement in the high electric-field regime for the mentioned model system. In spite of this, there remains a certain degree of doubt about how the FPA could handle a more realistic model of a concrete semiconductor with several possible scattering mechanisms and complicated band structure. The FPA considers transport, in the opposite regime of the ballistic one, as a certain diffusive-drifting “motion” of the carriers in the energy space and it is valid when $\tau (\vec{p})<<t<<\tau _{E},$ where $\tau (\vec{p})$ is the momentum relaxation time and $\tau _{E}$ is the energy relaxation time. The method is semiclassical by its own nature and applicable when the energy exchange between the carriers and the surrounding medium can be assumed quasicontinuous which excludes highly inelastic scattering processess. It is valid when the average carrier energy is much larger than the exchanged energy as in the case of high-field transport. The FPA has the advantage of being analytical, and, whenever it can be applied, saves computational time and allows a more transparent physical interpretation. In this report, we present results within the FPA for bulk Si and compare them with experimental data and previous Monte Carlo simulations (using the BTE). We show that the FPA leads to good results as compared with those data whatever several scattering mechanisms (six intervalley phonons between the Si $\Delta $ valleys in the conduction band) are taken into account. The intravalley acoustic phonons were ignored, and thus our results are reliable just for high enough temperatures. The evolution of the energy distribution function (DF) $f(E,t)$ is governed by the Fokker-Planck equations[@b1; @b2; @b3] $$\frac{\partial }{\partial t}f(E,t)+\frac{1}{N(E)}\frac{\partial }{\partial E}J(E,t)=0, \label{e1}$$ where $$J(E,t)=W(E)N(E)f(E,t)-\frac{\partial }{\partial E}\left[ D(E)N(E)f(E,t)\right] , \label{e2}$$ such that $f(E,t)N(E)$ gives the number of carriers at time $t$ with energies in the interval $(E,E+dE)$, while the function $N(E)$ represents the density of states (DOS). In Eq. (\[e2\]) $W(E)$ represents a certain “drift velocity” in energy space and in fact gives the rate of energy balance of the carrier, $D(E)$ is a kind of diffusion coefficient and $J(E,t) $ represents thus the carrier current density in energy space.[@b1; @b2; @b3] Under steady state conditions, the Eq. (\[e2\]) transforms into $$\frac{\partial }{\partial E}\left[ D(E)N(E)f(E)\right] =W(E)N(E)f(E). \label{e3}$$ The carriers interact with the phonons and the applied dc electric field $\vec{F}$. We assume a phonon reservoir in thermal equilibrium at the temperature $T$ and that the continuous exchange of phonons between the carriers and the bath does not affect the thermal equilibrium of the latter. Hence, the coefficients $W(E)$ and $D(E)$ are split as follows $$D(E)=D_{F}(E)+D_{ph}(E)\quad ,\quad W(E)=W_{F}(E)+W_{ph}(E). \label{e4}$$ The label ”$F$” (”$ph$”) denotes the electric field (phonon) contribution to these coefficients, whose explicit forms will be given below. Equation (\[e3\]) has the simple solution $$f(E)=\exp \left\{ \int \left[ \frac{W_{ph}(E)}{D_{F}(E)}dE\right] \right\} , \label{e5}$$ where $D_{ph}(E)$ was neglected. This approximation is very well fulfilled in all the cases of interest for us [@b3]. We consider transport of electrons in the Si conduction band (CB) in a high dc electric field regime ($F>10$ kV/cm). We take into account the six ellipsoidal energy valleys of Si at the $\Delta $ points of the Brillouin zone (along the $<100>$ direction). To be specific, let us take $\vec{F}=(0,0,F)$, where the $z$ axis is taken along one high symmetry direction, and denote by “$l$” (“$tr$”) the valleys with principal axis parallel (perpendicular) to $\vec{F}$. The explicit expressions for $D_{F}(E)$ and $W_{ph}(E)$ are[@b2; @b3] $$D_{F}(E)=\left\langle \tau (\vec{p})[q\vec{F}\cdot \nabla _{\vec{p}}\epsilon (\vec{p})]^{2}\right\rangle , \label{e8}$$ $$W_{ph}(E)=\hbar \omega \left[ 1/\tau _{abs}(\vec{p})-1/\tau _{em}(\vec{p})\right] , \label{e9}$$ where the brackets represent an average over the constant energy surface $\epsilon (\vec{p})=const$, $\tau (\vec{p})$ is the [total]{} relaxation time due to the electron-phonon scattering, and “[abs]{}” (“[em]{}”) denotes phonon absorption (emission) by the electron due to the several scattering mechanisms. A straightforward evaluation of Eq.(\[e8\]) leads to $$D_{F}^{j}(E)=\frac{2e^{2}F^{2}}{3m_{j}}\tau (\epsilon )\gamma (\epsilon )/(\gamma ^{\prime }(\epsilon ))^{2}\quad j=l,tr, \label{e11}$$ where $\gamma ^{\prime }(\epsilon )$ denotes the derivative of the function $\gamma (\epsilon )=\epsilon (1+\alpha \epsilon )$ responsible by the non-parabolicity of the band structure with $\epsilon =\epsilon (\vec{p})$ being the energy dispersion for a given valley. For $\tau (\epsilon )$ we shall consider the six intervalley phonons responsible for transitions between the equivalent $\Delta $-valleys of Si. Then the total relaxation time reads as $$\frac{1}{\tau (\epsilon )}=\sum_{i=1}^{6}C_{oi}\left[ n_{i}(T)\sqrt{\gamma (\epsilon +\hbar \omega _{i})}\|1+2\alpha (\epsilon +\hbar \omega _{i})\|\right.$$ $$\left. +(n_{i}(T)+1)\sqrt{\gamma (\epsilon -\hbar \omega _{i})}\|1+2\alpha (\epsilon -\hbar \omega _{i})\|\Theta (\epsilon -\hbar \omega _{i})\right] , \label{e12}$$ where $\Theta (\epsilon )$ is the step function, $n_{i}(T)$ is the phonon distribution function, and $C_{oi}=(m_{tr}m_{l}^{1/2}D_{oi}^{2})/(\sqrt{2}\pi \rho \hbar ^{3}\omega _{i})$ with $m_{tr}$ ($m_{l}$) being the transverse (longitudinal) effective mass, $\rho $ the semiconductor density, $\omega _{i}$ and $D_{oi}$ are the frequency and deformation-potential constant respectively for intervalley phonons of type $i$ [@b9]. In Eq.(\[e12\]), the first and second terms correspond to $1/\tau _{abs}^{i}(\epsilon )$ and $1/\tau _{em}^{i}(\epsilon )$ respectively. The phonon contribution can be written as $$W_{ph}(\epsilon )=\sum_{i=1}^{6}W_{ph}^{i}(\epsilon ). \label{e17}$$ where $W_{ph}^{i}(\epsilon )$ is given by Eq.(\[e9\]) for each $i.$ Intravalley optical phonons do not contribute to transition rates, because the corresponding transitions are forbidden by the selection rules and we assume that, for high $T$ and $E,$ the contribution of intravalley acoustic phonons should be neglected. Once, we have evaluated $W_{ph}(E)$ and $D_{F}^{j}(E)$, we have to calculate the integral in Eq.(\[e5\]) to obtain the two DF, $f_{l}(E)$ and $f_{tr}(E) $, corresponding to the $l$-$tr$ valleys respectively. Of course, the FPA is of practical use only in the case when such integration can be analytically performed which is not the case for expressions of $D_{F}^{j} $ and $1/\tau (E)$ given by Eqs.(\[e11\]) and (\[e12\]). So, we consider a single effective intervalley phonon with energy $\hbar \omega _{0}=0.0343\, $ eV, obtained from an average of different phonon frequencies given in Table VI of [@b9], and with constant $D_{o}$ obtained by the superposition of the different deformation-potential constants, but also including the number of final equivalent valleys for each kind of transition. With this approximation we obtain the following result $$\sum_{i=1}^{6}\frac{W_{ph}^{i}(E)}{D_{F}^{j}(E)}=\frac{3m_{j}\hbar \omega _{0}}{2e^{2}F^{2}}\sum_{i,k}C_{oi}C_{ok}\Phi (E,T),\quad j=l,tr, \label{e18}$$ with $$\Phi (E,T)=n^{2}(T)(E+1)(1+E_{0}(E+1))(1+2E_{0}(E+1))^{2}$$ $$-(n(T)+1)^{2}(E-1)(1+E_{0}(E-1))(1+2E_{0}(E-1))^{2}\Theta (E-1). \label{e19}$$ where hereafter $E$ is in units of $\hbar \omega _{0}$ and $E_{0}=\hbar \omega _{0}\alpha $. Considering the contributions from different valleys and using Eq.(\[e5\]), again applying the parameters from Table VI Ref.[@b9], we are led to: $$f(E)=\exp \left[ \beta _{j}\int \Phi (x,T)dx\right] , \label{e20}$$ with $\beta _{j}=(3m_{j}m_{d}^{3}D_{o}^{4})/4\pi ^{2}\rho ^{2}e^{2}F^{2}\hbar ^{4}$ and $D_{o}$ is the effective deformation-potential constant defined through $D_{o}^{4}=\sum D_{oi}^{2}D_{ok}^{2}$. We have estimated $D_{o}=12.09\times 10^{8}$ eV/cm and in all the above expressions the overlapping integral (see Ref.[@b9]) was taken equal to unity. However, for numerical computations it should be useful consider it as a fitting parameter. The integral involved in Eq.(\[e20\]) can be analytically performed in a straightforward way and the general structure of the DF has the form: $$f(E)=E^{A}(1+E_{0}E)^{B}\exp (C\cdot P(E)), \label{e22}$$ where $A$, $B$ and $C$ are parameters dependent on $T$ and $F$ and $P(E$) is a polynomial.. This structure is far from a Maxwellian one. The DF describes the stationary non-equilibrium configuration where an electron temperature $T_{e}$ cannot be defined. From Eq.(\[e22\]), we can immediately obtain the average electron energy, estimated as $E_{av}(T,F)=(E_{av}^{l}+2 E_{av}^{tr})/3$ with $$E_{av}^{j}=\left[ \int Ef_{j}(E)N(E)dE\right] /\int f(E)_{j}N(E)dE\quad j=l,\,tr. \label{e23}$$ In Fig. 1, the electric-field dependence of $E_{av}$ is depicted for different temperatures. As expected, we found a weak temperature dependence. We can see that the average electron energy increases for larger electric fields. Moreover, the condition $E_{av}>>\hbar \omega _{0}$ is fairly well accomplished, ensuring that the FPA is within its range of validity for the given temperatures. Our results cannot be expected to be correct for low temperatures (or too low carrier energies) because we neglect intravalley acoustic phonons. The drift velocity $v_{d}=(v_{dl}+2v_{dtr})/3$ can be also evaluated from[@b2; @b3] $$v_{dj}=\frac{2eF}{3m_{j}}\int \frac{\gamma (E)\tau (E)}{(\gamma ^{\prime }(E))^{2}}\left[ -\frac{df_{j}(E)}{dE}\right] N(E)dE/\int f(E)_{j}N(E)dE. \label{e25}$$ In Fig. 2, we show $v_{d}$ as a function of $F$ for two different temperatures. We see that the general behavior of the curve is qualitatively correct if compared with experimental results and Monte Carlo simulations. For a more quantitative comparison, we present in Fig. 3 our results together with the experimental data and those from Monte Carlo simulations[@b10; @b11]. As it can be seen, we obtained a good agreement with both results for $T=300$ K in the high electric-field regime. However, in the opposite limit, the results from FPA do not reproduce those from experiments and Monte Carlo calculations, as should be expected. For comparison with experimental data we have taken $\beta _{j}$ as a fitting parameter, an issue which can be reasonably understood considering the overlapping integral for the electron-phonon scattering probabilities. Another point to be stressed is that reliable results were obtained just for high temperatures. However this is not a limitation of the FPA itself but of our present calculations since we have ignored intravalley acoustic phonons. In conclusion, we have shown that transport problems can be tackled by the mathematically simple FPA even in the case of a concrete semiconductor. We can also emphasize that the possibility for achieving correct results from the FPA depends on the chance of performing good enough approximations for the relaxation processes, allowing the analytical evaluation of the integral in Eq.(\[e5\]) and still retaining the essential physical picture. The results, being acceptable just for the higher electric fields, in fact are very close to those of Monte-Carlo simulations. Additional calculations for T= 430 K also revealed good agreement with those of [@b10]. The saturation effect, however, is not predicted by FPA. For higher electric field intensities a slow but ever decreasing behaviour is achieved. We acknowledge financial support from the Fundação de Amparo à Pesquisa de São Paulo. F. C. is grateful to Departamento de Física, Universidade Federal de São Carlos, for hospitality. I.B.Levinson, Fiz.Tverd.Tela [6]{},2113(1965) \[Sov.Phys.Solid State [6]{},1665 (1965)\]. T.Kurosawa,J.Phys.Jpn. [20]{}, 937 (1965). E.Bringuier, Phys.Rev. [B 52]{}, 8092 (1995). E.Bringuier, Phys.Rev. [B 54]{}, 1799 (1996). E.Bringuier, Phil.Magazine [B 77]{}, 959 (1998). E.Bringuier, Am.J. of Phys. [66 ]{}, 995 (1998). E.Bringuier, Phys. Rev. [B 49]{}, 7974 (1994). B.K.Ridley , J.Phys. C [16]{}, 3373, (1983). C.Jacoboni and L.Reggiani,Rev.of Mod. Phys. [55]{}, 645 (1983). C.Jacoboni, R.Minder and G.Majni, J.Phys.Chem of Solids [36]{}, 1129 (1975). C.Canali, C.Jacoboni, F.Nava, G.Ottaviani and A.Alberigi-Quaranta, Phys.Rev [B 12]{}, 2265 (1975). Permanent Address: Depto. de Física Teórica, Universidad. de la Habana, Vedado 10400, Havana, Cuba.
--- abstract: 'We study Dirac neutrinos propagating in rotating background matter. First we derive the Dirac equation for a single massive neutrino in the noninertial frame, where matter is at rest. This equation is written in the effective curved space-time corresponding to the corotating frame. We find the exact solution of the Dirac equation. The neutrino energy levels for ultrarelativistic particles are obtained. Then we discuss several neutrino mass eigenstates, with a nonzero mixing between them, interacting with rotating background matter. We derive the effective Schrödinger equation governing neutrino flavor oscillations in rotating matter. The new resonance condition for neutrino oscillations is obtained. We also examine the correction to the resonance condition caused by the matter rotation.' address: \| Physics Faculty, National Research Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia;\ Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN), 142190 Troitsk, Moscow, Russia;\ Institute of Physics, University of São Paulo, CP 66318, CEP 05315-970 São Paulo, SP, Brazil\ $^$E-mail: maxdvo@izmiran.ru author: - 'Maxim Dvornikov$^$' title: Neutrino interaction with background matter in a noninertial frame --- Introduction ============ Nowadays it is commonly believed that neutrinos are massive particles and there is a nonzero mixing between different neutrino generations. These neutrino properties result in transitions between neutrino flavors, or neutrino oscillations. It is also known that various external fields, like electroweak interaction of neutrinos with background fermions, neutrino electromagnetic interaction, and neutrino interaction with a strong gravitational field, can also influence the process of neutrino oscillations. As shown in Ref. , noninertial effects in accelerated and rotating frames can affect neutrino propagation and oscillations. The consideration of the reference frame rotation is particularly important for astrophysical neutrinos emitted by a rapidly rotating compact star, e.g., a pulsar. For example, the possibility of the pulsar spin down by the neutrino emission and interaction with rotating matter was recently discussed in Ref. . It should be noted that besides elementary particle physics, various processes in noninertial frames are actively studied in condensed matter physics. For instance, the enhancement of the spin current in a semiconductor moving with an acceleration was recently predicted in Ref. . In the present work we summarize the results of Ref. , where the neutrino interaction with background matter in a rotating frame was studied. We assume that neutrinos can interact with background fermions by means of electroweak forces. We also take that neutrino mass eigenstates are Dirac particles. In our treatment we account for the noninertial effects since we find the exact solution of a Dirac equation for a neutrino moving in the curved space-time with a metric corresponding a rotating frame. This work is organized in the following way. In Sec. \[sec:NUMATFLS\], we start with the brief description of the neutrino interaction with background matter in Minkowski space-time. We consider matter moving with a constant velocity and discuss both neutrino flavor and mass eigenstates. Then, in Sec. \[sec:MASSNUNONIF\], we study background matter moving with an acceleration. The Dirac equation, including noninertial effects, for a neutrino interacting with such matter is written down in a frame where matter is at rest. In Sec. \[sub:ROTATION\], we solve the Dirac equation and find the neutrino energy spectrum for ultrarelativistic neutrinos moving in matter rotating with a constant angular velocity. Then, in Sec. \[sec:OSC\], we apply our results for the description of neutrino oscillations in rotating background matter. The effective Schrödinger equation governing neutrino oscillations is derived and the new resonance condition is obtained. We consider how the matter rotation can affect the resonance in neutrino oscillation in a realistic astrophysical situation. Finally, in Sec. \[sec:SUMMARY\], we summarize our results. Neutrino interaction with background matter\[sec:NUMATFLS\] =========================================================== In this section we describe the interaction of different neutrino flavors with background matter in a flat space-time. We discuss a general case of matter moving with a constant mean velocity and having a mean polarization. Then we consider the matter interaction of neutrino mass eigenstates, which are supposed to be Dirac particles. The interaction of the neutrino flavor eigenstates $\nu_{\alpha}$, $\alpha=e,\mu,\tau$, with background matter in the flat space-time is described by the following effective Lagrangian[@GiuKim07]: $$\label{eq:Lf} \mathcal{L}_{\mathrm{eff}} = -\sum_{\alpha}\bar{\nu}_{\alpha}\gamma_{\mu}^{\mathrm{L}}\nu_{\alpha}\cdot f_{\alpha}^{\mu},$$ where $\gamma_{\mu}^{\mathrm{L}}=\gamma_{\mu}(1-\gamma^{5})/2$, $\gamma^{\mu}=(\gamma^{0},\bm{\gamma})$ are the Dirac matrices, and $\gamma^{5}=\mathrm{i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. The interaction Lagrangian in Eq. (\[eq:Lf\]) is derived in the mean field approximation using the effective external currents $f_{\alpha}^{\mu}$ depending on the characteristics of background matter as[@DvoStu02] $$\label{eq:flavnueffp} f_{\alpha}^{\mu} = \sqrt{2}G_{\mathrm{F}}\sum_{f} \left( q_{\alpha,f}^{(1)}j_{f}^{\mu}+q_{\alpha,f}^{(2)}\lambda_{f}^{\mu} \right),$$ where $G_{\mathrm{F}}$ is the Fermi constant and the sum is taken over all background fermions $f$. Here $$\label{eq:jms} j_{f}^{\mu}=n_{f}u_{f}^{\mu},$$ is the hydrodynamic current and $$\label{eq:lambdams} \lambda_{f}^{\mu} = n_{f} \left( (\bm{\zeta}_{f}\mathbf{u}_{f}), \bm{\zeta}_{f}+\frac{\mathbf{u}_{f}(\bm{\zeta}_{f}\mathbf{u}_{f})}{1+u_{f}^{0}} \right),$$ is the four vector of the matter polarization. In Eqs. (\[eq:jms\]) and (\[eq:lambdams\]), $n_{f}$ is the invariant number density (the density in the rest frame of fermions), $\bm{\zeta}_{f}$ is the invariant polarization (the polarization in the rest frame of fermions), and $u_{f}^{\mu}=\left(u_{f}^{0},\mathbf{u}_{f}\right)$ is the four velocity. To derive Eqs. (\[eq:flavnueffp\])-(\[eq:lambdams\]) it is crucial that background fermions have constant velocity. Only in this situation one can make a boost to the rest frame of the fermions where $n_{f}$ and $\bm{\zeta}_{f}$ are defined. The explicit form of the coefficients $q_{\alpha,f}^{(1,2)}$ in Eq. (\[eq:flavnueffp\]) can be found in Ref. . Nowadays it is experimentally confirmed that the flavor neutrino eigenstates are the superposition of the neutrino mass eigenstates, $\psi_{i}$, $i=1,2,\dotsc$, $$\label{eq:nupsi} \nu_{\alpha}=\sum_{i}U_{\alpha i}\psi_{i},$$ where $\left(U_{\alpha i}\right)$ is the unitary mixing matrix. The transformation in Eq. (\[eq:nupsi\]) diagonalizes the neutrino mass matrix. Only using the neutrino mass eigenstates we can reveal the nature of neutrinos, i.e. say whether they are Dirac or Majorana particles. Despite the great experimental efforts to shed light upon the nature of neutrinos, this issue still remains open. Here we shall suppose that $\psi_{i}$ correspond to Dirac fields. The effective Lagrangian for the interaction of $\psi_{i}$ with background matter can be obtained using Eqs. (\[eq:Lf\]) and (\[eq:nupsi\]), $$\label{eq:Lg} \mathcal{L}_{\mathrm{eff}} = - \sum_{ij} \bar{\psi}_{i}\gamma_{\mu}^{\mathrm{L}}\psi_{j}\cdot g_{ij}^{\mu},$$ where $$\label{eq:gab} g_{ij}^{\mu} = \sum_{\alpha}U_{\alpha i}^{}U_{\alpha j}f_{\alpha}^{\mu},$$ is the nondiagonal effective potential in the mass eigenstates basis. Using Eq. (\[eq:Lg\]) one obtains that the corresponding Dirac equations for the neutrino mass eigenstates are coupled, $$\label{eq:Depsi} \left[ \mathrm{i}\gamma^{\mu}\partial_{\mu}-m_{i}-\gamma_{\mu}^{\mathrm{L}}g_{ii}^{\mu} \right] \psi_{i} = \sum_{j\neq i}\gamma_{\mu}^{\mathrm{L}}g_{ij}^{\mu}\psi_{j},$$ where $m_{i}$ is the mass of $\psi_{i}$. One can proceed in the analytical analysis of Eq. (\[eq:Depsi\]) if we exactly account for only the diagonal effective potentials $g_{ii}^{\mu}$. To take into account the r.h.s. of Eq. (\[eq:Depsi\]), depending on the nondiagonal elements of the matrix $\left(g_{ij}^{\mu}\right)$, with $i\neq j$, one should apply a perturbative method (see Sec. \[sec:OSC\] below). Massive neutrinos in noninertial frames\[sec:MASSNUNONIF\] ========================================================== In this section we generalize the Dirac equation for a neutrino interacting with a background matter to the situation when the velocity of the matter motion is not constant. In particular, we study the case of the matter rotation with a constant angular velocity. Then we obtain the solution of the Dirac equation and find the energy spectrum. If we discuss a neutrino mass eigenstate propagating in a nonuniformly moving matter, the expressions for $f_{\alpha}^{\mu}$ in Eqs. (\[eq:flavnueffp\])-(\[eq:lambdams\]) become invalid since they are derived under the assumption of the unbroken Lorentz invariance. The most straightforward way to describe the neutrino evolution in matter moving with an acceleration is to rewrite the Dirac equation for a neutrino in the noninertial frame where matter is at rest. In this case one can unambiguously define the components of $f_{\alpha}^{\mu}$. Assuming that background fermions are unpolarized, we find that in this reference frame $$\label{eq:f0nonin} f_{\alpha}^{0}=\sqrt{2}G_{\mathrm{F}}\sum_{f}q_{\alpha,f}^{(1)}n_{f}\neq0,$$ with the rest of the effective potentials being equal to zero. It is known that the motion of a test particle in a noninertial frame is equivalent to the interaction of this particle with a gravitational field. The Dirac equation for a massive neutrino moving in a curved space-time and interacting with background matter can be obtained by the generalization of Eq. (\[eq:Depsi\]) (see also Ref. ), $$\label{eq:Depsicurv} \left[ \mathrm{i}\gamma^{\mu}(x)\nabla_{\mu}-m \right] \psi = \frac{1}{2}\gamma_{\mu}(x)g^{\mu} \left[ 1-\gamma^{5}(x) \right] \psi,$$ where $\gamma_{\mu}(x)$ are the coordinate dependent Dirac matrices, $\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu}$ is the covariant derivative, $\Gamma_{\mu}$ is the spin connection, $\gamma^{5}(x) = -\tfrac{\mathrm{i}}{4!} E^{\mu\nu\alpha\beta} \gamma_{\mu}(x) \gamma_{\nu}(x) \gamma_{\alpha}(x) \gamma_{\beta}(x)$, $E^{\mu\nu\alpha\beta} = \tfrac{1}{\sqrt{-g}} \varepsilon^{\mu\nu\alpha\beta}$ is the covariant antisymmetric tensor in curved space-time, and $g=\det(g_{\mu\nu})$ is the determinant of the metric tensor $g_{\mu\nu}$. Note that in Eq. (\[eq:Depsicurv\]) we account for only the diagonal neutrino interaction with matter. That is why we omit the index $i$ in order not to encumber the notation: $m\equiv m_{i}$ etc. It should be noted that analogous Dirac equation was discussed in Ref. . We shall be interested in the neutrino motion in matter rotating with the constant angular velocity $\omega$. Choosing the corotating frame we get that only $g^{0} \equiv g^{0}_{ii}$ is nonvanishing, cf. Eqs. (\[eq:gab\]) and (\[eq:f0nonin\]). Neutrino motion in a rotating frame\[sub:ROTATION\] --------------------------------------------------- The interval in the rotating frame is[@LanLif94] $$\label{eq:mertrot} \mathrm{d}s^{2} = g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu} = (1-\omega^{2}r^{2})\mathrm{d}t^{2} - \mathrm{d}r^{2}-2\omega r^{2}\mathrm{d}t\mathrm{d}\phi - r^{2}\mathrm{d}\phi^{2}-\mathrm{d}z^{2},$$ where we use the cylindrical coordinates $x^{\mu}=(t,r,\phi,z)$. One can check that the metric tensor in Eq. (\[eq:mertrot\]) can be diagonalized, $\eta_{ab}=e_{a}^{\ \mu}e_{b}^{\ \nu}g_{\mu\nu}$, if we use the following vierbein vectors: $$\begin{aligned} \label{eq:vierb} e_{0}^{\ \mu}= & \left( \frac{1}{\sqrt{1-\omega^{2}r^{2}}},0,0,0 \right), \nonumber \\ e_{1}^{\ \mu}= & (0,1,0,0), \nonumber \\ e_{2}^{\ \mu}= & \left( \frac{\omega r}{\sqrt{1-\omega^{2}r^{2}}},0,\frac{\sqrt{1-\omega^{2}r^{2}}}{r},0 \right), \nonumber \\ e_{3}^{\ \mu}= & (0,0,0,1).\end{aligned}$$ Here $\eta_{ab}=\text{diag}(1,-1,-1,-1)$ is the metric in a locally Minkowskian frame. Let us introduce the Dirac matrices in a locally Minkowskian frame by $\gamma^{\bar{a}}=e_{\ \mu}^{a}\gamma^{\mu}(x)$, where $e_{\ \mu}^{a}$ is the inverse vierbein: $e_{\ \mu}^{a}e_{b}^{\ \mu}=\delta_{b}^{a}$. Starting from now, we shall mark an index with a bar to demonstrate that a gamma matrix is defined in a locally Minkowskian frame. As shown in Ref. , $\gamma^{5}(x) = \mathrm{i} \gamma^{\bar{0}} \gamma^{\bar{1}} \gamma^{\bar{2}} \gamma^{\bar{3}} = \gamma^{\bar{5}}$ does not depend on coordinates. After the straightforward calculation of the spin connection on the basis of Eq. (\[eq:vierb\]), the Dirac Eq. (\[eq:Depsicurv\]) can be rewritten as $$\begin{aligned} \label{eq:Derotf} [\mathcal{D} - & m]\psi = \frac{1}{2} \sqrt{1-\omega^{2}r^{2}}\gamma^{\bar{0}}g^{0} (1-\gamma^{\bar{5}})\psi, \nonumber \\ \mathcal{D} = & \mathrm{i} \frac{\gamma^{\bar{0}}+\omega r\gamma^{\bar{2}}}{\sqrt{1-\omega^{2}r^{2}}} \partial_{0} + \mathrm{i} \gamma^{\bar{1}} \left( \partial_{r}+\frac{1}{2r} \right) + \mathrm{i} \gamma^{\bar{2}} \frac{\sqrt{1-\omega^{2}r^{2}}}{r}\partial_{\phi} + \mathrm{i} \gamma^{\bar{3}}\partial_{z} \notag \\ & - \frac{\omega}{2(1-\omega^{2}r^{2})} \gamma^{\bar{3}}\gamma^{\bar{5}}.\end{aligned}$$ The analogous Dirac equation was recently derived in Ref. . Since Eq. (\[eq:Derotf\]) does not explicitly contain $t$, $\phi$, and $z$, its solution can be expressed as $$\label{eq:psitildepsi} \psi = \exp \left( -\mathrm{i}Et+\mathrm{i}J_{z}\phi+\mathrm{i}p_{z}z \right) \psi_{r},$$ where $\psi_{r}=\psi_{r}(r)$ is the spinor depending on the radial coordinate, $J_{z}=\tfrac{1}{2}-l$ (see, e.g., Ref. ), and $l=0,\pm1,\pm2,\dotsc$. In Eq. (\[eq:Derotf\]) one can neglect terms $\sim(\omega r)^{2}$. Indeed, if we study a neutrino in a rotating pulsar, then $r\lesssim10\thinspace\text{km}$ and $\omega\lesssim10^{3}\thinspace\text{s}^{-1}$. Thus $(\omega r)^{2}\lesssim1.1\times10^{-3}$ is a small parameter. Therefore Eq. (\[eq:Derotf\]) can be transformed to $$\begin{gathered} \label{eq:Depsisimp} \bigg[ \mathrm{i}\gamma^{\bar{1}} \left( \partial_{r}+\frac{1}{2r} \right) - \gamma^{\bar{2}} \left( \frac{J_{z}}{r}-\omega r E \right) + \gamma^{\bar{0}} \left( E-\frac{g^{0}}{2} \right) - \gamma^{\bar{3}}p_{z} \\ + \frac{g^{0}}{2}\gamma^{\bar{0}}\gamma^{\bar{5}} - \frac{\omega}{2}\gamma^{\bar{3}}\gamma^{\bar{5}} - m \bigg] \psi_{r}=0,\end{gathered}$$ where we keep only the terms linear in $\omega$. It should be noted that the term $\sim\omega\gamma^{\bar{3}}\gamma^{\bar{5}}$ in Eq. (\[eq:Depsisimp\]) is equivalent to the neutrino interaction with matter moving with an effective velocity. The solution of Eq. can be presented in the form[@Dvo14], $\psi_{r}^{\mathrm{L}} = (0,\eta)^\mathrm{T}$ and $\psi_{r}^{\mathrm{R}} = (\xi,0)^\mathrm{T}$, where $$\label{eq:etaxi} \eta = \left( \begin{array}{c} -\mathrm{i}C_{1}I_{N,s} \\ C_{2}I_{N-1,s} \end{array} \right), \quad \xi = \left( \begin{array}{c} C_{3}I_{N,s} \\ -\mathrm{i}C_{4}I_{N-1,s} \end{array} \right).$$ Here $N=0,1,2,\dotsc$, $s=N-l$, $I_{N,s}=I_{N,s}(\rho)$ is the Laguerre function, and $\rho = E \omega r^2$. The explicit form of the Laguerre function can be found, e.g., in Ref. . To derive Eq. we use the Dirac matrices in the chiral representation[@ItzZub80]. In the important case when $\omega\ll g^{0}$, the coefficients $C_i$, $i=1,\dots,4$, in Eq. are expressed in the following way[@Dvo14]: $$\begin{aligned} \label{eq:C1-4slowrot} C_{1}^{2} \approx & \frac{E_{A}\omega}{2\pi}\frac{E_{A}-p_{z}-g^{0}}{E_{A}-g^{0}}, \quad C_{3}^{2} \approx \frac{\omega}{2\pi} \left( E_{S}+p_{z} \right), \nonumber \\ C_{2}^{2} \approx & \frac{E_{A}\omega}{2\pi}\frac{E_{A}+p_{z}-g^{0}}{E_{A}-g^{0}}, \quad C_{4}^{2} \approx \frac{\omega}{2\pi} \left( E_{S}-p_{z} \right).\end{aligned}$$ It should be noted that the solutions presented in Eqs. (\[eq:etaxi\]) and (\[eq:C1-4slowrot\]) satisfy the normalization condition, $$\label{eq:normgen} \int\psi_{N,s,p_{z}}^{\dagger}(x) \psi_{N',s',p'_{z}}(x)\sqrt{-g}\mathrm{d}^{3}x = \delta_{NN'}\delta_{ss'} \delta \left( p_{z}-p'_{z} \right).$$ Here $\psi$ and $\psi_{r}$ are related by Eq. (\[eq:psitildepsi\]). The energy levels in Eq. are $$\begin{aligned} \label{eq:Energylev} \left[ E_{A}-2N\omega-g^{0} \right]^{2} = & (2N\omega)^{2}+4N\omega g^{0} + \left( p_{z}-\frac{\omega}{2} \right)^{2}, \nonumber \\ \left[ E_{S}-2N\omega \right]^{2} = & (2N\omega)^{2} + \left( p_{z}+\frac{\omega}{2} \right)^{2},\end{aligned}$$ where $E_{A}$ and $E_{S}$ are the energies of active and sterile neutrinos respectively. Comparing the expression for $E_{S}\approx2N\omega+\sqrt{(2N\omega)^{2}+p_{z}^{2}}$ with the energy of a neutrino in an inertial nonrotating frame $\sqrt{\mathbf{p}_{\perp}^{2}+p_{z}^{2}}$, where $\mathbf{p}_{\perp}$ is the momentum in the equatorial plane, we can identify $2N\omega$ inside the square root as $\|\mathbf{p}_{\perp}\|$. It should be also noted that the term $2N\omega$, which additively enters to both $E_{A}$ and $E_{S}$, is due to the noninertial effects for a Dirac fermion in a rotating frame[@HehNi90]. We can also get the corrections to the energy levels due to the nonzero mass, $E_{A,S}\to E_{A,S}+E_{A,S}^{(1)}$. On the basis of Eq. (\[eq:etaxi\]) and (\[eq:C1-4slowrot\]) one finds the expression for $E_{A,S}^{(1)}$ in the limit $\omega\ll g^{0}$, $$\label{eq:EAS1} E_{A}^{(1)} = \frac{m^{2}}{2 \left( E_{A}-2N\omega-g^{0} \right) }, \quad E_{S}^{(1)} = \frac{m^{2}}{2 \left( E_{S}-2N\omega \right) }.$$ If we discuss neutrinos moving along the rotation axis, then $2N\omega\ll\|p_{z}\|$. Using Eq. (\[eq:Energylev\]) we get the energy levels of active neutrinos in this case $$\label{eq:Eadecom} E_{A} = \|p_{z}\|+g^{0} \left( 1+\frac{2N\omega}{\|p_{z}\|} \right) + 2N\omega + \frac{2(N\omega)^{2}}{\|p_{z}\|} + \frac{m^{2}}{2\|p_{z}\|},$$ where we also keep the mass correction in Eq. (\[eq:EAS1\]). One can see in Eq. (\[eq:Eadecom\]) that $\|p_{z}\|+g^{0}+\frac{m^{2}}{2\|p_{z}\|}$ corresponds to the energy of a left-handed neutrino interacting with background matter in a flat space-time. The rest of the terms in Eq. (\[eq:Eadecom\]) are the corrections due to the matter rotation. Flavor oscillations of Dirac neutrinos in rotating matter\[sec:OSC\] ==================================================================== In this section we study the evolution of the system of massive mixed neutrinos in rotating matter. We formulate the initial condition for this system and derive the effective Schrödinger equation which governs neutrino flavor oscillations. Then we find the correction to the resonance condition owing to the matter rotation and estimate its value for a millisecond pulsar. We can generalize the results of Sec. \[sec:MASSNUNONIF\] to include different neutrino eigenstates. The interaction of neutrino mass eigenstates with background matter is nondiagonal, cf. Eq. (\[eq:Lg\]). Therefore the generalization of Eq. (\[eq:Derotf\]) for several mass eigenstates $\psi_{i}$ reads $$\label{eq:Derotfpsia} \left[ \mathcal{D}-m_{i} \right] \psi_{i} = \frac{1}{2}\gamma^{\bar{0}} g_{i}^{0} (1-\gamma^{\bar{5}}) \psi_{i} + \frac{1}{2}\gamma^{\bar{0}} \sum_{j\neq i} g_{ij}^{0}(1-\gamma^{\bar{5}})\psi_{j},$$ where $g_{i}^{0}\equiv g_{ii}^{0}$ and $g_{ij}^{0}$ are the time components of the matrix $\left(g_{ij}^{\mu}\right)$ given in Eq. (\[eq:gab\]), $m_{i}$ is the mass of $\psi_{i}$, and $\mathcal{D}$ can be found in Eq. (\[eq:Derotf\]). As in Sec. \[sec:MASSNUNONIF\], we omitted the term $(\omega r)^{2}\ll1$ in Eq. (\[eq:Derotfpsia\]). Note that Eq. (\[eq:Derotfpsia\]) is a generalization of Eq. (\[eq:Depsi\]) for a system of the neutrino mass eigenstates moving in a rotating frame. We shall study the evolution of active ultrarelativistic neutrinos and neglect neutrino-antineutrino transitions. In this case we can restrict ourselves to the analysis of two component spinors. The general solution of Eq. (\[eq:Derotfpsia\]) has the form, $$\label{eq:gensolosc} \eta_{i}(x) = \sum_{N,s} \int\frac{\mathrm{d}p_{z}}{\sqrt{2\pi}} a_{N,s,p_{z}}^{(i)} e^{\mathrm{i}p_{z}z+\mathrm{i}J_{z}\phi} u_{N,s,p_{z}}(r) e^{-\mathrm{i}E_{i}t},$$ where $u_{N,s,p_{z}}$ are the basis spinors and $a_{N,s,p_{z}}^{(i)} = a_{N,s,p_{z}}^{(i)}(t)$ are the $c$-number functions. The energy levels $E_{i}$ are given in Eq. (\[eq:Eadecom\]) with $m\to m_{i}$. Here we omit the subscript $A$ in order not to encumber the notation. Our goal is to find the coefficient $a_{N,s,p_{z}}^{(i)}=a_{N,s,p_{z}}^{(i)}(t)$. We neglect the small ratio $\omega/g_{i}^{0}$ in Eq. (\[eq:gensolosc\]). Considering the system of two neutrino mass eigenstates, $i=1,2$, parameterized with one mixing angle $\theta$, and choosing the appropriate initial condition[@Dvo14], on the basis of Eq. (\[eq:Derotfpsia\]) we get the effective Schrödinger equation for $\tilde{\Psi}^{\mathrm{T}}=(a_{1},a_{2})$,$$\label{eq:ScheqtildePsi} \mathrm{i}\frac{\mathrm{d}\tilde{\Psi}}{\mathrm{d}t} = \left( \begin{array}{cc} 0 & g_{12}^{0} \exp \left[ \mathrm{i} \left(E_{1}-E_{2} \right) t \right] \\ g_{12}^{0} \exp \left[ \mathrm{i} \left( E_{2}-E_{1} \right) t \right] & 0 \end{array} \right) \tilde{\Psi}.$$ Here we omitted all the indexes of $a_{i}$ besides $i=1,2$. It is convenient to introduce the modified effective wave function $\Psi=\mathcal{U}_{3}\tilde{\Psi}$, where $\mathcal{U}_{3}=\text{diag}\left(e^{\mathrm{i}\Omega t/2},e^{-\mathrm{i}\Omega t/2}\right)$, $\Omega=E_{1}-E_{2}$. Using Eq. (\[eq:ScheqtildePsi\]), we get for $\Psi$ $$\label{eq:ScheqPsi} \mathrm{i}\frac{\mathrm{d}\Psi}{\mathrm{d}t} = \left( \begin{array}{cc} \Omega/2 & g_{12}^{0} \\ g_{12}^{0} & -\Omega/2 \end{array} \right) \Psi.$$ Note that Eq. (\[eq:ScheqPsi\]) has the form of the effective Schrödinger equation one typically deals with in the study of neutrino flavor oscillations in background matter. If the transition probability for $\nu_{\alpha}\leftrightarrow\nu_{\beta}$ is close to one, i.e. $P_{\nu_{\beta}\to\nu_{\alpha}}=\left\|\left\langle \nu_{\alpha}(t)\|\nu_{\beta}(0)\right\rangle \right\|^{2}\approx1$, flavor oscillations of neutrinos are said to be at resonance. Using Eqs. (\[eq:nupsi\]), (\[eq:f0nonin\]), (\[eq:Eadecom\]), and , the resonance condition can be written as, $$\label{eq:rescondgen} \left( f_{\alpha}^{0}-f_{\beta}^{0} \right) \left( 1+\frac{2N\omega}{\|p_{z}\|} \right) + \frac{\Delta m^{2}}{2\|p_{z}\|}\cos2\theta=0,$$ where $\Delta m^{2}=m_{1}^{2}-m_{2}^{2}$ is the mass squared difference. Let us consider electroneutral background matter composed of electrons, protons, and neutrons. If we study the $\nu_{e}\to\nu_{\alpha}$ oscillation channel, where $\alpha=\mu,\tau$, we get that $f_{\nu_{\alpha}}^{0}=-\tfrac{1}{\sqrt{2}}G_{\mathrm{F}}n_{n}$ and $f_{\nu_{\beta}}^{0}\equiv f_{\nu_{e}}^{0}=\sqrt{2}G_{\mathrm{F}}\left(n_{e}-\tfrac{1}{2}n_{n}\right)$, where $n_{e}$ and $n_{n}$ are the densities of electrons and neutrons. Using Eq. (\[eq:rescondgen\]), we obtain that $$\label{eq:resnuenumu} \sqrt{2}G_{\mathrm{F}}n_{e} \left( 1+\frac{2N\omega}{\|p_{z}\|} \right) = \frac{\Delta m^{2}}{2\|p_{z}\|}\cos2\theta.$$ At the absence of rotation, $\omega=0$, Eq. (\[eq:resnuenumu\]) is equivalent to the Mikheyev-Smirnov-Wolfenstein resonance condition in background matter[@BleSmi13]. Let us evaluate the contribution of the matter rotation to the resonance condition in Eq. (\[eq:resnuenumu\]) for a neutrino emitted inside a rotating pulsar. We make a natural assumption that for a corotating observer neutrinos are emitted in a spherically symmetric way from a neutrinosphere. That is we should take that $l\approx0$ and $N\approx s$. Then the trajectory of a neutrino is deflected because of the noninertial effects and the interaction with background matter. The radius $\mathcal{R}$ of the trajectory can be found from $$\label{eq:trajrad} \mathcal{R}^{2} = 2\|p_{z}\|\omega \int_{0}^{\infty} r^{2}\|u_{N,s,p_{z}}(r)\|^{2}r\mathrm{d}r \approx \frac{2N}{\|p_{z}\|\omega},$$ where we take into account that $N\gg1$. We shall assume that $\mathcal{R}\sim R_{\mathrm{0}}$, where $R_{\mathrm{0}}=10\thinspace\text{km}$ is the pulsar radius. In this case neutrinos escape a pulsar. Taking that $\omega=10^{3}\thinspace\text{s}^{-1}$ and using Eq. (\[eq:trajrad\]), we get that the correction to the resonance condition in Eq. (\[eq:resnuenumu\]) is $\frac{2N\omega}{\|p_{z}\|}\approx\left(R_{\mathrm{0}}\omega\right)^{2}\approx10^{-3}$. The obtained correction to the effective number density is small but nonzero. This result corrects our previous statement[@DvoDib10] that a matter rotation does not contribute neutrino flavor oscillations. Conclusion\[sec:SUMMARY\] ========================= In conclusion we notice that we have studied the evolution of massive mixed neutrinos in nonuniformly moving background matter. The interaction of neutrinos with background fermions is described in frames of the Fermi theory (see Sec. \[sec:NUMATFLS\]). A particular case of the matter rotating with a constant angular velocity has been studied in Sec. \[sub:ROTATION\]. We have derived the Dirac equation for a weakly interacting neutrino in a rotating frame and found its solution in case of ultrarelativistic neutrinos, cf. Eqs. (\[eq:etaxi\]) and (\[eq:C1-4slowrot\]). The energy spectrum obtained in Eqs. (\[eq:Energylev\]) and (\[eq:EAS1\]) includes the correction owing to the nonzero neutrino mass. We have used the Dirac equation in a noninertial frame, cf. Eq. (\[eq:Depsicurv\]), as a main tool for the study of the neutrino motion in matter moving with an acceleration. To develop the quantum mechanical description of such a neutrino we have chosen a noninertial frame where matter is at rest. In this frame the effective potential of the neutrino-matter interaction is well defined. However, the wave equation for a neutrino turns out to be more complicated since one has to deal with noninertial effects. In Sec. \[sec:OSC\] we have generalized our results to include various neutrino generations as well as mixing between them. We have derived the effective Schrödinger equation which governs neutrino flavor oscillations. We have obtained the correction to the resonance condition in background matter owing to the matter rotation. Studying neutrino oscillations in a millisecond pulsar, we have obtained that the effective number density changes by $0.1\thinspace\%$ owing to the matter rotation. Despite the obtained correction is small, we may suggest that our results can have some implication to the explanation of great linear velocities of pulsars. It was suggested in Ref. that an asymmetry in neutrino oscillations in a magnetized pulsar can explain a great linear velocity of the compact star. An evidence for the alignment of the angular and the linear velocity vectors of pulsars was reported in Ref. . Therefore we may suggest that neutrino flavor oscillations in a rapidly rotating pulsar can contribute to its linear velocity. It should be noted that neutrino spin-flavor oscillations, including noninertial effects, in a rapidly rotating magnetized star were studied in Ref. in the context of the explanation of high linear velocities of pulsars. Finally, we mention that the Dirac equation for a fermion, electroweakly interacting with the rotating background matter, was recently solved[@Dvo15]. The vierbein vectors, different from these in Eq. , were used in Ref. . Comparing the energy levels obtained in Ref. with the results of the general analysis[@HehNi90], one concludes that the vierbein used in Ref. is more appropriate for the description of ultrarelativistic particles like neutrinos. Acknowledgments {#acknowledgments .unnumbered} =============== I am thankful to the organizers of the International Conference on Massive Neutrinos for the invitation and support, to S. P. Gavrilov for helpful comments, to FAPESP (Brazil) for the Grant No. 2011/50309-2, to the Tomsk State University Competitiveness Improvement Program and to RFBR (research project No. 15-02-00293) for partial support. [50]{} G. Lambiase, Neutrino oscillations in non-inertial frames and the violation of the equivalence principle. Neutrino mixing induced by the equivalence principle violation, Eur. Phys. J. C* 19, 553 (2001). M. Dvornikov and C. O. Dib, Spin-down of neutron stars by neutrino emission, Phys. Rev. D 82, 043006 (2010) \[arXiv:0907.1445\]. B. Basu and D. Chowdhury, Inertial effect on spin orbit coupling and spin transport, Ann. Phys. (N.Y.) 335, 47 (2013) \[arXiv:1302.1063\]. M. Dvornikov, Neutrino interaction with matter in a noninertial frame, J. High Energy Phys. 10 (2014) 053 \[arXiv:1408.2735\]. C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics (Oxford University Press, Oxford, 2007), pp. 137–179. M. Dvornikov and A. Studenikin, Neutrino spin evolution in presence of general external fields, J. High Energy Phys. 09 (2002) 016 \[hep-ph/0202113\]. A. A. Grib, S. G. Mamaev and V. M. Mostepanenko, Quantum Effects in Intense External Fields: Methods and Results not Related to the Perturbation Theory (Atomizdat, Moscow, 1980), pp. 13–15. D. Píriz, M. Roy and J. Wudka, Neutrino oscillations in strong gravitational fields, Phys. Rev. D 54, 1587 (1996) \[hep-ph/9604403\]. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Butterworth Heinemann, Amsterdam, 1994), 4th ed., pp. 329–330. K. Bakke, Rotating effects on the Dirac oscillator in the cosmic string spacetime, Gen. Relativ. Grav. 45, 1845 (2013) \[arXiv:1307.2847\]. P. Schluter, K.-H. Wietschorke and W. Greiner, The Dirac equation in orthogonal coordinate systems: I. The local representation, J. Phys. A: Math. Gen. 16, 1999 (1983). C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980), pp. 691–696. F. W. Hehl and W.-T. Ni, Inertial effects of a Dirac particle, Phys. Rev. D 42, 2045 (1990). M. Blennow and A. Yu. Smirnov, Neutrino propagation in matter, Adv. High Energy Phys. 2013, 972485 (2013) \[arXiv:1306.2903\]. A. Kusenko and G. Segrè, Velocities of pulsars and neutrino oscillations, Phys. Rev. Lett. 77, 4872 (1996) \[hep-ph/9606428\]. S. Johnston, G. Hobbs, S. Vigeland, M. Kramer, J. M. Weisberg and A. G. Lyne, Evidence for alignment of the rotation and velocity vectors in pulsars, Mon. Not. Roy. Astron. Soc. 364, 1397 (2005) \[astro-ph/0510260\]. G. 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--- abstract: 'The morphological dependence of the luminosity function is studied using a sample containing approximately 1500 bright galaxies classified into Hubble types by visual inspections for a homogeneous sample obtained from the Sloan Digital Sky Survey (SDSS) northern equatorial stripes. Early-type galaxies are shown to have a characteristic magnitude by 0.45 mag brighter than spiral galaxies in the $r^{\ast}$ band, consistent with the ‘universal characteristic luminosity’ in the $B$ band. The shape of the luminosity function differs rather little among different morphological types: we do not see any symptoms of the sharp decline in the faint end for the luminosity function for early-type galaxies at least 2 mag fainter than the characteristic magnitude, although the faint end behaviour shows a slight decline ($\alpha{\lesssim}-1$) compared with the total sample. We also show that a rather flat faint end slope for early-type galaxies is not due to an increasing mixture of the dwarf galaxies which have softer cores. This means that there are numerous faint early-type galaxies with highly concentrated cores.' author: - 'Osamu Nakamura, Masataka Fukugita, Naoki Yasuda, Jon Loveday, Jon Brinkmann, Donald P. Schneider, Kazuhiro Shimasaku, Mark SubbaRao' title: Luminosity Function of Morphologically Classified Galaxies in the Sloan Digital Sky Survey --- Introduction ============ The origin of morphology of galaxies is a long-standing issue, which could provide a key to discerning among models of the formation of galaxies. How galaxy morphology changes as a function of the lookback time is perhaps the prime approach to this problem, and the knowledge of the morphological dependence of the local luminosity function at zero redshift is the baseline. One specific example of the issues is whether the luminosity function of elliptical galaxies obeys the Schechter-type function with a rather flat faint end (e.g., Marzke et al. 1994; Kochanek et al. 2001), or the Gaussian function as inferred by Binggeli, Sandage & Tammann (1988) and more recently by Bernardi et al. (2002a). If the latter is correct, one would envisage galaxy morphology as a bulge-luminosity sequence (Dressler & Sandage 1983; Meisels and Ostriker 1984), which in turn reveals a clue about the formation of elliptical galaxies and bulges. There are also a number of uses of the morphology-dependent luminosity function (MDLF). We mention only one example: the frequency of gravitational lensing of quasar images is approximately proportional to the luminosity density of early-type galaxies rather than that of all galaxies (Fukugita & Turner 1991). The uncertainty in the MDLF is the largest source of error in predicting the frequency of gravitational lenses, and thus in inferring the cosmological constant from such analyses. The understanding of the MDLF is significantly poorer than that of the luminosity function for galaxies in general, which has undergone substantial progress in the latest years (Folkes et al. 1999; Blanton et al. 2001) by virtue of large galaxy samples. The traditional way to obtain MDLF is to use morphological classification based on visual inspections of the images (Binggeli et al. 1988; Loveday et al. 1992; Marzke et al. 1994; 1998; Kochanek et al. 2001). Some modern studies attempt to use spectroscopic features to classify galaxies into morphological types (Bromley et al. 1998; Folkes et al. 1999), which makes it possible to analyze large samples. Although a general correlation is known between spectroscopic and Hubble morphologies, the samples derived from the two methods are considerably different. In particular, the classification using spectroscopic features or colours is sensitive to small star formation activities now or in the near past in early-type galaxies, whilst Hubble morphology is insensitive to this process. The problem of automated classification always lies in the difficulty in finding quantitative measures that strongly correlate with the Hubble sequence based on visual inspections. In this paper we derive the MDLF based on visual classifications using a homogeneous bright galaxy sample from Sloan Digital Sky Survey (SDSS; York et al. 2000). The sample we use in this paper is small, but it is based on a homogeneous morphological classification with accurate photometry. The SDSS conducts both photometric (Gunn et al. 1998; Hogg et al. 2001; Pier et al. 2002) and spectroscopic surveys, and is producing a homogeneous data set, which is suitable to studies of galaxy statistics. The initial survey observations were made in the northern and southern equatorial stripes, and produced a galaxy catalogue to $r^{\ast}=22.5$ mag in five colour bands (Fukugita et al. 1996) with a photometric calibration using a new standard star network observed at USNO (Smith et al. 2002). Spectroscopic follow-up is made to 17.8 mag with accurately defined criteria for target selection (Strauss et al. 2002). Our study is limited to bright galaxies with $r^{\ast}\leq 15.9$ mag after Galactic extinction correction, since visual classifications sometimes cannot be made confidently beyond this magnitude with the SDSS imaging data. We have classified all galaxies satisfying this magnitude criterion in the northern equatorial stripe. The total number of galaxies in our sample is 1875, of which 1600 have spectroscopic information. The dominant part of the data we used are already published as an [Early Data Release]{} (EDR) (Stoughton et al. 2002). Our present work uses primarily the EDR but supplemented by observations which are not included in EDR to make the sample as complete as possible. Photometry of galaxies in this region is discussed in a galaxy number count paper of Yasuda et al. (2001), and the luminosity function is derived by Blanton et al. (2001), which also discuss spectroscopic details. The sample and the morphology classification ============================================ The region of the sky we consider is the northern equatorial stripe (SDSS photometry run numbers 752 and 756) for $145.15^\circ\leq \alpha({\rm J2000})\leq 235.97^\circ$ and $\|\delta({\rm J2000})\|\leq 1.27 ^\circ$, which is included in the EDR sample. The total area is 229.7 square deg. We apply Galactic extinction correction using the extinction map of Schlegel, Finkbeiner & Davis (1998) assuming $R_{r^{\ast}}=A_{r^{\ast}}/E(B-V)=2.75$, and select galaxies with the Petrosian magnitude ${r^{\ast}}_P\leq 15.9$ after the correction in the automatically-generated photometric catalogue (Stoughton et al. 2002). We use the extinction-corrected Petrosian magnitude throughout this paper. The photometric catalogue yields 2418 galaxy candidates with ${r^{\ast}}_P\leq 15.9$ if we follow the criteria given in Strauss et al. (2002). This sample still contains number of double stars and shredded galaxies due to deblending failures, which cannot be rejected by the automated algorithm. We obtain after visual inspection of all galaxy candidates 1875 galaxies, of which 1600 (85%) are included in the spectroscopic sample. Spectroscopy was made using 50 plugged plates with additional 41 plates that are centred in the neighbouring stripes. These plates cover 228.1 square deg. The confidence level for the redshift determination is mostly over 99%, but 9 galaxies are given low ($<85$%) confidence, which we omit from our sample. We also drop 38 galaxies which either contain multiple galaxies or have poor photometry due to deblending failures. This leaves 1553 galaxies. We note that there are some galaxies which are dropped in the primary galaxy selection in the photometric catalogue (Yasuda et al. 2001; Strauss et al. 2002) due to saturation flags caused by nearby bright stars or to other reasons. We estimate that we have probably missed about $\approx$88 galaxies in our field from the rate of missed galaxies given in Yasuda et al. So the overall sample completeness is estimated to be 79.5%. For more detailed discussion for the spectroscopic sample, see Blanton et al. (2001). All galaxies in our sample (1875) are classified into 7 morphological classes, $T=0$ (corresponding to E in the Hubble type), 1 (S0), 2 (Sa), 3 (Sb), 4 (Sc), 5 (Sd), and 6 (Im). Morphology classification is carried out by two of us (MF and ON) using the $g^{\ast}$ band image of each galaxy displayed on the SAOimage viewer, according to [Hubble Atlas of Galaxies]{} (Sandage 1961). We also refer to morphological types given by the [Third Reference Catalogue of Bright Galaxies]{} (de Vaucouleurs et al. 1991; RC3), so that our classification closely matches to the traditional scheme, although the RC3 classification, which is based on the photographic material, is occasionally incorrect when galaxies are viewed with the CCD image, with which we can look at the image with different levels of brightness and contrast. We give an index of $-1$ when we cannot assign a morphological type. The classification by the two independent visual inspections agrees to within $\Delta T\leq 1.5$ for most galaxies and a mean (0.5 step in $T$) is taken for our final classification. We reclassify galaxies into three groups of $T$, $0\le T\le 1.0$ (E-S0), $1.5\le T\le 3$ (S0/a-Sb), and $3.5 \le T\le 5$ (Sbc-Sd). The morphological distributions of the galaxies in the different samples are given in Table 1. The ratio of E-S0:S0/a-Sb:Sbc-Sd:Im $\simeq$ 0.40:0.34:0.24:0.02. This is a somewhat larger fraction of E-S0 compared to the value usually adopted due to our use of $r^{\ast}$ colour as the prime passband. For the same reason the fraction of Im galaxies is smaller by a factor of 2-3 than that from $B$ selected samples. In this work we do not divide the morphology into further detailed classes considering the uncertainty in visual classification, especially between E and S0 for fainter galaxies. We consider $T>5$ galaxies separately, since the spectroscopic target selection is biased against low surface brightness galaxies, and the relatively low quality of photometry for this class of galaxies makes the incompleteness significant; the completeness fraction of Im galaxies as read from Table 1 is only 54%, which is compared with $\approx$83% for other classes of galaxies. Along with a small Im fraction, our sample for an appropriate redshift range is too small to derive a reliable MDLF for Im galaxies. Morphology-Dependent Luminosity Functions ========================================= We show in Figure 1 the differential number counts of galaxies. The slope of the counts is slightly steeper than that of the Euclidean value, and is in agreement with previous studies (Yasuda et al. 2001). The counts of spectroscopic galaxies (indicated by the dashed curve) follow those of the photometric sample within one sigma of Poisson statistics, so that the completeness correction for the sample of the present paper does not depend on brightness. We use the recession velocity with respect to the Galactic Standard of Rest according to RC3. We select galaxies in the redshift range 3000 km s$^{-1}<cz< 36000$ km s$^{-1}$. The lower cutoff is imposed to avoid large effects from peculiar velocity flow, and the upper cutoff is practically the limit of our sample. We further impose a cut on apparent magnitude as $r^{\ast}\geq 13.2$ mag, since very bright galaxies are often dropped from spectroscopic targets. These selections exclude 71 galaxies from our sample, leaving 1482 galaxies used to estimate the MDLF. The redshift distributions of our galaxy sample are shown in Figure 2, where the curves show expectations for a homogeneous universe with the MDLF derived in this paper. We compute MDLFs for the samples with three methods: maximum-likelihood (ML) (Sandage, Tammann & Yahil 1979), step-wise maximum-likelihood (SWML) (Efstathiou, Ellis & Peterson 1988) and the $V_{\rm max}$ method. We take the step of luminosity to be 0.25 mag for SWML. We adopt $\Omega=0.3$ and $\lambda=0.7$ for cosmology, although the maximum redshift of our sample is $z=0.12$ and the results hardly depend on the cosmological parameters. The $K$ correction is taken from Fukugita, Shimasaku & Ichikawa (1995) with an interpolation with respect to $g^{\ast}-r^{\ast}$ colour for each galaxy. The results from the first two methods, ML and SWML, show a good agreement, but those from the $V_{\rm max}$ method differ from the former two in the faint end. This is a well-known effect generally ascribed to inhomogeneous galaxy distributions in the redshift space, as are visible in Figure 2. In Figure 3 we present the MDLF from ML and SWML in the $r^{\ast}$ passband, together with the absolute magnitude distribution of galaxies used in the analysis. The ML estimate assumes the Schechter function $$\phi(L)dL=\phi^\left({L\over L_}\right)^\alpha \exp\left[-\left({L\over L_}\right)\right]{dL\over L_} \ , \label{eq:schechter}$$ and the derived parameters are given in Table 2, where we take the Hubble constant $h=H_0/100$ km s$^{-1}$Mpc$^{-1}$. Only a crude estimate (with ML) is presented for the luminosity function for Im galaxies, since our sample is too small. We also present the results for the total sample, which includes not only galaxies with $T=0-6$, but also those could not be classified ($T=-1$). This is a bright-galaxy version of the analysis given by Blanton et al. (2001). In this table we also give the luminosity densities obtained by integrating (\[eq:schechter\]) over $L=0$ to $\infty$. The contours of one and two standard deviation errors calculated from the likelihood functions are shown in $\alpha-M^$ plane in Figure 4. We have also carried out a jack-knife error estimate, by dividing the sample into ten RA bins (width of $\sim$1.2 hr), in order to study the effect of the sample variance. The best-fit values for the subsamples all fall within the one-sigma ellipse given above, and the variance estimated from the jack-knife method is smaller than the error we quoted. So we adopt one-sigma of the fit for our final error estimate. We then assign an additional 0.05 mag error from the calibration of photometry (added in quadratures). For more discussion about errors and selection effects, see Blanton et al. (2001). The errors expected from a number of items seen in their analysis are significantly smaller than the statistical error we are concerned with here. We determine the normalization $\phi^$ of the MDLF following the method of Efstathiou et al. (1988) for each sample of morphologically-classified galaxies. We adopt the region of $M_{r^{\ast}}$ where the sample contains sufficient number of galaxies, dropping too bright ($M_{r^{\ast}}<M_{r^{\ast}}^-2$) and faint ($M_{r^{\ast}}>M_{r^{\ast}}^+1$) galaxies and those with high redshifts to avoid strong shot noise effects. We choose the redshift range to be $0.01\le z\le 0.075$, for which the selection function for the total sample is ${\gtrsim}0.14$. The numbers of galaxies used to determine $\phi^$ are given in Table 1 above. In Table 2 we give jack-knife errors for $\phi^$. The normalization significantly varies depending on the cutoff of the redshift range, reflecting the presence of large-scale structure, such as a clump seen between $z=0.07$ and 0.08 in Figure 2. The variation of the normalization by varying the upper cutoff between 0.07 and 0.08 is comparable to the jack-knife error we quoted. The normalizations (and errors) are then corrected for the sample incompleteness derived in Table 2 by comparing the spectroscopic sample with good-quality photometry and redshift determinations \[(d) in Table 1\] to the photometric sample \[(a) in Table 1\]. A small difference of areas covered by photometric and spectroscopic surveys is also taken into account. Furthermore, an extra correction factor of (1875+88)/1875=1.05 is multiplied to correct for the incompleteness of the photometric catalogue as discussed in section 2. We can see following features in our luminosity functions: \(i) The characteristic luminosity and the faint end slope of the total sample are consistent with the parameters derived by Blanton et al. (2001) within 1$-$1.3 sigma. The normalization, however, is significantly lower, corresponding to by 30% in the luminosity density, than that of Blanton et al. This is ascribed to the local deficit of galaxies in the northern equatorial stripe seen for $r^{\ast}<16$ mag, and is ascribed to large-scale structure, as discussed in Yasuda et al. (2001). We confirmed that the normalization rapidly approaches that of Blanton et al. when we take the limiting magnitude fainter; with $r^{\ast}<16.5$ mag, the luminosity density agrees with that of Blanton et al. within 10%. \(ii) The characteristic luminosity of early-type galaxies is more luminous than that of later-type galaxies by about 0.45 mag. This is consistent with the ‘universal characteristic luminosity’ known for the $B$ band (Tammann, Yahil & Sandage 1979), because we expect that $B-r^{\ast}$ colour differs by 0.4 mag between E and Sb (Fukugita et al. 1995). This implies that the universal characteristic luminosity in the $B$ band is an accidental effect. \(iii) The shape of the luminosity function of early-type galaxies is not much different from that of late-type galaxies, although some trend is seen that the number of early-type galaxies slightly declines ($\alpha{\lesssim}-1$) towards the faint end. This conclusion agrees with Marzke et al. (1994) for the $B$ band, and Kochanek et al. (2001) for the $K$ band, but does not agree with Loveday et al. (1992), which show an appreciable decline towards the faint end (see Zucca, Pozzetti & Zamorani 1994, which ascribe Loveday et al.’s result to a sample incompleteness). In particular, we do not see a sharp decline of the luminosity function, as inferred in Binggeli et al. (1988) and Bernardi et al. (2002a). The latter authors fit the luminosity function of early-type galaxies selected with photometric and spectroscopic parameters (Bernardi et al. 2002b)[^1] to a Gaussian function with a peak at $M_{r^{\ast}}=-20.38$ mag ($h=1$): their data go beyond the peak only slightly, and the turn-over is not conclusive. Our luminosity function, which goes down to $-$18.75 mag, does not show any turnover to this magnitude. \(iv) The luminosity function of late-type spirals (Sbc-Sd) does not exhibit an increase ($\alpha{\gtrsim}-1$) towards the faint end. Our late-type spiral galaxy sample shows an even faster decline compared to that of early-type spiral galaxies. We found that the luminosity function derived from $V_{\rm max}$ shows a somewhat faster increase ($\alpha=-1.16$) compared with those for other types, but this trend is not visible with the MDLF from the ML or SWML methods. We ascribe this larger $\alpha$ from the $V_{\rm max}$ method to a local effect of the galaxy distribution, as we mention below. In any case the steepening of the faint end slope does not occur up to the Im type. This might appear to contrast with the conventional belief that late-type galaxies have a steep slope. This is due to our exclusion of very late galaxies ($T>5$), and is consistent with Marzke et al. (1994), who found that only the Im luminosity function shows a steep faint end slope. \(vi) The Im type luminosity function shows a steep faint end slope, $\alpha\sim -1.9$, consistent with Marzke et al. It may be worth commenting that the absolute magnitude distributions of early- and late-type spiral galaxies shown in Figure 3 appear to indicate steeper faint end slope for the latter. This is in fact what we have obtained when we use the $V_{\rm max}$ method. This reflects the effect seen in Figure 2 that the morphological composition appears to change as a redshift \[i.e., the frequency of late-type spirals is high in nearby ($z<0.05$) sample\]. This effect disappears when we use the likelihood method to calculate the luminosity function [under the assumption that it is universal]{}. For practical uses of the MDLF presented here, the normalization should be multiplied by a factor of 1.29 to correct for the local deficit of galaxies in the northern equatorial stripe in brighter magnitudes. Morphological classification with the concentration index ========================================================= The luminosity function of early-type galaxies we derived does not show a conspicuous decline towards the faint end. One may suspect that that our E and S0 sample may contain increasingly more dwarf ellipticals and spheroidals, which are not separated from their giant counterparts in visual classifications, towards the faint end, and therefore the luminosity function of giant elliptical galaxies might actually decline. This point may be studied by using a concentration index, since early-type dwarfs usually have galaxy cores significantly softer than those of giant elliptical galaxies (e.g., Kormendy 1986). We define the (inverse) concentration index by the ratio of the two Petrosian radii $C=r_{50}/r_{90}$ measured in the $r^{\ast}$ band, where $r_{50}$ and $r_{90}$ are radii which correspond to the apertures that include 50% and 90% of the Petrosian flux. In Figure 5 we plot $C$ as a function of absolute magnitudes for E and S0 galaxies. The plot shows that there is no evident trend that fainter early-type galaxies have softer cores; most of the data points fall below $C<0.34$, which is a typical value that divides early and late types, down to $-$19 mag. Shimasaku et al. (2001) report that this $C$ parameter shows the strongest correlation with visually-classified morphology among simple photometrically-defined parameters (see also, Doi, Fukugita & Okamura 1993; Abraham et al. 1994; Blanton et al. 2001; Strateva et al. 2001; Bernardi et al. 2002b). We thus separate morphologies into early and late types according as $C<0.35$ or $C>0.35$, which corresponds to the devision at S0/a. The early-type galaxy sample (706 galaxies) thus defined shows a 82% completeness and is contaminated by late-type galaxies by 18% when we take the visually-classified sample as the reference. The late-type sample (713 galaxies) also shows a 82% completeness and a 18 % contamination from the opposite sample. This choice of $C$ minimizes the contamination of the opposite morphologies either way. The analysis is similar to what was already presented by Blanton et al. (2001), with the difference that they have used $C=0.43$ (which corresponds to Sb for bright galaxies) to divide the early- and late-type galaxy samples. Figure 6 shows the MDLF separated according to this $C$ index. The parameters of the Schechter function from the ML analysis are given in Table 3 above. The use of a different division at $C=0.34$, which is about the division at S0 galaxies, changes the MDLF only slightly. The feature of the luminosity functions is similar to what are derived from the visually-classified sample. The MDLF for early-type galaxies shows a characteristic luminosity brighter than that for late-types, and has a slightly declining faint-end shape while late-type galaxies show a flat faint end. No sharp decline of the luminosity function is visible at least two mag fainter than $M^$, or at least 2 mag fainter than the peak inferred by Bernardi et al. (2002a). Our result means that it is unlikely that the visually-classified sample is dominated by dwarf ellipticals that have soft cores in faint magnitudes. The luminosity function of late-type galaxies also shares the features of the visually-classified late-type galaxy sample. This analysis shows that the MDLFs using the concentration index are similar to those with visually-classified samples, although a somewhat smaller difference of characteristic luminosities of the two types represents the $\sim$20% contamination from the opposite types. Conclusions =========== Our sample is small and we may not be able to extract quantitatively robust parameters, yet we obtain a number of useful conclusions. The most important feature with our analysis is that we have used a homogeneous photometric catalogue with sharply defined selection criteria and a homogeneously morphologically-classified sample based on Hubble morphology of galaxies, rather than a sample classified by indicators using spectroscopic features or colours, which are sensitive to small star formation activities in the present or the near past. The first conclusion we have obtained is that the shape of the MDLF does not depend too strongly on the Hubble types. The characteristic luminosity of elliptical and S0 galaxies is brighter than that of spiral galaxies in the $r^{\ast}$ band. The amount of the difference in brightness is consistent with universal characteristic luminosity in the $B$ band, which was found by Tammann et al. (1979). The MDLF of early-type galaxies somewhat declines in the faint end, but does not exhibit a sharp decline, and this is not due to an increasing mixture of dwarf galaxies at least in the magnitude range we are concerned with. The conclusion is unchanged if we use the concentration index as a classifier of early-type galaxies. This indicates that there are many intrinsically faint elliptical galaxies, whose luminosities are fainter than those of bulges in spiral galaxies. The existence of numerous early-type galaxies with a hard core at small luminosities indicates that morphology is unlikely to be a sequence of the bulge luminosity as advocated by Dressler & Sandage (1983), and by Meisels and Ostriker (1984). Our conclusion also justifies the calculation of the strong gravitational lensing frequency of quasars using the standard Schechter function without introducing a cutoff in the luminosity function, which would affect the frequency of sub-arcsecond lensing. Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington. We would like to thank Sadanori Okamura for useful comments. 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G. et al. 2000, AJ, 120, 1579 Zucca, E., Pozzetti, L. and Zamorani, G. 1994, MNRAS, 269, 953 [lcccccc]{} & $0\le T\le 1$ & $1< T\le 3$ & $3< T\le 5$ & $5< T\le 6$ & $T=-1$ & total\ & (E & S0) & (S0/a-Sb) & (Sbc-Sd) & (Im) & (unclass.) &\ (a) photometric sample & 740 & 630 & 444 & 35 & 26 & 1875\ (b) spectroscopic sample & 630 & 545 & 381 & 23 & 21 & 1600\ (c) sample with good photometry & 617 & 539 & 373 & 21 & 12 & 1562\ (d) sample with $z$ ($\geq85$% CL) & 616 & 538 & 369 & 19 & 11 & 1553\ (e) sample used in MDLF & 597 & 518 & 350 & (10) & (7) & 1482\ (f) sample used to give $\phi^$ & 314 & 368 & 253 & (5) & & 894\ [lcccc]{} morphology & $M^*({\rm r^{\ast}})-5\log h$ & $\alpha$ & $\phi ^\ast$ & $\cal L({\rm r^{\ast}})$\ & & & $(0.01h^3 $Mpc$^{-3}$) & ($10^8h{\rm L}_\odot$ Mpc$^{-3}$)\ total & $-20.65\pm 0.12$ & $-1.10\pm 0.14$ & $1.43\pm0.21$ & 2.00\ $0\le T\le 1.0$ & $-20.75\pm 0.17$ & $-0.83\pm 0.26$ & $0.47\pm0.09$ & 0.62\ $1.5\le T\le 3 $ & $-20.30\pm 0.19$ & $-1.15\pm 0.26$ & $0.95\pm0.15$ & 1.00\ $3.5\le T\le 5 $ & $-20.30\pm 0.20$ & $-0.71\pm 0.26$ & $0.43\pm0.05$ & 0.37\ $5.5\le T\le 6 $ & $\sim -20.0$ & $-1.9$ & $\sim0.04$ &\ $C<0.35$ & $-20.62\pm 0.14$ & $-0.68\pm0.23$ & $0.67\pm0.12$ & 0.76\ $C>0.35$ & $-20.35\pm 0.19$ & $-1.12\pm0.18$ & $1.09\pm0.14$ & 1.17\ [^1]: These authors adopted selection criteria based on the concentration index ($C<0.4$; see below), the PCA spectral classification index for early-type galaxies, and the de Vaucouleurs- versus exponential-likelihood parameters which are produced from the photometric pipeline. Note that the last parameter correlates very weakly with visual morphology; see Shimasaku et al. 2001.
--- abstract: 'We define Casson-Gordon $\sigma$-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus $h$ and show that their slice genus is $h$, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding an annulus in the 4-ball.' address: \| Laboratoire I.R.M.A. Université Louis Pasteur\ Strasbourg, France title: On the slice genus of links --- Introduction ============ A knot in $S^3$ is slice if it bounds a smooth $2$-disk in the $4$-ball $B^4$. Levine showed [@Le] that a slice knot is algebraically slice, i.e. any Seifert form of a slice knot is metabolic. In this case, the Tristram-Levine signatures at the prime power order roots of unity of a slice knot must be zero. Levine showed also that the converse holds in high odd dimensions, i.e. any algebraically slice knot is slice. This is false in dimension $3$: Casson and Gordon [@CG1; @CG2; @G] showed that certain two-bridge knots in $S^3$, which are algebraically slice, are not slice knots. For this purpose, they defined several knot and 3-manifold invariants, closely related to the Tristram-Levine signatures of associated links. Further methods to calculate these invariants were developed by Gilmer [@Gi3; @Gi4], Litherland [@Li], Gilmer-Livingston [@GL], and Naik [@N]. Lines [@L] also computed some of these invariants for some fibered knots, which are algebraically slice but not slice. The slice genus of a link is the minimal genus for a smooth oriented connected surface properly embedded in $B^4$ with boundary the given link. The Murasugi-Tristram inequality (see Theorem \[MT\] below) gives a lower bound on the slice genus of a link in terms of the link’s Tristram-Levine signatures and related nullity invariants. The second author [@Gi1] used Casson-Gordon invariants to give another lower bound on the slice genus of a knot. In particular he gave examples of algebraically slice knots whose slice genus is arbitrarily large. We apply these methods to restrict the slice genus of a link. We study as an example a family of two component links, which have genus $h$ Seifert surfaces. Using Theorem \[main\], we show that these links cannot bound a smoothly embedded surface in $B^4$ with genus lower than $h$, while the Murasugi-Tristram inequality does not show this. In fact there are some links with the same Seifert form that bound annuli in $B^4$. We work in the smooth category. The second author was partially supported by NSF-DMS-0203486. Preliminaries ============= The Tristram-Levine signatures ------------------------------ Let $L$ be an oriented link in $S^3$, with $\mu$ components, and $\theta_S$ be the Seifert pairing corresponding to a connected Seifert surface $S$ of the link. For any complex number $\lambda$ with $\| \lambda \| =1$, one considers the hermitian form $\theta_S^\lambda := (1-\lambda) \theta_S + (1 - \overline{\lambda}) (\theta_S)^T$. The Tristram signature $\sigma_L(\lambda)$ and nullity $n_L(\lambda)$ of $L$ are defined as the signature and nullity of $\theta_S^\lambda$. Levine defined these same signatures for knots [@Le]. The Alexander polynomial of $L$ is $\Delta_L (t) := \text{Det} ( \theta_S - t (\theta_S)^T).$ As is well-known, $\sigma_L$ is a locally constant map on the complement in $S^1$ of the roots of $\Delta_L$ and $n_L$ is zero on this complement. If $\Delta_L = 0, $ it is still true that the signature and nullity are locally constant functions on the complement of some finite collection of points. The Murasugi-Tristram inequality allows one to estimate the slice genus of $L$, in terms of the values of $\sigma_L(\lambda)$ and $n_L(\lambda)$. [[@M; @T]]{} Suppose that $L$ is the boundary of a properly embedded connected oriented surface $F$ of genus $g$ in $B^4$. Then, if $\lambda$ is a prime power order root of unity, we have $$\|\sigma_L(\lambda)\| + n_L(\lambda) \leq 2 g + \mu - 1.$$ \[MT\] The Casson-Gordon $\sigma$-invariant ------------------------------------ In this section, for the reader convenience, we review the definition and some of the properties of the simplest kind of Casson-Gordon invariant. It is a reformulation of the Atiyah-Singer $\alpha$-invariant. Let $M$ be an oriented compact three manifold and $\chi \co H_1(M) \rightarrow\mathbb {C}^$ be a character of finite order. For some $q \in {\mathbb{N}}^$, the image of $\chi$ is contained a cyclic subgroup of order $q$ generated by $\alpha=e^{2i \pi /q}$. As $\operatorname{Hom}(H_1(M), {{C}}_q) =[M,B({{C}}_q)]$, it follows that $\chi$ induces $q$-fold covering of $M$, denoted $\widetilde{M}$, with a canonical deck transformation. We will denote this transformation also by $\alpha.$ If $\chi$ maps onto ${{C}}_q,$ the canonical deck transformation sends $x$ to the other endpoint of the arc that begins at $x$ and covers a loop representing an element of $(\chi)^{-1}(\alpha)$. As the bordism group $\Omega_3(B({{C}}_q))= {{C}}_q$, we may conclude that $n$ disjoint copies of $M$ , for some integer $n$, bounds bound a compact $4$-manifold $W$ over $B({{C}}_q)$. Note $n$ can be taken to be $q.$ Let $\widetilde{W}$ be the induced covering with the deck transformation, denoted also by $\alpha$, that restricts to $\alpha$ on the boundary. This induces a ${\mathbb{Z}}[{{C}}_q]$- module structure on $C_(\widetilde{W})$, where the multiplication by $\alpha \in {\mathbb{Z}}[{{C}}_q]$ corresponds to the action of $\alpha$ on $\widetilde{W}.$ The cyclotomic field ${\mathbb{Q}}({{C}}_q)$ is a natural ${\mathbb{Z}}[{{C}}_q]$-module and the twisted homology $H_^t(W;{\mathbb{Q}}({{C}}_q))$ is defined as the homology of $$C_(\widetilde{W}) \otimes_{{\mathbb{Z}}[{{C}}_q]} {\mathbb{Q}}({{C}}_q).$$ Since ${\mathbb{Q}}({{C}}_q)$ is flat over ${\mathbb{Z}}[{{C}}_q]$, we get an isomorphism $$H_^t (W;{\mathbb{Q}}({{C}}_q)) \simeq H_(\widetilde{W}) \otimes_{{\mathbb{Z}}[{{C}}_q]} {\mathbb{Q}}({{C}}_q).$$ Similarly, the twisted homology $H_^t(M;{\mathbb{Q}}({{C}}_q))$ is defined as the homology of $$C_(\widetilde{M}) \otimes_{{\mathbb{Z}}[{{C}}_q]} {\mathbb{Q}}({{C}}_q).$$ Let $\widetilde{\phi}$ be the intersection form on $H_2(\widetilde{W};{\mathbb{Q}})$ and define $$\phi_{\chi} (W)\co H_2^t(W;{\mathbb{Q}}({{C}}_q)) \times H_2^t(W;{\mathbb{Q}}({{C}}_q)) \to {\mathbb{Q}}({{C}}_q)$$ so that, for all $a,b$ in ${\mathbb{Q}}({{C}}_q)$ and $x,y$ in $H_2(\widetilde{W})$, $$\phi_{\chi} (W)(x \otimes a, y \otimes b) = \overline{a} b \sum_{i=1}^q \widetilde{\phi} (x,\alpha^i y) \overline{\alpha}^i,$$ where $a \rightarrow \bar{a}$ denotes the involution on ${\mathbb{Q}}({{C}}_q)$ induced by complex conjugation. The Casson-Gordon $\sigma$-invariant of $(M,\chi)$ and the related nullity are $$\sigma(M,\chi):= \frac{1}{n} \big( {\operatorname{Sign}}(\phi_{\chi}(W)) - {\operatorname{Sign}}(W) \big)$$ $$\eta(M,\chi):= {\dim} \ H^t_1(M;{\mathbb{Q}}({{C}}_q)).$$ \[sigma\] If $U$ is a closed 4-manifold and $\chi \co H_1(U) \rightarrow C_q$ we may define $\phi_{\chi}(U) $ as above. One has that modulo torsion the bordism group $\Omega_4(B({{C}}_q)) $ is generated by the constant map from $CP(2)$ to $B({{C}}_q).$ If $\chi$ is trivial, one has that ${\operatorname{Sign}}(\phi_{\chi}(U))= {\operatorname{Sign}}(U). $ Since both signatures are invariant under cobordism, one has in general that ${\operatorname{Sign}}(\phi_{\chi}(U))= {\operatorname{Sign}}(U).$ The independence of $\sigma(M,\chi)$ from the choice of $W$ and $n$ follows from this and Novikov additivity. One may see directly that these invariants do not depend on the choice of $q$. In this way Casson and Gordon argued that $\sigma(M,\chi)$ is an invariant. Alternatively one may use the Atiyah-Singer G-Signature theorem and Novikov additivity [@AS]. We now describe a way to compute $ \sigma(M,\chi)$ for a given surgery presentation of $(M,\chi)$. Let $K$ be an oriented knot in $S^3$. Let $A$ be an embedded annulus such that $\partial A=K \cup K^{\prime}$ with $lk(K,K^{\prime})=f$. A p-cable on $K$ with twist $f$* is defined to be the union of oriented parallel copies of $K$ lying in $A$ such that the number of copies with the same orientation minus the number with opposite orientation is equal to $p$. \[cable\] Let us suppose that $M$ is obtained by surgery on a framed link $L=L_1 \cup \dots \cup L_\mu$ with framings $f_1,\dots,f_\mu$. One shows that the linking matrix $\Lambda$ of $L$ with framings in the diagonal is a presentation matrix of $H_1(M)$ and a character on $H_1(M)$ is determined by ${\alpha}^{p_i}=\chi(m_{L_i}) \in {{C}}_q$ where $m_{L_i}$ denotes the class of the meridian of $L_i$. Let $\vec p= (p_1, {\dots}, p_\mu)$. We use the following generalization of a formula in [@CG2 Lemma (3.1)], where all $p_i$ are assumed to be $1$, that is given in [@Gi2 Theorem(3.6)]. \[surgeryformula\] Suppose $\chi$ maps onto $C_q$. Let $L^{\prime}$ with $\mu^\prime$ components be the link obtained from $L$ by replacing each component by a non-empty algebraic $p_i$-cable with twist $f_i$ along this component. Then, if $\lambda=e^{2i r \pi /q}$, for $(r,q)=1$, one has $$\sigma(M,\chi^r) = \sigma_{L^{\prime}}(\lambda) - \operatorname{Sign}(\Lambda) + 2 \frac{r(q-r)}{q^2} \vec p^\top \Lambda \vec p,$$ $$\eta(M,\chi^r) = \eta_{L^{\prime}}(\lambda)-\mu^\prime + \mu.$$ The following proposition collects some easy additivity properties of the $\sigma$-invariant and the nullity under the connected sum. \[add\] Suppose that $M_1,$ $M_2$ are connected. Then,\ for all $\chi_i \in H^1(M_i; C_q)$, $i=1,2$, we have $$\sigma(M_1\# M_2, \chi_1 \oplus \chi_2) = \sigma(M_1,\chi_1) + \sigma(M_2,\chi_2).$$ If both $\chi_i$ are non-trivial, then $$\eta(M_1\# M_2,\chi_1 \oplus \chi_2) = \eta(M_1,\chi_1) + \eta(M_2,\chi_2) + 1.$$ If one $\chi_i$ is trivial, then $$\eta(M_1\# M_2,\chi_1 \oplus \chi_2) = \eta(M_1,\chi_1)+ \eta(M_2,\chi_2).$$ \[s1s2\] For all $\chi \in H_1(S^1 \times S^2; C_q)$, we have $$\sigma(S^1 \times S^2, \chi) = 0$$ $$\text{If } \chi \neq 0 \text{, then } \eta(S^1 \times S^2,\chi)=0. \text{ If } \chi=0 \text{, then } \eta(S^1 \times S^2,\chi)= 1 .$$ Proposition \[s1s2\] for non-trivial $\chi$ can be proved for example by the use of Proposition \[surgeryformula\], since $S^1 \times S^2$ is obtained by surgery on the unknot framed $0$. However it is simplest to derive this result directly from the definitions. The Casson-Gordon $\tau$-invariant {#cgt} ---------------------------------- In this section, we recall the definition and some of the properties of the Casson-Gordon $\tau$-invariant. Let $C_\infty$ denote a multiplicative infinite cyclic group generated by $t.$ For $\chi^+ \co H_1(M) \to {{C}}_q \oplus \ C_\infty$, we denote $\bar{\chi} \co H_1(M) \to {{C}}_q $ the character obtained by composing $\chi^+$ with projection on the first factor. The character $\chi^+$ induces a ${{C}}_q \times C_\infty$-covering $\widetilde{M}_\infty$ of $M$. Since the bordism group $\Omega_3(B({{C}}_q \times C_\infty ))= {{C}}_q,$ bounds a compact $4$-manifold $W$ over $B({{C}}_q \times C_\infty)$ Again $n$ can be taken from to be $q$. If we identify ${\mathbb{Z}}[{{C}}_q \times C_\infty]$ with the Laurent polynomial ring ${\mathbb{Z}}[{{C}}_q][t,t^{-1}]$, the field ${\mathbb{Q}}({{C}}_q)(t)$ of rational functions over the cyclotomic field ${\mathbb{Q}}({{C}}_q)$ is a flat ${\mathbb{Z}}[{{C}}_q \times C_\infty]$-module. We consider the chain complex $C_(\widetilde{W}_\infty)$ as a ${\mathbb{Z}}[{{C}}_q \times C_\infty]$-module given by the deck transformation of the covering. Since $W$ is compact, the vector space $H_2^t(W;{\mathbb{Q}}({{C}}_q)(t)) \simeq H_2(\widetilde{W}_\infty) \otimes_{{\mathbb{Z}}[{{C}}_q][t,t^{-1}]} {\mathbb{Q}}({{C}}_q)(t)$ is finite dimensional. We let $J$ denote the involution on ${\mathbb{Q}}({{C}}_q)(t)$ that is linear over ${\mathbb{Q}}$ sends $t^i$ to $t^{-i}$ and $\alpha^i$ to $\alpha^{-i}.$ As in [@G], one defines a hermitian form, with respect to $J$, $$\phi_{\chi^+} \co H_2^t(W;{\mathbb{Q}}({{C}}_q)(t)) \times H_2^t(W;{\mathbb{Q}}({{C}}_q)(t)) \to {\mathbb{Q}}({{C}}_q)(t),$$ such that $$\phi_{\chi^+} (x \otimes a, y \otimes b) = J(a) \cdot b \cdot \sum_{i \in {\mathbb{Z}}} \sum_{j= 1} ^q \widetilde{\phi^+} (x,t^i \alpha^j y) \overline{\alpha}^j t^{-i}.$$ Here $\widetilde{\phi^+} $ denotes the ordinary intersection form on $\widetilde{W}_\infty.$ Let $\mathcal{W} ({\mathbb{Q}}({{C}}_q)(t))$ be the Witt group of non-singular hermitian forms on finite dimensional ${\mathbb{Q}}({{C}}_q)(t)$ vector spaces. Let us consider $H_2^t(W;{\mathbb{Q}}({{C}}_q)(t)) / (\text{Radical}( \phi_{\chi^+}))$. The induced form on it represents an element in $\mathcal {W}$ $ ({\mathbb{Q}}({{C}}_q)(t)),$ which we denote $w(W)$. Furthermore, the ordinary intersection form on $H_2(W;{\mathbb{Q}})$ represents an element of $\mathcal{W} ({\mathbb{Q}})$. Let $w_0(W)$ be the image of this element in $\mathcal{W} ({\mathbb{Q}}({{C}}_q)(t))$. The Casson-Gordon $\tau$-invariant of $(M,\chi^+)$ is $$\tau (M,\chi^+) : = \frac{1}{n} \big( w(W) - w_0(W) \big) \in \mathcal{W} ({\mathbb{Q}}({{C}}_q)(t)) \otimes {\mathbb{Q}}.$$ Suppose that $nM$ bounds another compact $4$-manifold $W^\prime$ over $B({{C}}_q \times C_\infty)$. Form the closed compact manifold over $B({{C}}_q \times C_\infty)$, $U := W \cup W^\prime$ by gluing along the boundary. By Novikov additivity, we get $w(U)-w_0(U)= \big( w(W) - w_0(W) \big) - \big( w(W^\prime) - w_0(W^\prime) \big)$. Using [@CF], the bordism group $\Omega_4(B({{C}}_q \times C_\infty)) $, modulo torsion, is generated by $CP(2)$, with the constant map to $B({{C}}_q \times C_\infty)$. We have that $w( CP(2))=w_0(CP(2))$. Since $w(U)$, and $w_0(U)$ only depend on the bordism class of $U$ over $B({{C}}_q \times C_\infty)$, it follows that $w(U)=w_0(U)$ and $ \tau (M,\chi^+)$ is independent of the choice of $W$. Using the above techniques, one may check $\tau (M,\chi^+)$ is independent of $n$. If $A \in \mathcal{W} ({\mathbb{Q}}({{C}}_q)(t)),$ let $A(t)$ be a matrix representative for $A$. The entries of $A(t)$ are Laurent polynomials with coefficients in ${\mathbb{Q}}({{C}}_q)$. If $\lambda$ is in $S^1 \subset \mathbb{C}$, then $A(\lambda)$ is hermitian and has a well defined signature $\sigma_\lambda(A)$. One can view $\sigma_\lambda(A)$ as a locally constant map on the complement of the set of the zeros of $\det A(\lambda)$. As in [@CG1], we re-define $\sigma_\lambda(A)$ at each point of discontinuity as the average of the one-sided limits at the point. We have the following estimate [@Gi3 Equation (3.1)]. \[firstest\] Let $\chi^+ \co H_1(M) \to {{C}}_q \oplus C_\infty$ and $\bar \chi \co H_1(M) \to {{C}}_q$ be $\chi^+$ followed by the projection to ${{C}}_q$. We have $$\| \sigma_{1} \big(\tau (M,\chi^+) \big) - \sigma(M,\bar {\chi}) \| \leq \eta(M,\bar{\chi}).$$ Linking forms ------------- Let $M$ be a rational homology 3-sphere with linking form $$l \co H_1(M) \times H_1(M) \to {\mathbb{Q}}/{\mathbb{Z}}.$$ We have that $l$ is non-singular, that is the adjoint of $l$ is an isomorphism $\iota \co H_1(M) \to \operatorname{Hom}(H_1(M) ,{\mathbb{Q}}/{\mathbb{Z}})$. Let $H_1(M)^$ denote $ \operatorname{Hom}(H_1(M), \mathbb{C}^). $ Let $\nu$ denote the map ${\mathbb{Q}}/{\mathbb{Z}}\rightarrow \mathbb{C}^$ that sends $\frac a b $ to $e^{\frac {2 \pi i a}{b}} .$ So we have an isomorphism $\jmath \co H_1(M) \to H_1(M)^$ given by $x \mapsto \nu \circ \iota(x).$ Let $\beta \co H_1(M)^ \times H_1(M)^* \rightarrow {\mathbb{Q}}/{\mathbb{Z}}$ be the dual form defined by $ \beta(\jmath x, \jmath y)= - l(x,y)$. The form $\beta$ is metabolic with metabolizer $H$ if there exists a subgroup $H$ of $H_1(M)^$ such that $H^\bot = H$. \[extend\] [[@Gi1]]{} If $M$ bounds a spin $4$-manifold $W$ then $\beta = \beta_1 \oplus \beta_2$ where $\beta_2$ is metabolic and $\beta_1$ has an even presentation with rank $\text{dim } H_2(W;{\mathbb{Q}})$ and signature Sign($W$). Moreover, the set of characters that extend to $H_1(W)$ forms a metabolizer for $\beta_2$. \[linking\] Link invariants {#LI} --------------- Let $L=L_1 \cup \dots \cup L_{\mu}$ be an oriented link in $S^3$. Let $N_2$ be the two-fold covering of $S^3$ branched along $L$ and $\beta_L$ be the linking form on $H_1(N_2)^$, see previous section. We suppose that the Alexander polynomial of $L$ satisfies $$\Delta_L(-1) \neq 0.$$ Hence, $N_2$ is a rational homology sphere. Note that if $ \Delta_L(-1) \neq 1$, then $H_1(N_2;{\mathbb{Z}})$ is non-trivial. For all characters $\chi$ in $H_1(N_2)^$, the Casson-Gordon $\sigma$-invariant of $L$ and the related nullity are (see Definition \[sigma\]): $$\sigma(L,\chi) := \sigma(N_2,\chi),$$ $$\eta(L,\chi):= \eta(N_2,\chi).$$ \[cg\] If $L$ is a knot, then Definition \[cg\] coincides with $\sigma(L,\chi)$ defined in [@CG1 p.183]. Framed link descriptions ======================== In this section, we study the Casson-Gordon $\tau$-invariants of the two-fold cover $M_2$ of the manifold $M_0$ described below. Let $S^3 - T(L)$ be the complement in $S^3$ of an open tubular neighborhood of $L$ in $S^3$ and $P$ be a planar surface with $\mu$ boundary components. Let $S$ be a Seifert surface for $L$ and $\gamma_i$ for $i=1, \dots, \mu$ be the curves where $S$ intersects the boundary of $S^3 - T(L)$. We define $M_0$ as the result of gluing $P \times S^1$ to $S^3 - T(L)$, where $P \times {1}$ is glued along the curves $\gamma_i$. Let $$ be a point in the boundary of $P$. A recipe for drawing a framed link description for $M_0$ is given in the proof of Proposition \[M0\]. $$H_1(M_0) \simeq {\mathbb{Z}}\oplus {\mathbb{Z}}^{\mu -1} \simeq \langle m \rangle \oplus \ {\mathbb{Z}}^{\mu -1},$$ where $m$ denotes the class of $* \times S^1$ in $P \times S^1$. \[M0\] Form a 4-manifold $X$ by gluing $P \times D^2$ to $D^4$ along $S^3$ in such a way that the total framing on $L$ agrees with the Seifert surface $S$. The boundary of this 4-manifold is $M_0$. We can get a surgery description of $M_0$ in the following way: pick $\mu - 1$ paths of $S$ joining up the components of $L$ in a chain. Deleting open neighborhoods of these paths in $S$ gives a Seifert surface for a knot $L^\prime$ obtained by doing a fusion of $L$ along bands that are neighborhoods of the original paths. Put a circle with a dot around each of these bands (representing a 4-dimensional $1$-handle in Kirby’s [@K] notation), and the framing zero on $L^\prime.$ This describes a handlebody decomposition of $X.$ One can then get a standard framed link description of $M_0$ by replacing the circle with dots with unknots $T_1,\dots,T_{\mu-1}$ framed zero. This changes the $4$-manifold but not the boundary. Note also that $lk(T_i,T_j)=0$ and $lk(T_i,L^\prime)=0$ for all $i=1, \dots, \mu-1$. Hence $H_1(M_0) \simeq {\mathbb{Z}}^\mu$ and $m$ represents one of the generators. We now consider an infinite cyclic covering $M_\infty$ of $M_0$, defined by a character $H_1(M_0) \to C_\infty= \langle t \rangle $ that sends $m$ to $t$ and the other generators to zero. Let us denote by $M_2$ the intermediate two-fold covering obtained by composing this character with the quotient map $C_\infty \to C_2$ sending $t$ to $-1$. Let $m_2$ denote the loop in $M_2$ given by the inverse image of $m$. A recipe for drawing a framed link description for $M_2$ is given in the proof of Remark \[surgM2\]. \[M2\] There is an isomorphism between $H_1(N_2)$ and the torsion subgroup of $H_1(M_2)$, which only depends on $L.$ Moreover $$H_1(M_2) \simeq H_1(N_2) \oplus {\mathbb{Z}}^\mu \simeq H_1(N_2) \oplus \langle m_2 \rangle \oplus {\mathbb{Z}}^{\mu-1}.$$ Let $R$ be the result of gluing $P \times D^2$ to $S^3 \times I$ along $L\times {1} \subset S^3 \times {1}$ using the framing given by the Seifert surface. Thus $R$ is the result of adding $\mu-1$ 1-handles to $S^3 \times I$ and then one 2-handle along $L'$, as in the proof above. Then $X$ in the proof above can be obtained by gluing $D^4$ to $R$ along $S^3 \times {0}.$ Since $D^2$ is the double branched cover of itself along the origin, $P \times D^2$ is the double branched cover of itself along $P \times {0}$. Let $R_2$ denote the double branched cover of $R$ that is obtained by gluing $P \times D^2$ to $N_2 \times I$ along a neighborhood of the lift of $L\times {1} \subset S^3 \times {1}.$ We have that $\partial R_2= -N_2 \sqcup M_2$, where $R_2$ is the result of adding $\mu-1$ 1-handles to $N_2 \times I$ and then one 2-handle along the lift $L'.$ Moreover this lift of $L'$ is null-homologous in $N_2.$ It follows that $H_1(R_2)$ is isomorphic to $H_1(N_2) \oplus {\mathbb{Z}}^{\mu -1},$ with the inclusion of $N_2$ into $R_2$ inducing an isomorphism $i_N$ of $H_1(N_2)$ to the torsion subgroup of $H_1(R_2).$ Turning this handle decomposition upside down we have that $R_2$ is the result of adding to $M_2 \times I$ one 2-handle along a neighborhood of $m_2$ and then $\mu-1$ 3-handles. It follows that $H_1(R_2) \oplus {\mathbb{Z}}= H_1(R_2) \oplus \langle m_2 \rangle $ is isomorphic to $H_1(M_2)$ with the inclusion of $M_2$ in $R_2 $ inducing an isomorphism $i_M$ of the torsion subgroup $H_1(M_2)$ to the torsion subgroup of $H_1(R_2).$ Thus $(i_M)^{-1} \circ i_N$ is an isomorphism from $H_1(N_2)$ to the torsion subgroup of $H_1(M_2)$ and this isomorphism is constructed without any arbitrary choices. \[surgM2\] We could have proved Proposition \[M0\] in a similar way to the proof of Proposition \[M2\]. We could have also proved Proposition \[M2\] (except for the isomorphism only depending on $L$) in a similar way to the proof of Proposition \[M0\] as follows. We can find a surgery description of $M_2$ from a surgery description of $N_2$. The procedure of how to visualize a lift of $L$ and the surface $S$ in $N_2$ is given in [@AK]. One considers the lifts of the paths chosen in the proof of Proposition \[M0\], on the lift of $S.$ One then fuses the components of the lift of $L$ along these paths, obtaining a lift of $L'.$ The surgery description of $M_2$ is obtained by adding to the surgery description of $N_2$ the lift of $L'$ with zero framing together with $\mu-1$ more unknotted zero-framed components encircling each fusion. The linking matrix of this link is a direct sum of that of $N_2$ and a $\mu \times \mu$ zero matrix. Let $i_T$ denote the inclusion of the torsion subgroup of $H_1(M_2)$ into $H_1(M_2),$ and let $\psi \co H_1(N_2) \rightarrow H_1(M_2)$ denote the monomorphism given by $i_T \circ (i_M)^{-1} \circ i_N.$ Let $\chi^+ \co H_1(M_2) \to {{C}}_q \oplus C_\infty.$ Let $\chi \co H_1(N_2) \to {{C}}_q $ be $\chi^+ \circ \psi$ composed with the projection to ${{C}}_q . $ We have that: $$\| \sigma_1(\tau(M_2,\chi^+)) -\sigma(L,\chi)\| \leq \eta(L,\chi) + \mu.$$ \[est\] If $L$ is a knot, then $\tau(M_2,\chi^+)$ coincides with $\tau(L,\chi)$ defined in [@CG1 p.189]. We use the surgery description of $M_2$ given in Remark \[surgM2\]. Let $P$ be given by the surgery description of $M_2$ but with the component corresponding to $L^\prime$ deleted. Hence, $$P = N_2 \sharp_{(\mu - 1)} S^1 \times S^2.$$ $\chi+$ induces some character $\chi^\prime$ on $H_1(P)$. According to Section \[cgt\], we let $\overline{\chi} \in H^1(M_2;{{C}}_q)$ and $\overline{\chi}^\prime \in H^1(P;{{C}}_q)$ denote the characters $\chi^+$ and $\chi^\prime$ followed by the projection ${{C}}_q \oplus C_\infty \to {{C}}_q$. Using Propositions \[add\] and \[s1s2\], one has that $$\sigma(P,\overline{\chi}^\prime) = \sigma(L,\chi) \text{ and } \eta(P,\overline{\chi}^\prime)=\eta(L,\chi) + \mu - 1.$$ Moreover, since $M_2$ is obtained by surgery on $L^\prime$ in $P$, it follows from [@Gi3 Proposition (3.3)] that $$\|\sigma(P,\overline{\chi}^\prime) - \sigma(M_2,\overline{\chi}) \|+\| \eta(M_2,\overline{\chi}) - \eta(P,\overline{\chi}^\prime) \| \leq 1 \text{ or }$$ $$\|\sigma(L,\chi) - \sigma(M_2,\overline{\chi}) \|+\| \eta(M_2,\overline{\chi}) - \eta(L,\chi) - \mu +1\| \leq 1.$$ Thus $$\|\sigma(L,\chi) - \sigma(M_2,\overline{\chi}) \| \leq \eta(L,\chi) + \mu - \eta(M_2,\overline{\chi}) .$$ Finally, one gets, by Theorem \[firstest\], $$\| \sigma_1(\tau(M_2,\chi^+)) -\sigma(L,\chi)\| \leq \| \sigma_1(\tau(M_2,\chi^+)) - \sigma(M_2,\overline{\chi}) \| + \| \sigma(M_2,\overline{\chi}) - \sigma(L,\chi)\|$$ $$\leq \eta(M_2,\overline{\chi})+ \eta(L,\chi) + \mu - \eta(M_2,\overline{\chi}) = \eta(L,\chi) + \mu.\eqno{\qed}$$ The slice genus of links ======================== See Section \[LI\] for notations. \[main\] Suppose $L$ is the boundary of a connected oriented properly embedded surface $F$ of genus $g$ in $B^4,$ and that $\Delta_L(-1) \neq 0$. Then, $\beta_L$ can be written as a direct sum $\beta_1 \oplus \beta_2$ such that the following two conditions hold: [1)]{}$\beta_1$ has an even presentation of rank $2g + \mu -1$ and signature $\sigma_L(-1)$, and $\beta_2$ is metabolic. [2)]{}There is a metabolizer for $\beta_2$ such that for all characters $\chi$ of prime power order in this metabolizer, $$\| \sigma(L,\chi)+\sigma_L(-1) \| \leq \eta(L,\chi) + 4g +3 \mu -2 .$$ We let $b_i(X)$ denote the ith Betti number of a space $X$. We have $b_1(F) = 2g + \mu -1.$ Let $W^\prime_0$, with boundary $M^\prime_0$, be the complement of an open tubular neighborhood of $F$ in $B^4$. By the Thom isomorphism, excision, and the long exact sequence of the pair $(B^4,W^\prime_0),$ $W^\prime_0$ has the homology of $S^1$ wedge $b_1(F)$ 2-spheres. Let $W^\prime_2$ with boundary $M^\prime_2$ be the two-fold covering of $W^\prime_0$. Note that if $F$ is planar, $M^\prime_0= M_0, $ and $M_2^\prime=M_2$ (see Section 3). Let $V_2$ be the two-fold covering of $B^4$ with branched set $F$. Note that $V_2$ is spin as $w_2(V_2)$ is the pull-up of a class in $H^2(B^4,{\mathbb{Z}}_2)$, by [@Gi5 Theorem 7], for instance. The boundary of $V_2$ is $N_2$. As in [@Gi1], one calculates that $b_2(V_2)=2g + \mu -1$. One has $\operatorname{Sign}(V_2)= \sigma_{L}(-1)$ by [@V]. By Lemma [\[extend\]]{}, $\beta_L$ can be written as a direct sum $\beta_1 \oplus \beta_2$ as in condition $1)$ above, such that the characters on $H_1(N_2)$ that extend to $H_1(V_2)$ form a metabolizer $H$ for $\beta_2$. We now suppose $\chi \in H$ and show that Condition $2)$ holds for $\chi.$ We also let $\chi$ denote an extension of $\chi$ to $H_1(V_2)$ with image some cyclic group ${{C}}_q$ where $q$ is a power of a prime integer (possibly larger than those corresponding to the character on $H_1(N_2)$). Of course $\chi \in H^1(V_2,{{C}}_q) $ restricted to $W_2^\prime$ extends $\chi $ restricted to $M_2^\prime$. We simply denote all these restrictions by $\chi$. Let $W^\prime_\infty$ denote the infinite cyclic cover of $W^\prime_0$. Note that $W^\prime_2$ is a quotient of this covering space. $\chi$ induces a ${{C}}_q$-covering of $V_2$ and thus of $W_2^\prime$. If we pull the ${{C}}_q$-covering of $W_2^\prime$ up to $W_\infty^\prime$, we obtain $\widetilde{W}_\infty^\prime$, a ${{C}}_q \times C_\infty$-covering of $W_2^\prime$. If we identify properly $F \times S^1$ in $M_2^\prime,$ this covering restricted to $F \times S^1$ is given by a character $H_1(F \times S^1) \simeq H_1(F) \oplus H_1(S^1) \to {{C}}_q \times C_\infty$ that maps $H_1(F)$ to zero in $C_\infty$, $H_1(S^1)$ to zero in ${{C}}_q$ and isomorphically onto $C_\infty$. For this note: since $\text{Hom}(H_1(F),{\mathbb{Z}})=H^1(F)= [F,S^1]$, we may define diffeomorphisms of $F \times S^1$ that induce the identity on the second factor of $H_1(F \times S^1) \approx H_1(F) \oplus {\mathbb{Z}},$ and send $(x,0)\in H_1(F) \oplus {\mathbb{Z}},$ to $(x,f(x))\in H_1(F) \oplus {\mathbb{Z}},$ for any $f \in \text{Hom}(H_1(F),{\mathbb{Z}}).$ As in [@Gi1], choose inductively a collection of $g$ disjoint curves in the kernel of $\chi$ that form a metabolizer for the intersection form on $H_1(F)/ H_1( \partial F)$. By taking a tubular neighborhood of these curves in $F$, we obtain a collection of $S^1 \times I$ embedded in $F$. Using these embeddings we can attach round 2-handles $(B^2 \times I) \times S^1$ along $(S^1 \times I) \times S^1$ to the trivial cobordism $M_2^\prime \times I$ and obtain a cobordism $\Omega$ between $M_2$ and $M_2^\prime$. Let $U= W_2^\prime \cup_{M_2^\prime} \Omega$ with boundary $M_2$. The ${{C}}_q \times C_\infty$-covering of $W^\prime_2$ extends uniquely to $U$. Note that $\Omega$ may also be viewed as the result of attaching round 1-handles to $M_2 \times I.$ As in [@Gi1], ${\operatorname{Sign}}(W^\prime_2)= {\operatorname{Sign}}(V_2)$. Since the intersection form on $\Omega$ is zero, we get ${\operatorname{Sign}}(U)={\operatorname{Sign}}(W_2^\prime)= {\operatorname{Sign}}(V_2)=\sigma_L(-1)$. The ${{C}}_q \times C_\infty$-covering of $\Omega$, restricted to each round $2$-handle is $q$ copies of $B^2 \times I \times {\mathbb{R}}$ attached to the trivial cobordism $\widetilde{M}_\infty^\prime \times I$ along $q$ copies of $S^1 \times I \times {\mathbb{R}}$. Using a Mayer-Vietoris sequence, one sees that the inclusion induces an isomorphism (which preserves the Hermitian form) $$H_2^t(U;{\mathbb{Q}}({{C}}_q)(t)) \simeq H_2^t(W_2^\prime;{\mathbb{Q}}({{C}}_q)(t)).$$ Thus, if $w(W_2^\prime)$ denotes the image of the intersection form on $H_2^t(W_2^\prime;{\mathbb{Q}}({{C}}_q)(t))$ in $\mathcal{W}$ $({\mathbb{Q}}({{C}}_q)(t))$, we get $\sigma_1(\tau(M_2, \chi^+)) = \sigma_1(w(W_2^\prime))-\sigma_{L}(-1)$. If $q$ is a prime power, we may apply Lemma 2 of [@Gi1] and conclude that $H_i(\widetilde{W}_\infty^\prime;{\mathbb{Q}})$ is finite dimensional for all $i \neq 2$. Thus, $H_i^t(W_2^\prime; {\mathbb{Q}}({{C}}_q)(t))$ is zero for all $i \neq 2$. Since the Euler characteristic of $W_2^\prime$ with coefficients in ${\mathbb{Q}}({{C}}_q)(t)$ coincides with those with coefficients in ${\mathbb{Q}}$, we get $\text{dim} \ H_2^t(W_2^\prime;{\mathbb{Q}}({{C}}_q)(t)) = \chi(W_2^\prime)= $ $ 2\chi(W_0^\prime)= 2(1-\chi(F))= 2 b_1(F)$. Thus $\| \sigma_1 (\tau (M_2,\chi^+)+\sigma_L(-1) \| \leq 2 b_1(F)$. Hence, $$\| \sigma(L,\chi) + \sigma_L(-1) \| \leq \| \sigma(L,\chi) - \sigma_1 (\tau (M_2,\chi)^+)\| + \| \sigma_1 (\tau (M_2,\chi)^+) + \sigma_L(-1) \|$$ $$\leq \eta(L,\chi) + \mu + 2 ( 2g + \mu - 1) = \eta(L,\chi) + 4g + 3 \mu - 2 \text{ by Theorem \ref{est}. }\eqno{\qed}$$ Examples ======== Let $L=L_1 \cup L_2$ be the link with two components of Figure 1 and $S$ be the Seifert surface of $L$ given by the picture. The squares with $K$ denote two parallel copies with linking number $0$ of an arc tied in the knot $K$. Note that $L$ is actually a family of examples. Specific links are determined by the choice of two parameters: a knot $K$ and a positive integer $h.$ Since $S$ has genus $h$, the slice genus of $L$ is at most $h$. ![The link $L$](figflg1.eps){width="8.4cm" height="5.4cm"} One calculates that $\sigma_{L}(\lambda) =1$, and $n_{L}(\lambda) = 0$ for all $\lambda$. Thus, the Murasugi-Tristram inequality says nothing about the slice genus of $L$. In fact, if $K$ is a slice knot, then one can surger this surface to obtain a smooth cylinder in the 4-ball with boundary $L$. Thus there can be no arguments based solely on a Seifert pairing for $L$ that would imply that the slice genus is non-zero. \[example\] If $\sigma_K(e^{2 i \pi / 3}) \ge 2h$ or $\sigma_K(e^{2 i \pi / 3}) \le -2h -2,$ then $L$ has slice genus $h$. Using [@AK], a surgery presentation of $N_2$ as surgery on a framed link of $2h+1$ components can be obtained from the surface $S$ (see Figure 2). Let $Q$ be the $3$-manifold obtained from the link pictured in Figure 2. Here $K'$ denotes $K$ with the string orientation reversed. Since $RP(3)$ is obtained by surgery on the unknot framed $2$, we get: $$N_2 = RP(3) \#_{h} Q.$$ ![Surgery presentation of $Q$](figflg2.eps){width="6cm" height="3.6cm"} The linking matrix of the framed link of the surgery presentation of $N_2$ is $\Lambda = [2] \bigoplus \oplus^h \begin{bmatrix} 0 & 3 \\ 3 & 0 \end{bmatrix}$. $\Lambda$ is a presentation matrix of $(H_1(N_2)^,\beta_{L})$; we obtain $$H_1(N_2)^ \simeq {\mathbb{Z}}_2 \bigoplus \oplus^{2h} {\mathbb{Z}}_3$$ and $\beta_{L}$ is given by the following matrix, with entries in ${\mathbb{Q}}/ {\mathbb{Z}}$: $$[1/2] \bigoplus \oplus^h \begin{bmatrix} 0 & 1/3 \\ 1/3 & 0 \end{bmatrix}.$$ By Theorem \[main\], if ${L}$ bounds a surface of genus $h-1$ in $B^4$, then $\beta_{L}$ must be decomposed as $\beta_1 \oplus \beta_2$ where: 1)$\beta_1$ has an even presentation matrix of rank $2h -1$, and signature $1$ (all we really need here is that it has a rank $2h-1$ presentation.) 2)$\beta_2$ is metabolic and for all characters $\chi$ of prime power order in some metabolizer of $\beta_2$, the following inequality holds: $$\| \sigma({L},\chi) + 1 \| - \eta({L},\chi) \leq 4h . \leqno{()}$$ As ${\mathbb{Z}}_2 \bigoplus \oplus^{2h} {\mathbb{Z}}_3 $ does not have a rank $2h -1$ presentation, $\beta_2$ is non-trivial. As metabolic forms are defined on groups whose cardinality is a square, $\beta_2$ is defined on a group with no $2$-torsion. Thus the metabolizer contains a non-trivial character of order three satisfying $\beta_{L}(\chi,\chi) =0.$ The first homology of $Q$ is ${\mathbb{Z}}_3 \oplus {\mathbb{Z}}_3$, generated by, say, $m_1$ and $m_2$, positive meridians of these components. Each of these components is oriented counterclockwise. We first work out $\sigma(Q,\chi)$ and $\eta(Q,\chi)$ for characters of order three. Let $\chi_{(a_1,a_2)}$ denote the character on $H_1(Q)$ sending $m_j$ to $e^{\frac{2i \pi a_j}{3}}$, where the $a_j$ take the values zero and $\pm1.$ We use Proposition \[surgeryformula\] to compute $\sigma({Q},\chi_{(1,0)})$ and $\eta({Q},\chi_{(1,0)})$ assuming that $K$ is trivial. For this, one may adapt the trick illustrated on a link with $2$ twists between the components [@Gi2 Fig (3.3), Remark (3.65b)]. In the case $K$ is the unknot, we obtain $$\sigma({Q},\chi_{(1,0)}) = 1 \quad \text{ and } \quad \eta({Q},\chi_{(1,0)}) = 0.$$ It is not difficult to see that inserting the knots of the type $K$ changes the result as follows (note that $K$ and $K^\prime$ have the same Tristram-Levine signatures): $$\sigma(Q,\chi_{(1,0)}) = 1 + 2 \sigma_K(e^{2 \pi i / 3}) \quad \text{ and } \quad \eta(Q,\chi_{(1,0)}) = 0.$$ These same values hold for the characters $\chi_{(-1,0)}$ and $\chi_{(0,\pm 1)}$ by symmetry. Using Proposition \[surgeryformula\] $$\sigma(Q,\pm \chi_{(1,1)}) = -1 -24/9 + 4 \sigma_K(e^{2 \pi i / 3}) , \quad \quad \eta(Q,\pm \chi_{(1,1)}) = 0$$ $$\sigma(Q,\pm \chi_{(1,-1)}) = 4 + 24/9 + 4 \sigma_K(e^{2 \pi i/ 3}) \quad \text{ and } \quad \eta(Q,\pm \chi_{(1,-1)}) = 1.$$ One also has $$\sigma(Q,\chi_{(0,0)}) = 0 \quad \text{ and } \quad \eta(Q,\chi_{(0,0)}) = 0.$$ Any order three character on $N_2$ that is self annihilating under the linking form is given as the sum of the trivial character on $RP(3)$ and characters of type $\chi_{(0,0)}$, $\chi_{(\pm 1,0)}$ and $\chi_{(0,\pm 1)}$ on $Q$ and characters of type $\pm \chi_{(1,1)} + \pm \chi_{(1,-1)}$ on $Q \# Q$. Using Proposition \[add\], one can calculate $\sigma(L,\chi)$ and $\eta(L,\chi)$ for all these characters $\chi$. It is now a trivial matter to check that for every non-trivial character with $\beta (\chi,\chi)=0$, the inequality (\) is not satisfied. [CG2]{} , Math. Ann. 252, 111-131 (1980). , Ann. of Math. (2) 87, 546-604 (1968). , Ergebnisse der Mathematik und ihrer Grenzgebiete, [33]{}, Springer-Verlag, (1964). , Progr. Math., [62]{}, A La Recherche de la Topologie Perdue, Birkhauser, Boston, MA, 181–199 (1986). , Proc. Symp. in Pure Math. XXX, [2]{}, 39-53 (1978). , Invent. Math. 66, 191-197 (1982). , Trans. Amer. Math. Soc. 264, 353-380 (1981). , Quart. J. Math. Oxford 34, 305-322 (1983). , Comment. Math. Helv. 68, 1-19 (1993). , Math. Ann. 295 ([4]{}), 643–659 (1993). , Topology [31]{} , ([3]{}), 475–492 (1992). , Knot theory (Proc. Sem., Plans-sur-Bex, 1977), Lecture Notes in Math., 685, Springer Verlag, Berlin, 1-60 (1978). , Lecture Notes in Math 1374 Springer Verlag, Berlin (1989). , [Knot cobordism groups in codimension two]{}, Comment. Math. Helv. 44 229–244 (1969). , Knots, braids and singularities (Plans-sur-Bex, 1982), 147–173, Monogr. Enseign. Math., 31, Enseignement Math., Geneva, (1983). , Four-Manifold Theory (Durham, N.H., 1982), Contemp. Math., [35]{}, Amer. Math. Soc., Providence, RI, 327–362 (1984). , Trans. Amer. Math. Soc. 117 , 387–422 (1965). , J. Knot Theory Ramifications 5, 661–677 (1996). , Proc, Camb. Philos. Soc., 66, 251-264 (1969). , Math. USSR-Izv. 7 1239–1256 (1973).
--- abstract: 'The contribution of stars in galaxies to cosmic reionisation depends on the star formation history in the Universe, the abundance of galaxies during reionisation, the escape fraction of ionising photons and the clumping factor of the inter-galactic medium (IGM). We compute the star formation rate and clumping factor during reionisation in a cosmological volume using a high-resolution hydrodynamical simulation. We post-process the output with detailed radiative transfer simulations to compute the escape fraction of ionising photons. Together, this gives us the opportunity to assess the contribution of galaxies to reionisation self-consistently. The strong mass and redshift dependence of the escape fraction indicates that reionisation occurred between $z=15$ and $z=10$ and was mainly driven by proto-galaxies forming in dark-matter haloes with masses between $10^7 \, \mathrm{M}_{\odot}$ and $10^8 \, \mathrm{M}_{\odot}$. More massive galaxies that are rare at these redshifts and have significantly lower escape fractions contribute less photons to the reionisation process than the more-abundant low-mass galaxies. Star formation in the low-mass haloes is suppressed by radiative feedback from reionisation, therefore these proto-galaxies only contribute when the part of the Universe they live in is still neutral. After $z\sim10$, massive galaxies become more abundant and provide most of the ionising photons. In addition, we find that Population (Pop) III stars are too short-lived and not frequent enough to have a major contribution to reionisation. Although the stellar component of the proto-galaxies that produce the bulk of ionising photons during reionisation is too faint to be detected by the [James Webb Space Telescope]{} (JWST), these sources are brightest in the $\mathrm{H}\alpha$ and Ly-$\alpha$ recombination lines, which will likely be detected by JWST in deep surveys.' author: - \| Jan-Pieter Paardekooper$^{1}$[^1], Sadegh Khochfar$^{1}$ and Claudio Dalla Vecchia$^{1}$\ $^{1}$Max Planck Institute for extraterrestrial Physics, PO Box 1312, Giessenbachstr., 85741 Garching, Germany date: 'Accepted \\\. Received \\\; in original form \\\' title: 'The First Billion Years project - IV: Proto-galaxies reionising the Universe' --- \[firstpage\] radiative transfer -– methods: numerical -– galaxies: dwarf -– galaxies: high-redshift -– cosmology: theory Introduction ============ One of the major challenges in modern cosmology is identifying the nature of the sources responsible for reionising the Universe. The Gunn-Peterson trough in the spectra of high-redshift quasars indicates that the IGM was highly ionised at $z<6$ [@Fan:2006he]. The integrated Thomson (electron) scattering optical depth to the surface of last scattering, $\tau_{\mathrm{e}}$, suggests that reionisation was well underway by $z \approx 10.5$ [@2011ApJS..192...18K]. It remains uncertain which sources transformed the IGM into its highly ionised state. The most likely candidates are stars in galaxies [@2008ApJ...682L...9F]. The contribution of galaxies to reionisation depends critically on the star formation history in the Universe, the abundance of reionising sources, the clumping factor of the IGM and the fraction of ionising photons that escapes into the IGM, the so-called escape fraction. High-redshift surveys that are probing the star formation rate of galaxies at $6 \lesssim z \lesssim 8$ show that an escape fraction of more than $\sim 30\%$ is needed for the observed galaxy population to produce enough photons to keep the Universe ionised [@2010Natur.468...49R; @2012ApJ...752L...5B]. Although there is some evidence for a redshift evolution of the escape fraction, such high escape fractions are not observed in the local Universe [@BlandHawthorn:1999jm; @Deharveng:2001gr; @Heckman:2001fn] and at higher redshifts [@Iwata:2009dy; @2011ApJ...736...18N; @2012ApJ...751...70V]. All the massive galaxies targeted by observations have average escape fractions less than 20%, although samples are small. With this escape fraction, the ionising emissivity of the observed galaxy population is insufficient to maintain reionisation. However, the bulk of the star formation rate at these high redshifts likely arises in galaxies below the detection limit of current observational facilities [@2012ApJ...754...46T]. If the escape fraction evolves with redshift, these galaxies could provide the majority of photons for reionisation [@Trenti:2010fe; @2012MNRAS.423..862K]. Numerical studies [@Gnedin:2008ib; @Wise:2009fn; @Razoumov:2010bh; @Yajima:2010fb] find escape fractions between $\sim 0 - 1$, with possible redshift or mass dependence. In part the large differences between studies may be caused by numerical issues, because the radiative transfer simulations are computationally challenging, making approximations necessary. However, most studies targeted only a few objects and different studies focussed on different mass galaxies at different redshifts, making comparison difficult. In simulations of idealised, isolated galaxies @Paardekooper:2011cz found the main constraint on the escape fraction to be the dense gas in the star-forming regions, which provides an explanation for the large spread in escape fractions reported in previous studies. In addition, they found that the escape fraction can vary over several orders of magnitude over the lifetime of the galaxy, making it necessary to determine the contribution of galaxies to reionisation not only sampling the mass function, but also over a wide range of redshifts. In this letter we present results on the escape fraction and ionising emissivity from a large statistical sample of proto-galaxies in a high-resolution cosmological, hydrodynamical simulation. In combination with the derived clumping factor of the IGM in the simulated volume we assess self-consistently the contribution of stars in galaxies to cosmic reionisation. Method ====== In the standard cold dark matter paradigm, at the relevant redshifts for reionisation most ionising radiation is produced by stars in proto-galaxies forming in dark matter haloes of $< 10^9 \, \mathrm{M}_{\odot}$ [@Barkana:2001tz; @Choudhury:2008dm]. We compute the ionising photon production and escape fraction in proto-galaxies in this mass range using the [First Billion Years]{} (FiBY) simulation suite (Khochfar et al. in prep.; Dalla Vecchia et al. in prep.). The simulation we use contains $2 \times 684^3$ dark matter and gas particles in a comoving volume of 4 Mpc on the side, with a gas-particle mass of 1250 $M_{\odot}$. At redshift 6, the simulation reproduces the observed mass function of galaxies and star formation rates (Khochfar et al. in prep.). For the FiBY simulation we use a modified version of the [OWLS]{} code [@Schaye:2010jl]. Star formation follows a pressure law [@2008MNRAS.383.1210S], where we assume population (Pop) III stars form at metallicities $Z < 10^{-4} \, \mathrm{Z}_{\odot}$ (with $\mathrm{Z}_{\odot} = 0.02$) and Pop II stars at higher metallicities. Supernova feedback is modelled by injecting thermal energy that is efficiently converted into kinetic energy without the need to turn off radiative cooling temporarily [@2012MNRAS.426..140D]. Feedback from reionisation is modelled as a uniform UV-background following @2001cghr.confE..64H by switching from collisional to photo-ionisation equilibrium cooling tables. Gas above a density threshold of $n_{\mathrm{shield}} = 0.01 \, \mathrm{cm}^{-3}$ is modelled to shield against ionising radiation [@2011AAS...21734501N]. We assume that reionisation takes place within the bounds set by WMAP [@2011ApJS..192...18K], starting around redshift $12$ and ending around redshift $9$. This is consistent with the computations of the ionising emissivity from proto-galaxies that we present in this letter. We will show that between redshift $12$ and $9$ the proto-galaxies in the simulation produce enough ionising photons to reionise the computational volume. We have extracted all haloes from this simulation that contain at least 1 star, 1000 dark matter particles and 100 gas particles for post-processing with radiative transfer simulations. This results in more than 11000 haloes between $z=20$ and $z=6$. We determine the escape fraction with an updated version of the [SimpleX]{} radiative transfer code [@Paardekooper:2010iu], computing the absorption of the ionising radiation by both hydrogen and helium atoms in 10 frequency bins until the photons reach the virial radius of the halo. We will discuss these simulations in more detail in a forthcoming paper (Paardekooper et al. in prep). The luminosity and spectra of the star particles are computed from stellar synthesis models for both Pop III [@Raiter:2010hs] and Pop II [@2003MNRAS.344.1000B] stars. Combined with the escape fraction, we obtain the number of ionising photons that every proto-galaxy in our simulation contributes to cosmic reionisation. To first order the reionisation process can be modelled by equating the number of photons produced per baryon to the number of recombinations in the ionised IGM [e.g. @Madau:1999kl]. The volume fraction of ionised hydrogen, $Q_\mathrm{H \, II}$ is then given by $$ \frac{\mbox{d} Q_\mathrm{H \, II} }{\mbox{d}t} = \frac{ \dot{N}_{\mathrm{ion}}}{\bar{n}_{\mathrm{H,}0}} - Q_\mathrm{H \, II} \, C \, \bar{n}_{\mathrm{H,}0} \, \alpha(T) \, (1+z)^3,$$ where $\dot{N}_{\mathrm{ion}}$ is the total number of ionising photons available for reionisation per second per comoving Mpc, $\bar{n}_{\mathrm{H,}0} = 1.90641 \times 10^{-7} \, \mathrm{cm}^{-3}$ is the current mean number density of hydrogen in the IGM, $\alpha(T)$ is the recombination coefficient of hydrogen, which is a function of the IGM temperature $T$, $C \equiv \langle n_{\mathrm{H}}^2 \rangle / \langle n_{\mathrm{H}} \rangle^2$ is the clumping factor of the gas in the IGM and $z$ is the redshift. We assume that the ionised gas in the IGM has a temperature of 20,000 K, while we compute the clumping factor of the IGM gas from the simulation, using only gas with overdensity $\Delta < 100$, thus excluding gas inside dark matter haloes (because recombinations in that gas are already taken into account in the escape fraction calculations). We find a redshift-dependent clumping factor between 1.5 and 6.5, consistent with previous studies [@Pawlik:2009id; @2012ApJ...747..100S]. We compute the Thomson optical depth, which is the quantity measured by the WMAP satellite, by integrating $Q_\mathrm{H \, II}$ over all redshifts: $$ \tau_{\mathrm{e}} = \int_0^{z_{\mathrm{rec}}} \mbox{d}z \left\| \frac{ \mbox{d} t}{ \mbox{d} z} \right\| c \, Q_\mathrm{H \, II}(z) \, \bar{n}_{\mathrm{H,}0} \, (1+z)^3 \, \sigma_{\mathrm{T}},$$ with $\sigma_{\mathrm{T}}$ the cross section for Thomson scattering and $c$ the speed of light. The contribution of proto-galaxies to cosmic reionisation ========================================================= ![The escape fraction of ionising photons as function of redshift. The black solid line represents the escape fraction averaged over all haloes, while the blue dashed, green dotted and red dash-dot lines denote the escape fraction averaged over proto-galaxies in haloes with virial masses of $10^7 \, \mathrm{M}_{\odot}$, $10^8 \, \mathrm{M}_{\odot}$ and $10^9 \, \mathrm{M}_{\odot}$, respectively. The grey area represents the standard deviation of the mean. []{data-label="fig_fEsc"}](fig1.pdf){width="\columnwidth"} In Fig. \[fig\_fEsc\] we show the average escape fraction as function of redshift for proto-galaxies in haloes of different masses. The average escape fraction rises with time, but proto-galaxies inside a certain mass halo at the same redshift may have very different escape fractions, for example due to a different formation history or environment, which causes the large standard deviation in the mean. Due to the efficiency of stellar feedback in clearing away the gas from the dense sites of star formation, the escape fraction in $10^7 \, \mathrm{M}_{\odot}$ haloes is higher than in the $10^8 \, \mathrm{M}_{\odot}$ haloes. Proto-galaxies in haloes with masses above $10^9 \, \mathrm{M}_{\odot}$ have, at all redshifts, significantly lower escape fractions than their counterparts in lower mass haloes, due to their larger and denser gas content. Haloes with masses below $10^8 \, \mathrm{M}_{\odot}$ dominate the ionising photon budget at redshifts higher than 10, due to their high escape fractions and high abundance. After redshift 10, the ionising emissivity of the proto-galaxies in these haloes drops as a result of suppression of star formation by the uniform UV-background. The background heating only suppresses star formation in haloes that do not contain enough dense gas to shield against the radiation. This counteracts the effect of the high escape fraction, since ionising radiation is mainly produced by massive, young stars and suppression of star formation results in little or no ionising radiation being produced (see Fig. \[fig\_Q\]). Low-mass haloes therefore only contribute to the ionising photon budget when the part of the Universe they live in is still neutral, while more massive haloes have enough dense gas to shield against the external radiation and continue to form stars. ![The contribution of proto-galaxies to reionisation. In all panels the black solid lines represent all haloes, while the blue dashed and green dotted lines show the contribution of haloes with masses below $10^7 \, \mathrm{M}_{\odot}$ and $10^8 \, \mathrm{M}_{\odot}$, respectively. The red dash-dot lines represent the contribution of Pop III stars. [Top panel:]{} The halo mass below which 50% of the ionising photons is produced, $M_{0.5}$, at each redshift. The grey area represents the range of halo masses in which star formation is taking place. [Middle panel:]{} The optical depth for Thomson scattering, $\tau_{\mathrm{e}}$ as function of redshift. The data point represents the value of $\tau_{\mathrm{e}}$ as found by measurements with the WMAP satellite [@2011ApJS..192...18K]. [Bottom panel:]{} The volume filling fraction of ionised hydrogen, $Q_{\mathrm{H\,II}}$, as function of redshift. []{data-label="fig_Q"}](fig2.pdf){width="\columnwidth"} In Fig. \[fig\_Q\] we show the evolution of $Q_\mathrm{H \, II}$ and $\tau_\mathrm{e}$ with redshift. Reionisation is complete at redshift 10.5, with a duration (defined as the redshift interval in which $Q_\mathrm{H \, II}$ changes from 20% to 80 %) of $\Delta z \approx 2.1$. This is before the lower limit of the end of reionisation set by the measurements of quasar spectra. The dashed curve shows the reionisation history including only proto-galaxies residing in $< 10^7 \, \mathrm{M}_{\odot}$ haloes, which do not produce enough photons to reionise the Universe. The bulk of photons is produced by proto-galaxies in haloes with masses between $10^7 \, \mathrm{M}_{\odot}$ and $10^8 \, \mathrm{M}_{\odot}$. Reionisation is only delayed by $\Delta z \approx 0.5$ if we exclude all sources in haloes with masses larger than $10^8 \, \mathrm{M}_{\odot}$. The integrated Thomson scattering optical depth that we find in our model is $\tau_\mathrm{e} = 0.096$, which is well within the error bars from the WMAP measurement [@2011ApJS..192...18K]. This shows that the reionisation history we find is consistent with the two main observational constraints of reionisation, the absorption features in high-redshift quasars and the Thomson optical depth as observed by WMAP. The contribution of Pop III stars to reionisation is negligible in our simulations. Although these sources produce copious amounts of ionising photons [@Schaerer:2002bm], they are short-lived and not abundant enough to contribute significantly to reionisation. The contribution of Pop III stars to the total photon budget is exceeded by metal-enriched Pop II stars at redshifts below 15. We thus conclude that although reionisation started with the appearance of the first Pop III stars, they did not contribute significantly to reionisation on global scale. In the top panel in Fig. \[fig\_Q\] we show the halo mass below which 50% of the ionising photons are produced as function of redshift. At redshifts higher than $10$, half of the ionising photons are produced by proto-galaxies in haloes with masses between $10^7 \, \mathrm{M}_{\odot}$ and $10^8 \, \mathrm{M}_{\odot}$. After this redshift, haloes with higher masses take over the photon production, because star formation is suppressed in the lower mass haloes due to the UV-background. ![The cumulative number of ionising photons per baryon as function of redshift. Different colors denote the contribution of proto-galaxies below a certain stellar mass. In our computation full reionisation requires $1.6$ photons per baryon, which is represented by the black dotted line.[]{data-label="fig_NIonBaryon"}](fig3.pdf){width="\columnwidth"} To get a better picture of the mass of galaxies mostly contributing to reionisation, we show in Fig. \[fig\_NIonBaryon\] the cumulative number of photons per baryon produced by proto-galaxies below a certain stellar mass as function of redshift. Given our estimates of the ionising emissivity and the clumping factor, reionisation requires approximately 1.6 photons per baryon. Proto-galaxies with stellar masses below $10^6 \, \mathrm{M}_{\odot}$ have produced this number of photons by redshift 10, with most ionising photons being produced by proto-galaxies with stellar masses between $10^5 \, \mathrm{M}_{\odot}$ and $10^6 \, \mathrm{M}_{\odot}$. The observability of the sources of reionisation ================================================ ![The observability of the sources of reionisation in the UV continuum and three recombination lines. In all plots colors are similar to the colors in Fig. \[fig\_NIonBaryon\]. The shaded area denotes the maximum and minimum brightness and the orange area covers the range detectible by the relevant instrument on JWST. []{data-label="fig_obs"}](fig4.pdf){width="\columnwidth"} One of the science goals of the [James Webb Space Telescope*]{} (JWST) is to study the galaxy population during reionisation [@2006SSRv..123..485G]. The deepest NIRCam survey will search for high-redshift objects with exposure times of $10^6$ seconds, resulting in a flux limit of $1.4 \, \mathrm{nJy}$ at $2 \, \mu\mathrm{m}$. This can be converted into AB magnitude using $$m_{\mathrm{AB}} = 31.4 - 2.5\log(f_{\nu}),$$ resulting in a magnitude limit of $m_{\mathrm{AB}} = 31.0$. In addition, JWST will search for recombination line radiation from the first galaxies using the MIRI and NIRspec instruments. At $5.6 \, \mu \mathrm{m}$, the limiting flux of the MIRI instrument is $23$nJy or $m_{\mathrm{AB}} = 28$, while NIRSpec will measure line intensities down to $2 \times 10^{-19} \, \mathrm{erg} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}$ with a resolution $R=1000$. We determine the flux that the NIRCam survey would receive from the proto-galaxies in our simulation by computing the spectra of the Pop II and Pop III stars for radiation with energy below the ionisation energy of hydrogen. In Fig. \[fig\_obs\] we show the redshifted flux from the sources in the $2 \, \mu\mathrm{m}$ band, although due to the flatness of the spectrum in this wavelength range the results do not depend sensitively on the choice of wavelength. Reionisation is driven by proto-galaxies with stellar masses around $10^6 \, \mathrm{M}_{\odot}$. At redshift 10.5, when the proto-galaxies have produced enough photons to reionise the volume, these sources are not brighter than $f_{\nu} = 0.09 \,\,\mathrm{nJy}$ or $m_{\mathrm{AB}} = 34$ , making it impossible for JWST to observe them. If we rescale the flux limits according to $f_{\mathrm{lim}} \propto 1/\sqrt{t_{\mathrm{exp}}}$, exposure times of $2 \times 10^8 \mathrm{s}$ are necessary to observe these sources. Since we do not account for dust attenuation between the proto-galaxy and the observer, we can only give upper limits of the flux. In addition, we show in Fig. \[fig\_obs\] the brightness of the proto-galaxies in our simulation in three emission lines, $\mathrm{H}\alpha$, Ly-$\alpha$ and $\mathrm{He II} \, 1640 \mbox{\AA}$. The flux in these lines is given by [@Johnson:2009gb] $$f(\lambda_{\mathrm{obs}}) = \frac{\ell_{\mathrm{em}} \lambda_{\mathrm{em}} (1+z) R}{4 \pi c D_{\mathrm{L}}^2(z)},$$ where $\ell_{\mathrm{em}}$ is the luminosity along the line of sight, $D_{\mathrm{L}}^2(z)$ is the luminosity distance at redshift $z$, $c$ is the light speed and $R=\lambda/\Delta \lambda$ is the spectral resolution. Assuming that the galaxy is unresolved, we compute $\ell_{\mathrm{em}}$ for every proto-galaxy by summing up the contribution to the emissivity from every gas particle. JWST will observe the $\mathrm{H}\alpha$ line with the MIRI instrument. At redshift $12$ and below, proto-galaxies of $M_{\star} \ge 10^6 \, \mathrm{M}_{\odot}$ are bright enough to be detected with MIRI. The Ly-$\alpha$ and $\mathrm{He II} \, 1640 \mbox{\AA}$ lines will be observed with the NIRSpec instrument. The greater sensitivity of the NIRSpec instrument could in principle make it possible to observe the Ly-$\alpha$ emission line from all sources with $M_{\star} \ge 10^4 \, \mathrm{M}_{\odot}$ at redshift 20 and below. However, this line is difficult to detect at these high redshifts due to the large Gunn-Peterson optical depth. Scattering off interstellar neutral gas could make the line observable even through a fully neutral IGM [@2010MNRAS.408..352D]. The flux in the $\mathrm{He II} \, 1640 \mbox{\AA}$ line is always lower than the $\mathrm{H}\alpha$ flux. It is therefore most likely that the sources of reionisation will be observed in the $\mathrm{H}\alpha$ or Ly-$\alpha$ line. Conclusions and discussion ========================== We have presented high-resolution cosmological simulations of galaxy formation, from which we calculated the ionising photon production and escape fraction of a large statistical sample of galaxies. From these simulations we computed the contribution of proto-galaxies to cosmic reionisation in a self-consistent way. Our main findings are: - Reionisation is primarily driven by proto-galaxies in dark-matter haloes with masses between $10^7 \, \mathrm{M}_{\odot}$ and $10^8 \, \mathrm{M}_{\odot}$, which have very high escape fractions, because supernova feedback efficiently acts to clear away gas from the sites of star formation. - Star formation in these haloes is suppressed by UV-feedback, these proto-galaxies therefore only contribute to reionisation when the part of the Universe they live in is still neutral. - After reionisation the Universe is kept ionised by massive galaxies with lower escape fractions, that start appearing more frequently. - We find a strong mass and redshift dependence of the escape fraction that suggests that galaxies above the observational limits of present surveys do not contribute enough photons to drive reionisation. - Pop III stars do not contribute significantly to the reionisation process. - There is great prospect that JWST will observe the proto-galaxies that reionised the Universe in the $\mathrm{H}\alpha$ and Ly-$\alpha$ recombination lines. In our galaxy formation simulation radiative feedback on the halo gas from sources within the halo is neglected. Radiative feedback is capable of evacuating gas from haloes with $M_{\mathrm{vir}} \lesssim 10^7 \, \mathrm{M}_{\odot}$ [@Wise:2009fn], thereby suppressing further star formation in the halo. We find that supernova feedback has the same effect, albeit with a short delay. Since the time it takes for the dense, cold gas in the halo to be converted into stars ($\sim 1 \, \mathrm{Gyr}$) is much larger than the lifetime of the massive stars that end their lives as supernova ($\sim 1 \, \mathrm{Myr}$ for pair-instability supernovae), this delay will not affect our results significantly. We model reionisation with a uniform UV-background, disregarding the contribution from local sources. We could therefore be underestimating the suppression of star formation in haloes that are not able to shield against the ionising radiation. This likely occurs in haloes with masses $M_{\mathrm{vir}} \lesssim 10^9 \, \mathrm{M}_{\odot}$ [@Okamoto:2008ha; @Hasegawa:2012wu]. Since this will only affect proto-galaxies in a region of the Universe that has already been ionised, this will not change our conclusions. However, it could delay the completion of reionisation due to suppression of clustered low-mass sources and reduce the number of photons available for reionisation. Assuming all star formation in haloes with $M_{\mathrm{vir}} < 10^9 \, \mathrm{M}_{\odot}$ is suppressed after reionisation brings the cumulative number of ionising photons per baryon at redshift 6 in our simulation close to the observed value [@Bolton:2007gc]. The resolution in our simulation is around 6 physical pc at redshift 15, which is high enough to resolve regular and giant molecular clouds at the sites of star formation. However, we do not resolve the birth cloud of the stellar population, which would lower the escape fraction. Since this equally applies to all haloes in the volume, it does not affect our conclusion on the mass range that contributes most photons to reionisation. The effect of a lower escape fraction can be estimated by scaling the curves in Fig. \[fig\_NIonBaryon\] accordingly. The volume of our simulation is 4 Mpc, which could possibly bias our results. The initial conditions for the cosmological simulation were chosen to avoid biased regions with high $\sigma$ peaks. This is reflected in the mass function of dark matter haloes in our volume, which is in agreement with the average of the Sheth-Tormen mass function [@Sheth:2002fn]. Although we cannot accurately sample the high-mass end of the halo mass function, the strong mass-dependence we find for the escape fraction indicates that the contribution from these rare high-mass sources is small. Our results show that reionisation is initially a local process driven by the appearance of many low-mass haloes which do not cluster strongly and ionise their neighbourhood until [[H$\,$ii]{} ]{}regions overlap and cover larger cosmic volumes. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Jarrett Johnson and the TMoX group for helpful discussions and Jim Dunlop and Chael Kruip for comments on an earlier draft. C.D.V. acknowledges support by Marie Curie Reintegration Grant FP7-RG-256573. Barkana R., Loeb A., 2001, Physics Reports, 349, 125 Bland-Hawthorn J., Maloney P. R., 1999, , 510, L33 Bolton J. S., Haehnelt M. G., 2007, , 382, 325 Bouwens R. J., Illingworth G. D., Oesch P. A., Trenti M., Labbe I., Franx M., Stiavelli M., Carollo C. M., van Dokkum P., Magee D., 2012, , 752, L5 Bruzual G., Charlot S., 2003, , 344, 1000 Choudhury T. R., Ferrara A., Gallerani S., 2008, , 385, L58 Dalla Vecchia C., Schaye J., 2012, , 426, 140 Deharveng J.-M., Buat V., Le Brun V., Milliard B., Kunth D., Shull J. M., Gry C., 2001, , 375, 805 Dijkstra M., Wyithe J. S. B., 2010, , 408, 352 Fan X., Carilli C. L., Keating B., 2006, , 44, 415 Faucher-Giguere C.-A., Lidz A., Hernquist L., Zaldarriaga M., 2008, , 682, L9 Gardner J. P., Mather J. C., Clampin M., Doyon R., Greenhouse M. A., Hammel H. B., Hutchings J. B., Jakobsen P., Lilly S. J., Long K. S., Lunine J. I., Mccaughrean M. J., Mountain M., Nella J., Rieke G. H., Rieke M. J., Rix H.-W., Smith E. P., Sonneborn G., Stiavelli M., Stockman H. S., Windhorst R. A., Wright G. S., 2006, Space Science Reviews, 123, 485 Gnedin N. Y., Kravtsov A. V., Chen H.-W., 2008, , 672, 765 Haardt F., Madau P., 2001, "Clusters of galaxies and the high redshift universe observed in X-rays, p. 64 Hasegawa K., Semelin B., 2012, arXiv:1209.4143 Heckman T. M., Sembach K. R., Meurer G. R., Leitherer C., Calzetti D., Martin C. L., 2001, , 558, 56 Iwata I., Inoue A. K., Matsuda Y., Furusawa H., Hayashino T., Kousai K., Akiyama M., Yamada T., Burgarella D., Deharveng J.-M., 2009, , 692, 1287 Johnson J. L., Greif T. H., Bromm V., Klessen R. S., Ippolito J., 2009, , 399, 37 Komatsu E., Smith K. M., Dunkley J., Bennett C. L., Gold B., Hinshaw G., Jarosik N., Larson D., Nolta M. R., Page L., Spergel D. N., Halpern M., Hill R. S., Kogut A., Limon M., Meyer S. S., Odegard N., Tucker G. S., Weiland J. L., Wollack E., Wright E. L., 2011, , 192, 18 Kuhlen M., Faucher-Giguere C.-A., 2012, , 423, 862 Madau P., Haardt F., Rees M. J., 1999, , 514, 648 Nagamine K., Choi J., Yajima H., 2011, American Astronomical Society, 217, 34501 Nestor D. B., Shapley A. E., Steidel C. C., Siana B., 2011, , 736, 18 Okamoto T., Gao L., Theuns T., 2008, , 390, 920 Paardekooper J.-P., Kruip C. J. H., Icke V., 2010, , 515, A79 Paardekooper J.-P., Pelupessy F. I., Altay G., Kruip C. J. H., 2011, , 530, A87 Pawlik A. H., Schaye J., van Scherpenzeel E., 2009, , 394, 1812 Raiter A., Schaerer D., Fosbury R. A. E., 2010, , 523, A64 Razoumov A. O., Sommer-Larsen J., 2010, , 710, 1239 Robertson B. E., Ellis R. S., Dunlop J. S., McLure R. J., Stark D. P., 2010, , 468, 49 Schaerer D., 2002, , 382, 28 Schaye J., Dalla Vecchia C., 2008, , 383, 1210 Schaye J., Dalla Vecchia C., Booth C. M., Wiersma R. P. C., Theuns T., Haas M. R., Bertone S., Duffy A. R., McCarthy I. G., Voort F. v. d., 2010, , 402, 1536 Sheth R. K., Tormen G., 2002, , 329, 61 Shull J. M., Harness A., Trenti M., Smith B. D., 2012, , 747, 100 Tanvir N. R., Levan A. J., Fruchter A. S., Fynbo J. P. U., Hjorth J., Wiersema K., Bremer M. N., Rhoads J., Jakobsson P., O’Brien P. T., Stanway E. R., Bersier D., Natarajan P., Greiner J., Watson D., Castro-Tirado A. J., Wijers R. A. M. J., Starling R. L. C., Misra K., Graham J. F., Kouveliotou C., 2012, , 754, 46 Trenti M., Stiavelli M., Bouwens R. J., Oesch P., Shull J. M., Illingworth G. D., Bradley L. D., Carollo C. M., 2010, , 714, L202 Vanzella E., Guo Y., Giavalisco M., Grazian A., Castellano M., Cristiani S., Dickinson M., Fontana A., Nonino M., Giallongo E., Pentericci L., Galametz A., Faber S. M., Ferguson H. C., Grogin N. A., Koekemoer A. M., Newman J., Siana B. D., 2012, , 751, 70 Wise J. H., Cen R., 2009, , 693, 984 Yajima H., Choi J.-H., Nagamine K., 2010, , 412, 411 \[lastpage\] [^1]: E-mail: jppaarde@mpe.mpg.de
--- author: - \| AAAI Press\ Association for the Advancement of Artificial Intelligence\ 2275 East Bayshore Road, Suite 160\ Palo Alto, California 94303\ bibliography: - 'references.bib' title: 'Supplementary material for learning generative networks from off-target samples ' --- Proof for theorems =================== We first have the following lemma, that readily follows from the Proposition 1 and Theorem 1 of Goodfellow et al [@GoodfellowNIPS2014]. For a fixed generator $G$, the optimal discriminator $\hat{\D}$ that achieves the minimum of equation (6) is $$\hat{\D}^(\x) = \frac{\hatw(\x)q(\x)}{\hatw(\x)q(\x) + \pg(\x)}$$ and the $\pg$ that achieves the maximum of of $\hat{D}^$ is $\pg = \hatw(\x)q(\x)$ Recall the definition of $\rho$, $$\rho = \sup_{x \in \suppp} \frac{q(\x)}{p(\x)}$$ We first prove the upper-bound on KL divergence between $p$ and $\pg$. If $w(\x) \geq \epsilon\ \forall x\in \suppq$, and $J(\hatw) \leq \epsilon^2$, then $$\KL(p\|\|\pg) \leq \log \Big(\frac{1}{1-\epsilon \rho}\Big)$$ $$\begin{aligned} \KL(p\|\|\pg) &= \int_\x p(\x) \log \frac{p(\x)}{\hatw(\x)q(\x)}d\x\\\end{aligned}$$ Since $J(\hatw)\leq \epsilon^2$, we have $\|\hatw(\x) - w(\x)\|\leq \epsilon$, and $\forall \x\in \suppq$, $\hat{w}(\x) \geq w(\x) - \epsilon$. Decreasing the denominator increases the value of the whole expression, we may take $ w(\x)-\epsilon$ to upper bound the $\KL(p\|\|\pg)$. $$\begin{aligned} \KL(p\|\|\pg) &\leq \int_\x p(\x) \log \frac{p(\x)}{q(\x)(w(\x)-\epsilon)}\\ &= -\int_\x p(\x) \log \frac{q(\x)w(\x) - q(\x)\epsilon}{p(\x)}\\ &= -\int_\x p(\x) \log \Big( 1 - \frac{q(\x)}{p(\x)}\epsilon \Big)\\ &\leq -\int_\x p(\x) \log \Big( 1 - \rho\epsilon \Big)\end{aligned}$$ where the last line follows from the definition of $\rho$, and decreasing the quantity inside log increases the value of the whole expression. Now this is equivalent to $$\begin{aligned} \KL(p\|\|\pg) \leq -log(1-\rho\epsilon)\end{aligned}$$ Notice that $0\leq\rho \epsilon\leq 1$ because $\rho = \sup_{ x \in \suppp} \frac{1}{w(\x)}$ and $\epsilon < w(\x)$. We now prove the upper-bound on reverse-KL divergence between $p$ and $\pg$. If $J(\hatw) \leq \epsilon^2$, then $$KL(p_G \|\| p) \leq (1 + \epsilon) \log (1 + \epsilon\rho)$$ By Lemma 1, $p_G(\x) = \hat{w}(\x) q(\x)$, so $$KL(p_G \|\| p) = \int_\x \hat{w}(\x) q(\x) \log \frac{\hat{w}(\x) q(\x)}{p(\x)} d\x$$ Since $J(\hatw)\leq \epsilon^2$, we have $\|\hatw(\x) - w(\x)\|\leq \epsilon$, and $\forall x\in \suppq$, $\hat{w}(\x) \leq w(\x) + \epsilon$. So, $$\begin{aligned} KL(p_G \mid p) & \leq \int_\x (w(\x) + \epsilon)q(\x) \log \frac{(w(\x) + \epsilon) q(\x)}{p(\x)} d\x \\ & \leq \int_\x (w(\x)q(\x) + \epsilon q(\x)) \log \frac{w(\x)q(\x) + \epsilon q(\x)}{p(\x)} d\x \\ & \leq \int_\x (p(\x) + \epsilon q(\x)) \log \frac{p(\x) + \epsilon q(\x)}{p(\x)} d\x \\ & \leq \int_\x (p(\x) + \epsilon q(\x)) \log \left(1 + \epsilon\frac{q(\x)}{p(\x)}\right) d\x\\ & \leq \int_\x (p(\x) + \epsilon q(\x)) \log (1 + \epsilon\rho) d\x\\ & \leq \log (1 + \epsilon\rho) \int_\x) (p(\x) + \epsilon q(\x)) d\x\\ & \leq (1 + \epsilon) \log (1 + \epsilon\rho)\\\end{aligned}$$ Details of experiments ====================== We have used Keras[^1], a deep learning software package, to implement our NNs. Architecture for experiments ----------------------------- For the first domain, $D$, $G$, and $\hatw$ takes in as input the number of objects to be placed and the shape of the object. They do not conditioned on a state. For the second domain, they take poses of objects as an input. They do not condition on the objects already placed. For the last domain, they take poses of all the objects on the table as input. For the first domain, we use 3 layers of dense network for $D$,$G$, and $\hatw$. For the second and last domain, we use convolutional layers, as the location of each pose in the vector is unimportant. Specifically, for $D$, we have a input convolutional layer that has filter size 2 by 2, with stride of 2 by 1, and 256 filters. This is followed by a max pooling layer whose size is 2 by 1, and then this conv-max pooling is repeated one more time. Then, it is followed by two more conv layers whose number of filters is 256, and the filter size is 3 by 1. Then, the output gets merged with the action input, which is then followed by three dense layers each with size 32, 256, and 32. Lastly, It uses sigmoid activation in its output layer. The generator has exactly the same architecture, except that the last three dense layers have 32, 32, and 32 as its number of nodes. It has linear output layer. $\hatw$ has exactly the same architecture as the generator The maximum number of epochs for training $D$ and $G$ is 500, with batch size of 32. We used Adam with learning rate 0.001 for $D$ and $G$, and Adadelta with learning rate 1.0 for $\hatw$. [^1]: <https://github.com/fchollet/keras>
--- author: - Alexander E Patkowski title: A Strange Integral Equation and Some Integrals Associated with Number Theory --- Introduction ============ In Titchmarsh \[14, pg.35\], we find a famous relation connecting Fourier cosine transforms with Mellin transforms $$\int_{0}^{\infty}f(t)\lambda(t)\cos(xt)dt=\frac{1}{2i\sqrt{y}}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\phi(s-\frac{1}{2})\phi(\frac{1}{2}-s)(s-1)\Gamma(1+\frac{s}{2})\pi^{-\frac{s}{2}}\zeta(s)y^sds,$$ where $f(t)=\|\phi(it)\|^2,$ $\zeta(s)$ is the Riemann zeta function \[6, 14\], $\Gamma(s)$ is the gamma function, $\xi(s):=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s),$ $\lambda(t):=\xi(\frac{1}{2}+it)$ (Riemann’s $\Xi$ function), and $y=e^{x}.$ Recall \[6, 14\] the famous Riemann $\xi$ function satisfies the functional equation $$\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)=\frac{1}{2}s(s-1)\pi^{-\frac{1-s}{2}}\Gamma(\frac{1-s}{2})\zeta(1-s).$$ As usual, we put $\Re(s)=\sigma,$ $\Im(s)=t,$ for $s\in\mathbb{C}.$ Many authors \[4, 5, 8, 9, 12, 13, 14\] have utilized equation (1.1) and its variations to obtain many interesting relations among other special functions, as well as results on $\lambda(t).$ See also \[10, 11\] for some interesting ideas on integrals related to Riemann’s $\xi$ function. The most widely cited example in the literature appears to be \[14, eq.(2.16.1)\] (for example, see \[2, 7\]), and its variant \[14, eq.(2.16.2)\]: $$\Lambda(x):=\int_{0}^{\infty}\frac{\lambda(t)}{t^2+\frac{1}{4}}\cos(xt)dt=\frac{\pi}{2}\left(e^{x/2}-2e^{-x/2}\psi(e^{-2x})\right).$$ Here we have a slightly modified Jacobi theta function $\psi(x)=\sum_{n\ge1}e^{-\pi n^2 x}.$ As usual, we define the Laplace transform to be $$\mathcal{L}(f)(s):=\int_{0}^{\infty}f(t)e^{-st}dt.$$ If we assume $\Re(s)>1,$ we may write $$\begin{aligned} \mathcal{L}(\Lambda(x))(s-\frac{1}{2})\\ &=\int_{0}^{\infty}e^{-(s-\frac{1}{2})x}\int_{0}^{\infty}\frac{\lambda(t)}{t^2+\frac{1}{4}}\cos(xt)dtdx\\ &=(s-\frac{1}{2})\int_{0}^{\infty}\frac{\lambda(t)}{(t^2+\frac{1}{4})(t^2+(s-\frac{1}{2})^2)}dt\\ &=\frac{\pi}{2}\int_{0}^{\infty}e^{-sx}\left(e^{x}-2\psi(e^{-2x})\right)dx\\ &=\frac{\pi}{2}\left(\frac{1}{s-1}-2\int_{0}^{\infty}e^{-sx}\psi(e^{-2x})\right)dx\\ &=\frac{\pi}{2}\left(\frac{1}{s-1}+2\lim_{r\rightarrow0^{+}}\int_{1}^{r}t^{s-1}\psi(t^2)dt\right)\\ &=\frac{\pi}{2}\left(\frac{1}{s-1}-x^{-\frac{s}{2}}\sum_{n\ge1}\lim_{r\rightarrow0^{+}}\int_{r}^{x}t^{\frac{s}{2}-1}e^{-\pi n^2 t/x}dt\right)\\ &=\frac{\pi}{2}\left(\frac{1}{s-1}-\pi^{-\frac{s}{2}}\sum_{n\ge1}n^{-s}\gamma(\frac{s}{2},\pi n^2)\right)\\ &=\frac{\pi}{2}\left(\frac{1}{s-1}-\pi^{-\frac{s}{2}}\sum_{n\ge1}n^{-s}\left(\Gamma(\frac{s}{2})-\Gamma(\frac{s}{2},\pi n^2)\right)\right)\\ &=\frac{\pi}{2}\left(\frac{1}{s-1}-\frac{2\xi(s)}{s(s-1)}+\pi^{-s/2}\sum_{n\ge1}n^{-s}\Gamma(\frac{s}{2},\pi n^2)\right)\end{aligned}$$ In the line (1.10) we made a change of variables $x=-\log t,$ and in subsequent lines employed the definition of the incomplete gamma functions $$\gamma(s,x)=\int_{0}^{x}t^{s-1}e^{-t}dt, ~~~~~~~~\mbox{and}~~~~~~~~ \Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}dt,$$ where $\Gamma(s)=\gamma(s,x)+\Gamma(s,x).$ Note that by an instance $F(y)=e^{-ay^2}$ of the Müntz formua \[14, eq.(2.11.1)\], for $0<\sigma<1,$ $$\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\Gamma(\frac{s}{2})\zeta(s)(\sqrt{a}x)^{-s}ds=\sum_{n\ge1}e^{-an^2x^2}-\frac{1}{2x}\sqrt{\frac{\pi}{a}}.$$ Multiplying through by $x^{v-1},$ with $\Re(v)>1,$ and integrating over the interval $[0, z],$ $z>0,$ we obtain $$\frac{z^v}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\Gamma(\frac{s}{2})\zeta(s)\frac{(\sqrt{a}z)^{-s}}{(v-s)}ds=\frac{a^{-v/2}}{2}\sum_{n\ge1}n^{-v}\gamma(\frac{v}{2},a zn^2)-\frac{z^{v-1}}{2(v-1)}\sqrt{\frac{\pi}{a}},$$ again for $0<\sigma<1.$ (Note this integration is justified on the left side, since if $\Re(v-s)>0,$ then $0^{v-s}=0.$) On the other hand the integral on the left-hand side of (1.16) is precisely $\Upsilon(v)$ defined in Theorem 1, when $a=\pi,$ $z=1,$ $\sigma=\frac{1}{2}.$ Hence we have arrived twice at the following result. When $\Re(s)>1,$ we have $$\Upsilon(s):=(s-\frac{1}{2})\int_{0}^{\infty}\frac{\lambda(t)}{(t^2+\frac{1}{4})(t^2+(s-\frac{1}{2})^2)}dt=\frac{\pi}{2}\left(\frac{1}{s-1}-\frac{2\xi(s)}{s(s-1)}+\pi^{-s/2}\sum_{n\ge1}n^{-s}\Gamma(\frac{s}{2},\pi n^2)\right).$$ Or equivalently, the Riemann $\xi$-function satisfies the integral equation $$\int_{0}^{\infty}f(\frac{1}{2}+it)K(s,t)dt=\frac{\pi}{2}\left(\frac{-2f(s)}{s(s-1)}+g(s)\right),$$ with kernal $K(s,t)=(t^2+\frac{1}{4})^{-1}(t^2+(s-\frac{1}{2})^2)^{-1}(s-\frac{1}{2}),$ where $g(s)$ may be explicitly computed for any solution $f$ of (1.18). The assertion that $g(s)$ may be explicitly computed follows directly from Mellin transforms in the proof of Theorem 1. From Titchmarsh \[14, pg. 257\] we have $\lambda(t)\ll t^Ae^{-\frac{\pi}{4}t},$ which gives us $$\Upsilon(s)\ll \int_{0}^{\infty}\frac{t^Ae^{-\frac{\pi}{4}t}}{(t^2+\frac{1}{4})(t^2+(s-\frac{1}{2})^2)}dt<+\infty.$$ Clearly we have that $\Upsilon(s)=-\Upsilon(1-s).$ Upon noticing this fact, we may use (1.16) (or equivalently (1.17)) to obtain a well-known result concerning $\zeta(s).$ Computing the residue $R_{s=0}=\zeta(0)/v=-1/(2v)$ out from the integral, we may then extract the series $\frac{1}{2}\pi^{-(1-v)/2}\sum_{n\ge1}n^{-(1-v)}\Gamma(\frac{1-v}{2},\pi n^2)$ to obtain the following expansion \[1, pg.256, eq.(30)\] as a direct corollary to Theorem 1: $$\pi^{-s/2}\zeta(s)\Gamma(\frac{s}{2})=\frac{1}{s(s-1)}+\pi^{-s/2}\sum_{n\ge1}n^{-s}\Gamma(\frac{s}{2},\pi n^2)+\pi^{-(1-s)/2}\sum_{n\ge1}n^{-(1-s)}\Gamma(\frac{1-s}{2},\pi n^2).$$ Imaginary quadratic forms and the associated integral equation ============================================================== Following \[6, pg.511\] put $$L_{K}(s, \chi)=\sum_{\mathfrak{a}}\chi(\mathfrak{a})(N\mathfrak{a})^{-s},$$ where $\sigma>1,$ and $\chi$ maps the class group $\mathfrak{H}$ to the complex plane $\mathbb{C}.$ In \[8, 9\], we find N. S. Koshlyakov investigating integrals related to $L$-functions associated with number fields as well as \[12, 14\]. We consider his work coupled with that of the ideas in the introduction. Let $D$ denote a discriminant with respect to a primitive ideal $\mathfrak{a}.$ Then we have the functional equation \[6, eq.(22.51)\] $$\Omega_K(s,\chi)=\Omega_K(1-s,\chi),$$ where $\Omega_K(s,\chi)=(2\pi)^{-s}\Gamma(s)\|D\|^{\frac{s}{2}}L_K(s,\chi).$ An important integral representation relevant to our study, which continues $L_K(s,\chi)$ to the entire complex plane, is given by Hecke \[6, eq.(22.52)\]: $$\Omega_K(s,\chi)=\frac{\|\mathfrak{H}\|\delta(\chi)}{\bar{D}s(s-1)}+\int_{1}^{\infty}(t^{s-1}+t^{-s})\sum_{\mathfrak{a}}\chi(\mathfrak{a})e^{-2\pi tN\mathfrak{a}/\sqrt{\|D\|}}dt,$$ where $\mathfrak{\|H\|}$ is the class number, and $\bar{D}$ is $6$ if $D=-3,$ $-4$ if $D=4,$ and $2$ if $D<-4.$ We shall prove an equivalent form of this integral representation toward the end using Theorem 2. Put $\Omega_K(\frac{1}{2}+it,\chi)=\mathfrak{O}_K(t),$ and note that $$\int_{0}^{\infty}f(t)\mathfrak{O}_K(t)\cos(xt)dt=\frac{1}{2i\sqrt{y}}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\phi(s-\frac{1}{2})\phi(\frac{1}{2}-s)(2\pi)^{-s}\|D\|^{s/2}\Gamma(s)L_K(s,\chi)y^sds.$$ N.S. Koshlyakov appears to be the first to consider instances of this type of general integral with $L$-functions associated with number fields (see, especially, \[9, pg.217–220\]). We may easily evaluate the case $\phi(s)=1,$ and so $f(t)=1,$ for all $t.$ Consider $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}(2\pi)^{-s}\|D\|^{s/2}\Gamma(s)L_K(s,\chi)y^sds,$$ for $c>1,$ and move the line of integration to $c=\frac{1}{2}.$ Since $L_K(s,\chi)$ has a simple pole at $s=1$ when $\chi$ is trivial, we compute \[6, pg.512, eq.(22.50)\] $$R_{s=1}=\frac{\|\mathfrak{H}\|\delta(\chi)}{\bar{D}},$$ (where $\delta(\chi)=0$ when $\chi$ is non-trivial, $1$ otherwise) and obtain $$\int_{0}^{\infty}\mathfrak{O}_K(t)\cos(xt)dt=\frac{\pi}{2}\left(\frac{\|\mathfrak{H}\|\delta(\chi)e^{x/2}}{\bar{D}}-e^{-x/2}\Psi(e^{-x})\right),$$ where $\Psi(x)=\sum_{\mathfrak{a}}\chi(\mathfrak{a})e^{-2\pi(N\mathfrak{a})x/\sqrt{\|D\|}}.$ (Note that (2.5) is precisely $\Psi(y^{-1}).$) Applying the same concepts as in our introduction, we may obtain the following result from (2.6). For $\Re(s)>1,$ we have that $$\mathfrak{\Upsilon}(s)=\int_{0}^{\infty}\mathfrak{O}_K(t)\bar{K}(s,t)dt=\frac{\pi}{2}\left(\frac{\|\mathfrak{H}\|\delta(\chi)}{\bar{D}(s-1)}-\Omega_K(s,\chi)+\sum_{\mathfrak{a}}\chi(\mathfrak{a})\left(\frac{\sqrt{\|D\|}}{2\pi N\mathfrak{a}}\right)^s\Gamma(s,\frac{2\pi N\mathfrak{a}}{\sqrt{\|D\|}})\right),$$ with kernel $\bar{K}(s,t)=(s-\frac{1}{2})(t^2+(s-\frac{1}{2})^2)^{-1}.$ Due to the kernel, we again have $\mathfrak{\Upsilon}(s)=-\mathfrak{\Upsilon}(1-s),$ and as a direct corollary we have \[6, pg. 512, eq.(22.54)\] $$\Omega_K(s,\chi)=\frac{\|\mathfrak{H}\|\delta(\chi)}{\bar{D}s(s-1)}+\sum_{\mathfrak{a}}\chi(\mathfrak{a})\left(\frac{\sqrt{\|D\|}}{2\pi N\mathfrak{a}}\right)^s\Gamma(s,\frac{2\pi N\mathfrak{a}}{\sqrt{\|D\|}})+\sum_{\mathfrak{a}}\chi(\mathfrak{a})\left(\frac{\sqrt{\|D\|}}{2\pi N\mathfrak{a}}\right)^{1-s}\Gamma(1-s,\frac{2\pi N\mathfrak{a}}{\sqrt{\|D\|}}).$$ Other Applications and Comments for Further Research ==================================================== Here we discuss some results related to (1.3) that we hope will encourage further interest. First we recall for $0<\Re(s)<1,$ the well-known formula $$\int_{0}^{\infty}t^{s-1}\cos(xt)dt=\frac{\Gamma(s)\cos(\frac{\pi}{2}s)}{x^s}.$$ Hence for $0<\Re(s)<1,$ $$\int_{0}^{\infty}t^{s-1}\left(\int_{0}^{\infty}\cos(xt)(2\psi(x^2)-\frac{1}{x})dx\right)dt=\frac{2\xi(s)\Gamma(s)\cos(\frac{\pi}{2}s)}{s(s-1)}.$$ This gives us ($0<c<1$) $$2\psi(x^2)-\frac{1}{x}=\int_{0}^{\infty}\cos(xt)\left(\frac{1}{2\pi i }\int_{c-i\infty}^{c+i\infty}\frac{2\xi(s)\Gamma(s)\cos(\frac{\pi}{2}s)t^{-s}}{s(s-1)}ds\right)dt=\int_{0}^{\infty}\cos(xt)\dot{I}(t)dt,$$ say. So we may now compute (using (1.3)) $$\frac{\lambda(t)}{t^2+\frac{1}{4}}=\int_{0}^{\infty}\cos(xt)e^{-x/2}\left(\int_{0}^{\infty}\cos(e^{-x}u)\dot{I}(u)du\right)dx.$$ It is also possible to prove (3.2) using Parseval’s theorem for Mellin transforms and the functional equation (1.2). Using the work from \[3\] with the above observations, we consider the following result. See \[3, Definition 1.1\] for a criterion for an entire function to belong to the Laguerre-Pólya class. The Riemann Hypothesis is equivalent to the statement that the integral $$\ddot{I}(t):=\int_{0}^{\infty}\cos(xt)e^{-x/2}\left(\int_{0}^{\infty}\cos(e^{-x}u)\dot{I}(u)du\right)dx,$$ has only real zeros. Consequently, the Riemann Hypothesis is true if and only if $\ddot{I}(t)$ is in the Laguerre-Pólya class. The proof uses the integral (3.4) coupled with comparing the kernel of $\ddot{I}(t),$ given by $$\ddot{k}(x):=e^{-x/2}\left(\int_{0}^{\infty}\cos(e^{-x}u)\dot{I}(u)du\right)dx,$$ to what is considered an ‘admissible’ kernel according to the definition given in \[3, Definition 1.2\]. For example, it is easily verified that $\ddot{k}(x)$ is even by the functional equation for the Jacobi theta function (and (3.3)), $\ddot{k}(x)>0$ for all $x\in\mathbb{R},$ but $\frac{d}{dx}\ddot{k}(x)>0$ for all $x>0.$ Hence the kernel is monotone increasing for $x>0.$ The fact that the kernel satisfies these properties tells us that $\ddot{I}(t)$ is a real entire function, and it is possible to have only real zeros. The remainder of the proof utilizes the well-known criterion that the Riemann hypothesis is equivalent to the statement that all the zeros of $\lambda(t)$ are real, together with \[3, Definition 1.1\]. At this point we are left with some questions regarding the solutions of Theorem 1. First, in light of equation (1.18) resembling a “modified" Fredholm integral of the second kind, does the integral equation in Theorem 1 admit application of a kind of Fredholm theory? Can anything new be said by attempting a Neumann series solution? More interesting relations may be obtained by expanding $L_K(s,\chi)$ in (2.4) into a sum of Epstein zeta functions, with formulas related to the work in \[8, 9, 15\]. This would allow us to obtain integrals related to a theta functions of the form $\sum_{n,m\in\mathbb{Z} }\chi_{n,m}q^{an^2+bnm+cm^2},$ $4ac-b^2>0,$where $q=e^{-\pi x},$ $x>0.$ It may also be of interest to look at the integral $\ddot{I}(t)$ by explicitly computing $\dot{I}(u)$ using the Residue theorem. [9]{} J.M. Borwein D. M. Bradley, R. E. Crandall, Computational strategies for the Riemann zeta function, Journal of Computational and Applied Mathematics 121 (2000) 247–296. J. B. Conrey and A. Ghosh, Turán inequalities and zeros of Dirichlet series associated with certain cusp forms, Trans. Amer. Math. Soc. 342 (1994), 407-419 G. Csordas, Fourier Transforms of Positive Definite Kernels and the Riemann $\xi$-function, Computational Methods and Function Theory, Volume 15, Issue 3, pp 373–391 (2015). A. Dixit, Series transformations and integrals involving the Riemann $\Xi$-function, J. Math. Anal. Appl., 368, (2010), 358–373. A. Dixit Character analogues of Ramanujan type integrals involving the Riemann $\Xi$-function, Pacific J. Math., 255, No. 2 (2012), 317–348 H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. H. Ki, The Riemann $\Xi$-function under repeated differentiation, J. of Number Theory, Volume 120, Issue 1, September 2006, Pages 120–131. N. S. Koshlyakov, Investigation of some questions of the analytic theory of a rational and quadratic field. I Izv. Akad. Nauk SSSR Ser. Mat., 18:2 (1954), 113–144. N. S. Koshlyakov, Investigation of some questions of the analytic theory of a rational and quadratic field. II (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 18 No. 3, 213–260 (1954). A. Kuznetsov, Integral representations for the Dirichlet L-functions and their expansions in Meixner–Pollaczek polynomials and rising factorials, Integral Transforms and Special Functions 18 (2007), No. 11-12, 809–817. A. Kuznetsov, Expansion of the Riemann $\Xi$ function in Meixner-Pollaczek polynomials, Canadian Math. Bulletin 51 (2008), No. 4, 561–569. S. Ramanujan, New expressions for Riemann’s functions $\xi(s)$ and $\Xi(t)$, Quart. J. Math., 46: 253–260, 1915. R. Spira, The Integral Representation for the the Riemann $\Xi$ function, J. of Number Theory, Volume 3, Issue 4, November 1971, Pages 498–501 E. C. Titchmarsh, The theory of the Riemann zeta function, Oxford University Press, 2nd edition, 1986. K. S. Williams and Zhang Nan-Yue, On the Epstein zeta function, Tamkang J. Math. 26 (1995), 165-176. 1390 Bumps River Rd.\ Centerville, MA 02632\ USA\ E-mail: alexpatk@hotmail.com
--- abstract: 'Since the launch of the Large Area Telescope (LAT) on board the Fermi spacecraft in June 2008, the number of observed $\gamma$-ray pulsars has increased dramatically. A large number of these are also observed at radio frequencies. Constraints on the viewing geometries of 5 of 6 $\gamma$-ray pulsars exhibiting single-peaked $\gamma$-ray profiles were derived using high-quality radio polarization data [@Weltevrede]. We obtain independent constraints on the viewing geometries of 6 by using a geometric emission code to model the Fermi LAT and radio light curves (LCs). We find fits for the magnetic inclination and observer angles by searching the solution space by eye. Our results are generally consistent with those previously obtained [@Weltevrede], although we do find small differences in some cases. We will indicate how the $\gamma$-ray and radio pulse shapes as well as their relative phase lags lead to constraints in the solution space. Values for the flux correction factor ($f_\Omega$) corresponding to the fits are also derived (with errors).' address: - '$^1$ Centre for Space Research, North-West University, Potchefstroom Campus, Private Bag X6001, Potchefstroom 2520, South Africa' - '$^2$ Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA' - '$^3$ Department of Physics, University of Maryland, College Park, MD 20742, USA' author: - 'A S Seyffert$^1$, C Venter$^1$, T J Johnson$^{2,3}$ and A K Harding$^2$' title: 'Constraining viewing geometries of pulsars with single-peaked $\gamma$-ray profiles using a multiwavelength approach' --- Introduction ============ We study 6 pulsars (J0631+1036, J0659+1414, J0742$-$2822, J1420$-$6048, J1509$-$5850, and J1718$-$3825) detected by Fermi LAT, all exhibiting single-peaked pulse profiles within current statistics at energies $> 0.1$ GeV. These pulsars are also detected in the radio band. Both of these properties aid in constraining the possible solution space for the respective pulsar geometries, as derived from predictions of their light curves (LCs). A study of the pulsars’ geometric parameters, the inclination and observer angles $\alpha$ and $\zeta$, has been performed [@Weltevrede] using the radio polarization and LC data. They constrained the solution space for $\alpha$ and $\beta$ of 5 of the individual pulsars, with the impact angle $\beta=\zeta-\alpha$, using fits to these radio data as well as predictions for the value of the half opening angle, $\rho$, of the radio beam derived from the radio pulse width (e.g., [@Gil84]). The aim of this study is to obtain similar constraints on $\alpha$ and $\zeta$ of the same pulsars using an independent, multiwavelength approach. We use a geometric pulsar emission code (e.g., [@Venter09]) to model the Fermi LAT and radio LCs, and fit the predicted radio and $\gamma$-ray profiles to the data concurrently, thereby significantly constraining $\alpha$ and $\zeta$. This also allows us to test the consistency of the various approaches used to infer the pulsar geometry. Model {#sec:model} ===== We use an idealized picture of the pulsar system, wherein the magnetic field has a retarded dipole structure [@Deutsch55] and the $\gamma$-ray emission originates in regions of the magnetosphere (referred to as ‘gaps’) where the local charge density is sufficiently lower than the Goldreich-Julian charge density [@GJ69]. These gaps facilitate particle acceleration and radiation. We assume that there are constant-emissivity emission layers embedded within the gaps in the pulsar’s corotating frame. The location and geometry of these emission layers determine the shape of the $\gamma$-ray LCs, and there exist multiple models for the geometry of the magnetosphere describing different possible gap configurations. We included two models for the $\gamma$-ray emission regions in this study, namely the Outer Gap (OG) and Two-Pole Caustic (TPC) models. In both the OG and TPC models (the Slot Gap model [@Arons83] may serve as physical basis for the latter), emission is produced by accelerated charged particles moving within narrow gaps along the last open magnetic field lines. In the OG model [@CHR86], radiation originates above the null charge surface (where the Goldreich-Julian charge density changes sign) and interior to the last open magnetic field lines. The TPC gap starts at the stellar surface and extends along the last open field lines up to near the light cylinder[^1], where the corotation speed approaches the speed of light [@Dyks03]. The special relativistic effects of aberration and time-of-flight delays, which become important in regions far from the stellar surface (especially near the light cylinder), together with the curvature of the magnetic field lines, cause the radiation to form caustics (accumulated emission in narrow phase bands). These caustics are detected as peaks in the observed $\gamma$-ray LC [@Dyks03; @Morini83]. We used an empirical radio cone model [@Story07], where the beam diameter, width, and altitude are functions of the pulsar period $P$, its time derivative $\dot{P}$, and the frequency of observation $\nu$, in order to obtain predictions for the radio LCs. This is different from the approach taken by [@Weltevrede], as we have a different prescription for the radio emission altitude and $\rho$, and we do not use the polarization data. Method ====== Using the above geometric models, we generated LCs as a function of $\alpha$ and $\zeta$, keeping the gap widths and extents constant. Due to the large size of the ($\alpha$,$\zeta$)-space at a $1^\circ$ resolution, it is impractical to search blindly for the optimum LC by eye. We therefore started by generating a so-called atlas of LCs with a $10^\circ$ resolution for each pulsar and first identified candidate LC fits. We then performed refined searches in the regions of ($\alpha$,$\zeta$)-space around those candidate fits at a $5^\circ$ and later $1^\circ$ resolution. This approach of generating all the LCs in a candidate region and comparing them directly, allowed us to obtain constraints on $\alpha$ and $\zeta$. We then inferred contours for $\alpha$ and $\zeta$ for each pulsar and for each geometric model. We lastly used these contours to find the corresponding values of $f_\Omega$ (with errors; see Section \[sec:fom\]) predicted by the OG and TPC models. Obtaining the contours {#sec:ObCon} ---------------------- To obtain an ($\alpha$,$\zeta$)-contour delineating the region of plausible LC fits, we start at the candidate solution ($5^\circ$ resolution) and first move away from it in steps of $1^\circ$ in $\alpha$. At each step we then step through $\zeta$ until the upper and lower bounds of the contour are located for the fixed $\alpha$. This procedure is repeated until no further acceptable LCs are found. As an example, the red regions A and B in the left panel of Figure \[fig:boundConts\] indicate the resulting contours obtained in this manner for PSR J1509$-$5850 for the TPC case. ![\[fig:boundConts\]Left: The ($\alpha$,$\zeta$)-contours (regions A and B) for radio and $\gamma$-ray LC fits of PSR J1509$-$5850 for the TPC model (red). The dark grey and light grey regions correspond to double-peaked and single-peaked radio LCs respectively. The solid lines indicate approximately the locations of the boundaries to the contours due to: (a) the relative phase lag between the radio and $\gamma$-ray peaks being too large, (b) the bridge emission’s relative intensity being too low, and (c,d) the $\gamma$-ray peak being too narrow. The dashed line (bounded area) indicates the resulting errors on $\alpha$ and $\zeta$, with the green block indicating our best fit. Right: Representative LC fits corresponding to the contours A and B.](J1509-5850_tpcContoursV2_CountoursBoundsFitbox.png){width="20.5pc"} ![\[fig:boundConts\]Left: The ($\alpha$,$\zeta$)-contours (regions A and B) for radio and $\gamma$-ray LC fits of PSR J1509$-$5850 for the TPC model (red). The dark grey and light grey regions correspond to double-peaked and single-peaked radio LCs respectively. The solid lines indicate approximately the locations of the boundaries to the contours due to: (a) the relative phase lag between the radio and $\gamma$-ray peaks being too large, (b) the bridge emission’s relative intensity being too low, and (c,d) the $\gamma$-ray peak being too narrow. The dashed line (bounded area) indicates the resulting errors on $\alpha$ and $\zeta$, with the green block indicating our best fit. Right: Representative LC fits corresponding to the contours A and B.](J1509-5850_TPC_topLeft_representative.png "fig:"){width="16pc"} ![\[fig:boundConts\]Left: The ($\alpha$,$\zeta$)-contours (regions A and B) for radio and $\gamma$-ray LC fits of PSR J1509$-$5850 for the TPC model (red). The dark grey and light grey regions correspond to double-peaked and single-peaked radio LCs respectively. The solid lines indicate approximately the locations of the boundaries to the contours due to: (a) the relative phase lag between the radio and $\gamma$-ray peaks being too large, (b) the bridge emission’s relative intensity being too low, and (c,d) the $\gamma$-ray peak being too narrow. The dashed line (bounded area) indicates the resulting errors on $\alpha$ and $\zeta$, with the green block indicating our best fit. Right: Representative LC fits corresponding to the contours A and B.](J1509-5850_TPC_bottomRight_representative.png "fig:"){width="16pc"} Oftentimes we find two disjoint solution contours, with one of the two sometimes producing a better LC fit. The example shown here is one of these cases, as can be seen from the representative LCs shown in Figure \[fig:boundConts\]. Contour B yields a considerably better fit than contour A, and thus we can ignore contour A in favour of contour B as indicated by the dotted line in Figure \[fig:boundConts\]. The values of $\alpha$ and $\zeta$ reported in are those at the centre of this dotted box, while the errors on these values are conservatively chosen to encompass the dimensions of the box. In this case the resulting values are $\alpha=(61^\circ\pm5^\circ)$ and $\zeta=(44^\circ\pm7^\circ)$. Finding $f_\Omega$ {#sec:fom} ------------------ It is important to be able to convert the observed energy flux of a pulsar to its all-sky luminosity. The flux correction factor $f_\Omega$ is used for this purpose. It is a highly model-dependent parameter, and allows us to determine what fraction of the all-sky luminosity we are observing from a pulsar if the geometry is known ($\alpha$ and $\zeta$), for a given model. It is also crucial in deriving the $\gamma$-ray efficiency of a pulsar, and is defined as [@Watters09] $$f_\Omega(\alpha,\zeta_E)=\frac{ \iint {F_\gamma} \left( \alpha , \zeta , \phi \right) \sin \zeta d\zeta d\phi }{ 2 \int {F_\gamma} \left( \alpha , {\zeta}_E , \phi \right) d \phi },$$ with $F_\gamma$ being the photon flux per solid angle (‘intensity’), and $\zeta_E$ the Earth line-of-sight. The value of $f_\Omega$ is typically taken to be 1, meaning that the observed energy flux is assumed to be equal to the average energy flux over the entire sky. Due to the OG model predicting emission over a relatively smaller region of phase space, its associated $f_\Omega$ values are typically smaller than those of the TPC model for the same $\alpha$ and $\zeta$. (Note, we assume that $F_\gamma(\zeta,\phi)/F_{\rm \gamma,tot}$ is approximately equal to the ratio of observed energy flux vs. total energy flux.) We can now use the ($\alpha$,$\zeta$)-contours obtained by the method discussed in Section \[sec:ObCon\] to constrain the value of $f_\Omega$ by computing $f_\Omega(\alpha,\zeta)$ and overplotting those contours (see Figure \[fig:consOnF\]). ![\[fig:consOnF\]Contour plot of $f_\Omega$ as function of $\alpha$ and $\zeta$ implied by the TPC model for PSR J1509-5850. The red area indicates the best-fit $(\alpha,\zeta)$-contour for the radio and $\gamma$-ray LCs.](J1509-5850_fOmegaContour_tpc_1deg_withContoursV2.png){width="25pc"} Results {#sec:results} ======= We infer values for $\alpha$ and $\zeta$ for each pulsar with typical errors of $\sim5^\circ$ (see ). The tabulated values are the average values of $\alpha$, $\zeta$, and $\beta$ implied by the solution contours, while the errors are chosen conservatively so as to include the full (non-rectangular) contour. We note that once a solution is found at a particular $\alpha$ and $\zeta$, it is worthwhile to study the LC at the position $\alpha'=\zeta$ and $\zeta'=\alpha$ (i.e., when the angles are interchanged). Such a complementary solution may provide a good fit in some cases (see the alternative solutions listed in ), due to the symmetry of the model as well the symmetry in pulsar geometry (i.e., $\|\beta\|$ remaining constant under this transformation). As mentioned in Section \[sec:model\], we see a shift in best-fit contours, with the TPC model consistently yielding smaller $\alpha$ and $\zeta$ values compared to those obtained using the OG model due to a different gap geometry. This is because similar phaseplots are obtained when a larger $\alpha$ is chosen for the OG model than for the TPC model. Reproduction of the radio pulse shape then necessitates a larger value of $\zeta$ for the OG fit, since the contour is constrained to appear in one of the grey regions (as determined by peak multiplicity; see Figure \[fig:boundConts\]). The net effect of this is that the OG contours are found farther from the origin at $\alpha=\zeta=0$ along the grey bands than the corresponding TPC contours. [llcccc]{} Pulsar & Model & $\alpha~(^\circ)$ & $\zeta~(^\circ)$ & $\beta~(^\circ)$ & $f_\Omega$ J0631$+$1036 & OG & $74\pm5\0$ & $67\pm4\0$ & $-\06\pm2\0\m$ & $0.93\pm0.06$ & TPC & $71\pm6\0$ & $66\pm7\0$ & $-\05\pm3\0\m$ & $1.04\pm0.04$ J0659$+$1414 & OG & $59\pm3\0$ & $48\pm3\0$ & $-12\pm5\0\m$ & $1.16\pm0.53$ & TPC & $50\pm4\0$ & $39\pm4\0$ & $-13\pm6\0\m$ & $1.64\pm0.04$ & TPC$^$ & $38\pm1\0$ & $50\pm4\0$ & $\m11\pm4\0\m$ & $1.63\pm0.05$ J0742$-$2822 & OG & $86\pm3\0$ & $71\pm5\0$ & $-16\pm6\0\m$ & $0.99\pm0.10$ & OG$^$ & $71\pm6\0$ & $86\pm4\0$ & $\m16\pm6\0\m$ & $0.81\pm0.09$ & TPC & $64\pm8\0$ & $80\pm4\0$ & $\m15\pm6\0\m$ & $0.88\pm0.41$ J1420$-$6048 & OG & $67\pm5\0$ & $45\pm7\0$ & $-22\pm9\0\m$ & $0.77\pm0.13$ & TPC & $64\pm6\0$ & $43\pm8\0$ & $-21\pm9\0\m$ & $0.90\pm0.10$ & TPC$^$ & $42\pm5\0$ & $63\pm5\0$ & $\m21\pm9\0\m$ & $0.77\pm0.06$ J1509$-$5850 & OG & $66\pm4\0$ & $50\pm7\0$ & $-18\pm8\0\m$ & $0.77\pm0.11$ & TPC & $61\pm5\0$ & $44\pm7\0$ & $-18\pm8\0\m$ & $0.89\pm0.10$ J1718$-$3825 & OG & $67\pm6\0$ & $48\pm6\0$ & $-19\pm8\0\m$ & $0.76\pm0.12$ & TPC & $61\pm5\0$ & $43\pm6\0$ & $-19\pm8\0\m$ & $0.86\pm0.07$ & TPC$^$ & $42\pm6\0$ & $62\pm5\0$ & $\m19\pm7\0\m$ & $0.83\pm0.10$ Our results for $f_\Omega$ for each of the pulsars (and models) are given in Table \[tab:results\]. Discussion and Conclusions ========================== The good constraints on $\alpha$ and $\zeta$, obtained using the method described above, emphasizes the merit of a multiwavelength approach. For example, the light grey regions in Figure \[fig:boundConts\] indicate single-peaked radio profile LC solutions. These would represent typical constraints one would be able to derive for $\alpha$ and $\zeta$ when considering only the radio profile shapes. Similarly, when only considering the (single-peaked) $\gamma$-ray LCs, one would find relatively large ($\alpha$,$\zeta$)-contours [@Watters09]. The requirement of fitting both the radio and $\gamma$-ray profile shape, as well as their relative phase lag, results in much smaller contours, as shown in Figure \[fig:boundConts\]. Generally, our best-fit ($\alpha$,$\zeta$) compare favourably with those inferred by [@Weltevrede] for the first three pulsars (see Table \[tab:results\]). However, comparison is hampered by uncertainties in estimating the half opening angle, $\rho$, which sensitively influences the optimal solutions obtained by [@Weltevrede]. Even a small error of $5^\circ$ on $\rho$ leads to relatively large errors on the allowed $\alpha$, so that our best fits would then be included in their inferred parameter ranges. Direct comparison of our results with those of [@Weltevrede] for PSR J1420$-$6048 and PSR J1718$-$3825 is not feasible due to different approaches for estimating the radio pulse width, $W$. We only model the most prominent radio peak, while [@Weltevrede] measures $W$ as the width of the total two-peaked profile. Implementing a factor 2 difference in $W$ will change the $\rho$-contours and thus their best-fit solution considerably, improving the agreement between the two approaches. Comparison with [@Weltevrede] is not possible for PSR J1509-5850 as the lack of polarization data inhibited inference of a best-fit ($\alpha$,$\beta$) by [@Weltevrede] in this case. The next step for this kind of study is to apply a mathematically rigorous method of determining the best-fit LC solutions as a function of many free model parameters to the six pulsars modelled in this paper. An example is the Markov chain Monte Carlo method which has been successfully applied to millisecond pulsar LCs [@Johnson11]. CV is supported by the South African National Research Foundation. AKH acknowledges support from the NASA Astrophysics Theory Program. CV, TJJ, and AKH acknowledge support from the Fermi Guest Investigator Program as well as fruitful discussions with Patrick Weltevrede. References {#references .unnumbered} ========== [12]{} Weltevrede P et al. 2010 [Astrophys. J.]{} [708]{} 1426–41 Gil J, Gronkowski P and Rudnicki W 1984 [Astron. Astrophys.]{} [132]{} 312–16 Venter C, Harding A K and Guillemot L 2009 [Astrophys. J.]{} [707]{} 800–22 Deutsch A J 1955 [Ann. d’Astrophys.]{} [18]{} 1–10 Goldreich P and Julian W H 1969 [Astrophys. J.]{} [157]{} 869–80 Arons J 1983 [Astrophys. J.]{} [266]{} 215–41 Cheng K S, Ho C and Ruderman M 1986 [Astrophys. J.]{} [300]{} 500–21 Dyks J and Rudak B 2003 [Astrophys. J.]{} [598]{} 1201–6 Morini M 1983 [Mon. Not. R. Astron. Soc.]{} [202]{} 495–510 Story S A, Gonthier P L and Harding A K 2007 [Astrophys. J.]{} [671]{} 713–26 Watters K P, Romani R W, Weltevrede P and Johnston S 2009 [Astrophys. J.]{} [695]{} 1289–1301 Johnson T J, Harding A K and Venter C 2011, in preparation [^1]: The slight difference in transverse polar position of the gaps with respect to the magnetic axis (as motivated by physical models) will result in a systematic shift in best-fit ($\alpha$,$\zeta$)-contours between the two models. See Section \[sec:results\].
--- abstract: 'We study distributed optimization to minimize a global objective that is a sum of smooth and strongly-convex local cost functions. Recently, several algorithms over undirected and directed graphs have been proposed that use a gradient tracking method to achieve linear convergence to the global minimizer. However, a connection between these different approaches has been unclear. In this paper, we first show that many of the existing first-order algorithms are in fact related with a simple state transformation, at the heart of which lies the $\mc{AB}$ algorithm. We then describe distributed heavy-ball, denoted as $\mc{AB}m$, i.e., $\mc{AB}$ with momentum, that combines gradient tracking with a momentum term and uses nonidentical local step-sizes. By simultaneously implementing both row- and column-stochastic weights, $\mc{AB}m$ removes the conservatism in the related work due to doubly-stochastic weights or eigenvector estimation. $\mc{AB}m$ thus naturally leads to optimization and average-consensus over both undirected and directed graphs, casting a unifying framework over several well-known consensus algorithms over arbitrary strongly-connected graphs. We show that $\mathcal{AB}m$ has a global $R$-linear rate when the largest step-size is positive and sufficiently small. Following the standard practice in the heavy-ball literature, we numerically show that $\mc{AB}m$ achieves accelerated convergence especially when the objective function is ill-conditioned.' author: - 'Ran Xin, Student Member, IEEE, and Usman A. Khan, Senior Member, IEEE [^1]' bibliography: - 'sample.bib' title: 'Distributed heavy-ball: A generalization and acceleration of first-order methods with gradient tracking' --- Distributed optimization, linear convergence, first-order method, heavy ball method, momentum. Introduction {#s1} ============ We consider distributed optimization, where $n$ agents collaboratively solve the following problem: $$\min_{\mb{x}\in\mathbb{R}^n}F(\mb{x}) \triangleq \frac{1}{n}\sum_{i=1}^{n}f_i(\mb{x}),$$ and each local objective, $f_i:\mathbb{R}^p\rightarrow\mathbb{R}$, is smooth and strongly-convex. The goal of the agents is to find the global minimizer of the aggregate cost via only local communication with their neighbors. This formulation has recently received great interest with applications in e.g., machine learning [@forero2010consensus; @distributed_Boyd; @raja2016cloud; @wai2018multi], control [@jadbabaie2003coordination], cognitive networks, [@distributed_Mateos; @distributed_Bazerque], and source localization [@distributed_Rabbit; @safavi2018distributed]. Early work on this topic builds on the seminal work by Tsitsiklis in [@DOPT1] and includes Distributed Gradient Descent (DGD) [@uc_Nedic] and distributed dual averaging [@duchi2012dual] over undirected graphs. Leveraging push-sum consensus [@ac_directed0], Refs. [@opdirect_Tsianous; @opdirect_Nedic] extend the DGD framework to directed graphs. Based on a similar concept, Refs. [@D-DGD; @D-DPS] propose Directed-Distributed Gradient Descent (D-DGD) for directed graphs that is based on surplus consensus [@ac_Cai1]. In general, the DGD-based methods achieve sublinear convergence at $\mathcal{O}\left(\frac{\log k}{\sqrt{k}}\right)$, where $k$ is the number of iterations, because of the diminishing step-size used in the iterations. The convergence rate of DGD can be improved with the help of a constant step-size but at the expense of an inexact solution [@DGD_Yuan; @balancing]. Follow-up work also includes augmented Lagrangians [@ADMM_Wei; @ADMM_Mota; @ADMM_Shi; @ESOM], which shows exact linear convergence for smooth and strongly-convex functions, albeit requiring higher computation at each iteration. To improve convergence and retain computational simplicity, fast first-order methods that do not (explicitly) use a dual update have been proposed. Reference [@DNC] describes a distributed Nesterov-type method based on multiple consensus inner loops, at $\mathcal{O}\left(\frac{\log k}{k^2}\right)$ for smooth and convex functions, with bounded gradients. EXTRA [@EXTRA] uses the difference of two consecutive DGD iterates to achieve an $\mathcal{O}\left(\frac{1}{k}\right)$ rate for arbitrary convex functions and a $Q$-linear rate for strongly-convex functions. DEXTRA [@DEXTRA] combines push-sum [@ac_directed0] and EXTRA [@EXTRA] to achieve an $R$-linear rate over directed graphs given that a constant step-size is carefully chosen in some interval. Refs. [@exactdiffusion1; @exactdiffusion2] apply an adapt-then-combine structure [@diffusion] to EXTRA [@EXTRA] and generalize the symmetric weights in EXTRA to row-stochastic, over undirected graphs. Noting that DGD-type methods are faster with a constant step-size, recent work [@AugDGM; @harness; @add-opt; @diging; @linear_row; @FROST; @AB; @dnesterov; @jakovetic2018unification; @SUCAG] uses a constant step-size and replaces the local gradient, at each agent in DGD, with an estimate of the global gradient. A method based on gradient tracking was first shown in [@AugDGM] over undirected graphs, which proposes Aug-DGM (that uses nonidentical step-sizes at the agents) with the help of dynamic consensus [@DAC] and shows convergence for smooth convex functions. When the step-sizes are identical, the convergence rate of Aug-DGM was derived to be $\mathcal{O}\left(\frac{1}{k}\right)$ for arbitrary convex functions and $R$-linear for strongly-convex functions in [@harness]. ADD-OPT [@add-opt] extends [@harness] to directed graphs by combining push-sum with gradient tracking and derives a contraction in an arbitrary norm to establish an $R$-linear convergence rate when the global objective is smooth and strongly-convex. Ref. [@diging] extends the analysis in [@harness; @add-opt] to time-varying graphs and establishes an $R$-linear convergence using the small gain theorem [@control]. In contrast to the aforementioned methods [@AugDGM; @harness; @add-opt; @diging], where the weights are doubly-stochastic for undirected graphs and column-stochastic for directed graphs, FROST [@linear_row; @FROST] uses row-stochastic weights, which have certain advantages over column-stochastic weights. Ref. [@jakovetic2018unification] unifies EXTRA [@EXTRA] and gradient tracking methods [@AugDGM; @harness] in a primal-dual framework over static undirected graphs. More recently, Ref. [@dnesterov] proposes distributed Nesterov over undirected graphs that also uses gradient tracking and shows a convergence rate of $\mathcal{O}((1-{c}{\mathcal{Q}^{-\frac{5}{7}}})^k)$ for smooth, strongly-convex functions, where $\mathcal{Q}$ is the condition number of the global objective. Refs. [@NEXT; @sonata], on the other hand, consider gradient tracking in distributed non-convex problems, while Ref. [@SUCAG] uses second-order information to accelerate the convergence. Of significant relevance here is the $\mathcal{AB}$ algorithm [@AB], also appeared later in [@the_copy_work_2], which can be viewed as a generalization of distributed first-order methods with gradient tracking. In particular, the algorithms over undirected graphs in Refs. [@AugDGM; @harness] are a special case of $\mc{AB}$ because the doubly-stochastic weights therein are replaced by row- and column- stochastic weights. $\mathcal{AB}$ thus is naturally applicable to arbitrary directed graphs. Moreover, the use of both row- and column-stochastic weights removes the need for eigenvector estimation[^2], required earlier in [@add-opt; @diging; @linear_row; @FROST]. Ref. [@AB] derives an $R$-linear rate for $\mathcal{AB}$ when the objective functions are smooth and strongly-convex. In this paper, we provide an improved understanding of $\mc{AB}$ and extend it to the $\mc{AB}m$ algorithm, a distributed heavy-ball method, applicable to both undirected and directed graphs. We now summarize the main contributions: 1. We show that many of the existing accelerated first-order methods are either a special case of $\mc{AB}$ [@AugDGM; @harness], or can be adapted from its equivalent forms [@diging; @add-opt; @linear_row; @FROST]. 2. We propose a distributed heavy-ball method, termed as $\mc{AB}m$, that combines $\mc{AB}$ with a heavy-ball (type) momentum term. To the best of our knowledge, this paper is the first to use a momentum term based on the heavy-ball method in distributed optimization. 3. $\mc{AB}m$ employs nonidentical step-sizes at the agents and thus its analysis naturally carries to nonidentical step-sizes in $\mc{AB}$ and to the related algorithms in [@AugDGM; @harness; @diging; @add-opt; @linear_row; @FROST]. 4. We cast a unifying framework for consensus over arbitrary graphs that results from $\mc{AB}m$ and subsumes several well-known algorithms [@ac_Cai1; @ac_row]. On the analysis front, we show that $\mc{AB}$ (without momentum) converges faster as compared to the algorithms over directed graphs in [@add-opt; @diging; @linear_row; @FROST], where separate iterations for eigenvector estimation are applied nonlinearly to the underlying algorithm. Towards $\mc{AB}m$, we establish a global $R$-linear convergence rate for smooth and strongly-convex objective functions when the largest step-size at the agents is positive and sufficiently small. This is in contrast to the earlier work on non-identical step-sizes within the framework of gradient tracking [@AugDGM; @digingun; @digingstochastic; @lu2018geometrical], which requires the heterogeneity among the step-sizes to be sufficiently small, i.e., the step-sizes are close to each other. We also acknowledge that similar to the centralized heavy-ball method [@polyak1964some; @polyak1987introduction], dating back to more than 50 years, and the recent work [@ghadimi2015global; @IAGM; @lessard2016analysis; @drori2014performance; @polyak2017lyapunov; @IGM; @sHB], a global acceleration can only be shown via numerical simulations. Following the standard practice, we provide simulations to verify that $\mathcal{AB}m$ has accelerated convergence, the effect of which is more pronounced when the global objective function is ill-conditioned. We now describe the rest of the paper. Section \[s2\] provides preliminaries, problem formulation, and introduces distributed heavy-ball, i.e., the $\mc{AB}m$ algorithm. Section \[s3\] establishes the connection between $\mc{AB}$ and related algorithms. Section \[s4\] includes the main results on the convergence analysis, whereas Section \[s6\] provides a family of average-consensus algorithms that result naturally from $\mc{AB}m$. Finally, Section \[s7\] provides numerical experiments and Section \[s8\] concludes the paper. Basic Notation: We use lowercase bold letters to denote vectors and uppercase letters for matrices. The matrix, $I_n$, is the $n\times n$ identity, whereas $\mb{1}_n$ ($\mb{0}_n$) is the $n$-dimensional column vector of all ones (zeros). For an arbitrary vector, $\mb{x}$, we denote its $i$th element by $[\mb{x}]_i$ and its largest and smallest element by $[\mb{x}]_{\max}$ and $[\mb{x}]_{\min}$, respectively. We use $\mbox{diag}(\mb{x})$ to denote a diagonal matrix that has $\mb{x}$ on its main diagonal. For two matrices, $X$ and $Y$, $\mbox{diag}\left(X,Y\right)$ is a block-diagonal matrix with $X$ and $Y$ on its main diagonal, and $X\otimes Y$ denotes their Kronecker product. The spectral radius of a matrix, $X$, is represented by $\rho(X)$. For a primitive, row-stochastic matrix, $A$, we denote its left and right eigenvectors corresponding to the eigenvalue of $1$ by $\bs{\pi}_r$ and $\mb{1}_n$, respectively, such that $\bs{\pi}_r^\top\mb{1}_n = 1$; similarly, for a primitive, column-stochastic matrix, $B$, we denote its left and right eigenvectors corresponding to the eigenvalue of $1$ by $\mb{1}_n$ and $\bs{\pi}_c$, respectively, such that $\mb{1}_n^\top\bs{\pi}_c = 1$. For a matrix $X$, we denote $X_\infty$ as its infinite power (if it exists), i.e., $X_\infty =\lim_{k\rightarrow\infty}X^k.$ From the Perron-Frobenius theorem [@matrix], we have $A_\infty=\mb{1}_n\bs{\pi}_r^\top$ and $B_\infty=\bs{\pi}_c\mb{1}_n^\top$. We denote $\left\\|\cdot\right\\|_\mathcal{A}$ and $\left\\|\cdot\right\\|_\mathcal{B}$ as some arbitrary vector norms, the choice of which will be clear in Lemma \[contra\], while $\left\\|\cdot\right\\|$ denotes the Euclidean matrix and vector norms. Preliminaries and Problem Formulation {#s2} ===================================== Consider $n$ agents connected over a directed graph, $\mc{G}=(\mc{V},\mc{E})$, where $\mc{V}=\{1,\cdots,n\}$ is the set of agents, and $\mc{E}$ is the collection of ordered pairs, $(i,j),i,j\in\mc{V}$, such that agent $j$ can send information to agent $i$, i.e., $j\rightarrow i$. We define $\mc{N}_i^{{\scriptsize \mbox{in}}}$ as the collection of in-neighbors of agent $i$, i.e., the set of agents that can send information to agent $i$. Similarly, $\mc{N}_i^{{\scriptsize \mbox{out}}}$ is the set of out-neighbors of agent $i$. Note that both $\mc{N}_i^{{\scriptsize \mbox{in}}}$ and $\mc{N}_i^{{\scriptsize \mbox{out}}}$ include agent $i$. The agents solve the following problem: $$\begin{aligned} \mbox{P1}: \quad\min_{\mb{x}\in\mathbb{R}^n}F(\mb{x})\triangleq\frac{1}{n}\sum_{i=1}^nf_i(\mb{x}),\nonumber\end{aligned}$$ where each $f_i:\mbb{R}^p\rightarrow\mbb{R}$ is known only to agent $i$. We formalize the set of assumptions as follows. \[asp1\] The graph, $\mc{G}$, is strongly-connected. \[asp2\] Each local objective, $f_i$, is $\mu_i$-strongly-convex, i.e., $\forall i\in\mc{V}$ and $\forall\mb{x}, \mb{y}\in\mbb{R}^p$, we have $$f_i(\mb{y})\geq f_i(\mb{x})+\nabla f_i(\mb{x})^\top(\mb{y}-\mb{x})+\frac{\mu_i}{2}\\|\mb{x}-\mb{y}\\|^2,$$ where $\mu_i\geq0$ and $\sum_{i=1}^{n}\mu_i>0$. \[asp3\] Each local objective, $f_i$, is $l_i$-smooth, i.e., its gradient is Lipschitz-continuous: $\forall i\in\mc{V}$ and $\forall\mb{x}, \mb{y}\in\mbb{R}^p$, we have, for some $l_i>0$, $$\qquad\\|\mb{\nabla} f_i(\mb{x})-\mb{\nabla} f_i(\mb{y})\\|\leq l_i\\|\mb{x}-\mb{y}\\|.$$ Assumptions \[asp2\] and \[asp3\] ensure that the global minimizer, $\mb{x}^\in\mbb{R}^p$, of $F$ exists and is unique [@nesterov2013introductory]. In the subsequent analysis, we use $\mu \triangleq \frac{1}{n}\sum_{i=1}^{n}\mu_i$ and $l \triangleq \frac{1}{n}\sum_{i=1}^{n} l_i$, as the strong-convexity and Lipschitz-continuity constants, respectively, for the global objective, $F$. We define $\ol{l}\triangleq\max_il_i$. We next describe the heavy-ball method that is credited to Polyak and then introduce the distributed heavy-ball method, termed as the $\mc{AB}m$ algorithm, to solve Problem P1. Heavy-ball method {#hbm} ----------------- It is well known [@polyak1987introduction; @nesterov2013introductory] that the best achievable convergence rate of the gradient descent algorithm, $$\mb{x}_{k+1} = \mb{x}_{k} - \alpha\nabla F\left(\mb{x}_{k}\right),$$ is $\mc{O}((\tfrac{\mc{Q}-1}{\mc{Q}+1})^k)$, where $\mc{Q}\triangleq\tfrac{l}{\mu}$ is the condition number of the objective function, $F$. Clearly, gradient descent is quite slow when $\mc{Q}$ is large, i.e., when the objective function is ill-conditioned. The seminal work by Polyak [@polyak1964some; @polyak1987introduction] proposes the following heavy-ball method: $$\label{HB} \mb{x}_{k+1} = \mb{x}_{k} - \alpha\nabla F(\mb{x}_{k}) + \beta(\mb{x}_{k}-\mb{x}_{k-1}),$$ where $\beta\left(\mb{x}_{k}-\mb{x}_{k-1}\right)$ is interpreted as a “momentum” term, used to accelerate the convergence process. Polyak shows that with a specific choice of $\alpha$ and $\beta$, the heavy-ball method achieves a local* accelerated rate of $\mc{O}((\tfrac{\mc{\sqrt{Q}}-1}{\mc{\sqrt{Q}}+1})^k)$. By local, it is meant that the acceleration can only be analytically shown when $\\|\mb{x}_0-\mb{x}^\\|$ is sufficiently small. Globally, i.e., for arbitrary initial conditions, only linear convergence is established, while an analytical characterization of the acceleration is still an open problem, see related work in [@ghadimi2015global; @IGM; @IAGM; @sHB; @polyak2017lyapunov]. Numerical analysis and simulations are often employed to show global acceleration, i.e., it is possible to tune $\alpha$ and $\beta$ such that the heavy-ball method is faster than gradient descent [@drori2014performance; @lessard2016analysis]. Distributed heavy-ball: The $\mathcal{AB}m$ algorithm ----------------------------------------------------- Recall, that our goal is to solve Problem P1 when the agents, possessing only local objectives, exchange information over a strongly-connected directed graph, $\mc{G}$. Each agent, $i\in\mc{V}$, maintains two variables: $\mb{x}^{i}_k$, $\mb{y}^{i}_k\in\mbb{R}^p$, where $\mb{x}^i_k$ is the local estimate of the global minimizer and $\mb{y}^i_k$ is an auxiliary variable. The $\mc{AB}m$ algorithm, initialized with arbitrary $\mb{x}^i_0$’s, $\mb{x}_{-1}^i=\mb 0_p$ and $\mb{y}^i_0=\nabla f_i(\mb{x}^i_0),\forall i\in\mc{V}$, is given by[^3]: \[AB\] $$\begin{aligned} \mb{x}^i_{k+1}=&\sum_{j=1}^{n}a_{ij}\mb{x}^{j}_k-\alpha_i\mb{y}^i_k + \beta_i \left( \mb{x}_k^i - \mb{x}_{k-1}^i \right), \label{ABa} \\ \mb{y}^i_{k+1}=&\sum_{j=1}^{n}b_{ij}\mb{y}^j_k+\nabla \label{ABb} f_i\big(\mb{x}^i_{k+1}\big)-\nabla f_i\big(\mb{x}^i_k\big), \end{aligned}$$ where $\alpha_i\geq 0$ and $\beta_i\geq 0$ are respectively the local step-size and the momentum parameter adopted by agent $i$. The weights, $a_{ij}$’s and $b_{ij}$’s, are associated with the graph topology and satisfy the following conditions: $$\begin{aligned} a_{ij}&=\left\{ \begin{array}{rl} >0,&j\in\mc{N}_i^{{\scriptsize \mbox{in}}},\\ 0,&\mbox{otherwise}, \end{array} \right. \quad \sum_{j=1}^na_{ij}=1,\forall i\begin{color}{black},\end{color} \end{aligned}$$ $$\begin{aligned} b_{ij}&=\left\{ \begin{array}{rl} >0,&i\in\mc{N}_j^{{\scriptsize \mbox{out}}},\\ 0,&\mbox{otherwise}, \end{array} \right. \quad \sum_{i=1}^nb_{ij}=1,\forall j. \end{aligned}$$ Note that the weight matrix, $A=\{a_{ij}\}$, in Eq. is RS (row-stochastic) and the weight matrix, $B=\{b_{ij}\}$ in Eq. is CS (column-stochastic), both of which can be implemented over undirected and directed graphs alike. Intuitively, Eq. tracks the average of local gradients, $\frac{1}{n}\sum_{i=1}^{n}\nabla f_i(\mb{x}^i_k)$, see [@AugDGM; @harness; @add-opt; @diging; @linear_row; @FROST; @AB; @dnesterov; @jakovetic2018unification; @DAC], and therefore Eq. asymptotically approaches the centralized heavy-ball, Eq. , as the descent direction $\mb{y}^i_k$ becomes the gradient of the global objective. Vector form: For the sake of analysis, we now write $\mc{AB}m$ in vector form. We use the following notation: $$\begin{aligned} \mb{x}_k\triangleq \left[ \begin{array}{c} \mb{x}_{k}^1\\ \vdots\\ \mb{x}_{k}^n \end{array} \right],~\mb{y}_k\triangleq \left[ \begin{array}{c} \mb{y}_{k}^1\\ \vdots\\ \mb{y}_{k}^n \end{array} \right],~ \nabla\mb{f}(\mb{x}_k)\triangleq \left[ \begin{array}{c} \nabla f_1(\mb{x}_{k}^1)\\ \vdots\\ \nabla f_n(\mb{x}_{k}^n) \end{array} \right],\end{aligned}$$ all in $\mathbb{R}^{np}$. Let $\bs\a$ and $\bs\b$ define the vectors of the step-sizes and the momentum parameters, respectively. We now define augmented weight matrices, $\mc{A},\mc{B}$, and augmented step-size and momentum matrices, $D_{\bds{\alpha}},D_{\bds{\beta}}$: $$\begin{aligned} \mc{A} &\triangleq A \otimes I_p,\qquad D_{\bds{\alpha}}\triangleq \mbox{diag}(\bs\a)\otimes I_p,\\ \mc{B} &\triangleq B \otimes I_p,\qquad D_{\bds{\beta}}\triangleq\mbox{diag}(\bs\b)\otimes I_p, \end{aligned}$$ all in $\mathbb{R}^{np\times np}$. Using the notation above, $\mc{AB}m$ can be compactly written as: \[ABmv\] $$\begin{aligned} \mb{x}_{k+1} &= \mc{A}\mb{x}_k - D_{\bds{\alpha}}\mb{y}_k + D_{\bds{\beta}} \left(\mb{x}_k-\mb{x}_{k-1}\right) , \label{ABmva}\\ \mb{y}_{k+1} &= \mc{B}\mb{y}_k + \nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k), \label{ABmvb}.\end{aligned}$$ We note here that when $\beta_i=0,\forall i$, $\mc{AB}m$ reduces to $\mc{AB}$ [@AB], albeit with two distinguishing features: (i) the algorithm in [@AB] uses an identical step-size, $\alpha$, at each agent; and (ii) Eq. in [@AB] is in an adapt-then-combine form. Connection with existing first-order methods {#s3} ============================================ In this section, we provide a generalization of several existing methods that employ gradient tracking [@AugDGM; @harness; @add-opt; @diging; @linear_row; @FROST] and show that $\mc{AB}$ lies at the heart of these approaches. To proceed, we rewrite the $\mc{AB}$ updates below (without momentum) [@AB]. \[ABv\] $$\begin{aligned} \mb{x}_{k+1} &= \mc{A}\mb{x}_k - \alpha\mb{y}_k, \label{ABva}\\ \mb{y}_{k+1} &= \mc{B}\mb{y}_k + \nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k) \label{ABvb}.\end{aligned}$$ Since $\mc{AB}$ uses both RS and CS weights simultaneously, it is natural to ask how are the optimization algorithms that require the weight matrices to be doubly-stochastic (DS) [@EXTRA; @AugDGM; @harness; @diging], or only CS [@add-opt; @diging], or only RS [@linear_row; @FROST], are related to each other. We discuss this relationship next. Optimization with DS weights: Refs. [@AugDGM; @harness; @diging] consider the following updates, termed as Aug-DGM in [@AugDGM] and DIGing in [@diging]: \[harness\] $$\begin{aligned} \mb{x}_{k+1} &= \mc{W}\mb{x}_k -\alpha\mb{y}_k, \label{harness_a}\\ \mb{y}_{k+1} &= \mc{W}\mb{y}_k + \nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k), \label{harness_b} \end{aligned}$$ where $\mc{W} = W \otimes I_p$, and $W$ is a DS weight matrix. Clearly, to obtain DS weights, the underlying graph must be undirected (or balanced) and thus the algorithm in Eqs. is not applicable to arbitrary directed graphs. That $\mc{AB}$ generalizes Eqs. is straightforward as the DS weights naturally satisfy the RS requirement in the top update and the CS requirement in the bottom update, while the reverse is not true. Similarly, we note that a related algorithm, EXTRA [@EXTRA], is given by $$\mb{x}_{k+1} = (I+\mc{W})\mb{x}_{k} - \wt{\mc{W}}\mb{x}_{k-1} - \alpha\left(\nabla\mb{f}(\mb{x}_k)-\nabla\mb{f}(\mb{x}_{k-1})\right),$$ where the two weight matrices, $\mc{W}$ and $\wt{\mc{W}}$, must be symmetric and satisfy some other stringent requirements, see [@EXTRA] for details. Eliminating the $\mb{y}_k$-update in $\mc{AB}$, we note that $\mc{AB}$ can be written in the EXTRA format as follows: $$\begin{aligned} \nonumber \mb{x}_{k+1} =& \left(I+(\mc{A+B}-I)\right)\mb{x}_k \\\label{ABmEXTRA} &- \left(\mc{B A}\right)\mb{x}_{k-1} - \alpha \left(\nabla \mb{f}(\mb{x}_{k})-\nabla \mb{f}(\mb{x}_{k-1})\right).\end{aligned}$$ It can be seen that the linear convergence of $\mc{AB}$ does not follow from the analysis in [@EXTRA] as $\mc{A+B}-I$ and $\mc{BA}$ are not necessarily symmetric. Analysis of the $\mc{AB}$ algorithm, therefore, generalizes that of EXTRA to non-doubly-stochastic and non-symmetric weight matrices. Optimization with CS weights: We now relate $\mc{AB}$ to ADD-OPT/Push-DIGing that only require CS weights [@add-opt; @diging]. Since $\mc{B}$ is already CS in $\mc{AB}$, it suffices to seek a state transformation that transforms $\mc{A}$ from RS to CS, while respecting the graph topology. To this aim, let us consider the following transformation on the $\mb{x}_k$-update in $\mc{AB}$: $\wt{\mb{x}}_k\triangleq\Pi_r\mb{x}_k,$ where $\Pi_r \triangleq \mbox{diag}(n\bpi_r)\otimes I_p$ and $\bpi_r$ is the left-eigenvector of the RS weight matrix, $A$, corresponding to the eigenvalue $1$. The resulting transformed $\mc{AB}$ is given by \[AB\_addopt\] $$\begin{aligned} \label{AB_addopta} \wt{\mb{x}}_{k+1} &= \wt{\mc{B}}~\wt{\mb{x}}_k - \alpha\Pi_r\mb{y}_k,\\\label{AB_addoptb} \mb{x}_{k+1} &= \left(\mbox{diag}(n\bpi_r)\otimes I_p\right)^{-1}\wt{\mb{x}}_{k+1},\\\label{AB_addoptc} \mb{y}_{k+1} &= \mc{B}\mb{y}_k + \nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k), \end{aligned}$$ where it is straightforward to show that $\mc{\wt{B}}= \Pi_r\mc{A}\Pi_r^{-1}$ is now CS and $\wt{\mc{B}}\left(\bpi_r\otimes I_p\right)=\bpi_r \otimes I_p$. In order to implement the above equations, two different CS matrices ($\wt{\mc{B}}$ and $\mc{B}$) suffice, as long as they are primitive and respect the graph topology. The second update requires the right-eigenvector of the CS matrix used in the first update, i.e., $\wt{\mc{B}}$. Since this eigenvector is not known locally to any agent, ADD-OPT/Push-DIGing [@add-opt; @diging] propose learning this eigenvector with the following iterations: $\mb{w}_{k+1}=\wt{\mc{B}}\mb{w}_{k},\mb{w}_0=\mb{1}_{np}$. The algorithms provided in [@add-opt; @diging] essentially implement Eqs. , albeit with two differences: (i) the same CS weight matrix is used in all updates; and, (ii) the division in Eq. is replaced by the estimated component, $\mb{w}_{k+1}^i$, of the left-eigenvector at each agent. This nonlinearity causes stability issues in ADD-OPT/Push-DIGing, whereas their convergence compared to $\mc{AB}$ is slower because such an eigenvector estimation is not needed in the latter on the account of using the RS weights. Furthermore, the local step-sizes are now given by $n\alpha[\bpi_r]_i$ that shows that ADD-OPT/Push-DIGing should work with nonidentical step-sizes. Optimization with RS weights: The state transformation technique discussed above also leads to an algorithm from $\mc{AB}$ that only requires RS weights. Since $\mc{A}$ in $\mc{AB}$ is RS, a transformation now is imposed on the $\mb{y}_k$-update and is given by $\wt{\mb{y}}_k\triangleq \Pi_c^{-1}\mb{y}_k$, where $\Pi_c \triangleq \mbox{diag}(\bpi_c)\otimes I_p,$ and $\bpi_c$ is the right-eigenvector of the CS weight matrix, $B$, corresponding to the eigenvalue $1$. Equivalently, $\mc{AB}$ is given by $$\begin{aligned} \mb{x}_{k+1} &= \mc{A}\mb{x}_k -\alpha\Pi_c\wt{\mb{y}}_k, \label{AB_frosta}\\ \wt{\mb{y}}_{k+1} &= \wt{\mc{A}}\wt{\mb{y}}_k + \Pi_c^{-1}\left(\nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k)\right) \label{AB_frostb}, \end{aligned}$$ where $\wt{\mc{A}}=\Pi_c^{-1}\mc{B}\Pi_c$ is now RS and $\left(\bpi_c^\top\otimes I_p\right)\wt{\mc{A}}=\bpi_c^\top\otimes I_p$. Since the above form of $\mc{AB}$ cannot be implemented because $\bpi_c$ is not locally known, an eigenvector estimation is used in FROST [@linear_row; @FROST] and the division in Eq. is replaced with the appropriate estimated component of $\bpi_c$. The observations on different weight matrices in the two updates, nonidentical step-sizes, stability, and convergence made earlier for ADD-OPT/Push-DIGing are also applicable here. In conclusion, the $\mc{AB}$ algorithm has various equivalent representations and several already-known protocols can in fact be derived from these representations. In a similar way, $\mc{AB}m$ leads to protocols that add momentum to Aug-DGM, ADD-OPT/Push-DIGing, and FROST. We will revisit the relationship and equivalence cast here in Sections \[s6\] and \[s7\]. In Section \[s6\], we will show that both $\mc{AB}$ and $\mc{AB}m$ naturally provide a non-trivial class of average-consensus algorithms, a special case of which are [@ac_row] and surplus consensus [@ac_Cai1]. In Section \[s7\], we will compare these algorithms numerically. Convergence Analysis {#s4} ==================== We now start the convergence analysis of the proposed distributed heavy-ball method, $\mc{AB}m$. In the following, we first provide some auxiliary results borrowed from the literature. Auxiliary Results ----------------- The following lemma establishes contractions with RS and CS matrices under arbitrary norms [@AB]; note thacontraction in the Euclidean norm is not applicable unless the weight matrix is DS as in [@harness; @diging]. A similar result was first presented in [@add-opt] for CS matrices, and later in [@linear_row; @FROST] for RS matrices. \[contra\] Consider the augmented weight matrices $\mc{A}$ and $\mc{B}$. There exist vector norms, denoted as $\left\\|\cdot\right\\|_\mc{A}$ and $\left\\|\cdot\right\\|_\mc{B}$, such that $\forall\mb{x}\in\mbb{R}^{np}$, $$\begin{aligned} \label{A_ctr} \left\\|\mc{A}\mb{x}-\mc{A}_\infty\mb{x}\right\\|_\mc{A}&\leq\sigma_\mc{A}\left\\|\mb{x}-\mc{A}_\infty\mb{x}\right\\|_\mc{A},\\\label{B_ctr} \left\\|\mc{B}\mb{x}-\mc{B}_\infty\mb{x}\right\\|_\mc{B}&\leq\sigma_\mc{B}\left\\|\mb{x}-\mc{B}_\infty\mb{x}\right\\|_\mc{B}, \end{aligned}$$ where $0<\sigma_\mc{A}<1$ and $0<\sigma_\mc{B}<1$ are some constants. The next lemma from [@AB] states that the sum of $\mb{y}_k^i$’s preserves the sum of local gradients. This is a direct consequence of the dynamic consensus [@DAC] employed with CS weights in the $\mb{y}_k$-update of $\mc{AB}m$. \[sum\] $(\mb{1}_n^\top \otimes I_p) \mb{y}_k = (\mb{1}_n^\top \otimes I_p) \nabla\mb{f}(\mb{x}_k),\forall k$. The next lemma is standard in the convex optimization theory [@bertsekas1999nonlinear]. It states that the distance to the optimizer contracts at each step in the standard gradient descent method. \[centr\_d\] Let $F$ be $\mu$-strongly-convex and $l$-smooth. For $0<\alpha<\frac{2}{l}$, we have $$\left\\|\mb{x}-\alpha\nabla F(\mb{x})-\mb{x}^\right\\| \leq\sigma_F\left\\|\mb{x}-\mb{x}^\right\\|,$$ where $\sigma_F=\max\left(\left\|1-\mu \alpha\right\|,\left\|1-l\alpha \right\|\right)$. Finally, we provide a result from nonnegative matrix theory. \[rho\](Theorem 8.1.29 in [@matrix]) Let $X\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mb{x}\in\mathbb{R}^{n}$ be a positive vector. If $X\mb{x}<\omega\mb{x}$ with $\omega>0$, then $\rho(X)<\omega$. Main results {#s5} ------------ The convergence analysis of $\mc{AB}m$ is based on deriving a contraction relationship between the following four quantities: $\\|\mb{x}_{k+1}-\mc{A}_\infty\mb{x}_{k+1}\\|_\mc{A}$, the consensus error in the network; $\\|\mc{A}_\infty\mb{x}_{k+1}-\mb{1}_n \otimes \mb{x}^\\|$, the optimality gap; $\left\\|\mb{x}_{k+1}-\mb{x}_k\right\\|$, the state difference; and $\\|\mb{y}_{k+1}-\mc{B}_\infty\mb{y}_{k+1}\\|_\mc{B}$, the (biased) gradient estimation error. We will establish an LTI-system inequality where the state vector is the collection of these four quantities and then develop the convergence properties of the corresponding system matrix. Before we proceed, note that since all vector norms on finite-dimensional vector spaces are equivalent [@matrix], there exist positive constants $c_{\mathcal{A}\mathcal{B}},c_{\mathcal{B}\mathcal{A}},c_{2\mathcal{A}},c_{\mathcal{A}2},c_{2\mathcal{B}},c_{\mathcal{B}2}$ such that $$\begin{aligned} \\|\cdot\\|_\mathcal{A} &\leq c_{\mathcal{A}\mathcal{B}}\\|\cdot\\|_\mathcal{B},~~ \\|\cdot\\| \leq c_{2\mathcal{A}}\\|\cdot\\|_\mathcal{A},~~ \\|\cdot\\|_\mathcal{A} \leq c_{\mathcal{A}2}\\|\cdot\\|, \\ \\|\cdot\\|_\mathcal{B} &\leq c_{\mathcal{B}\mathcal{A}}\\|\cdot\\|_\mathcal{A}, ~~\\|\cdot\\| \leq c_{2\mathcal{B}}\\|\cdot\\|_\mathcal{B}, ~~\\|\cdot\\|_\mathcal{B} \leq c_{\mathcal{B}2}\\|\cdot\\|.\end{aligned}$$ We also define $\ol{\alpha}\triangleq\left[\bs{\alpha}\right]_{\max}$ and $\ol{\beta}\triangleq\left[\bs{\beta}\right]_{\max}$. In the following, we first provide an upper bound on the estimate, $\mb{y}_k$, of the gradient of the global objective that will be useful in deriving the aforementioned LTI system. \[y\] The following inequality holds, $\forall k$: $$\begin{aligned} \\|\mb{y}_k\\| \leq&~ c_{2\mc{A}}\ol{l}\left\\|\mc{B}_{\infty}\right\\|\\|\mb{x}_k-\mc{A}_\infty\mb{x}_k\\|_\mc{A} + c_{2\mc{B}}\\|\mb{y}_k-\mc{B}_\infty\mb{y}_k\\|_\mc{B}\\ &+ \ol{l}\left\\|\mc{B}_{\infty}\right\\|\\|\mc{A}_\infty\mb{x}_k-\mb{1}_n \otimes \mb{x}^\\|. \end{aligned}$$ Recall that $\mc{B}_{\infty}= (\bs{\pi}_c\otimes I_p)(\mb{1}_n^\top\otimes I_p).$ We have $$\label{inte_3} \left\\|\mb{y}_k\right\\| \leq c_{2\mc{B}}\left\\|\mb{y}_k-\mc{B}_\infty\mb{y}_k\right\\|_\mc{B} + \left\\|\mc{B}_\infty\mb{y}_k\right\\|.$$ We next bound $\left\\|\mc{B}_\infty\mb{y}_k\right\\|$: $$\begin{aligned} \label{inte_4} \\|\mc B_\infty\mb{y}_k\\| =&~ \\|(\bs{\pi}_c\otimes I_p)(\mb{1}_n^\top\otimes I_p)\nabla\mb{f}(\mb{x}_k)\\|\nonumber,\\ =&~ \\|\bs{\pi}_c\\|\left\\|{\tsum}_{i=1}^{n}\nabla f_i(\mb{x}_k^i)-{\tsum}_{i=1}^{n}\nabla f_i(\mb{x}^)\right\\| \nonumber,\\ \leq&~\\|\bs{\pi}_c\\|~\ol{l}~\sqrt{n}\\|\mb{x}_k-\mb{1}_n\otimes \mb{x}^\\|, \nonumber\\ \leq&~c_{2\mc{A}}~\ol{l}~\left\\|\mc{B}_{\infty}\right\\| \\|\mb{x}_k-\mc A_\infty\mb{x}_k\\|_\mc{A} \nonumber\\ &+~\ol{l}~\left\\|\mc{B}_{\infty}\right\\|\\|\mc A_\infty\mb{x}_k-\mb{1}_n \otimes \mb{x}^\\|, \end{aligned}$$ where the first inequality uses Jensen’s inequality and the last inequality uses the fact that $\left\\|\mc{B}_{\infty}\right\\|=\sqrt{n}\\|\bs{\pi}_c\\|$. The lemma follows by plugging Eq. into Eq. . In the next Lemmas \[xc\]-\[yc\], we derive the relationships among the four quantities mentioned above. We start with a bound on $\\|\mb{x}_{k+1}-\mc{A}_\infty\mb{x}_{k+1}\\|_\mc{A}$, the consensus error in the network. \[xc\] The following inequality holds, $\forall k$: $$\begin{aligned} \\|\mb{x}&_{k+1}-\mc{A}_\infty\mb{x}_{k+1}\\|_\mc{A} \\ \leq&\left(\sigma_{\mc{A}}+\ol{\alpha} c_{\mc{A}2}c_{2\mc{A}}~\ol{l}\left\\|I_{np}-\mc{A}_\infty\right\\|\left\\|\mc{B}_{\infty}\right\\|\right) \left\\|\mb{x}_{k}-\mc{A}_\infty\mb{x}_{k}\right\\|_\mc{A} \\ &+ \ol{\alpha} c_{\mc{A}2}~\ol{l}\left\\|I_{np}-\mc{A}_\infty\right\\| \left\\|B_{\infty}\right\\|\\|\mc{A}_\infty\mb{x}_k-\mb{1}_n \otimes \mb{x}^\\| \\ &+ \ol\beta c_{\mc{A}2}\left\\|I_{np}-\mc{A}_\infty\right\\| \left\\|\mb{x}_k-\mb{x}_{k-1}\right\\|\nonumber\\ &+ \ol{\alpha}c_{\mc{A}2}c_{2\mc{B}}\left\\|I_{np}-\mc{A}_\infty\right\\|\left\\|\mb{y}_k-\mc{B}_\infty\mb{y}_k\right\\|_\mc{B}.\end{aligned}$$ First, note that $\mc{A}_{\infty}\mc{A}=\mc{A}_{\infty}$. Following the $\mb{x}_k$-update of $\mc{AB}m$ in Eq. and using the one-step contraction property of $\mc{A}$ from Lemma \[contra\], we have: $$\begin{aligned} \big\\|\mb{x}&_{k+1}-\mc{A}_\infty\mb{x}_{k+1}\big\\|_\mc{A} \nonumber\\ =& \big\\|\mc{A}\mb{x}_k -D_{\bds{\alpha}}\mb{y}_k + D_{\bds{\b}} (\mb{x}_k-\mb{x}_{k-1}) \\&-A_\infty\big(\mc{A}\mb{x}_k -D_{\bds{\alpha}}\mb{y}_k +D_{\bds{\b}} (\mb{x}_k-\mb{x}_{k-1})\big)\big\\|_\mc{A}, \nonumber\\ \leq&~\sigma_{\mc{A}} \left\\|\mb{x}_{k}-\mc{A}_\infty\mb{x}_{k}\right\\|_\mc{A} + \ol{\alpha}~c_{\mc{A}2}\left\\|I_{np}-\mc{A}_\infty\right\\| \left\\|\mb{y}_k\right\\| \\&+ \ol{\beta}~c_{\mc{A}2}\left\\|I_{np}-\mc{A}_\infty\right\\| \left\\|\mb{x}_k-\mb{x}_{k-1}\right\\|, \end{aligned}$$ and the proof follows from Lemma \[y\]. Next, we derive a bound for $\left\\|\mc{A}_\infty\mb{x}_{k+1}-\mb{1}_n\otimes \mb{x}^\right\\|$, which can be interpreted as the optimality gap between the network accumulation state, $\mc{A}_\infty\mb{x}_k$, and the global minimizer, $\mb{1}_n\otimes\mb{x}^$. \[xo\] The following inequality holds, $\forall k$, when $0<\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c<\frac{2}{nl}$: $$\begin{aligned} \label{2} \\|\mc{A}_\infty\mb{x}&_{k+1}-\mb{1}_n \otimes \mb{x}^\\| \nonumber\\ \leq&~\ol{\alpha}\left(\bpi_r^\top\bpi_{c}\right)n\ol{l}c_{2\mc{A}}\left\\|\mb{x}_{k}-\mc{A}_\infty\mb{x}_{k}\right\\|_\mc{A} \nonumber\\ &+\lambda \left\\| \mc{A}_{\infty}\mb{x}_k -\mb{1}_n \otimes \mb{x}^* \right\\| \nonumber\nonumber\\ &+ \beta \\|\mc{A}_{\infty}\\| \\|\mb{x}_k-\mb{x}_{k-1}\\| \nonumber \\ &+ \ol{\alpha} c_{2B}\\|\mc{A}_\infty\\|\left\\|\mb{y}_k-\mc{B}_{\infty}\mb{y}_k\right\\|_\mc{B}, \end{aligned}$$ where [$\lambda=\max\left\{\left\|1-\mu n \bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c\right\|, \left\|1-ln\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c \right\|\right\}. $]{} Recall the $\mb{x}_k$-update of $\mc{AB}m$ in Eq. , we have that $$\begin{aligned} \label{21} \\|\mc{A}_\infty\mb{x}&_{k+1}-\mb{1}_n \otimes \mb{x}^\\| \nonumber\\ =& \left\\|\mc{A}_\infty\big(\mc{A}\mb{x}_k -D_{\bds{\alpha}}\mb{y}_k + D_{\bds\b} (\mb{x}_k-\mb{x}_{k-1})\big)-\mb{1}_n \otimes \mb{x}^\right\\|, \nonumber\\ =&~\big\\|\mc{A}_\infty\big(\mc{A}\mb{x}_k -D_{\bds{\alpha}}\mb{y}_k + (D_{\bds{\alpha}}-D_{\bds{\alpha}})\mc{B}_{\infty}\mb{y}_k \nonumber\\ &~~+ D_{\bds\b} (\mb{x}_k-\mb{x}_{k-1})\big)-\mb{1}_n \otimes \mb{x}^\big\\| \nonumber,\\ \leq& \left\\|\mc{A}_{\infty}\mb{x}_k-\mc{A}_{\infty}D_{\bds{\alpha}}\mc{B}_{\infty}\nabla\mb{f}\left(\mb{x}_k\right)-\left(\mb{1}_n \otimes I_p\right) \mb{x}^ \right\\| \nonumber\\ &+ \ol\beta \\|\mc{A}_{\infty}\\| \\|\mb{x}_k-\mb{x}_{k-1}\\| \nonumber\\ &+ \ol{\alpha} c_{2B}\\|\mc{A}_\infty\\|\left\\|\mb{y}_k-\mc{B}_{\infty}\mb{y}_k\right\\|_{\mc{B}}, \end{aligned}$$ where in the last inequality, we use $\mc{B}_{\infty}\mb{y}_k=\mc{B}_{\infty}\nabla\mb{f}\left(\mb{x}_k\right)$ adapted from Lemma \[sum\]. Since the last two terms in Eq. match the last two terms in Eq. , what is left is to bound the first term. Before we proceed, define $$\begin{aligned} \wt{\mb{x}}_k &\triangleq (\bds{\pi}^\top_r \otimes I_p)\mb{x}_k, \\ \nabla\mb{f}\left((\mb{1}_n\otimes I_p)\wt{\mb{x}}_k\right) &\triangleq \left[\nabla f_1(\wt{\mb{x}}_k)^\top,\cdots,\nabla f_n(\wt{\mb{x}}_k)^\top\right]^\top,\end{aligned}$$ and note that $$\begin{aligned} &\mc{A}_{\infty}D_{\bds{\alpha}}\mc{B}_{\infty} \\ =& \left(\mb{1}_n \otimes I_p\right)\left(\bds{\pi}^\top_r \otimes I_p\right) \left(\mbox{diag}(\bds{\alpha})\otimes I_p\right) \left(\bds{\pi}_c \otimes I_p\right)\left(\mb{1}_n^\top \otimes I_p\right) \nonumber\\ =& \left(\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c\right)\left(\mb{1}_n\otimes I_p\right)\left(\mb{1}_n^\top\otimes I_p\right). \end{aligned}$$ Now we bound the first term in Eq. . We have [$$\begin{aligned} \\|\mc{A}&_{\infty}\mb{x}_k-\mc{A}_{\infty}D_{\bds{\alpha}}\mc{B}_{\infty}\nabla\mb{f}(\mb{x}_k)-\left(\mb{1}_n \otimes I_p\right) \mb{x}^* \\| \nonumber \\ =&\Big\\| \left(\mb{1}_n \otimes I_p\right)\Big(\wt{\mb{x}}_k-(\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c)(\mb{1}_n^\top\otimes I_p) \nabla\mb{f}(\mb{x}_k)-\mb{x}^* \Big) \Big\\| \nonumber,\\ \leq& \left\\| \left(\mb{1}_n \otimes I_p\right)\left(\wt{\mb{x}}_k -n (\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c) \nabla F(\wt{\mb{x}}_k)-\mb{x}^* \right) \right\\| \nonumber\\ &+\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c\left\\|\left(\mb{1}_n \otimes I_p\right)\big(n\nabla F(\wt{\mb{x}}_k)- (\mb{1}_n^\top\otimes I_p) \nabla\mb{f}(\mb{x}_k)\big) \right\\| \nonumber,\\ \triangleq&~s_1 + s_2, \nonumber \end{aligned}$$]{}and we bound $s_1$ and $s_2$ next. Using Lemma \[centr\_d\], we have that if $0<\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c<\frac{2}{nl}$, $$\begin{aligned} \label{s1b} s_1 &= \sqrt{n} \left\\|\wt{\mb{x}}_k -n (\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c) \nabla F(\wt{\mb{x}}_k)-\mb{x}^\right\\| \nonumber,\\ &\leq \sqrt{n}\lambda \left\\|\wt{\mb{x}}_k -\mb{x}^\right\\| \nonumber,\\ &= \lambda \left\\| \mc{A}_{\infty}\mb{x}_k -\mb{1}_n \otimes \mb{x}^* \right\\|, \end{aligned}$$ where [$\lambda=\max\left\{\left\|1-\mu n \bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c\right\|, \left\|1-ln\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c \right\|\right\}. $]{} We next bound $s_2$. Since [$\nabla F(\wt{\mb{x}}_k)=\frac{1}{n}(\mb{1}_n^\top\otimes I_p)\nabla\mb{f}(\wt{\mb{x}}_k)$]{}, $$\begin{aligned} \label{23} s_2 & \leq \left(\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c\right)n\left\\|\nabla\mb{f}\left((\mb{1}_n\otimes I_p)\wt{\mb{x}}_k\right)-\nabla\mb{f}(\mb{x}_k)\right\\| \nonumber,\\ &\leq \left(\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c\right)n\ol{l}c_{2\mc{A}}\left\\|\mb{x}_{k}-\mc{A}_\infty\mb{x}_{k}\right\\|_\mc{A}, \nonumber\\ &\leq \ol{\alpha}\left(\bpi_r^\top\bpi_{c}\right)n\ol{l}c_{2\mc{A}}\left\\|\mb{x}_{k}-\mc{A}_\infty\mb{x}_{k}\right\\|_\mc{A}, \end{aligned}$$ and the lemma follows from Eqs. , , and . The next step is to bound the state difference, $\left\\|\mb{x}_{k+1}-\mb{x}_k\right\\|$. \[m\] The following inequality holds, $\forall k$: $$\begin{aligned} \label{3} \\|\mb{x}&_{k+1}-\mb{x}_{k}\\| \nonumber\\ \leq&~\left(c_{2\mc{A}}\left\\|\mc{A}-I_{np}\right\\|+\ol{\alpha}c_{2\mc{A}}\ol{l}\left\\|\mc{B}_{\infty}\right\\|\right)\left\\|\mb{x}_k-\mc{A}_\infty\mb{x}_k\right\\|_\mc{A} \nonumber\\ &+\ol{\alpha}\ol{l}\left\\|\mc{B}_{\infty}\right\\|\\|\mc{A}_\infty\mb{x}_k-\mb{1}_n \otimes \mb{x}^\\| \nonumber\\ &+\ol{\beta}\left\\|\mb{x}_{k}-\mb{x}_{k-1}\right\\| +\ol{\alpha}c_{2\mc{B}}\\|\mb{y}_k-\mc{B}_\infty\mb{y}_k\\|_\mc{B}. \nonumber\end{aligned}$$ Note that $\mc{A}\mc{A}_{\infty}=\mc{A}_{\infty}$ and hence $\mc{A}\mc{A}_{\infty}-\mc{A}_{\infty}$ is a zero matrix. Following the $\mb{x}_k$-update of $\mc{AB}m$, we have: [$$\begin{aligned} \\|\mb{x}&_{k+1}-\mb{x}_{k}\\| \nonumber\\ =& \left\\| \mc{A}\mb{x}_k -D_{\bds{\alpha}}\mb{y}_k + D_{\bs\b} (\mb{x}_k-\mb{x}_{k-1}) -\mb{x}_k \right\\| \nonumber,\\ =& \left\\| (\mc{A}-I_{np})(\mb{x}_k-\mc{A}_{\infty}\mb{x}_k) -D_{\bds{\alpha}}\mb{y}_k + D_{\bs\b} (\mb{x}_k-\mb{x}_{k-1}) \right\\| \nonumber,\\ \leq&~c_{2\mc{A}}\left\\|\mc{A}-I_{np}\right\\|\left\\|\mb{x}_k-\mc{A}_{\infty}\mb{x}_k\right\\|_\mc{A}+\ol{\beta}\left\\|\mb{x}_{k}-\mb{x}_{k-1}\right\\| + \ol{\alpha} \left\\|\mb{y}_k\right\\| , \nonumber\end{aligned}$$]{} and the proof follows from Lemma \[y\]. The final step in formulating the LTI system is to write $\left\\|\mb{y}_{k+1}-\mc{B}_{\infty}\mb{y}_{k+1}\right\\|$, the biased gradient estimation error, in terms of the other three quantities. We call this biased to make a distinction with the unbiased gradient estimation error: $\left\\|\mb{y}_{k+1}-\mc{W}_\infty\mb{y}_{k+1}\right\\|$, where $\mc{W}$ is doubly-stochastic. \[yc\] The following inequality holds, $\forall k$: $$\begin{aligned} \\|\mb{y}&_{k+1}-\mc{B}_{\infty}\mb{y}_{k+1}\\| \nonumber\\ =&\Big(c_{2\mc{A}}c_{\mc{B}2}~\ol{l}\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mc{A}-I_{np}\right\\| \nonumber\\ &+\ol{\alpha}c_{2\mc{A}}c_{\mc{B}2}~\ol{l}^2\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mc{B}_{\infty}\right\\|\Big)\left\\|\mb{x}_k-\mc{A}_\infty\mb{x}_k\right\\|_\mc{A} \nonumber\\ &+\ol{\alpha} c_{\mc{B}2}~\ol{l}^{2}\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mc{B}_{\infty}\right\\|\\|\mc{A}_\infty\mb{x}_k-\mb{1}_n \otimes \mb{x}^\\| \nonumber\\ &+\ol\beta c_{\mc{B}2}~\ol{l}\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mb{x}_{k}-\mb{x}_{k-1}\right\\|\nonumber\\ &+\Big(\sigma_\mc{B}+\ol{\alpha} c_{\mc{B}2}c_{2\mc{B}}~\ol{l}\left\\|I_{np}-\mc{B}_{\infty}\right\\|\Big)\left\\|\mb{y}_k-\mc{B}_\infty\mb{y}_k\right\\|_\mc{B} \nonumber. \nonumber\end{aligned}$$ Note that $\mc{B}_{\infty}\mc{B}=\mc{B}_{\infty}$. From Eq. , we have: [$$\begin{aligned} \\|&\mb{y}_{k+1}-\mc{B}_\infty\mb{y}_{k+1}\\|_\mc{B} \nonumber\\ =&~\big\\|\mc{B}\mb{y}_k+\nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k) \nonumber\\ &-\mc{B}_{\infty}\big(\mb{y}_k+\nabla \mb{f}(\mb{x}_{k+1})-\nabla \mb{f}(\mb{x}_k)\big)\big\\|_\mc{B} \nonumber\\ \leq&~\sigma_\mc{B}\\|\mb{y}(k)-\mc{B}_\infty\mb{y}(k)\\|_\mc{B} +c_{\mc{B}2}\ol{l}\left\\|I_{np}-\mc{B}_{\infty}\right\\|\\|\mb{x}_{k+1}-\mb{x}_k\\|_2, \nonumber\end{aligned}$$]{}where in the inequality above we use the contraction property of $\mc{B}$ from Lemma \[contra\]. The proof follows by applying the result of Lemma \[m\] to the inequality above. With the help of the Lemmas \[xc\]-\[yc\], we now present the main result of this paper, i.e., the $\mc{AB}m$ algorithm converges to the global minimizer at a global $R$-linear rate. \[R\] Let $0<\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c<\frac{2}{nl}$, then the following LTI inequality holds entry-wise: $$\label{LMI} \mb{t}_{k+1} \leq J_{\bds{\alpha},\ol\beta}\mb{t}_{k},$$ where $\mb{t}_{k}\in\mathbb{R}^4$ and $J_{\bds{\alpha},\ol\b}\in\mathbb{R}^{4\times 4}$ are respectively given by: [$$\begin{aligned} \mb{t}_k&=\left[ \begin{array}{c} \left\\|\mb{x}_{k}-\mc{A}_\infty\mb{x}_{k}\right\\|_\mc{A} \\ \left\\|\mc{A}_\infty\mb{x}_{k}-\mb{1}_n \otimes \mb{x}^\right\\| \\ \left\\|\mb{x}_{k}-\mb{x}_{k-1}\right\\| \\ \left\\|\mb{y}_{k}-\mc{B}_\infty\mb{y}_{k}\right\\|_\mc{B} \end{array} \right], \nonumber\\ J_{\bds{\alpha},\ol{\beta}}&=\left[ \begin{array}{cccc} \sigma_\mc{A}+a_1\ol{\alpha} & a_2\ol{\alpha} &\ol{\beta} a_3 &a_4\ol{\alpha}\\ a_5\ol{\alpha} & \lambda & \ol{\beta} a_6 & a_7\ol{\alpha}\\ a_8+a_9\ol{\alpha}& a_{10}\ol{\alpha} & \ol{\beta} & a_{11} \ol{\alpha}\\ a_{12}+a_{13}\ol{\alpha}& a_{14}\ol{\alpha}& \ol{\beta} a_{15} & \sigma_\mc{B} + a_{16}\ol{\alpha} \end{array} \right], \nonumber\end{aligned}$$]{} and the constants $a_i$’s in the above expression are [$$\begin{aligned} a_1 &=& c_{\mc{A}2}c_{2\mc{A}}\ol{l}\left\\|I_{np}-\mc{A}_\infty\right\\| \left\\|\mc{B}_{\infty}\right\\|, \\ a_2 &=& c_{\mc{A}2}\ol{l}\left\\|I_{np}-\mc{A}_\infty\right\\| \left\\|\mc{B}_{\infty}\right\\|, \\ a_3 &=& c_{\mc{A}2}\left\\|I_{np}-\mc{A}_\infty\right\\|, \\ a_4 &=& c_{\mc{A}2}c_{2\mc{B}}\left\\|I_{np}-\mc{A}_\infty\right\\|,\\ a_5 &=& nc_{2\mc{A}}\left(\bpi_r^\top\bpi_c\right)\ol{l}, \\ a_6 &=& \\|\mc{A}_\infty\\|, \\ a_7 &=& c_{2\mc{B}}\\|\mc{A}_\infty\\|, \\ a_8 &=& c_{2\mc{A}}\left\\|\mc{A}-I_{np}\right\\|, \\ a_{9} &=& c_{2\mc{A}}\ol{l}\left\\|\mc{B}_{\infty}\right\\|, \\ a_{10} &=& \ol{l}\left\\|\mc{B}_{\infty}\right\\|, \\ a_{11} &=& c_{2\mc{B}}, \\ a_{12} &=& c_{\mc{B}2}c_{2\mc{A}}\ol{l}\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mc{A}-I_{np}\right\\|, \\ a_{13} &=& c_{\mc{B}2}c_{2\mc{A}}\ol{l}^2\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mc{B}_{\infty}\right\\|, \\ a_{14} &=& c_{\mc{B}2}\ol{l}^2\left\\|I_{np}-\mc{B}_{\infty}\right\\|\left\\|\mc{B}_{\infty}\right\\|, \\ a_{15} &=& c_{\mc{B}2}\ol{l}\left\\|I_{np}-\mc{B}_{\infty}\right\\|, \\ a_{16} &=& c_{\mc{B}2}c_{2\mc{B}}\ol{l}\left\\|I_{np}-B_{\infty}\right\\|. \end{aligned}$$]{}When the largest step-size, $\ol{\alpha}$, satisfies [$$\begin{aligned} 0<\ol{\alpha}& < \min\Bigg\{\frac{1}{nl\bpi_r^\top \bpi_{c}},\frac{\delta_3-\delta_1a_8}{a_9\delta_1+a_{10}\delta_2+a_{11}\delta_4}, \nonumber\\ &~~~\frac{(1-\sigma_B)\delta_4-\delta_{1}a_{12}}{a_{13}\delta_1+a_{14}\delta_2+a_{14}\delta_4},\frac{(1-\sigma_B)\delta_4-\delta_{1}a_{12}}{a_{13}\delta_1+a_{14}\delta_2+a_{14}\delta_4}\Bigg\} \label{a}\end{aligned}$$]{}and when the largest momentum parameter, $\ol{\beta}$, satisfies [$$\begin{aligned} 0\leq\beta& < \min\Bigg\{ \frac{\delta_1(1-\sigma_A)-\left(a_1\delta_1+a_2\delta_2+a_4\delta_4\right)\ol{\alpha}}{a_3\delta_3}, \nonumber\\ &\frac{\Big(\delta_2\mu[\bpi_r]_{\min}[\bpi_c]_{\min}-\left(a_5\delta_1+a_7\delta_4\right)\Big)\ol{\alpha}}{a_6\delta_3}, \nonumber\\ &\frac{\delta_3-\delta_1a_8-(a_9\delta_1+a_{10}\delta_2+a_{11}\delta_4)\ol{\alpha}}{\delta_3}, \nonumber\\ &\frac{(1-\sigma_B)\delta_4-\delta_{1}a_{12}-(a_{13}\delta_1+a_{14}\delta_2+a_{14}\delta_4)\ol{\alpha}}{a_{15}\delta_3} \Bigg\}, \label{b}\end{aligned}$$]{}where $\delta_1,\delta_2,\delta_3,\delta_4$ are arbitrary constants such that [$$\begin{aligned} \left\{ \begin{array}{lll} \delta_1 &<& \max \left\{ \frac{\delta_3}{a_8},\frac{(1-\sigma_B)\delta_4}{a_{12}} \right\}, \\ \delta_2 &>& \frac{a_5\delta_1+a_7\delta_4}{\mu[\bpi_r]_{\min}[\bpi_c]_{\min}}, \\ \delta_3 &>& 0, \\ \delta_4 &>& 0, \end{array} \right. \nonumber\end{aligned}$$]{}then $\rho(J_{\bds{\alpha},\ol\beta})<1$ and thus $\\|\mb{x}_k-\mb{1}_n\otimes\mb{x}^\\|$ converges to zero linearly at the rate of $\mc{O}(\rho(J_{\bds{\alpha},\ol\beta}))^k$. It is straightforward to verify Eq. by combining Lemmas \[xc\]-\[yc\]. The next step is to find the range of $\ol{\alpha}$ and $\ol{\beta}$ such that $\rho(J_{\bds{\alpha},\ol\beta})<1$. In the light of Lemma \[rho\], we solve for a positive vector $\bds{\delta}=[\delta_1,\delta_2,\delta_3,\delta_4]^\top$ and the range of $\ol{\alpha}$ and $\ol{\beta}$ such that the following inequality holds: $$J_{\bds{\alpha},\ol\beta}\bds{\delta}<\bds{\delta},$$ which is equivalent to the following four conditions: [$$\begin{aligned} a_3\delta_3\beta &<\delta_1(1-\sigma_A)-\left(a_1\delta_1+a_2\delta_2+a_4\delta_4\right)\ol{\alpha}, \label{i1}\\ a_6\delta_3\beta &<\delta_2-\delta_2\lambda-\left(a_5\delta_1+a_7\delta_4\right)\ol{\alpha}, \label{i2}\\ \delta_3\beta&<\delta_3-\delta_1a_8-(a_9\delta_1+a_{10}\delta_2+a_{11}\delta_4)\ol{\alpha}, \label{i3}\\ a_{15}\delta_3\beta &< (1-\sigma_B)\delta_4 -\delta_{1}a_{12}-(a_{13}\delta_1+a_{14}\delta_2+a_{14}\delta_4)\ol{\alpha}. \label{i4}\end{aligned}$$]{}Recall $\lambda$ in Lemma \[xo\], when $\ol{\alpha}<\frac{1}{nl\bpi_r^\top\bpi_c}$, we have $$\begin{aligned} \lambda = 1-\mu n\bpi_r^\top\mbox{diag}(\bds{\alpha})\bpi_c \leq 1 - \mu n[\bpi_r]_{\min}[\bpi_c]_{\min}\ol{\alpha}.\end{aligned}$$ Therefore, the third condition in Eq. is satisfied when $$\begin{aligned} a_6\delta_3\beta < \delta_2\mu n[\bpi_r]_{\min}[\bpi_c]_{\min}\ol{\alpha}-\left(a_5\delta_1+a_7\delta_4\right)\ol{\alpha}. \label{i2'}\end{aligned}$$ For the right hand side of the Eq. , , and to be positive, each one of these equations needs to satisfy the conditions we give below. We first choose arbitrary positive constants, $\delta_3$ and $\delta_4$, then pick $\delta_1$ satisfying Eqs. and , and finally choose $\delta_2$ according to Eq. . Note that $\delta_1,\delta_2,\delta_3,$ and $\delta_4$ are chosen to ensure that the upper bounds on $\ol{\alpha}$ are all positive. Subsequently, from Eqs. , , and , together with the requirement that $\ol{\alpha}<\frac{1}{nl\bpi_r^\top \bpi_{c}}$, we obtain the upper bound on the largest step-size, $\ol{\alpha}$. Finally, the original four conditions in Eqs. , , and lead to an upper bound on $\ol{\beta}$, and the theorem follows. Remark 1: In Theorem \[R\], we have established the $R$-linear rate of $\mc{AB}m$ when the largest step-size, $\ol{\alpha}$, and the largest momentum parameter, $\ol{\beta}$, respectively follow the upper bounds described in Eq. and Eq. . Note that $\delta_1,\delta_2,\delta_3,\delta_4$ therein are tunable parameters and only depend on the network topology and the objective functions. The upper bounds for $\ol\alpha$ and $\ol\beta$ may not be computable for arbitrary directed graphs as the contraction coefficients, $\sigma_{\mc{A}}$, $\sigma_{\mc{B}}$, and the norm equivalence constants may be unknown. However, when the graph is undirected, we can obtain computable bounds for $\ol\alpha$ and $\ol\beta$, as developed in [@harness; @dnesterov] for example. The upper bound on $\ol{\beta}$ also implies that if the step-sizes are relatively large, only small momentum parameters can be picked to ensure stability. Remark 2: The nonidentical step-sizes in gradient tracking methods [@AugDGM; @harness] have previously been studied in [@AugDGM; @digingstochastic; @digingun; @lu2018geometrical]. These works rely on some notion of heterogeneity among the step-sizes, defined respectively as the relative deviation of the step-sizes from their average, $\frac{\\|(I-W)\bs{\alpha}\\|}{\\|W\bs{\alpha}\\|}$, in [@AugDGM; @digingstochastic], and as the ratio of the largest to the smallest step-size, ${[\bds{\alpha}]_{\max}}/{[\bds{\alpha}]_{\min}}$, in [@digingun; @lu2018geometrical]. The authors then show that when the heterogeneity is sufficiently small and when the largest step-size follows a bound that is a function of the heterogeneity, the proposed algorithms converge to the global minimizer. It is worth noting that sufficiently small step-sizes do not guarantee sufficiently small heterogeneity in both of the above definitions. In contrast, the upper bound on the largest step-size in this paper, Eq. , is independent of any notion of heterogeneity and only depends on the objective functions and the network topology. Each agent therefore locally picks a sufficiently small step-size without any coordination. Based on the discussion in Section \[s3\], our approach thus improves the analysis in [@AugDGM; @digingstochastic; @digingun; @lu2018geometrical]. Besides, Eq. allows the existence of zero step-sizes among the agents as long as the largest step-size is positive and is sufficiently small. Remark 3: To show that $\mc{AB}m$ has an $R$-linear rate for sufficiently small $\ol\alpha$ and $\ol\beta$, one can alternatively use matrix perturbation analysis as in [@AB] (Theorem 1). However, it does not provide explicit upper bounds on $\ol\alpha$ and $\ol\beta$ in closed form. Average-Consensus from $\mc{AB}m$ {#s6} ================================= In this section, we show that $\mc{AB}m$ subsumes a novel average-consensus algorithm over strongly-connected directed graphs. To show this, we choose the objective functions as$$\wt{f}_i(\mb{x})=\tfrac{1}{2}\\|\mb{x}-\bds{\upsilon}_i\\|^2,\quad \forall i.$$ Clearly, the minimization of $\wt{F}=\sum_{i=1}^{n} \wt{f}_i$ is now achieved at $\mb{x}^=\tfrac{1}{n} \sum_{i=1}^{n}\bds{\upsilon}_i$. The $\mc{AB}m$ algorithm, Eq. , thus naturally leads to the following average-consensus algorithm, termed as $\mc{AB}m$-$\mc{C}$, with $\nabla\mb{f}(\mb{x}_{k+1})-\nabla\mb{f}(\mb{x}_k)=\mb{x}_{k+1}-\mb{x}_k$; for the sake of simplicity, we choose $\alpha_i=\alpha,\beta_i=\beta,\forall i$: $$\begin{aligned} \mb{x}_{k+1} &= (\mc{A}+\b I)\mb{x}_k - \a\mb{y}_k - \b\mb{x}_{k-1}, \\ \mb{y}_{k+1} &= (\mc{A} + \b I - I) \mb{x}_k + (\mc{B}-\a I)\mb{y}_k - \b\mb{x}_{k-1}.\end{aligned}$$ Its local implementation at each agent $i$ is given by: $$\begin{aligned} \mb{x}_{k+1}^i =& \sum_{j\in\mc{N}_i\setminus i}a_{ij}\mb{x}_k^j + (a_{ii}+\b)\mb{x}_k^i - \a\mb{y}^i_k - \b\mb{x}_{k-1}^i, \\ \mb{y}_{k+1}^i =& \sum_{j\in\mc{N}_i\setminus i}a_{ij}\mb{x}_k^j + (a_{ii}+\b-1)\mb{x}_k^i\\&+ \sum_{j\in\mc{N}_i\setminus i}b_{ij}\mb{y}_k^j + (b_{ii}-\alpha)\mb{y}_k^i - \b\mb{x}_{k-1}^i,\end{aligned}$$ where $\mb{x}^i_0=\bds\upsilon_i$ and $\mb{y}_i^0 = 0,~\forall i$. From the analysis of $\mc{AB}m$, an $R$-linear convergence of $\mc{AB}m$-$\mc{C}$ to the average of $\bds{\upsilon}_i$’s is clear from Theorem \[R\]. It may be possible to make concrete rate statements by studying the spectral radius of the following system matrix: $$\label{scm} \left[\begin{array}{c} \mb{x}_{k+1}\\ \mb{y}_{k+1}\\ \mb{x}_{k} \end{array} \right] = \left[\begin{array}{ccc} \mc A+\b I & -\a I & -\b I\\ \mc A+\b I - I & \mc B-\a I & -\b I\\ I & 0 & 0\\ \end{array} \right] \left[\begin{array}{c} \mb{x}_{k}\\ \mb{y}_{k}\\ \mb{x}_{k-1} \end{array} \right].$$ However, this analysis is beyond the scope of this paper. We note that when $\b=0$, the above equations still converge to the average of $\bds{\upsilon}_i$’s according to Theorem \[R\]. What is surprising is that, with $\beta=0$, $\mc{AB}m$-$\mc{C}$ reduces to $$\begin{aligned} \label{sc} \left[\begin{array}{c} \mb{x}_{k+1}\\ \mb{y}_{k+1} \end{array} \right] = \left[\begin{array}{ccc} \mc A & -\a I\\ \mc A - I & \mc B-\a I \end{array} \right] \left[\begin{array}{c} \mb{x}_{k}\\ \mb{y}_{k} \end{array} \right],\end{aligned}$$ which is surplus consensus [@ac_Cai1], after a state transformation with $\mbox{diag}\left(I, -I\right)$; in fact, any state transformation of the form $\mbox{diag}(I, \wt{I})$ applies here as long as $\wt{I}$ is diagonal (to respect the graph topology) and invertible. More importantly, compared with surplus consensus [@ac_Cai1], $\mc{AB}m$-$\mc{C}$ uses information from the past iterations. This history information is in fact the momentum from a distributed optimization perspective, which may lead to accelerated convergence as we will numerically show in Section \[s7\]. Following this discussion, choosing the local functions as $\wt f_i$’s in [@AugDGM; @harness], or in ADD-OPT [@add-opt; @diging], or in FROST [@linear_row; @FROST], we get average-consensus with only DS, CS, or RS weights. The protocol that results directly from $\mc{AB}$ is surplus consensus, while the one resulting directly from FROST was presented in [@ac_row]. With the analysis provided in Section \[s3\], we see that the algorithm in [@ac_row] is in fact related to surplus consensus after a state transformation. Clearly, accelerated* average-consensus based exclusively on either row- or column-stochastic weights can be abstracted from the discussion herein, after adding a momentum term. Numerical Experiments {#s7} ===================== We now provide numerical experiments to illustrate the theoretical findings described in this paper. To this aim, we use two different graphs: an undirected graph, $\mc{G}_1$, and a directed graph, $\mc{G}_2$. Both graphs have $n=500$ agents and are generated using nearest neighbor rules and then we add less than $0.05\%$ random links. The number of edges in all cases is less $4\%$ of the total possible edges. Since the graphs are randomly generated across experiments, two sample graphs are shown in Fig. \[graph\], without the self-edges and random links for visual clarity. We generate DS weights using the Laplacian method: $W=I-\tfrac{1}{\max_i \deg_i+1} L$, where $L$ is the graph Laplacian and $\deg_i$ is the degree of node $i$. Additionally, we generate RS and CS weights with the uniform weighting strategy: $a_{ij}=\tfrac{1}{\|\mc{N}_j^{{\scriptsize \mbox{in}}}\|}$ and $b_{ij}=\tfrac{1}{\|\mc{N}_j^{{\scriptsize \mbox{out}}}\|},\forall i,j$. We note that both weighting strategies are applicable to undirected graphs, while only the uniform strategy can be used over directed graphs. Logistic Regression ------------------- We first consider distributed logistic regression: each agent $i$ has access to $m_i$ training data, $(\mb{c}_{ij},y_{ij})\in\mathbb{R}^p\times\{-1,+1\}$, where $\mb{c}_{ij}$ contains $p$ features of the $j$th training data at agent $i$, and $y_{ij}$ is the corresponding binary label. The agents cooperatively minimize $F=\sum_{i=1}^nf_i(\mb{b},c)$, to find $\mb{b}\in\mbb{R}^p,c\in\mbb{R}$, with each private loss function being [$$f_i(\mb{b},c)=\sum_{j=1}^{m_i}\ln\left[1+\exp\left(-\left(\mb{b}^\top\mb{c}_{ij}+c\right)y_{ij}\right)\right]+\frac{\lambda}{2}\\|\mb{b}\\|_2^2,$$]{}where $\frac{\lambda}{2}\\|\mb{b}\\|_2^2$ is a regularization term used to prevent over-fitting of the data. The feature vectors, $\mb{c}_{ij}$’s, are randomly generated from a Gaussian distribution with zero mean and the binary labels are randomly generated from a Bernoulli distribution. We plot the average of residuals at each agent, $\frac{1}{n}\sum_{i=1}^{n}\\|\mb{x}_i(k)-\mb{x}^\\|_2$, and first compare the performance of the following over undirected graphs in Fig. \[log\_re\] (Left): $\mc{AB}m$ with RS and CS weights; $\mc{AB}m$ with DS weights; distributed optimization based on gradient tracking from [@AugDGM; @harness; @diging], with DS weights; EXTRA from [@EXTRA]; and, centralized gradient descent. Next, we compare the performance similarly over directed graphs* in Fig. \[log\_re\] (Right). Here, the algorithms with doubly-stochastic weights [@EXTRA; @AugDGM; @harness; @diging] are not applicable, and instead we compare $\mc{AB}m$ with $\mc{AB}$ [@AB], ADD-OPT/Push-DIGing [@add-opt; @diging], and centralized gradient descent. The weight matrices are chosen as we discussed before and the algorithm parameters are hand-tuned for best performance (except for gradient descent where the optimal step-size is given by $\alpha=\tfrac{2}{\mu+l}$). We note that momentum improves the convergence when compared to applicable algorithms without momentum, while ADD-OPT/Push-DIGing are much slower because of the eigenvector estimation, see Section \[s3\] for details. Distributed Quadratic Programming --------------------------------- We now compare the performance of the aforementioned algorithms over different condition numbers of the global objective function, chosen to be quadratic, i.e., $F=\sum_i \mb{x}^\top Q_i\mb{x} + \mb{b}_i^\top\mb{x}$, where $Q_i\in\mbb{R}^{p\times p}$ is diagonal and positive-definite. The condition number $\mc{Q}$ of $F$ is given by the ratio of the largest to the smallest eigenvalue of $Q\triangleq\sum_{i=1}^{n}Q_i$. We first provide the performance comparison over undirected graphs in Fig. \[Qun\], and then provide the results over directed graphs in Fig. \[Qdi\]. In all of these experiments, we have hand-tuned the algorithm parameters for best performance. For small condition numbers, we note that gradient descent is quite fast and the distributed algorithms suffer from a relatively slower fusion over the graphs. Recall that the optimal convergence rate of gradient decent is $\mc{O}((\tfrac{\mc{Q}-1}{\mc{Q}+1})^k)$. When the condition number is large, gradient descent is quite conservative allowing fusion to catch up. Finally, we note that $\mc{AB}m$, with momentum, outperforms the centralized gradient descent when the condition number is large. This observation is consistent with the existing literature, see e.g., [@polyak1964some; @polyak1987introduction; @IAGM; @lessard2016analysis; @drori2014performance]. $\mc{AB}m$ and Average-Consensus -------------------------------- We now provide numerical analysis and simulations to show that $\mc{AB}m$-$\mc{C}$, in Eq. , possibly achieves acceleration when compared with surplus-consensus, in Eq. . To explain our choice of $\alpha$ and $\b$, we first note that the power limit of the system matrix in Eq. , denoted as $\mc{H}$, is [@ac_Cai1]: $$\begin{aligned} \lim_{k\rightarrow\infty}\mc{H}^k = \mc{H}_\infty = \left[\begin{array}{ccc} \mc{W}_\infty & -\mc{W}_\infty\\ 0_{np\times np} & 0_{np\times np} \end{array} \right],\end{aligned}$$ where $\mc{W}_\infty=(\frac{1}{n}\mb{1}_n\mb{1}_n^\top)\otimes I_p$. It is straightforward to show that $\mc{H}^k-\mc{H_\infty} = \left(\mc{H}-\mc{H}_\infty\right)^k.$ Similarly, for the augmented system matrix, $\wt{\mc{H}}$, in Eq. , we observe that and it can be verified that $\mc{\wt{H}}^k-\mc{\wt{H}_\infty} = (\mc{\wt{H}}-\mc{\wt{H}}_\infty)^k.$ We therefore use grid search [@nesterov2013introductory] to choose the optimal $\alpha^$ in $\mc{H}$ and the optimal $\wt{\alpha}^$ and $\wt{\beta}^$ in $\mc{\wt{H}}$, which respectively minimize $\rho(\mc{H}-\mc{H}_\infty)$ and $\rho(\mc{\wt{H}}-\mc{\wt{H}}_\infty)$. Numerically, we observe that it may be possible for the minimum of $\rho(\mc{\wt{H}}-\mc{\wt{H}}_\infty)$ to be smaller than that of $\rho\left(\mc{H}-\mc{H}_\infty\right)$. The convergence speed comparison between $\mc{AB}m$-$\mc{C}$ and surplus consensus [@ac_Cai1] is shown in Fig \[smc\] over a directed graph, $\mc{G}_2$. Conclusions {#s8} =========== In this paper, we provide a framework for distributed optimization that removes the need for doubly-stochastic weights and thus is naturally applicable to both undirected and directed graphs. Using a state transformation based on the non-$\mb{1}_n$ eigenvector, we show that the underlying algorithm, $\mc{AB}$, based on a simultaneous application of both RS and CS weights, lies at the heart of several algorithms studied earlier that rely on eigenvector estimation when using only CS (or only RS) weights. We then propose the distributed heavy-ball method, termed as $\mc{AB}m$, that combines $\mc{AB}$ with a heavy-ball (type) momentum term. To the best of our knowledge, this paper is the first to use a momentum term based on the heavy-ball method in distributed optimization. We show that $\mc{AB}m$ subsumes a novel average-consensus algorithm as a special case that unifies earlier attempts over directed graphs, with potential acceleration due to the momentum term. [Ran Xin]{} received his B.S. degree in Mathematics and Applied Mathematics from Xiamen University, China, in 2016, and M.S. degree in Electrical and Computer Engineering from Tufts University in 2018. Currently, he is a Ph.D. student in the Electrical and Computer Engineering department at Tufts University. His research interests include optimization theory and algorithms. [Usman A. Khan]{} has been an Associate Professor of Electrical and Computer Engineering (ECE) at Tufts University, Medford, MA, USA, since September 2017, where he is the Director of Signal Processing and Robotic Networks* laboratory. His research interests include statistical signal processing, network science, and distributed optimization over autonomous multi-agent systems. He has published extensively in these topics with more than 75 articles in journals and conference proceedings and holds multiple patents. Recognition of his work includes the prestigious National Science Foundation (NSF) Career award, several NSF REU awards, an IEEE journal cover, three best student paper awards in IEEE conferences, and several news articles. Dr. Khan joined Tufts as an Assistant Professor in 2011 and held a Visiting Professor position at KTH, Sweden, in Spring 2015. Prior to joining Tufts, he was a postdoc in the GRASP lab at the University of Pennsylvania. He received his B.S. degree in 2002 from University of Engineering and Technology, Pakistan, M.S. degree in 2004 from University of Wisconsin-Madison, USA, and Ph.D. degree in 2009 from Carnegie Mellon University, USA, all in ECE. Dr. Khan is an IEEE senior member and has been an associate member of the Sensor Array and Multichannel Technical Committee with the IEEE Signal Processing Society since 2010. He is an elected member of the IEEE Big Data special interest group and has served on the IEEE Young Professionals Committee and on IEEE Technical Activities Board. He was an editor of the IEEE Transactions on Smart Grid from 2014 to 2017, and is currently an associate editor of the IEEE Control System Letters. He has served on the Technical Program Committees of several IEEE conferences and has organized and chaired several IEEE workshops and sessions. [^1]: The authors are with the ECE Department at Tufts University, Medford, MA; [`ran.xin@tufts.edu, khan@ece.tufts.edu`]{}. This work has been partially supported by an NSF Career Award \# CCF-1350264. [^2]: Simultaneous application of both row- and column-stochastic weights was first employed for average-consensus in [@ac_Cai1] and towards distributed optimization in [@D-DPS; @D-DGD], albeit without gradient tracking. [^3]: We note that several variants of this algorithm can be extracted by considering an adapt-then-combine update, e.g., $\sum_{j=1}^{n}b_{ij}(\mb{y}^j_k+\nabla f_i(\mb{x}^i_{k+1}\big)-\nabla f_i\big(\mb{x}^i_k))$, see [@AB], instead of the combine-then-adapt update that we have used here in Eq. . The momentum term in Eq. can also be integrated similarly. We choose one of the applicable forms and note that extensions to other cases follow from this exposition and the subsequent analysis.
--- abstract: 'We present an efficient and very flexible numerical fast Fourier-Laplace transform, that extends the logarithmic Fourier transform (LFT) introduced by Haines and Jones \[Geophys. J. Int. 92(1):171 (1988)\] for functions varying over many scales to nonintegrable functions. In particular, these include cases of the asymptotic form $f(\nu\to0)\sim\nu^a$ and $f(\|\nu\|\to\infty)\sim\nu^b$ with arbitrary real $a>b$. Furthermore, we prove that the numerical transform converges exponentially fast in the number of data points, provided that the function is analytic in a cone $\|\Im{\nu}\|<\theta\|\Re{\nu}\|$ with a finite opening angle $\theta$ around the real axis and satisfies $\|f(\nu)f(1/\nu)\|<\nu^c$ as $\nu\to 0$ with a positive constant $c$, which is the case for the class of functions with power-law tails. Based on these properties we derive ideal transformation parameters and discuss how the logarithmic Fourier transform can be applied to convolutions. The ability of the logarithmic Fourier transform to perform these operations on multiscale (non-integrable) functions with power-law tails with exponentially small errors makes it the method of choice for many physical applications, which we demonstrate on typical examples. These include benchmarks against known analytical results inaccessible to other numerical methods, as well as physical models near criticality.' author: - Johannes Lang - Bernhard Frank bibliography: - 'biblio\_v2.bib' title: 'Fast logarithmic Fourier-Laplace transform of nonintegrable functions' --- Introduction ============ In physics, one is often confronted with the need to Fourier transform or convolve functions that are either only numerically available or whose exact transformation is not known. Since the reinvention of the fast Fourier transform (FFT) by Cooley and Tukey [@FFT1965], which reduces the numerical cost for both of these operations from $\mathcal{O}(N^2)$ to $\mathcal{O}(N \log_2{N})$, where $N$ denotes the number of grid points, the FFT has been established as the standard method for most situations. However, it necessarily requires an equidistant grid, which is quite inconvenient for many applications in theoretical physics. There, one frequently has to deal with slowly (i.e. algebraically) decaying functions, while the opposite limit of small arguments contains a lot of physical information. An example is provided by Green’s functions in many-body problems with short-range interactions [@zwer14varenna]. To implement an FFT under such circumstances, it is necessary to use a fine grid for small arguments that extends to very high frequencies, which is of course not very practicable due to the huge number of required data points. Consequently, a number of alternative methods have been introduced in the literature: Sometimes, sufficient knowledge about the asymptotic behavior at large arguments can be gained, subtracted and treated separately, such that the remainder of the function under consideration decays fast enough to be amenable to the application of an FFT [@BDMC2011; @BDMC2013a; @BDMC2013b]. More often, however, it is necessary to waive the advantages of the FFT in favor of a more flexible sampling, specifically adapted to the problem. This, however requires to apply a discrete Fourier transform (DFT) with $\mathcal{O}(N^2)$ numerical complexity [@Num_recipes]. A combination of the best of both worlds, i.e. an $N \log_2{N}$ scaling on a logarithmic grid, which is able to cover all physically relevant orders of magnitude, has first been proposed by Haines and Jones in form of the logarithmic Fourier transform (LFT), which they have applied in a geophysical context [@LFT1988]. In its original form however, the LFT is only applicable under very restrictive assumptions on the properties of the function $f(\nu)$ under consideration (e.g. $f(0)=0$) and on the allowed range of the trade-off parameter, which is necessary to adjust the LFT according to the asymptotics of $f(\nu)$.\ The aim of this work is to present a generalized version of the logarithmic Fourier-Laplace transformation that in particular applies to functions with nonintegrable power-law tails. We give the corresponding definition in section \[sec:def\] and show how the original restrictions can be lifted to extend to generalized functions [@GelfandBook]. Moreover, in section \[sec:convergence\] we give a proof that the LFT converges exponentially fast in the number of grid points used for the numerical evaluation, provided the function satisfies certain analyticity conditions. Furthermore, we discuss how the theorem can be applied for practical purposes and in particular show that functions with algebraic tails are perfectly amenable to the LFT. In section \[sec:idealTOP\], we find an ideal set of the trade-off parameters, based on the asymptotic behavior of the input data and extend the excellent performance of the LFT to convolutions in section \[sec:Convo\]. In section \[sec:Ex\] we provide several classes of mathematical examples highlighting the advantages of LFTs over FFTs and discuss possible optimizations. Finally, we show in section \[sec:physEx\] how the LFT can be applied to typical multiscale problems in physics on the example of a density-density correlation function and a simple variant of mode-coupling theory. We conclude in section \[sec:con\]. Definition {#sec:def} ========== Mathematical Formulation ------------------------ Following the standard convention in the physics literature, we define the Fourier transform of a function $\hat{f}(t)$ in the time domain as $$\begin{aligned} \label{eq:FT} f(\nu)=\mathcal{F}(\hat{f})(\nu)=\int_{-\infty}^\infty dt \hat{f}(t)e^{i \nu t}\,,\end{aligned}$$ while the inverse transform to frequency $\nu$ is given by $$\begin{aligned} \label{eq:ILFT} \hat{f}(t)=\mathcal{F}^{-1}(f)(t)=\int_{-\infty}^\infty\frac{d\nu}{2\pi}f(\nu)e^{-i\nu t}\,,\end{aligned}$$ for both $f, \hat{f} \in L^1[\mathbb{R} , \mathbb{C}] $. In the following, we utilize the LFT to extend the set of argument functions to include certain distributions, the precise properties of which we state below. We introduce the logarithmic frequency and time coordinates $\omega$ and $\tau$ via $$\begin{aligned} \label{eq:IFT} \nu=\sigma \bar{\nu} e^{\omega}\quad\text{and}\quad t=\eta \bar{t} e^{\tau}\,,\end{aligned}$$ where $\sigma=\pm 1 = \eta$ are necessary to distinguish between the positive and negative real axis, while the prefactors $\bar{\nu}$ and $\bar{t}$ are required for dimensional purposes and will be set to unity in the remainder of this paper. With these definitions, the inverse Fourier transform can be written as a convolution for every $t\in\mathbb{R}$: $$\begin{aligned} \label{eq:convo} \begin{split} &\hat{f}(\eta \|t\|)= e^{-k \tau}\\ & \left. \times\!\!\sum_{\sigma=\pm 1}\! \int\! \frac{d\omega}{2\pi}f(\sigma e^{\omega})e^{k(\omega+\tau)-i \sigma\eta \exp{(\omega+\tau)}}e^{(1-k)\omega} \right\|_{\tau =\ln \|t\|}\!, \end{split}\end{aligned}$$ where $k \in \mathbb{R}$ denotes the trade-off parameter [@LFT1988]. By the help of the convolution theorem of Fourier analysis (see also Eq. below), the integral in can be reformulated in terms of the product of two Fourier transforms $$\begin{aligned} \label{eq:LFTpre} \hat{f}(\eta \|t\|)= \frac{e^{-k \tau}}{2\pi}\sum_{\sigma=\pm 1}\mathcal{F}_{s\to\tau}\left[\mathcal{F}_{\omega\to s}\left(f(\sigma e^{\omega})e^{(1-k)\omega}\right)(s)\mathcal{F}^{-1}_{x\to s}\left(e^{k x-i\sigma \eta \exp{(x)}}\right)(s)\right]\left(\tau=\ln\|t\|\right)\,,\end{aligned}$$ provided that $k$ is chosen such that each of the three Fourier integrals converges, the conditions for which we will detail now.\ Since we ultimately aim for a numerical implementation, the LFT can in general only be applied if $$\begin{aligned} F_\sigma(\omega):=f(\sigma e^{\omega})e^{(1-k)\omega} \in L^1\, , \end{aligned}$$ such that the Fourier transformation $$\begin{aligned} g_\sigma(s):=\mathcal{F}_{\omega\to s}\left(f(\sigma e^{\omega})e^{(1-k)\omega}\right)(s) \end{aligned}$$ exists in the integral sense of Eq. . Regarding the original function $f(\nu)$ this statement is equivalent to $$\begin{aligned} \label{eq:integrability} \int_{0}^{\infty} d\nu \left\|f(\sigma\nu)\right\| \|\nu\|^{-k} < \infty\,. \end{aligned}$$ In the particular case of a power-law behavior, i.e. $f(\nu)\to\nu^a$ for $\|\nu\|\to 0$ and $f(\nu)\to\nu^b$ for $\|\nu\|\to\infty$, the trade-off parameter has to be chosen according to $$\begin{aligned} \label{eq:condition} 1+b<k<1+a \,.\end{aligned}$$ As a result, for theses functions the LFT even admits a pole of $f$ located at the origin or a branch cut beginning just there, as well as nonintegrable, algebraically growing asymptotics, provided that they can be controlled by an appropriate value of $k$. Applying the definition of the $\Gamma$ function the $f$-independent inverse Fourier transform in Eq. can be formally rewritten as $$\begin{aligned} \begin{split} h_{\sigma\eta}(s):=&\mathcal{F}^{-1}_{x\to s}\left(e^{k x- i\eta \sigma \exp{(x)}}\right)(s)\\=&\frac{1}{2\pi} \left(i\sigma\eta\right )^{is-k}\Gamma(k-i s)\,, \end{split}\end{aligned}$$ for $k \in \mathbb{R}\setminus \mathbb{Z}_0^-$, where the exclusion of nonpositive integers is due to the poles of the Gamma function $\Gamma(k-is)$. We point out that this result has to be considered as the analytic continuation of the integral representation $$\begin{aligned} \int dx\; e^{k x -i \eta \sigma \exp\left(x\right)} e^{- i s x} =\left(i\sigma\eta\right )^{is-k}\Gamma(k-i s) \, \end{aligned}$$ that, indeed, only holds if $0<k<1$, as emphasized by Haines and Jones [@LFT1988].\ Finally, we have to consider the transformation $\mathcal{F}_{s \to \tau}(g_\sigma(s) h_{\sigma\eta}(s))$ from the auxiliary variable $s$ to $\tau$ in Eq. . Since in any practical implementation the factor $g_\sigma(s)$ will only be known in an approximate, discretized form, no analytic continuation can be applied and we have to demand that $g_\sigma \cdot h_{\sigma\eta} \in L^1$. Given the asymptotics of the product [@frei05book] $$\begin{aligned} \label{eq:asymptotics} \left\|\Gamma(k-is)(i\sigma \eta)^{is-k}\right\|\!\propto\! \begin{cases} \!\!\sqrt{2 \pi} \|s\|^{k-1/2} e^{- \pi \|s\|}\!\!\!\! & \sigma \eta s \to \infty \\ \!\!\sqrt{2 \pi} \|s\|^{k-1/2} \!\!\!\!& \sigma \eta s \to -\infty \end{cases}, \end{aligned}$$ we conclude that $g_\sigma \cdot h_{\sigma\eta} \in L^1$ requires $g_\sigma$ to satisfy $\lim_{\|s\| \to \infty}\|s\|^{k+1/2} g_\sigma(s)=0$. According to the lemma of Riemann-Lebesgue for differentiable functions [@koer89book], the latter condition is fulfilled if $F_\sigma(\omega)$ is at least $$\begin{aligned} n:=\max(0, \lceil k+1/2\rceil) \label{eq:n} \end{aligned}$$ times differentiable with the derivatives $F^{({{\color{blue}l}} )}_\sigma(\omega) \in L_1$, for $0\leq l \leq n$. With respect to the original function $f(\nu)$ this implies that $f^{(n)}(\nu)$ exists, while the integrability condition on $F^{(n)}(\omega)$ reduces to Eq. , as can be shown by partial integration. All in all, the logarithmic Fourier transform reads $$\begin{aligned} \hat{f}(\eta \|t\|)= \frac{e^{-k \tau}}{(2\pi)^2}\sum_{\sigma=\pm 1}\mathcal{F}_{s\to\tau}\left[\mathcal{F}_{\omega\to s}\left(f(\sigma e^{\omega})e^{(1-k)\omega}\right)(i\sigma\eta)^{is-k}\Gamma(k-i s)\right]\left(\tau=\ln\|t\|\right)\,, \label{eq:LFT}\end{aligned}$$ which can be applied with a given value of the trade-off parameter $k \in\mathbb{R} \setminus \mathbb{Z}_0^-$ to all functions $f(\nu)$, that satisfy the summability criterion and are $n$ times differentiable, with $n$ set by Eq. . The computation of the Fourier transformation $f(\sigma\|\nu\|)=\mathcal{F}(\hat f(t ))$ via the LFT follows analogously, with the same conditions on $\hat f(t )$. It yields the same result as in Eq. , yet, with the replacements $f \to \hat{f}$, $\sigma \leftrightarrow \eta$, $\tau \leftrightarrow \omega$ and $(i\sigma\eta)^{is-k}\to (-i\sigma\eta)^{is-k}$, as well as an additional factor of $2\pi $ on the right-hand side of Eq. . We remark that the LFT can readily be generalized to Fourier-Laplace transforms of the form $$\begin{aligned} \mathcal{FL}(f)(t)=\int_{-\infty}^\infty\frac{d\nu}{2\pi}f(\nu)\exp(e^{i\phi}\nu t)\,,\end{aligned}$$ with $\phi \in [0,2\pi[$. The result in still holds, merely with the factor $(i\sigma\eta)^{is-k}$ substituted by $(e^{i\phi}\sigma\eta)^{is-k}$. Half-sided transforms, which correspond to the standard definition in case of the Laplace transformation, are simply obtained by restricting the outermost sum of Eq. to $\sigma=1$. Clearly, in most applications Laplace transforms, that is $\phi=\pi$, involve quickly converging integrals. Therefore we will focus on the most critical case of Fourier transforms ($\phi=\pi/2$) and inverse Fourier transforms ($\phi=3\pi/2$), where the transformation kernel entails no exponential suppression of large arguments. Nonetheless, we stress, that even for exponentially decaying integrals the logarithmic transforms are orders of magnitude faster than equidistant grids as is highlighted by the trivial example $f(\nu)=e^{-\|\nu\|}$ in section \[sec:Ex\]. After having discussed the mathematical framework of the LFT let us briefly comment on the role of $k$ (see also Ref. [@LFT1988]). The term trade-off parameter refers to the fact that $k>0$ ($k<0$) suppresses both the integrand of the Fourier transformation $\mathcal{F}_{\omega \to s}$ in the definition (cf. also Eq. ) for large $\omega \to \infty$ (small $\omega \to -\infty$) and the result in the $\tau \to \infty$ ($\tau \to -\infty$) limit, which corresponds to $t \to \pm \infty$ ($t \to 0$), due to the overall prefactor. Simultaneously, the convergence in the opposite limits is diminished. This dependence on $k$ can be utilized to tune the properties of the LFT to suit the asymptotics of $f$.\ A different perspective on the LFT is opened by the interpretation of the trade-off parameter as a shift of the final integration over $s$ to a contour in the complex plane. The discussion of the LFT in terms of contour integrals, which are sensitive to the analytic structure of $f$, is crucial to understand the convergence properties of the LFT detailed in section \[sec:convergence\] Numerical implementation ------------------------ So far, we have only utilized exact analytical reformulations of the problem. However, upon introducing exponential grids with index $\;n\in\{1,2,...,N\}$ in both frequency and time $$\begin{aligned} \label{eq:grids} \begin{split} \nu_n&= e^{\omega_n} \quad \text{and} \quad \nu_{-n}=- e^{\omega_n} \quad\text{with} \quad\omega_n=\Delta \omega \left(n +\omega_s\right)\;\\ t_n&= e^{\tau_n} \quad \,\text{and} \,\quad t_{-n}=- e^{\tau_n}\,\quad \text{with}\,\quad \tau_n=\Delta\tau \left(n +\tau_s\right) \end{split}\end{aligned}$$ and discretizing the auxiliary space via $s_n=\Delta s( n + s_s)$, the usefulness of the form is immediately revealed: The equidistant grids in $\tau$ and $\omega$ make it possible to take advantage of the efficient FFT algorithm – even for Laplace transforms, where fast algorithms otherwise require more elaborate methods from approximation theory [@Rokhlin1988; @Strain1992] – while covering low frequencies and short times with a high density of points, as opposed to a reduced sampling density at large arguments. Since in many physical applications the high-energy or frequency range shows an algebraic behavior, this covering of the frequency (momentum) and time (position) domain will be very favorable under many circumstances. Important physical examples include generic correlation functions in frequency and momentum space, while in a critical theory algebraic tails appear in the position and time argument. For instance the momentum distribution $n(k)$ of ultracold Fermions in the vicinity of an open-channel dominated Feshbach resonance [@chin10Feshbach] obeys the Tan energy theorem [@Tan08energy]: For momenta $k$ that exceed any intrinsic inverse length scale $n(k)$ decays like $\mathcal C/k^4$, where $\mathcal C$ is the observable Tan contact density [@kuhn11contact; @hoin13contact]. On the other hand, the phase transition to the superfluid is signaled by an instability of the pair propagator in the low-momentum limit. In this system, the LFT has been applied to study the phase diagram in the presence of a finite spin imbalance [@Frank2018]. Furthermore, similar challenges arise in the efficient simulation of analog low/high pass filters [@Christensen1990], in the context of signal processing as well as in the numerical solution of differential equations [@Boyd2001].\ In addition to the convenient distribution of points, the grid acquires a high degree of flexibility as the step sizes $\Delta \omega, \Delta \tau$ and $\Delta s$ and the linear shifts $\omega_s$ and $\tau_s$, that play the role of the prefactors $\bar{t}$ and $\bar{\nu}$, together with $s_s$ can be chosen at will. This allows to adjust the method to the asymptotics of various functions as is shown in section \[sec:Ex\]. The standard choice for all the shifts is $-N/2$ in order to cover positive and negative exponents equally. We will return to the question of how to determine the ideal transformation parameters in section \[sec:idealTOP\]. Before continuing, we remark however, that functions with important features on intermediate scales which cannot be considered as part of the asymptotics of small or large arguments (not even by using the entire set of parameters available in Def. ), will yield no advantage over an ordinary FFT. Such cases appear for double-peak structures whose centers are too far apart to be scaled to the high grid-density at $\omega \to 0$ without including an impractically large $N$. Similarly, functions that oscillate uniformly on all scales with a fixed frequency $\bar{\omega}$ will be inevitably undersampled by the given grid at frequencies $\omega \gtrsim \ln(\bar{\omega}/\Delta\omega)$. Convergence properties {#sec:convergence} ====================== Theoretical perspective {#sec:proof} ----------------------- Now we address the issue of how efficiently a function $f(\nu)$ that obeys the properties stated below Eq. can be sampled and then Fourier transformed on the exponential grid. This requires to answer the question of how quickly the sum $$\begin{aligned} \label{eq:LFTsum} \begin{split} \hat{f}_N(t_{\eta n})=\; & e^{-k \tau_n}\!\!\sum_{\sigma=\pm 1}\! \sum_{l=1}^N\frac{\Delta s}{2\pi}e^{i s_l \tau_n}(i\sigma \eta )^{i s_l -k}\Gamma(k-is_l)\\ &\! \!\qquad \qquad\cdot\sum_{m=1}^N \frac{\Delta \omega}{2\pi} f(\sigma e^{\omega_m})e^{(1-k)\omega_m}e^{i\omega_m s_l} \, \end{split}\end{aligned}$$ representing the numerical, discrete approximation on the grids defined in Eq. converges towards the exact integral as $N \to \infty$. First of all, we note that Eq. indeed approaches the LFT from Eq. . To see this one has to consider the limits of the largest values $\|\omega_{\pm N}\|, \|s_{\pm N}\| \to \infty$ at vanishing stepsizes $\Delta \omega, \Delta s \to 0.$ Taking the latter limit yields well-defined integrals on finite intervals, since all terms represent measurable functions. In particular, the sum in the second line can be interpreted as Fourier coefficient $F_l$ of the periodic function $F_\sigma(\omega)$ with period $2N\Delta\omega$. The differentiability of $F_\sigma(\omega)$ then implies the asymptotic behavior $F_l \lesssim C s^{(n+1)}$ with a positive constant $C$ [@koer89book], such that the limit $\|\omega_{\pm N}\|, \|s_{\pm N}\| \to \infty$ exists and by its uniqueness we recover the definition of the LFT. Beyond the mere existence, we now show that under conditions satisfied in many relevant application, the LFT converges exponentially fast in the number of grid points.\ Theorem: Let $f(\sigma e^\omega)$ be a function that is analytic in a closed strip of width $R^{(1)}>0$ around $\bar{\mathbb{R}}$, i.e. the affinely extended real axis of the logarithmic argument $\omega$, and whose asymptotic behavior can be controlled by a suitable choice of the trade-off parameter $k$, such that $F_\sigma(\omega)\in\mathcal{S}(\mathbb{R})$, where $\mathcal{S}$ denotes the space of Schwartz functions [@GelfandBook]. Then the deviation of the approximation Eq. from the exact expression Eq. vanishes exponentially in the number of grid points $N$.\ Proof: The rate of this convergence will not depend on the exact values of the centers of the grids $\omega_s$, $\tau_s$ and $s_s$. To keep the notation simple, we will in the following assume them to be given by integers. Obviously, $ \mathcal{S} \subset L^1$ and the Schwartz functions satisfy the differentiability condition by definition. Furthermore, the integral $$\begin{aligned} \label{eq:I1} I^\sigma_1(s) = \int \frac{d\omega}{2\pi}f(\sigma e^{\omega})e^{(1-k)\omega}e^{i s\omega} \, ,\end{aligned}$$ is finite and itself a Schwartz function since the integrand is an element of $\mathcal{S}(\mathbb{R})$. By virtue of the Paley-Wiener theorem [@titc75book] $I^\sigma_1(s)$ is analytic in a strip around the real $s$-axis, whose width $R^{(2)}>0$ is determined by the asymptotic decrease of $F_\sigma(\omega)$, which at least is exponential. Furthermore, the truncation error due to the finite summation interval, scales like $\|F_\sigma(\omega_{\pm N})\|$ and thus in any case merely gives rise to exponential corrections. Therefore, we consider right away the infinite sum. The latter can be replaced by a contour integral around the imaginary axis in the mathematically positive direction, which reads [^1] $$\begin{aligned} \label{eq:S1} \begin{split} & S_1^\sigma(s) =\sum_{m \in \mathbb{Z}} \frac{\Delta\omega}{2\pi}f(\sigma e^{\omega_n})e^{(1-k)\omega_n}e^{i s\omega_n} \\ & = \begin{cases} \oint\frac{d z}{2\pi i}f(\sigma e^{-i z})(1+n_B(z))e^{-(1-k+is)iz} & \Re(s)<0 \\ \oint\frac{d z}{2\pi i}f(\sigma e^{-i z})(n_B(z))e^{-(1-k+is)iz} & \Re(s)>0 \end{cases} \,. \end{split}\end{aligned}$$ Here $n_B(z)=1/(\exp(2\pi z/\Delta\omega)-1)$ is the Bose-Einstein distribution with “inverse temperature” $\beta_\omega=2\pi/\Delta\omega$, whose simple poles at $\beta_\omega \omega_n$ make sure that one recovers the original series with the help of the residue theorem. Subtracting the exact integral $I_1^\sigma(s)$, which is also taken along the imaginary axis, one obtains for the difference $$\begin{aligned} \label{eq:E1} \begin{split} &E_1^\sigma(s)= S^\sigma_1(s)-I^\sigma_1(s)=\\& =\; \int_{R^{(1)}-i \infty}^{R^{(1)}-+ i \infty} \frac{d z}{2\pi i}f(\sigma e^{-i z})(n_B(z))e^{-(1-k+is)iz} \\ &+ \;\int_{-R^{(1)}+i \infty}^{-R^{(1)}- i \infty} \frac{d z}{2\pi i}f(\sigma e^{-i z})(1+n_B(z))e^{-(1-k+is)iz} \, , \end{split}\end{aligned}$$ where we have made use of Cauchy’s theorem to deform the integration contour such that it remains within boundary of the analytic domain of $f(\sigma \exp(\omega))$ (cf. Fig. \[fig:Boundary\]). ![(Color online) The original sum in $S_1^\sigma(s)$, evaluated at the green dots, is replaced by an integration contour along the imaginary axis (blue dashed line) and then shifted by a finite real part $\pm R^{(1)}$. The closest approach of a non-analyticity of $f(\sigma e^{-iz})$ – here symbolized by a red dot for a pole and a red zigzag line for a branch cut – to the imaginary axis determines $R^{(1)}$.[]{data-label="fig:Boundary"}](Path_Color.pdf){width="0.9\columnwidth"} Extracting the dominant exponential behavior we can write for the error $$\begin{aligned} \label{eq:E1final} E_1^\sigma(s)= e^{R^{(1)}(-\beta_\omega+\|s\|)} F^\sigma_1(s,R^{(1)})\, .\end{aligned}$$ The function $F_1$ arises from the remaining integrals and is the Fourier transformation of an integrable function, which in particular implies that the exponential prefactor indeed yields the leading asymptotic behavior for $\|s\| \to \infty$ due to the lemma of Riemann and Lebesgue. Moreover, by increasing $\beta_\omega \sim N$ the error becomes exponentially small in the number of grid points, provided $\|s\|<\beta_\omega$. From a theoretical perspective, we can take the limit $N \to \infty$ and achieve exponential convergence uniformly in $s$. Next, one has to investigate the convergence properties of the remaining sum over the auxiliary variable $s$ in . First, we focus on the properties of the exact integral $$\begin{aligned} \label{eq:I2} I_2^\sigma(\tau) = \int \frac{ds}{2 \pi} \Gamma(k-is)(i\sigma \eta)^{is-k} I_1^\sigma(s) e^{i s\tau} \, . \end{aligned}$$ The asymptotics of the product gives at most rise to an algebraic growth, while for $\|s\| \to \infty$ the integral $I_1^\sigma(s)\in\mathcal{S}$ decays exponentially fast, thus rendering $I_2^\sigma(\tau)$ well-defined. To estimate the error arising from replacing the analytic expression in Eq. by a discretized numerical approximation we consider the difference $$\begin{aligned} \label{eq:E2} \begin{split} E^\sigma_2(\tau)=&I^\sigma_2(\tau)\\&-\sum_{\|l\|\leq N} \frac{\Delta s}{2\pi}\Gamma(k-is_l)e^{i s_l \tau}(-i\sigma \eta)^{i s_l-k}S^\sigma_1(s_l) \, . \end{split}\end{aligned}$$ Since the difference $E_1^\sigma(s) = S_1^\sigma(s) -I_1^\sigma(s)$ becomes exponentially small with decreasing $\Delta\omega$, we can replace the sum by the exact integral $I_1^\sigma(s)$. Furthermore, the sum can be extended to $l \in \mathbb{Z}$ because the truncation to $\|l\| \leq N$ neglects only terms that are exponentially suppressed due to the asymptotics of $I_1^\sigma(s)$ that, provided $s_{\pm N}$ are large enough, determines the exponential tails of $S_1^\sigma(s)$. Then, the error $E_2^\sigma(\tau)$ can be treated analogously to Eq. in terms of complex contour integrals by introducing the Bose distribution $n_B(z)$, which now involves the inverse temperature $\beta_s=2\pi/\Delta s$. In this step we have to analytically continue the integral $I_1^\sigma(s \to z)$, which is in general not known in closed form for genuine complex arguments $z$. Yet, performing this continuation numerically is not required as we need the expression only on a formal level to determine the error. Like above, we shift the contour integrals into the complex plane to the fixed finite real parts $\pm R^{(2)}$ within the width of the analytic domain of $I_1^\sigma(z)$, but have to take into account the simple poles of the $\Gamma$ function located at the nonpositive integers enclosed by the modified contour. Altogether, we can estimate the error by $$\begin{aligned} \label{eq:E2new} E^\sigma_2(\tau)= e^{R^{(2)}(-\beta_s +\|\tau\|)}F_2^\sigma(\tau, R^{(2)})+E^\sigma_{2,\Gamma}(\tau)\, .\end{aligned}$$ Once again $F_2^\sigma(\tau, R^{(2)})$ is the Fourier transform of an integrable function and does not overcome the leading exponential such that the first term vanishes exponentially for all $\|s\|$ in the limit $\beta_s \sim 1/\Delta s \sim N\to\infty$. The second term summarizes the contributions from the residues $\text{Res}(\Gamma,-m) = (-1)^m/m!$ for $m \in \mathbb{N}_0$ and reads $$\begin{aligned} \label{eq:E2Gamma} \begin{split} &E^\sigma_{2 \Gamma}(\tau) = -\!\!\sum_{\substack{m=\lceil -k \rceil\\ \land m \geq 0}}^{\lfloor R^{(2)}-k\rfloor} \frac{(-1)^m}{m!} \frac{e^{(k+m)\tau}(i \sigma \eta)^m I_1^\sigma(-i(m+k))}{e^{\beta_s(k+m)}-1} \\ &\quad\;\; -\!\! \sum_{\substack{m=\lceil -R^{(2)}-k \rceil\\ \land m \geq 0}}^{\lfloor -k \rfloor} \frac{(-1)^m}{m!} \frac{e^{(k+m)\tau}(i\sigma \eta)^m I_1^\sigma(-i(k+m))}{1-e^{-\beta_s(k+m)}}\, . \end{split}\end{aligned}$$ Note that the Bose functions control the exponential function $\exp((k+m)\tau)$ via $\beta_s$ in analogy to the first term in Eq. . In total, we observe that both $E_1^\sigma$ and $E_2^\sigma$ decrease exponentially in the limit $\Delta_\omega,\Delta s \to 0$ for all $s$, while all intermediate steps do not violate this scaling. $\square$\ Note that for $\phi\in]\pi/2,3\pi/2[$, which includes the half-sided Laplace transform, convergence is even faster. This, however, takes its toll, when considering the inverse transform, where the exponential growth of $\|\Gamma(k-is)\|^{-1}\sim e^{\pi\|s\|/2}$ in the case of the Laplace transform severely magnifies errors and limits the useful interval in $s$ and thereby the attainable precision. For the special case of $k=1/2$, this has been analyzed in detail by Epstein and Schotland [@Epstein2008], who, given a noise $\delta$ on the input, also derive a bound on the maximum resolution $\Delta\omega$ and precision $\epsilon$ of the numeric inverse Laplace transform: $$\begin{aligned} \label{eq:laplace} 1\leq\frac{2\Delta\omega}{\pi^2}\ln{\left(\sqrt{2\pi}\frac{\epsilon}{\delta}\right)}\;.\end{aligned}$$ As we will see below, this bound can be significantly improved by the use of the theorem in Sec. \[sec:proof\] in combination with the knowledge of $R^{(1)}$. Practical aspects ----------------- Regarding the implementation of the LFT for practical purposes, it is helpful to relate the analytic properties of the function $f$ in the logarithmic argument $\omega$ to the original argument $\nu$, since $f(\nu)$ corresponds to the form that is given in most applications. From this we find bounds for $R^{(1)}$ for some relevant classes of functions. First of all, the analyticity of $f(\sigma e^\omega)$ implies that $f(\nu)$ is also analytic for $\nu \in \mathbb{R}\setminus\{0\}$. Moreover, the exponential decay of $F_\sigma(\omega\to\pm\infty)$ requires that a constant $c_\sigma>0$ exists, such that $$\begin{aligned} \label{eq:ana_asympt} \left\|f(\sigma\|\nu\|)f\left(\frac{\sigma}{\|\nu\|}\right)\right\|\underset{\nu\to 0}{<}\|\nu\|^{c_\sigma}\;.\end{aligned}$$ To connect the analyticity properties of $f$ with respect to the variables $\nu$ and $\omega$, we first note that the analytic strip of width $R^{(1)}$ in $\omega$ is equivalent to the statement that for each $\omega_0 \in \bar{\mathbb{R}}$ there is an $R^\sigma_{\omega_0}>0$, such that the Taylor series of $f(\sigma e^\omega)$ $$\begin{aligned} \label{eq:Taylor} f(\sigma e^\omega)= \sum_{n=0}^{\infty} \left(\frac{d^n}{d \omega^n} f(\sigma e^\omega)\right)_{\omega_0} \frac{(\omega-\omega_0)^n}{n!}\end{aligned}$$ converges for all $\|\omega-\omega_0\| \leq R^\sigma_{\omega_0}$. The width of the strip is given by $$\begin{aligned} \label{eq:R1} R^{(1)} = \inf_{\substack{\omega_0 \in \bar{\mathbb{R}}\\ \sigma= \pm 1}} R_{\omega_0}^\sigma\, . \end{aligned}$$ To estimate the minimal size of the analytic domain of $f(\nu)$, we first determine the image of the line $\omega_0 + i \lambda R^{(1)} $, with $\lambda \in [-1,1]$ centered around the real $\omega_0$ under the mapping . This yields circular sectors of radius $\nu_0=\exp{\omega_0}$ that are symmetric around the real $\sigma \nu$ half-axis and centered at $\nu=0$. The half opening angle is given by $\min(\pi/2,R^{(1)})$. The restriction of the angle arises from the separation of the variable $\nu$ with respect to $\sigma = \text{sign}( \Re(\nu))$. Taking the union over all $\omega_0 \in \mathbb{R}$, which yields the domain of analyticity of $f(\nu)$ gives rise to an infinite cone $\|\Im(\nu)\|<\tan{(\theta)}\|\Re(\nu)\|$ with opening angle $$\begin{aligned} \theta(R^{(1)}) = \min(\pi/2,R^{(1)}) >0 \, . \end{aligned}$$ Note that for asymptotically large linear-frequency arguments the required width in $\nu$ space grows linearly, but admits nonanalyticities close to the origin. This is to be expected since nonsmooth variations of $f(\nu)$ that are related to nonanalyticities are very well captured by the exponentially dense grid in the vicinity of the origin. On the other hand, the same behavior at large frequencies will cause deteriorated numerical results due to severe undersampling of sharp features.\ In view of these arguments it becomes apparent that algebraic functions with the asymptotic behavior $f(\nu \to 0)\sim \nu^a$ and $f(\nu \to \infty)\sim \nu^b$ with $a>b$ are ideal candidates for the application of the LFT, since they trivially satisfy condition and the transformation $\nu \to \sigma \exp(\omega)$ removes any nonanalyticity located at the origin, which arises from any $a \in \mathbb{R}\setminus \mathbb{N}_0$. We recall that the above considerations do not make any reference to the integrability properties of $f(\nu)$, which in case of algebraic functions can be controlled by the trade-off parameter. Another class of asymptotic behavior is given by exponential functions, which we discuss with the help of the simple example of a single, dominant exponent $f(\nu) \sim \exp(\alpha (\sigma \nu)^c)$ for $\sigma \nu \to 0, \infty$, with $c \in \mathbb{R} \setminus \{0\}$ and $\alpha \in \mathbb{C}\setminus\{0\}$ to include oscillating functions. The prefactor is allowed to contain an arbitrary algebraic function to which the problem would be reduced for $\alpha=0$ or $c=0$. After the mapping to $\omega = \Re \omega + i \Im \omega$ we have $$\begin{aligned} \left\|e^{\alpha(\sigma \nu )^c}\right\| = \left\| e^{\exp(c\, \Re \omega )\left(\Re \alpha \cos(c\, \Im \omega )- \Im \alpha \sin(c\,\Im \omega)\right) }\right\| \, . \end{aligned}$$ For real $\omega$ the limit $c\, \Re \omega \to \infty$ requires $\Re \alpha <0$, since otherwise the inner Fourier transform $\mathcal F_{\omega \to s}$ of the LFT in Eq. is not defined. Note that the trade-off parameter cannot be used to remedy the super-exponential growth for $\Re \alpha >0$, which is also beyond the scope of the notion of generalized Fourier transformations [@GelfandBook]. Furthermore, we observe that even if $\Re \alpha <0$, the boundary of the analytic strip cannot overcome the constraint $$\begin{aligned} R^{(1)} < \left\| \frac{1}{c} \arctan \left(\frac{\Re \alpha}{\Im \alpha}\right) \right\| \end{aligned}$$ because $f(\sigma e^\omega)$ diverges exponentially for $\Im \omega$ beyond that value if $c\, \Re \omega \to \infty$. Considering an oscillatory example $f(\nu) \sim e^{i \nu}$, we obtain $R^{(1)}=0$, irrespective of an integrable algebraic prefactor. Therefore, the convergence of the LFT is degraded to an algebraic one on a fundamental level as mentioned at the end of Sec. \[sec:def\].\ More generally the width of the analytic strip $R^{(1)}$ can be determined from the positions of singularities of $f(\nu)$ in the complex plane. Modeling $f(\nu)$ in the vicinity of a nonanalyticity at $\hat \nu = \nu_0 \pm i R^\sigma_{\nu_0}$, where $\nu_0 \in \mathbb{R} \setminus \{0\}$, $\sigma = \text{sgn}(\nu_0)$ and $R^\sigma_{\nu_0} >0$, in the form $$\begin{aligned} f(\nu) = \lambda (\nu-\nu_0 \mp i R^\sigma_{\nu_0})^\alpha \, , \end{aligned}$$ with $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{R} \setminus \mathbb{N}_0$, we find for the $k^{\text{th}}$ derivative $$\begin{aligned} \|f^{(k)}(\nu_0)\| = \|\lambda\| \prod_{j=0}^{k-1}(\alpha-j) \|R^\sigma_{\nu_0}\|^{\alpha-k} \, . \end{aligned}$$ This form can be used to extract $R_{\nu_0}^\sigma$ and therefore $R^{(1)}$ and is accessible even for numerical data via finite difference approximations.\ Regarding the application of the LFT in practice, we note that it can be implemented with any existing library of FFTs. However, one additionally has to evaluate the Gamma function for complex arguments. Fortunately, these do not depend on $f$ and can be tabulated if several transformations have to be performed. In any case, all modern programming languages include fast algorithms (e.g. Spouge’s approximation [@spou94]) to compute $\Gamma$ and the LFT will, therefore, not be severely slowed down compared to an FFT with the same number of data points. Optimal parameter choices {#sec:idealTOP} ========================= So far, we have shown that the LFT converges exponentially fast towards the exact Fourier transform if the analytic structure of $f$ satisfies the conditions of the above theorem. However, we have not yet made any rigorous statements regarding how many points have to be used to reach the desired precision or how the trade-off parameters should be adjusted to obtain an optimal convergence. As we have seen in section \[sec:proof\], the differences between the exact Fourier transform and the LFT approximant are well known, such that generic statements about the ideal parameter choices, as well as significant improvements to the results, can be made.\ In the following, we will focus mainly on algebraic functions, whose convergence properties can be influenced by the trade-off parameter in contrast to exponential functions. First, we recall that the only restriction on $k$ is given by the convergence of the inner integral in the LFT , thus the trade-off parameter has to be chosen according to $1+b<k<1+a$ and $\notin \mathbb Z_0^-$. From a numerical perspective one has to keep in mind that $k$ has to avoid nonpositive integers by a finite margin to bypass the poles of the $\Gamma$ function. In practice, a deviation of 0.01 turns out to be sufficient. Apart from this constraint it is desirable to choose $k$ close to $k_{\text{opt}}=1+(a+b)/2$ that symmetrizes the asymptotic behavior of the integrand in $I^\sigma_1(s)$ (see Eq. ) on both ends of the $\omega$-interval, such that the smallest truncation errors are achieved for the standard value for the centers of the grids (see Eq. ). With this trade-off parameter, using $N$ data points, a truncation error no larger than $\epsilon$ requires $$\begin{aligned} \label{eq:cond1} \Delta\omega=\frac{4}{(b-a)N}\ln(\epsilon)\,.\end{aligned}$$ As argued below equation the decay of the integrand in is dominated by $S_1^\sigma(s)$, the asymptotic behavior of which can be estimated from the contour integral , which gives rise to the asymptotic behavior $S_1^\sigma(s)\sim e^{-R^{(1)}\|s\|}$. However, approximately at $s_\text{max}=\pi/\Delta\omega$ this function drops below the error $E_1^\sigma(s)$ from Eq. . To compute the remaining Fourier transform with truncation errors that are consistent with the previous steps one consequently demands, that $S_1^\sigma(s_\text{max})<\epsilon$ from which one concludes $$\begin{aligned} \Delta\omega=-\frac{\pi R^{(1)}}{\ln(\epsilon)}\,.\end{aligned}$$ Together with this fixes the lower bound of points necessary for an absolute accuracy of roughly $\epsilon e^{-k_\text{opt}\tau}=\epsilon e^{-(1+(a+b)/2)\tau}$ to $$\begin{aligned} \label{eq:minN} N=\frac{4}{\left(a-b\right)\pi R^{(1)}}\ln^2(\epsilon)\,.\end{aligned}$$ Remarkably this scales only logarithmically with the desired precision. We emphasize that this statement holds in general, even if the optimal choice of the trade-off parameter is prohibited. In this case only the prefactor increases. However, one might suspect that $N$ will be drastically increased by the requirement $\Delta s \ll 1$ in order to control the error $E_2^\sigma(\tau)$, according to equation . Since the closest non-analyticity of $S_1^\sigma(z)$ to the real axis appears at a distance $R^{(2)} \simeq\min\{k-1-b,1+a-k\}$, we infer from the asymptotics $I_2^\sigma(\tau) \sim \exp(-R^{(2)}\tau)$ and $E_2^\sigma(\tau)\sim \exp(-(\beta_s + \|\tau\|)R^{(2)})$ that it makes only sense to include values $ \|\tau_n\| \leq \tau_{\max} = \pi/\Delta s =\beta_s/2$.Therefore, $\Delta s \approx (b-a)\pi/\ln{(\epsilon)}$ is possible.\ In addition, the error $E_{2\Gamma}^{\sigma}(\tau)$ due to the proximity between the integration contour and poles of the $\Gamma$ function does not influence $\Delta s$ and apart from a prefactor their $\tau$-dependence is exactly known, as can be seen from Eq. . Thus, if the numerical error in the time domain is dominated by these contributions one can fit the residues in $E^\sigma_{2 \Gamma}$ to the limits $\tau \to \tau_{\pm N}$. In these regimes only numerical noise remains, because by virtue of the lemma by Riemann and Lebesgue the exact function $I_2^\sigma(\tau)$ has decreased below the desired precision threshold. This procedure works particularly well, since the exponential terms are known exactly (see also the examples in section \[sec:Ex\]). Moreover, depending on $k$, only a few dominant terms have to be subtracted, while the remaining terms are negligible due to their strictly monotonically decreasing exponents. Note that this discussion does not include round-off errors. These will give rise to an additional limitation of the attainable precision, since the finite accuracy of the internal numerical operations sets a bound to the possible precision of the LFT and in particular determines how well the decay of $I_2^\sigma(\tau)$ for $\|\tau\| \to \infty$ can be resolved. Furthermore, the final multiplication with $e^{-k\tau}$ also affects the error estimate. For negative values of $k$ exceptional precision can be achieved in the regime of small $\tau$. For strongly negative values of $k$, however, these come at the price of enhanced errors at large arguments. As long as round-off errors are ignored, these are a minor problem, since the final Fourier transform in will typically decay to zero at large arguments, which allows to remove the previously discussed systematic errors from $E_{2\Gamma}^\sigma$ (cf. section \[sec:Ex\]). In practice, round-off errors, unfortunately, dominate $I^\sigma_2(\tau)$ at large arguments, which sets a lower boundary to the useful interval of the trade-off parameter. Positive values of $k$ on the other hand, which are necessary to treat non-integrable functions, that is those with $b>-1$, will result in undesirably enhanced errors near the origin in the image space (here $t$). Removal of these errors will typically involve fitting the asymptotic behavior to the numerical data within the range of $t$-values, where $\hat{f}(t)>\epsilon e^{-k\tau}$ can be satisfied and extrapolating it towards $t \to 0$. The dynamical compression, that is the diminishing length of this $t$-interval as $k$ increases, is the price to pay for numerically Fourier transforming non-integrable functions.\ If more than just the minimal number of data points necessary for a given precision are available, one can use them to compensate the enhanced truncation errors and perform several transformations with different trade-off parameters. Larger values of $k$ increase precision for $\tau>0$, while smaller values enhance the accuracy at negative $\tau$. If $k$ can be varied over a wide interval without creating too large truncation errors, this procedure can, for example, be used to mitigate the impact of dynamical compression (see section \[sec:Ex\]). To improve the results further one can use $\omega_s$ to shift the list of $\omega_n$ points to optimally sample the asymptotics of the function. For an arbitrary value of $k$ that is compatible with the convergence requirements the condition $F_\sigma(\omega_{\pm N})<\epsilon$ translates to $$\begin{aligned} \begin{split} \omega_s & = \frac{\ln(\epsilon)}{\left(a+1-k\right)\Delta \omega} \\ N & = -\frac{a-b}{\left(b+1-k\right)\left(a+1-k\right) \pi R^{(1)}}\ln^2(\epsilon)\, , \end{split} \end{aligned}$$ which reduces to equation and $\omega_s=-N/2$ if $k=k_\text{opt}$. Similarly, $s_s$ can be used to find the best distribution of the auxiliary space points $s_n$.\ Finally, if a precision close to the round-off limit is required, $e^{-k \tau}$ cannot be orders of magnitude larger than $\hat{f}(\eta e^\tau)$ but instead should stay as close as possible to $-\ln(\hat{f}(\eta e^\tau))/\tau$ in the range of $\tau$ arguments of interest.\ In case of the inverse Laplace transform $\hat{f}(t)\to f(\nu)$ we can use the same procedure for the optimization of the transformation parameters on an input with multiplicative noise of amplitude $\delta$. Given a good estimate of $R^{(1)}$, which equals the width of the analytic strip of $\hat{f}(e^{\tau})$ minus $\pi/2$, this allows us to enhance the relative precision of the result near $\nu=1$ to $\delta^{R^{(1)}/(\pi/2+R^{(1)})}$, where, as opposed to Eq. , no constraint on $\Delta\omega$ is required. Note that only for $R^{(1)}\ll 1$ one is struck by the dreaded exponential enhancement of errors. LFT Convolutions {#sec:Convo} ================ One of the most important applications of Fourier transforms in theoretical physics relies on the efficient calculation of convolutions. Due to the convolution theorem $$\begin{aligned} \label{eq:convTheorem} \int\frac{d\nu'}{2\pi}f(\nu')g(\nu-\nu')=\mathcal{F}(\hat{f}(t)\hat{g}(t))(\nu)\end{aligned}$$ the computational complexity $\propto N^2$ of the direct discretized evaluation of the integral can be reduced to $\propto N \log_2(N)$ when using the FFT algorithm. There are, however, many situations, where the convolution theorem may not be utilized. For example, if one of the factors $f( \nu)$ or $g( \nu)$ decays too slowly to be integrable (that is no faster than $1/\nu$), its Fourier transform can no longer be understood as an integral and the identity cannot be used to improve performance, since the FFT will fail to correctly determine either $\hat{f}(t)$ or $\hat{g}(t)$. Despite these complications, the convolution may still be defined as an ordinary integral (at least as long as the product $f( \nu)g( \nu)$ decays faster than $1/\nu$) and numerical evaluation is cumbersome but straightforward.\ Here the LFT really excels. On the one hand, it can be evaluated much faster than any direct evaluation of the convolution (even if performed on an optimized grid) due to its superior scaling in the number of data points. On the other, it is able to cope with non-integrable functions as long as a suitable trade-off parameter exists. In other words, the LFT is much less plagued by convergence problems than the FFT. Slowly decaying functions, as we have already pointed out in the last section, can be numerically transformed with the LFT at the price of dynamical compression that inevitably reduces the signal-to-noise contrast at small arguments in the image space. However, for convolutions this problem is slightly less pronounced: If possible, setting the trade-off parameter of the final $\tau \to \omega$ back-transform in to $k_\text{back}=1-k_1-k_2$, where $k_1$ and $k_2$ are the optimized parameters for the transforms of $f(\eta e^{\omega})$ and $g(\eta e^{\omega})$, respectively, renders the result unaffected by any dynamical compression in $\tau$ because the noise level in the time domain stays constant at $\mathcal{O}{(\epsilon)}$, as the problematic exponential prefactors cancel. In the exotic case of strongly divergent integrals, this value of $k_\text{back}$ may not be useful if $e^{k_\text{back}\omega_\text{max}}$ becomes much larger than the expected result, in which case more data points and the less aggressive, symmetric choice $k_{1}=k_{2}=k_\text{back}$ are typically better suited (see last example in \[sec:Ex\]). Examples and optimizations {#sec:Ex} ========================== To benchmark the LFT and to illustrate the role of the transformation parameters, in particular of $k$, we compute the Fourier transforms for several examples and compare the results to the exact solutions. Without loss of generality, we focus on functions that are centered around the origin and that vary on a characteristic scale of unity. Deviations from that behavior can be remedied by preprocessing the function with a variable transformation which combines a shift of the original argument followed by rescaling it with a proper $\bar\nu$.\ Let us begin with a benign example, a Lorentzian curve $$\begin{aligned} \label{eq:ex1} f(\nu)=\frac{1}{1+\nu^2}\,,\end{aligned}$$ which could also be transformed with an ordinary fast Fourier transform. However, the slow convergence of the integral implies that reaching a global precision of $\epsilon = 10^{-12}$ with the FFT requires roughly $10^{13}$ data points, which exceeds numerical feasibility by several orders of magnitude. In contrast, to achieve the same accuracy with the LFT only a little more than 300 points suffice, as can be deduced from Eq. with $R^{(1)}=\pi/2$. Indeed, Fig. \[fig:1\] has been obtained with 360 points and the (quasi-)optimal parameter $k=-1/100$, since $k_{\text{opt}}=0$ is prohibited by the $\Gamma$ function. Setting up the LFT in this way, the precision is no longer limited by the finite resolution, but by double-precision floating point arithmetic and error-propagation therein. In addition, the interval in $t$ can be chosen arbitrarily by adjusting $\Delta\tau$ and $\tau_s$ with no influence on the error level. As discussed in Sec. \[sec:idealTOP\], due to the proximity to the pole of the $\Gamma$ function the last point $\hat{f}(t_\text{max})$ has to be subtracted which corresponds to the constant that arises from the leading $m=0$ contribution to $\exp(-k \tau)E_{2 \Gamma}^\sigma(\tau)$, see Eq. . The same procedure, which amounts to nothing else than a trivial subtraction of a one-parameter fit has been used for all other plots (except Fig. \[fig:conv1\]) as well.\ ![(Color online) Fourier transform of $f(\nu)=\frac{1}{1+\nu^2}$ with $\Delta\omega=\Delta\tau=1/6$, $\Delta s=1/10$, $k=-1/100$ and symmetric intervals $\omega_s=s_s=\tau_s=-N/2$ on $N=360$ data points. The red line represents the analytical result $e^{-\|t\|}/(2\pi)$, while the numerical data is shown in black and the difference between the two in blue.[]{data-label="fig:1"}](fig1.pdf){width="0.9\columnwidth"} A significantly more demanding example (on the branch where $\sqrt{-1}=i$) is given by the function $$\begin{aligned} f(\nu)=\frac{\sqrt{-\nu}}{\nu+i}\,,\end{aligned}$$ whose Fourier transform has to be understood in the sense of tempered distributions. For positive arguments $t>0$ it reads $$\begin{aligned} \hat{f}(t)=\frac{(1-i)}{\sqrt{2}}e^{-t}\,.\end{aligned}$$ According to section \[sec:idealTOP\], the optimized trade-off parameter is close to $k=1$, which removes the divergent behavior from the numerical integrals at the price of reduced precision at very small values of $t$. To demonstrate how the ideal choice of $k$ might depend on the data range of interest in the image space, Fig. \[fig:2\] depicts the result for the ideal trade-off parameter $k=1.01$ and the suboptimal $k=0.71$. In order to achieve errors of $10^{-12}$ at $\tau=0$ in both cases, which requires roughly $N=600$ in the optimized setting, the grid size has been increased to $N=1000$. In agreement with the general discussion, values of $k$ larger than $1$ reduce errors at large arguments, while those smaller than unity increase precision close to $t=0$.\ ![(Color online) Numerical Fourier transform of $f(\nu)=\frac{\sqrt{-\nu}}{\nu+i}$ (black line) with $N=1000$ points, $\Delta\omega=1/5$, $\Delta s=2/45$, $\Delta\tau=1/20$, $s_s=\tau_s=-N/2$ and two different trade-off parameters. In green the difference between the exact analytical function $\hat{f}(t)$ (red line) and the numerical result with $k=1.01$ and $\omega_s=-N/2$ is shown, while the blue line depicts the same error but for the less aggressive $k=0.71$ and $\omega_s=-200$, which leads to smaller errors in the limit $t \to 0$ but to enhanced noise for $t \to \infty$, as discussed in the main text.[]{data-label="fig:2"}](fig2b.pdf){width="0.9\columnwidth"} The next example shows $$\begin{aligned} f(\nu)=\ln(\nu^2+1)\,,\end{aligned}$$ which transforms into $$\begin{aligned} \hat{f}(t)=\frac{e^{-\|t\|}}{\|t\|}\end{aligned}$$ and is again correctly described by the LFT on only 560 points (see Fig. \[fig:3\]). However, the divergence of $f(\nu)$ as $\nu \to \infty$ results in an even stronger dynamical suppression than before. The well-behaved function $f(\nu)=e^{-\|\nu\|}$, which in fact corresponds to the inverse transformation of the very first example , could also be treated by an FFT. However, the same accuracy as demonstrated in Fig. \[fig:4\] with $N=480$ on the logarithmic grid would require more than a million data points on a linear grid, illustrating that even for the most benign functions the LFT can outperform the direct application of an FFT. ![(Color online) Fourier transform of $f(\nu)=\ln(\nu^2+1)$ with $N=560$, $\Delta\omega=1/7$, $\Delta s=1/14$, $\Delta \tau=1/21$, $k=2.05$ and $\omega_s=s_s=\tau_s=-N/2$. Color coding is the same as in Fig. \[fig:1\].[]{data-label="fig:3"}](fig3.pdf){width="0.9\columnwidth"} ![(Color online) Fourier transform of $f(\nu)=e^{-\|\nu\|}$ with $N=480$, $\Delta\omega=1/15$, $\Delta s=2/21$, $\Delta \tau=1/12$, $k=-3/10$, $\omega_s=-420$ and $s_s=\tau_s=-N/2$. Color coding is the same as in Fig. \[fig:1\].[]{data-label="fig:4"}](fig4.pdf){width="0.9\columnwidth"} The convolution of $$\begin{aligned} f(\nu)=\frac{1}{-i+\nu}\end{aligned}$$ with itself, that appears frequently in the evaluation of Feynman diagrams with non-relativistic propagators [@abri75; @fett71], cannot be treated by FFTs, as the integral over $f(\nu)$ does not exist. Using the LFT remedies this issue by the help of the trade-off parameter. For instance, using the optimized value of $k$ given in section \[sec:idealTOP\], a constant error of roughly $10^{-12}$, which is limited only by the internal floating point precision, is obtained with only $N=560$ points. In Fig. \[fig:conv1\] the two leading contributions to $E^\sigma_{2 \Gamma}$ from Eq. for the transformation from $t$ to $\nu$ with $m=0,1$ were subtracted by fitting the two corresponding parameters $I_1^\sigma(-i(k+m))$ to the high-frequency range near $\nu=10^{28}$. ![(Color online) Convolution of $f(\nu)=\frac{1}{\nu-i}$ with itself, here $N=560$, $\Delta\omega=1/4$, $\Delta s=5/76$, $\Delta \tau=1/8$, $k_a=k_b=0.51$, $k_\text{back}=-0.02$, $\tau_s=-440$ and $s_s=\omega_s=-N/2$ were used. Color coding is the same as in Fig. \[fig:1\].[]{data-label="fig:conv1"}](conv2.pdf){width="0.9\columnwidth"} ![(Color online) Convolution of $f(\nu)=\ln(\nu^2+1)$ with itself, here $N=560$, $\Delta\omega=1/7$, $\Delta s=1/14$, $\Delta \tau=1/14$, $k_a=k_b=k_\text{back}=8/5$, $\omega_s=-154$, $\tau_s=-495.6$ and $s_s=-N/2$ were used. Color coding is the same as in Fig. \[fig:1\].[]{data-label="fig:conv2"}](conv1.pdf){width="0.9\columnwidth"} Encouraged by these results, one can try to convolve some more exotic functions, for example $f(\nu)=\ln(\nu^2+1)$ with itself. In this case neither the convolution as an integral, nor the product of the distributions in the time domain is in general well-defined [@GelfandBook]. Nevertheless, employing a cutoff $e^{-\delta \|\nu\|}$, with $\delta >0$, in the logarithmic frequency space and sending $\delta$ to zero at the end one finds analytically $$\begin{aligned} (f\star f)(\nu)=2\ln\left(1+\frac{\nu^2}{4}\right)-2\nu \arctan{\frac{\nu}{2}}\, .\end{aligned}$$ As Fig. \[fig:conv2\] demonstrates, this result is again very accurately recovered by means of an LFT with $N=560$. Here, due to the large frequency interval used, round-off errors multiplied by $e^{-k_\text{back}\omega}$ are the limiting factor at large $\nu$. This affects the choice of trade-off parameters for the forward and backward LFTs, which is expected on general grounds, as remarked at the end of section \[sec:Convo\] on convolutions. Furthermore, the first sub-leading divergence $\propto e^{2\omega}$ due to the next pole $(m=2)$ of the $\Gamma$ function beyond the constant has been fitted with a single parameter against the raw result at $\omega_N$ and subtracted. Application to physical examples {#sec:physEx} ================================ Following the purely mathematical discussion, we now provide physical examples to highlight the real-world advantages of the LFT. These demonstrations are deliberately chosen to be simple, yet of relevance to current research and with apparent generalizations to more challenging problems. Polarization function --------------------- The polarization function of the one-dimensional Bose gas is given by [@Mahan_book; @Kamenev_book] $$\begin{aligned} \label{eq:Lindhard} \Pi(\omega,q)=-\int\frac{dk}{2\pi}\frac{n_B(\xi_k)-n_B(\xi_{k-q})}{\omega-\xi_k+\xi_{k-q}+i0^{+}}\end{aligned}$$ with $n_B(k)=1/(e^{\beta \xi_k}-1)$ the Bose-Einstein distribution for the inverse temperature $\beta=1/(k_B T)=1$ and the free dispersion $\xi_k=k^2/(2m)-\mu$ with momentum $k$, mass $m=1/2$ and chemical potential $\mu$. $\Pi(\omega,q)$ describes density-density correlations and in general has no known closed expression. One therefore has to rely on a numerical evaluation of the integral. In case of the fugacity $z=\exp{(\beta\mu)}$ approaching unity from below, the density fluctuations in the regime of long wave lengths proliferate, which is reflected in the singular behavior $n_B(0)\sim-1/(\beta\mu)$. As a consequence, the direct evaluation of the polarization function becomes numerically expensive. However, precisely these low temperature correlation functions are a common ingredient in quantum many-body theories, in particular: quantum critical transport [@sach11], response near phase transitions in ultracold atoms [@endr12] and Bose gases in optical cavities [@rits13]. In the following, we will show that an efficient evaluation of $\Pi(\omega,q)$ with unrivaled precision is possible by means of the LFT. The integral in Eq. can be rewritten as a convolution, which allows for an efficient treatment that requires only two one-dimensional (half-sided) Fourier transforms: $$\begin{aligned} \!\Pi(\omega,q)&=\sum_{\eta=\pm}\frac{i\eta}{2q}\sigma\!\left(\frac{\omega+\eta q^2}{2q}\right)\\ \sigma(y)&=\mathcal{F}^{-1}_{x\to y}\left[\theta(-x)\mathcal{F}_{k\to x}\left(n_B(k)\right)(x)\right](y)\,.\label{eq:Lindhard_Fourier}\end{aligned}$$ The computation of the polarization function on a two-dimensional $(\omega,q)$-grid of size $N\times N$ therefore requires only $\mathcal{O}(N^2)$ operations for the evaluation of $\sigma(y(\omega,q))$ in contrast to $\mathcal{O}(N^3)$ for the direct approach. In fact, the actual fast Fourier transform in Eq. results only in subleading corrections to the overall complexity. The main advantage of the LFT over other Fourier transforms, however, lies in its accuracy. We highlight this in Fig. \[fig:lindhard\] by comparing the absolute error obtained for an ordinary FFT, the LFT and a discrete Fourier transform (DFT) in combination with a cubic spline interpolation. All algorithms use the same number of data points as well as optimized transformation parameters, with the DFT and LFT operating on a common grid. For all relevant values the LFT outperforms the other methods by several orders of magnitude, furthermore, in contrast to the FFT a much larger interval can be sampled. ![(Color online) Comparison of the absolute error in the evaluation of $\sigma(y)$ with $\mu=-1/10$ using an FFT (red) a DFT following a cubic spline interpolation (blue) and the LFT (green). For comparison we also show $\sigma(y)$ in black. The parameters of the LFT are $N=336$, $\Delta\omega=1/14$, $\Delta s=1/4$, $\Delta \tau=1/11$, $k=-9/10$, $\omega_s=-196$, $\tau_s=-234$ and $s_s=-N/2$ for the first and the same values, except $\Delta s=1/6$ and $k=1/20$, for the second transform. The same grid is then also used for the DFT. The FFT is run on a grid with the same number of points and a lattice spacing $\delta x\approx 0.07$ and $\delta y\approx 0.01$ for $x$ and $y$ in Eq. .[]{data-label="fig:lindhard"}](lindhard.pdf){width="0.9\columnwidth"} Glass transition ---------------- The glassy, mechanically rigid state of amorphous materials and its realization by supercooling liquids has been studied for a very long time [@debe01]. Nevertheless, a theoretical description of this state is difficult since one has to deal with density fluctuations on various length scales and very slow relaxation processes, as observed in experiments [@stef94]. One theoretical approach to this problem is mode coupling theory [@reic05; @goet08] which is based on an effective equation of motion for the dynamical structure factor $S(\mathbf k ,t)$. In the following, we illustrate how the LFT, which by construction is capable of dealing with multi-scale problems, can be applied to these kinds of models. Here we focus on a simplified variant of mode-coupling theory due to Leutheusser [@leut84] and Bengtzelius et al. [@beng84] and remark on the advantageous properties of the LFT for more generic problems of this kind. To study the relaxation of density distortions one introduces an effective temporal order-parameter $\Phi(t)$, whose phenomenological time evolution is given by [@leut84] $$\begin{aligned} \label{eq:EOM} \ddot{\Phi}(t) + \gamma \dot{\Phi}(t) + \Omega_0^2 \Phi(t) =- 4 \Omega_0^2 \lambda \int_0^{t} d\tau \; \Phi^2(\tau) \dot{\Phi}(\tau- t) \, .\end{aligned}$$ The left side of the equation is a simple harmonic oscillator with damping rate $\gamma$ and frequency $\Omega_0$. The correlations responsible for the glass transition are incorporated in the memory integral on the right-hand side and weighted by the dimensionless, positive coupling constant $\lambda$. The initial conditions $\Phi(0)=1$ and $\dot{\Phi}(0)=0$ model the original deviation from the equilibrium state $\Phi \equiv 0$. Despite its simplicity, the above equation takes the basic properties of the glass transition into account which we briefly review before presenting the solution based on the LFT. The physical order parameter distinguishes between two phases via the long-time limit $\Phi(t \to \infty) =c$: The ergodic phase is characterized by perfect relaxation corresponding to $c=0$, in contrast to the glass phase where the initial distortion never disappears completely and thus $c>0$ . To gain further insight into the phase diagram we apply the half-sided Fourier transformation $$\begin{aligned} \label{eq:halfFT} \hat\Phi(\nu) = \int_0^\infty dt\,\Phi(t) e^{i \nu t} \, . \end{aligned}$$ Furthermore, we reparametrize the function $\Phi(t) = \delta \Phi(t) +c$ to obtain the asymptotics $$\begin{aligned} \label{eq:ansatz} \delta \Phi(t) \to \begin{cases} 1-c & t\to 0^+ \\ 0 & t \to \infty \end{cases}\, ,\end{aligned}$$ irrespective of the phase. The constant value $c$ gives rise to a term proportional to $\delta(\nu)$ in the Fourier transform of Eq. that has to be canceled to satisfy $\delta \Phi(t \to \infty)=0$. This is the case if $$\begin{aligned} c(\lambda) = \begin{cases} 0 & \lambda < 1 \\ \cfrac{1+\sqrt{1-1/\lambda}}{2} & \lambda \geq 1 \end{cases}\,,\end{aligned}$$ which not only determines the asymptotic value of the order parameter but also identifies the critical coupling for the glass transition $\lambda_c=1$. As has been shown in Refs. [@leut84; @beng84] by analytic means, the approach to the phase boundary is characterized by a divergent low-frequency limit of $\delta \hat\Phi(\nu)$ that follows the power law $$\begin{aligned} \label{eq:critExp} \delta \hat\Phi(\nu = 0) \sim \begin{cases} (1-\lambda)^\mu\!\! & \text{for } \lambda \to 1^- \,\text{with } \mu = 1.76498...\\ (\lambda-1)^{\mu^\prime}\!\!\! & \text{for } \lambda \to 1^+\, \text{with } \mu^\prime = 0.76498... \end{cases}\end{aligned}$$ In the ergodic phase the half-sided Fourier transform of Eq. yields the self-consistent relation $$\begin{aligned} \label{eq:EOMnormal} \delta\hat\Phi(\nu) = - \frac{1}{i \nu + \displaystyle \frac{\Omega_0^2}{i \nu - \gamma - 4 \lambda \Omega_0^2 \mathcal{F}_{t\to\nu}[\delta \Phi^2(t)](\nu)}} \, .\end{aligned}$$ Similarly, in the glass phase one obtains a quadratic equation with the solution $$\begin{aligned} \label{eq:EOMglass} \begin{split} \delta \hat\Phi(\nu)& = \frac{- B \pm\sqrt{B^2 -4 A C}}{2 A}\\ A & = 8 c \lambda \Omega^2_0 i \nu \\ B & = \nu^2 +i \gamma \nu +\Omega_0^2\left[1-4 \lambda c^2 + 4 \lambda i \nu \mathcal{F}_{t\to\nu}[\delta \Phi^2(t)](\nu)\right] \\ C & = (1-c) \left[-i \nu + \gamma +4 \lambda \Omega_0^2 \mathcal{F}_{t\to\nu}[\delta \Phi^2(t)](\nu)\right] \, , \end{split}\end{aligned}$$ where in the first line one has to choose the branch that yields a positive $\Re \hat\Phi(\nu)$, since this function represents a retarded, bosonic correlation function [@fett71]. The equations or can be solved in an iterative manner, which requires repeated Fourier transformations between the time and frequency spaces. To reliably compute the critical behavior in the vicinity of the glass transition, however, one has to include very long times, especially to capture the extremely slow relaxation when approaching the instability from the ergodic phase. Yet, this is exactly the scenario the LFT has been devised for. Note that the formulation of Eq. in terms of $\delta \Phi$ is not only helpful for analyzing the problem in further detail, but in order to apply the LFT it is also mandatory because $\delta \Phi$ satisfies the condition . Figure \[fig:glass\] shows $\Phi(t)$ at the values $\lambda =1 \pm 2^{-21}\approx 1\pm 4.7\cdot 10^{-7}$ in the immediate vicinity of the glass transition. In the ergodic phase the plateau, which characterizes the so-called regime of $\beta$-relaxation, reaches times of $\mathcal O(10^{12})$ before the final $\alpha$-relaxation to $\Phi=0$ sets in. In addition to the global features of the dynamics, the LFT reproduces the critical exponents from Eq. with a numerical error on the order of $10^{-4}$ (see inset of Fig. \[fig:glass\]). To achieve such small errors one can profoundly benefit from the flexibility of the LFT: While the LFTs have to operate on identical $\omega$ and $\tau$ grids, irrespective of the direction of the transformation, the auxiliary $s$ space can be sampled for the two directions $\omega \to \tau$ and $\tau \to \omega$ independently. Moreover, one can introduce two LFTs for $\Re \delta \Phi(\nu)$ and $\Im \delta \Phi(\nu)$ separately without altering the overall computational cost. In total, the numerical effort to obtain $\Phi(t)$ at all times for a given $\lambda$ scales like $N_{\text{it}} \cdot N \log N$ where $N_{\text{it}}$ denotes the number of iterations needed to reach convergence. Using optimized trade-off parameters and a grid of length $N=2400$ suffices to produce the results shown in Fig. \[fig:glass\]. The small number of data points used in the LFT allows to find the converged order parameter in a couple of seconds, even close to the phase transition, where $N_{\text{it}} \sim 10^4$. ![(Color online) Time dependence of the order parameter for $\lambda = 1-2^{-21}$ (black) in the ergodic phase and for $\lambda=\lambda_c+2^{-21}$ (red) in the glass phase at $\gamma=1 = \Omega_0$. Inset: data points represent the numerical results for $\delta\hat\Phi(0)$. The straight lines are the analytical results for the critical exponent below (black) and above (red) the critical point. All results have been obtained with $N=2400$.[]{data-label="fig:glass"}](glass_full.pdf){width="0.9\columnwidth"} A direct numerical solution of the integro-differential equation on a discretized time axis requires step sizes $\Delta t \lesssim 1$ independent of the magnitude of $t$ in order to compute the cancellations between the various terms with sufficient precision. These are responsible for the slow evolution and the formation of the plateau over six orders of magnitude in time. Larger $\Delta t$ would lead to an instability of the numerical solution that diverges away from the physical $\Phi(t)$. Due to the scaling $\mathcal O (N_{\text{dir}}^2)$ of a direct approach and limited step size it is numerically completely unfeasible to reach times of order $10^{14}$. This, however, is necessary to reliably determine $\Phi(\omega = 0)$ and consequently the critical exponents [^2]. Regarding more complicated versions of mode-coupling theory that resolve the dependence on the length scales, thereby considering the structure factor $S(\mathbf{k}, t)$ instead of $\Phi(t)$, the grid size of $N \sim 10^3$ used here is still small enough to incorporate a second argument without running into memory limitations. If the coupling between different wave vectors can be written in terms of convolutions, which usually is the case [@reic05; @goet08], the LFT can also be applied to simplify the spatial dependence. Very similar problems appear in the context of approximate equations of motions of correlation functions in quantum field theory, which typically include algebraic decays in frequency and momentum space. As mentioned earlier, an example of the application of the LFT in the context of ultracold Fermi gases can be found in Ref. [@Frank2018]. Conlcusion {#sec:con} ========== We have shown rigorously that the LFT can be used to numerically transform nonintegrable functions, as long as their asymptotics can be controlled by the trade-off parameter. Furthermore, we have proven that one can achieve exponential convergence in the number of data points if the function is analytic in a cone with finite opening angle around the real axis in the original argument. Finally, we have given several examples that benchmark the superior convergence of the LFT compared to the FFT including functions that have to be considered within the concept of generalized Fourier transformations.\ Acknowledgments The authors thank Wilhelm Zwerger for fruitful discussions and comments on the manuscript and Christian Johansen for pointing out erroneous factors of $2\pi$. This work has been supported by the Nanosystems Initiative Munich (NIM). [^1]: This method of rewriting sums in terms of contour integrals is also used to evaluate Matsubara sums in the context of finite temperature quantum field theory [@abri75; @altl2010book; @fett71] [^2]: It is worth to mention, that in combination with a partially analytical solution, step sizes $\Delta t \gg 1$ are possible. The implementation of such a method is, however, far more complicated and time-consuming than the simple, direct approach presented here [@Goetze1988].
--- address: 'Pasadena, California' author: - Zeyu Guo bibliography: - 'ref.bib' title: '$\mathcal{P}$-schemes and Deterministic Polynomial Factoring over Finite Fields' --- 18[makeindex .nlo -s nomencl.ist -o .nls]{} I would like to thank my advisor, Chris Umans, for his patient guidance and continual encouragement throughout my graduate studies. His passion, enthusiasm, and dedication for research are truly inspiring. I am very fortunate to have such a great teacher as my advisor. I am indebted to Michael Aschbacher, who taught me a graduate algebra course, and Matthias Flach, who taught me a course on algebraic number theory. The knowledge I learned from their courses is crucial for the work presented in this thesis. I also want to thank Leonard Schulman, Anand Kumar Narayanan, Manuel Arora, and Jenish Mehta for many helpful conversations. In particular, I am grateful to Manuel Arora for explaining to me his work on $m$-schemes. Finally, I want to thank my family and friends for their continual support and encouragement.
--- bibliography: - 'bibthese.bib' ---
--- abstract: 'Phonon interactions in solid-state photonics systems cause intrinsic quantum decoherence and often present the limiting factor in emerging quantum technology. Due to recent developments in nanophotonics, exciton–cavity structures with very strong light–matter coupling rates can be fabricated. We show that in such structures, a new regime emerges, where the decoherence is completely suppressed due to decoupling of the dominant phonon process. Using a numerically exact tensor network approach, we perform calculations in this non-perturbative, non-Markovian dynamical regime. Here, we identify a strategy for reaching near-unity photon indistinguishability and also discover an interesting phonon-dressing of the exciton–cavity polaritons in the high-$Q$ regime, leading to multiple phonon sidebands when the light–matter interaction is sufficiently strong.' author: - 'Emil V. Denning' - 'Matias Bundgaard-Nielsen' - Jesper Mørk title: ' Electron–phonon decoupling due to strong light–matter interactions ' --- The development of scalable solid-state quantum technology is challenged by lattice vibrations, i.e. phonons, which even at zero temperatature deteriorates the quantum coherence [@fu2009observation; @iles2017phonon]. The interaction of electrons and phonons thus leads to remarkable features in the optical emission spectrum, such as broad spectral sidebands and incoherent scattering [@brash2019light; @tran2016quantum; @christiansen2017phonon; @doherty2013nitrogen; @selig2016excitonic]. This is detrimental to the optical coherence and important to circumvent for applications in quantum technology. It also presents an open quantum system with rich physics, operating in a regime of pronounced non-Markovian dynamics [@carmele2019non]. Recent developments in nanophotonics have opened up the possibility of creating dielectric nanocavities with deep subwavelength confinement of light [@choi2017self], leading to light–matter interaction strengths otherwise far beyond reach [@hu2018experimental; @wang2018maximizing]. Moreover, experiments have demonstrated very high coupling strengths between a plasmonic nanocavity and a pristine transition metal dichalcogenide monolayer [@qin2020revealing; @geisler2019single; @kleemann2017strong]. These developments open the door to a new regime of nanophotonic electron–phonon interactions, where the light–matter coupling rate is comparable to or larger than the dominating phonon frequencies in the environment. In this paper, we study theoretically this new regime of cavity quantum electrodynamics and discover new and important effects. The comparability of phononic and optical time scales makes calculations of the dynamical properties highly challenging and has demanded extensive development of non-perturbative and non-Markovian theoretical methods [@morreau2020phonon; @morreau2019phonon; @vagov2011real; @kaer2010non; @kaer2013microscopic]. To solve this outstanding problem, we have implemented a numerically exact and computationally efficient tensor network formulation [@jorgensen2019exploiting; @strathearn2018efficient]. Furthermore, we make use of a variational polaron perturbation theory to derive analytical results that explain the dynamical decoupling process. As an example, we consider a nanocavity containing a semiconductor quantum dot, which is coupled to the longitudinal acoustic phonon modes of the host lattice. With this example, we show that the spectral phonon sideband can be completely suppressed, when the nanocavity is in the low-$Q$ Purcell regime and the light–matter interaction strength exceeds the phonon cutoff frequency. Additionally, we predict a novel effect in the high-$Q$ limit, where each of the exciton polariton peaks in the spectrum is dressed with an individual phonon sideband, demonstrating non-perturbative dynamics, where polaritons and polarons occur at an equal footing. Our analysis is based on a generic system consisting of a localised exciton state, $\ket{X}$, a cavity mode with annihilation operator $a$ and a vibrational environment with phonon annihilation operators $\{b_\mathbf{k}\}$. The exciton–phonon coupling is described by the Hamiltonian [@mahan2013many] $$\begin{aligned} H_{ep} = \dyad{X}\sum_\mathbf{k} \hbar (g_{\mathbf{k}} b_\mathbf{k} + g_{\mathbf{k}}^* b_\mathbf{k}^\dagger ),\end{aligned}$$ where $\{g_{\mathbf{k}}\}$ are the exciton–phonon coupling strengths. The influence of the vibrational environment is characterised by the spectral density, $J(\nu) = \sum_\mathbf{k} \abs{g_\mathbf{k}}^2\delta(\nu-\nu_\mathbf{k})$, where $\nu_\mathbf{k}$ is the frequency of the phonon mode with momentum $\mathbf{k}$. For any realistic physical system, this spectral density has a cutoff frequency, $\xi$, such that $J(\nu)\simeq 0$ for $\nu \gg \xi$. This cutoff frequency is related to the length scale of the exciton wavefunction and the properties of available phonon modes in the material [@alkauskas2014first; @nazir2016modelling; @mahan2013many]. The light–matter interaction is governed by the Hamiltonian $H_0 = \hbar\delta a^\dagger a + \hbar g(\dyad{0}{X} a^\dagger + \dyad{X}{0} a)$, where $\delta = \omega_c-\omega_X$ is the cavity–exciton detuning, $g$ is the light–matter coupling strength and $\ket{0}$ is the electronic ground state. Furthermore, cavity losses with a rate $\kappa$ and exciton dephasing with a temperature-dependent rate $\gamma^(T)$ are treated through the Lindblad formalism [@breuer2002theory; @lindblad1976generators] as Markovian effects [@reigue2017probing; @tighineanu2018phonon; @muljarov2004dephasing]. To describe the optical emission properties of the system, we initialise it in the exciton state with zero photons in the cavity and calculate the spectral correlation function of the emitted photons as the system relaxes, $S(\omega,\omega')=\kappa\ev{a^\dagger(\omega)a(\omega')} = \kappa\int_{-\infty}^\infty\dd{t}\int_{-\infty}^\infty\dd{t'}e^{-i(\omega t-\omega't')}\ev{a^\dagger(t')a(t)}$. From this spectral function, we can calculate the emission spectrum as $S(\omega,\omega)$ [@steck2007quantum]. In addition, it provides access to the coherence properties of the emitted photons, for example their indistinguishability [@kiraz2004quantum], $\mathcal{I}=[\int\dd{\omega}S(\omega,\omega)]^{-2}\int\dd{\omega}\int\dd{\omega'}\abs{S(\omega,\omega')}^2$, which quantifies the interference visibility of two subsequently emitted photons. ![Illustration of phonon-mediated optical emission processes. [a.]{} In the Purcell regime, the exciton decays and emits a photon (orange arrow). During this process, a phonon wavepacket (blue wiggly arrow) might be emitted or absorbed, resulting in a photon with lower or higher energy. [b.]{} In the strong light–matter coupling regime, a phonon wavepacket can be emitted either by relaxation from the upper polariton to the lower one (downwards wiggly arrow), or when one of the polaritons decays to the ground state. [c.]{} In the phonon decoupling regime, where the polariton splitting, $2g$, exceeds the phonon cutoff frequency, $\xi$, the phonon sidebands on the two polaritons do not overlap and are hence spectrally resolved.[]{data-label="fig:phonon-effects"}](figure1.pdf){width="\columnwidth"} There are three main parameter regimes of this system: In the Purcell regime (Fig. \[fig:phonon-effects\]a), where $2g<\kappa$, the exciton decays and emits a photon into the cavity with a rate of $\Gamma_p=4g^2/\kappa$. In this process, a phonon wavepacket may be emitted or absorbed, generating a broad sideband in the emission spectrum. At low temperatures, $k_BT\ll\hbar\xi$, the sideband is asymmetric and red-detuned from the zero-phonon line, reflecting that phonon emission dominates over phonon absorption [@krummheuer2002theory]. In the strong coupling regime (Fig. \[fig:phonon-effects\]b), the coupling strength exceeds the decay, $2g>\kappa$, but is still well below the phonon cutoff frequency. Here, the exciton and cavity form hybrid polaritons, $\ket{\pm}=\ket{1,0}\pm\ket{0,X}$ (where $\ket{n,e}$ denotes a $n$-photon cavity state and electronic state $e\in\{0,X\}$) that are spectrally well-resolved and split by a frequency of $2g$. The dominating decoherence mechanism in this regime arises from a resonant transition from the upper polariton to the lower polariton under the emission of a phonon wavepacket with energy $\sim 2\hbar g$. If the temperature is sufficiently high to populate the phonon modes, the reverse process can also take place by phonon absorption. At low temperatures, the phonon emission process, $\ket{+}\rightarrow\ket{-}$, dominates, and a spectral polariton asymmetry can be observed, because photons are thus predominantly emitted from the lower polariton state [@roy2015quantum; @denning2020phonon; @morreau2019phonon]. Since the polariton splitting is small compared to the phonon cutoff frequency, the sideband seen in the Purcell regime is not resolved into contributions from the two polaritons. Increasing the coupling strength further leads to a regime of phonon decoupling (Fig. \[fig:phonon-effects\]c), where $2g$ exceeds the phonon cutoff frequency. Due to this, there are no phonon modes with sufficiently high energy to drive polariton transitions, and this decoupling leads to a recovery of the quantum coherence. Additionally, the spectral symmetry between the polariton peaks is restored and the polaritons are now so far separated that the individual phonon-polariton sidebands are spectrally resolved. ![image](figure2.pdf){width="\textwidth"} Calculating the temporal correlation function entering $S(\omega,\omega')$ is a technically demanding task due to the non-Markovian interactions with the phonon environment. Our approach, based on a tensor-network representation of the phonon influence functional, is described in the Supplementary Material. To illustrate the three different regimes in Fig. \[fig:phonon-effects\], we use a semiconductor quantum dot in a nanocavity as an example. Here, the phonon cutoff frequency is typically on the order of a few $\mathrm{ps^{-1}}$ [@denning2020phonon], and the spectral density is $J(\nu) = \alpha \nu^3\exp{-(\nu/\xi)^2}$, where $\alpha$ is an overall phonon coupling strength [@nazir2016modelling]. The optical emission spectra for parameters corresponding to the three characteristic parameter regimes are shown in Fig. \[fig:spectra\]. The spectra in the upper panels are calculated for a temperature of $T=4\mathrm{\; K}$, and in the lower panels for $T=150\mathrm{\; K}$. In the Purcell regime (Fig. \[fig:spectra\]a-b), the spectrum exhibits a narrow zero-phonon line dressed by a broad phonon sideband, which is asymmetric in the low-temperature limit. In addition, thermal phonon scattering and dephasing broadens the zero-phonon line at higher temperatures. In the strong coupling regime (Fig. \[fig:spectra\]c-d), the polariton peaks are asymmetric at low temperature, and the polariton peaks are dressed by a single phonon sideband. In the regime of phonon decoupling (Fig. \[fig:spectra\]e-f), the polaritons are split beyond the phonon cutoff frequency, and thus the polaritons are dressed by spectrally resolved sidebands. Furthermore, the polariton symmetry in the spectrum is recovered. To understand the complex behaviour of the system, we apply a variational polaron transformation to the Hamiltonian. This transformation is generated by the operator $V=\dyad{X}\sum_\mathbf{k}f_\mathbf{k}(b_\mathbf{k}^\dagger - b_\mathbf{k})/\nu_\mathbf{k}$ and transforms the Hamiltonian as $H_\mathrm{v} = e^{V}He^{-V}$. Here, $f_\mathbf{k}$ are variational coefficients, which are determined such that the Feynman-Bogoliubov bound on the free energy is minimized [@nazir2016modelling; @mccutcheon2011general; @gomez2018solid]. In practise, this means that the transformation depends on the relative magnitude of $g$ and $\xi$. An important characteristic of the transformation is the variational renormalisation factor, $B_\mathrm{v}=\ev{e^{\pm V}}$, which depends on $g$ and takes a value between 0 and 1, such that $B_\mathrm{v}\simeq 1$ when $2g\gg\xi$ (see Fig. \[fig:indist\]a). The significance of $B_\mathrm{v}$ is two-fold: First, the light–matter interaction in the transformed Hamiltonian, $H_\mathrm{v}$, is renormalised as $g\rightarrow gB_\mathrm{v}$, meaning that the phonons reduce the effective coupling strength. Furthermore, when the exciton–cavity system is in the Purcell regime, $2g<\kappa$, and $\kappa > \xi$, the probability of generating a phonon wavepacket jointly with the emission of a photon is given by $1-B_\mathrm{v}^2$, i.e. the phonon sideband constitutes a fraction of $1-B_\mathrm{v}^2$ of the total emission spectrum; in the limit $g\rightarrow 0$, $B_\mathrm{v}^2$ reduces to the Franck-Condon factor [@iles2017phonon]. However, as shown in Fig. \[fig:spectra\]c-d, this branching ratio does not hold in the phonon decoupling regime, where the polariton peaks are dressed with a phonon sideband, even though $g$ is sufficiently large to ensure $B_\mathrm{v}\simeq 1$. Thus, the polaritonic phonon sidebands are a strongly non-perturbative effect that cannot be captured even by the optimal perturbation theory. In analogy with the coupling strength renormalisation, the variational transformation also shifts the exciton resonance by $R_\mathrm{v}=\sum_\mathbf{k}f_\mathbf{k}(f_\mathbf{k}-2g_\mathbf{k})/\nu_\mathbf{k}$. This effect is of minor importance, but needs to be taken into account when setting the cavity frequency to resonance with the exciton. ![ [a.]{} Phonon spectral density evaluated at $\nu=2g$ (green, left axis) and variational renormalisation factor, $B_\mathrm{v}$, (right axis) at $T=\mathrm{\; 4K}$ (blue) and $T=150\mathrm{\; K}$ (red) as a function of the light–matter coupling strength, $g$. The phonon cutoff frequency is indicated as $\xi/2$ by a solid black line. [b.]{} Photon indistinguishability as a function of light–matter coupling strength, $g$ for fixed cavity decay, $\kappa=0.5\mathrm{\;ps^{-1}}$ (blue line and open circles) and cavity decay rate pinned to the coupling strength, $\kappa=3g$ (orange line and dots) at $T=4\mathrm{\;K}$. [c.]{} Same as in b., but at $T=150\mathrm{\; K}$. The line signatures are the same as in panel b.[]{data-label="fig:indist"}](figure3.pdf){width="\columnwidth"} To investigate the overall influence of the phonons in the decoupling regime, the photon indistinguishability is shown in Fig. \[fig:indist\]b as a function of $g$. The blue line with open circles signify a configuration with fixed cavity decay rate, corresponding to the blue spectra in the upper panels of Fig. \[fig:spectra\]. Here, it is clearly seen that the impact of the phonon environment is most significant when $J(2g)$ is maximal, meaning that the scattering process from the upper polariton to the lower is resonantly enhanced, and the photon emission process is exposed to strong decoherence. However, when $2g$ exceeds the cutoff frequency, the indistinguishability converges to $\sim 0.95$, due to the persistent polariton phonon sidebands. Alternatively, the orange line with dots shows the indistinguishability in a Purcell-configuration, where $\kappa$ is pinned at $3g$, ensuring that the system never enters the strong coupling regime. Here, the phonon sideband can be completely eliminated, when $B_\mathrm{v}\rightarrow 1$ and the zero-phonon line broadens sufficiently to absorb the entire sideband. The difference between the polariton and Purcell regimes becomes even more pronounced in the high-temperature limit (Fig. \[fig:indist\]c), where the sideband is more dominating in the spectrum. Due to thermal phonon population, the exciton dephasing here is stronger, meaning that the increase in indistinguishability for the Purcell configuration is slower than for the low-temperature case. It is noteworthy that even at this high temperature, it is possible to achieve phonon decoupling and thus near-unity indistinguishability. We now turn our attention towards the phonon-induced polariton asymmetry in the spectrum that arises when the upper polariton decays to the lower polariton, which is the dominant dephasing mechanism in the strong coupling regime at low temperatures. In Fig. \[fig:polariton-asymmetry\], we show, as a function of $g$, the spectral asymmetry between the polariton peaks (solid line and open circles, left axis), calculated as $A=(S_--S_+)/(S_-+S_+)$, where $S_\pm := S(\omega_\pm,\omega_\pm)$ is the emission spectrum evaluated at the upper ($+$) and lower ($-$) polariton peak. As expected, the polariton symmetry is recovered in the limit $2g\gg\xi$. To support this finding, we use a master equation in the variational frame to derive the asymmetry-driving differential scattering rate from the upper to the lower polariton (see Supplementary Material), $$\begin{aligned} \Gamma_A = \frac{\pi}{2}J(2gB_\mathrm{v})[1-F^2(2gB_\mathrm{v})] + \epsilon(g), \end{aligned}$$ where $F(\nu)$ is the dimensionless variational displacement function, $F(\nu_\mathbf{k})=f_\mathbf{k}/g_\mathbf{k}$ and $\epsilon(g)$ is a small term that vanishes in the limit $2g\gg\xi$. This analytical scattering rate is also shown in Fig. \[fig:polariton-asymmetry\] (shaded area, right axis) and exhibits a similar behaviour as the polariton asymmetry. These findings show that the phonon-induced polariton scattering can indeed be eliminated in the phonon decoupling regime, because there are no available phonon modes with sufficiently high frequency to match the polariton energy difference. However, as shown in Figs. \[fig:spectra\] and \[fig:indist\], this does not mean that the phonons are fully decoupled in this regime, since the polaritonic phonon sidebands do not rely on resonant transitions, but occur due to vibrational dressing of the individual polaritons. ![Asymmetry of polariton peaks, $A$, as a function of coupling strength (solid line and open circles, left axis), overlayed with analytically calculated differential polariton scattering rate, $\Gamma_A$ (shaded area, right axis). []{data-label="fig:polariton-asymmetry"}](figure4.pdf){width="\columnwidth"} In conclusion, we have shown that the phonons in the environment of a localised exciton coupled to a nanocavity can be dynamically decoupled when the light–matter coupling is sufficiently strong. We have found that an effective decoupling occurs in the Purcell regime, where the zero-phonon transition occurs with a rate much higher than the phonon cutoff frequency. Furthermore, we have found that the phonon-induced polariton scattering in the strong light–matter coupling regime can be eliminated when the polariton splitting exceeds the phonon cutoff frequency. However, we also find a significant phonon-dressing of the individual polaritons that persists into the phonon decoupling regime, demonstrating the importance of operating in the Purcell regime. These principal observations only rely on the relative magnitude of the exciton–cavity coupling strength and the phonon cutoff frequency, and generally hold for any exciton–cavity system. #### Acknowledgements The authors thank Mathias R. Jørgensen for helpful discussions. This work was supported by the Danish National Research Foundation through NanoPhoton - Center for Nanophotonics, grant number DNRF147. 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--- abstract: \| Relational datasets are being generated at an alarmingly rapid rate across organizations and industries. Compressing these datasets could significantly reduce storage and archival costs. Traditional compression algorithms, e.g., gzip, are suboptimal for compressing relational datasets since they ignore the table structure and relationships between attributes. We study compression algorithms that leverage the relational structure to compress datasets to a much greater extent. We develop [[Squish]{}]{}, a system that uses a combination of Bayesian Networks and Arithmetic Coding to capture multiple kinds of dependencies among attributes and achieve near-entropy compression rate. [[Squish]{}]{}\ also supports user-defined attributes: users can instantiate new data types by simply implementing five functions for a new class interface. We prove the asymptotic optimality of our compression algorithm and conduct experiments to show the effectiveness of our system: [[Squish]{}]{} achieves a reduction of over 50% in storage size relative to systems developed in prior work on a variety of real datasets. author: - \| Yihan Gao\ \ \ Aditya Parameswaran\ \ \ title: '[[[Squish]{}]{}]{}: Near-Optimal Compression for Archival of Relational Datasets' --- =10000 = 10000 Acknowledgement {#acknowledgement .unnumbered} =============== We thank the anonymous reviewers for their valuable feedback. We acknowledge support from grant IIS-1513407 awarded by the National Science Foundation, grant 1U54GM114838 awarded by NIGMS and 3U54EB020406-02S1 awarded by NIBIB through funds provided by the trans-NIH Big Data to Knowledge (BD2K) initiative (www.bd2k.nih.gov), the Siebel Energy Institute, and the Faculty Research Award provided by Google. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies and organizations. [ ]{}
--- abstract: \| We have developed a numerical software library for collisionless $N$-body simulations named “Phantom-GRAPE” which highly accelerates force calculations among particles by use of a new SIMD instruction set extension to the x86 architecture, Advanced Vector eXtensions (AVX), an enhanced version of the Streaming SIMD Extensions (SSE). In our library, not only the Newton’s forces, but also central forces with an arbitrary shape $f(r)$, which has a finite cutoff radius $r_{\rm cut}$ (i.e. $f(r)=0$ at $r>r_{\rm cut}$), can be quickly computed. In computing such central forces with an arbitrary force shape $f(r)$, we refer to a pre-calculated look-up table. We also present a new scheme to create the look-up table whose binning is optimal to keep good accuracy in computing forces and whose size is small enough to avoid cache misses. Using an Intel Core i7–2600 processor, we measure the performance of our library for both of the Newton’s forces and the arbitrarily shaped central forces. In the case of Newton’s forces, we achieve $2 \times 10^9$ interactions per second with one processor core (or $75$ GFLOPS if we count $38$ operations per interaction), which is $20$ times higher than the performance of an implementation without any explicit use of SIMD instructions, and $2$ times than that with the SSE instructions. With four processor cores, we obtain the performance of $8 \times 10^9$ interactions per second (or $300$ GFLOPS). In the case of the arbitrarily shaped central forces, we can calculate $1 \times 10^9$ and $4 \times 10^9$ interactions per second with one and four processor cores, respectively. The performance with one processor core is $6$ times and $2$ times higher than those of the implementations without any use of SIMD instructions and with the SSE instructions. These performances depend only weakly on the number of particles, irrespective of the force shape. It is good contrast with the fact that the performance of force calculations accelerated by graphics processing units (GPUs) depends strongly on the number of particles. Substantially weak dependence of the performance on the number of particles is suitable to collisionless $N$-body simulations, since these simulations are usually performed with sophisticated $N$-body solvers such as Tree- and TreePM-methods combined with an individual timestep scheme. We conclude that collisionless $N$-body simulations accelerated with our library have significant advantage over those accelerated by GPUs, especially on massively parallel environments. address: 'Center for Computational Science, University of Tsukuba, 1–1–1, Tennodai, Tsukuba, Ibaraki 305–8577, Japan' author: - Ataru Tanikawa - Kohji Yoshikawa - Keigo Nitadori - Takashi Okamoto title: 'Phantom-GRAPE: numerical software library to accelerate collisionless $N$-body simulation with SIMD instruction set on x86 architecture' --- , , & Stellar dynamics ,Method: $N$-body simulations Introduction ============ Self-gravity is one of the most essential physical processes in the universe, and plays important roles in almost all categories of astronomical objects such as globular clusters, galaxies, galaxy clusters, etc. In order to follow the evolution of such systems, gravitational $N$-body solvers have been widely used in numerical astrophysics. Due to prohibitively expensive computational cost in directly solving $N$-body problems, many efforts have been made to reduce it in various ways. For example, several sophisticated algorithms to compute gravitational forces among many particles with reduced computational cost have been developed, such as Tree method [@Barnes86], PPPM method [@Hockney81], TreePM method [@Xu95], etc. Another approach is to improve the computational performance with the aid of additional hardware, such as GRAPE (GRAvity PipE) systems, special-purpose accelerators for gravitational $N$-body simulations [@Sugimoto90; @Makino03; @Fukushige05], and general-purpose computing on Graphics Processing Units (GPGPUs). GRAPE systems have been used for further improvement of existing $N$-body solvers such as Tree method [@Makino91], PPPM method [@Brieu95; @Yoshikawa05], TreePM method [@Yoshikawa05], P$^2$M$^2$ tree method [@Kawai04], and PPPT method [@Oshino11]. They have also adapted to simulation codes for dense stellar systems based on fourth-order Hermite scheme, such as [NBODY4]{} [@Johnson06], [NBODY1]{} [@Harfst07], [kira]{} [@Simon08], and [ GORILLA]{} [@Tanikawa09]. Recently, @Hamada07, @Simon07, @Gaburov09, and @Bedorf12 explored the capability of commodity graphics processing units (GPUs) as hardware accelerators for $N$-body simulations and achieved similar to or even higher performance than the GRAPE-6A and GRAPE-DR board. A different approach to improve the performance of $N$-body calculations is to utilize Streaming SIMD Extensions (hereafter SSE), a SIMD (Single Instruction, Multiple Data) instruction set implemented on x86 and x86\_64 processors. @Nitadori06 exploited the SSE and SSE2 instruction sets, and achieved speeding up of the Hermite scheme [@Makino92] in mixed precision for collisional self-gravitating systems. Although unpublished in literature, Nitadori, Yoshikawa, & Makino have also developed a numerical library for $N$-body calculations in single-precision for collisionless self-gravitating systems in which two-body relaxation is not physically important and therefore single-precision floating-point arithmetic suffices for the required numerical accuracy. Furthermore, along this approach, they have also improved the performance in computing arbitrarily-shaped forces with a cutoff distance, defined by a user-specified function of inter-particle separation. Such capability to compute force shapes other than Newton’s inverse-square gravity is necessary in PPPM, TreePM, and Ewald methods. It should be noted that GRAPE-5 and the later families of GRAPE systems have similar capability to compute the Newton’s force multiplied by a user-specified cutoff function [@Kawai00], and can be used to accelerate PPPM and TreePM methods for cosmological $N$-body simulations [@Yoshikawa05]. Based on these achievements, a publicly available software package to improve the performance of both collisional and collisionless $N$-body simulations has been developed, which was named “Phantom-GRAPE” after the conventional GRAPE system. A set of application programming interfaces of Phantom-GRAPE for collisionless simulations is compatible to that of GRAPE-5. Phantom-GRAPE is widely used in various numerical simulations for galaxy formation [@Saitoh08; @Saitoh09] and the cosmological large-scale structures [@Ishiyama08; @Ishiyama09a; @Ishiyama09b; @Ishiyama10; @Ishiyama11]. Recently, a new processor family with “Sandy Bridge” micro-architecture[^1] by Intel Corporation and that with “Bulldozer” micro-architecture[^2] by AMD Corporation have been released. Both of the processors support a new set of instructions known as Advanced Vector eXtensions (AVX), an enhanced version of the SSE instructions. In the AVX instruction set, the width of the SIMD registers is extended from 128-bit to 256-bit. We can perform SIMD operations on two times larger data than before. Therefore, the performance of a calculation with the AVX instructions should be two times higher than that with the SSE instructions if the execution unit is also extended to 256-bit. @Tanikawa11 (hereafter, paper I) developed a software library for [collisional]{} $N$-body simulations using the AVX instruction set in the mixed precision, and achieved a fairly high performance. In this paper, we present a similar library implemented with the AVX instruction set but for [collisionless]{} $N$-body simulations in single-precision. The structure of this paper is as follows. In section \[sec:avx\], we overview the AVX instruction set. In section \[sec:implementation\], we describe the implementation of Phantom-GRAPE. In section \[sec:accuracy\] and \[sec:performance\], we show the accuracy and performance, respectively. In section \[sec:summary\], we summarize this paper. The AVX instruction set {#sec:avx} ======================= In this section, we present a brief review of the Advanced Vector eXtensions (AVX) instruction set. Details of the difference between SSE and AVX is described in section 3.1 of paper I. AVX is a SIMD instruction set as well as SSE, and supports many operations, such as addition, subtraction, multiplication, division, square-root, approximate inverse-square-root, several bitwise operations, etc. In such operations, dedicated registers with 256-bit length called “YMM registers” are used to store the eight single-precision floating-point numbers or four double-precision floating-point numbers. Note that the lower 128-bit of the YMM registers have alias name “XMM registers”, and can be used as the dedicated registers for the SSE instructions for a backward compatibility. An important feature of AVX and SSE instruction sets is the fact that they have a special instruction for a very fast approximation of inverse-square-root with an accuracy of about 12-bit. Actually, this instruction is quite essential to improve the performance of the gravitational force calculations, since the most expensive part in the force calculation is an execution of inverse-square-root of squared distances of the particle pairs. As already discussed in @Nitadori06, the approximate values can be adopted as initial values of the Newton-Raphson iteration to improve the accuracy, and we can obtain 24-bit accuracy after one Newton-Raphson iteration. For collisionless self-gravitating systems, however, the accuracy of $\simeq$ 12 bits is sufficient because the accuracy of inverse-square-root does not affect the resultant force accuracy if one adopts an approximate $N$-body solver such as Tree, PPPM and TreePM methods. Therefore, we use the raw approximate instruction throughout this study. Since the present-day compilers cannot always detect concurrency of the loops effectively, and cannot fully resolve the mutual dependency among data in the code, it is quite rare that compilers generate codes with SIMD instructions in effective manners from codes expressed in high-level languages. For an efficient use of the AVX instructions, we need to program with assembly-languages explicitly or compiler-dependent intrinsic functions and data type extensions. In assembly-languages, we can manually control the assignment of YMM registers to computational data, and minimize the access to the main memory by optimizing the assignment of each register. In this work, we adopt an implementation of the AVX instructions using inline-assembly language with C expression operands, embedded in C-language, which is a part of language extensions of GCC (GNU Compiler Collection). Implementation {#sec:implementation} ============== Here, we describe the detailed implementation to accelerate $N$-body calculation using the AVX instructions. For a given set of positions ${\textbf{\textit{r}}}_i$ of $N$ particles, we try to accelerate the calculations of a gravitational force given as follows: $$\label{eq:newton_force} {\textbf{\textit{a}}}_i = \sum_{j=1}^N \frac{Gm_j ({\textbf{\textit{r}}}_j - {\textbf{\textit{r}}}_i)}{(\|{\textbf{\textit{r}}}_j - {\textbf{\textit{r}}}_i\|^2 +\epsilon^2)^{3/2}},$$ where $G$ is the gravitational constant, $m_j$ the mass of the $j$-th particle, and $\epsilon$ the gravitational softening length. In addition to that, we also try to accelerate the computations of central forces among particles with an arbitrary force shape $f(r)$ given by $$\label{eq:arbitrary_force} {\textbf{\textit{a}}}_i=\sum_{j=1}^N m_j f(\|{\textbf{\textit{r}}}_j-{\textbf{\textit{r}}}_i\|)\frac{{\textbf{\textit{r}}}_j-{\textbf{\textit{r}}}_i}{\|{\textbf{\textit{r}}}_j-{\textbf{\textit{r}}}_i\|},$$ where $f(r)$ specifies the shape of the force as a function of inter-particle separation $r$ with a cutoff distance $r_{\rm cut}$ (i.e. $f(r)=0$ at $r>r_{\rm cut}$). In the above expressions, particles with subscript “$j$” exert forces on those with subscript “$i$”. In the rest of this paper, the former are referred to as “$j$-particles”, and the latter as “$i$-particles” just for convenience. Since individual forces exerted by $j$-particles on $i$-particles can be computed independently, we can calculate forces exerted by multiple $j$-particles on multiple $i$-particles in parallel. As described in the previous section, the AVX instructions can execute operations of eight single-precision floating-point numbers on YMM registers in parallel. By utilizing this feature of the AVX instructions, the forces on four $i$-particles from two $j$-particles can be computed simultaneously in a SIMD manner. ![image](fig01.eps){width="15cm"} Structures for the particle data -------------------------------- In computing the forces on four $i$-particles from two $j$-particles, we assign the data of $i$- and $j$-particles to YMM registers in the way shown in Figure \[fig:assignment\]. Suppose that $a$ and $b$ in Figure \[fig:assignment\] are $x$-components of $i$- and $j$-particles, respectively. Subtracting data in the YMM register (1) of Figure \[fig:assignment\] from data in the YMM register (2) of Figure \[fig:assignment\], we simultaneously obtain $x$-components of eight relative positions $c$ in the YMM register (3) of Figure \[fig:assignment\]. In order to effectively realize such SIMD computations with the AVX instructions, we define the structures for $i$-particles, $j$-particles and the resulting forces and potentials shown in List \[list:structures\]. Before computing the forces on $i$-particles, the positions and softening lengths of $i$-particles are stored in the structure `Ipdata`, and the positions and masses of $j$-particles are in the structure `Jpdata`. The resulting forces are stored in the structure `Fodata`. Note that the structures `Ipdata` and `Fodata` contain the data of four $i$-particles, while the structure `Jpdata` has the data for a single $j$-particle. Note that the positions, softening lengths, and forces of $i$-particles in the structures `Ipdata` and `Fodata` are declared as arrays of four single-precision floating-point numbers. Thus, the data on each array can be suitably loaded onto, or stored from the lower 128-bit of one YMM register. The assignment of the $i$-particles data shown in (1) of Figure \[fig:assignment\] can be realized by loading the data of four $i$-particles onto the lower 128-bit of one YMM register, and copying the data to its upper 128-bit. As for $j$-particles, since the structure `Jpdata` consists of four single-precision floating-point numbers, we can load the positions and the masses of two $j$-particles in one YMM-register at one time if they are aligned on the 32-byte boundaries. By broadcasting the $n$-th element ($n=0,1,2$ and 3) in each of the lower and upper 128-bit to all the other elements, we can realize the assignment of the $j$-particle data as depicted in (2) of Figure \[fig:assignment\]. After executing the gravitational force loop over $j$-particles, the partial forces on four $i$-particles exerted by different sets of $j$-particles are stored in the upper and lower 128-bit of a YMM register. Operating sum reduction on the upper and lower 128-bit of the YMM register, and storing the results into its lower 128-bit, we can smoothly store the results into the structure `Fodata`. // structure for i-particles typedef struct ipdata{ float x[4]; float y[4]; float z[4]; float eps2[4]; } Ipdata, pIpdata; // structure for j-particles typedef struct jpdata{ float x, y, z, m; } Jpdata, pJpdata; // structure for the resulting forces // and potentials of i-particles typedef struct fodata{ float ax[4]; float ay[4]; float az[4]; float phi[4]; } Fodata, pFodata; Macros for inline assembly codes -------------------------------- For the readability of the source codes shown below, let us introduce some preprocessor macros which are expanded into inline assembly codes. The definitions of the macros used in this paper are given in List \[list:macros\]. For macros with two and three operands, the results are stored in the second and third one, respectively, and the other operands are source operands. In these macros, operands named `src`, `src1`, `src2`, and `dst` designate the data in XMM or YMM registers, and those named `mem`, `mem64`, `mem128`, and `mem256` are data in the main memory or the cache memory, where numbers after `mem` indicate their size and alignment in bits. Brief descriptions of these macros are summarized in Table \[tab:macros\]. More detailed explanation of the AVX instructions can be found in Intel’s website[^3]. #define VZEROALL asm("vzeroall"); #define VLOADPS(mem256, dst) \ asm("vmovaps %0, %"dst::"m"(mem256)); #define VSTORPS(reg, mem256) \ asm("vmovaps %"reg ", %0" ::"m"(mem256)); #define VLOADPS(mem128, dst) \ asm("vmovaps %0, %"dst::"m"(mem128)); #define VSTORPS(reg, mem128) \ asm("vmovaps %"reg ", %0" ::"m"(mem128)); #define VLOADLPS(mem64, dst) \ asm("vmovlps %0, %"dst ", %"dst::"m"(mem64)); #define VLOADHPS(mem64, dst) \ asm("vmovhps %0, %"dst ", %"dst::"m"(mem64)); #define VBCASTL128(src, dst) \ asm("vperm2f128 %0, %"src ", %"src \ ", %"dst " "::"g"(0x00)); #define VCOPYU128TOL128(src,dst) \ asm("vextractf128 %0, %"src ", %"dst \ " "::"g"(0x01)); #define VGATHERL128(src1,src2,dst) \ asm("vperm2f128 %0, %"src2 ", %"src1 \ ", %"dst " "::"g"(0x02)); #define VCOPYALL(src,dst) \ asm("vmovaps %0, %"src ", %"dst); #define VBCAST0(src, dst) \ asm("vshufps %0, %"src ", %"src \ ", %"dst " "::"g"(0x00)); #define VBCAST1(src, dst) \ asm("vshufps %0, %"src ", %"src \ ", %"dst " "::"g"(0x55)); #define VBCAST2(src, dst) \ asm("vshufps %0, %"src ", %"src \ ", %"dst " "::"g"(0xaa)); #define VBCAST3(src, dst) \ asm("vshufps %0, %"src ", %"src \ ", %"dst " "::"g"(0xff)); #define VMIX0(src1,src2,dst) \ asm("vshufps %0, %"src2 ", %"src1 \ ", %"dst " "::"g"(0x88)); #define VMIX1(src1,src2,dst) \ asm("vshufps %0, %"src2 ", %"src1 \ ", %"dst " "::"g"(0xdd)); #define VADDPS(src1, src2, dst) \ asm("vaddps " src1 "," src2 "," dst); #define VSUBPS(src1, src2, dst) \ asm("vsubps " src1 "," src2 "," dst); #define VSUBPS_M(mem256, src, dst) \ asm("vsubps %0, %"src ", %"dst \ " "::"m"(mem256)); #define VMULPS(src1, src2, dst) \ asm("vmulps " src1 "," src2 "," dst); #define VRSQRTPS(src, dst) \ asm("vrsqrtps " src "," dst); #define VMINPS(src1, src2, dst) \ asm("vminps " src1 ", " src2 "," dst); #define VPSRLD(imm, src1, src2) \ asm("vpsrld %0, %"src1 ", %"src2::"I"(imm)); #define VPSLLD(imm, src1, src2) \ asm("vpslld %0, %"src1 ", %"src2::"I"(imm)); #define PREFETCH(mem) \ asm("prefetcht0 %0"::"m"(mem)); `VZEROALL` zero out all the YMM registers. ------------------------------ ------------------------------------------------------------------------------------------------------------ `VLOADPS(mem256,dst)` load eight packed values in `mem256` to `dst`. `VSTORPS(src,mem256)` store eight packed values in `src` to `mem256`. `VLOADPS(mem128,dst)` load four packed values in `mem128` to `dst`. `VSTORPS(src,mem128)` store four packed values in `src` to `mem128`. `VLOADLPS(mem64,dst)` load two packed values in `mem64` to the lower 64-bit of the lower 128-bit in `dst`. `VLOADHPS(mem64,dst)` load two packed values in `mem64` to the upper 64-bit of the lower 128-bit in `dst`. `VBCASTL128(src,dst)` broadcast data in the lower 128-bit of `src` to the lower and upper 128-bit of `dst`. `VCOPYU128TOL128(src,dst)` copy the upper 128-bit in `src` to the lower 128-bit in `dst`. `VGATHERL128(src1,src2,dst)` copy the lower 128-bit in `src1` and `src2` to the upper 128-bit and lower 128-bit in `dst`, respectively. `VCOPYALL(src,dst)` copy 256-bit data from `src` to `dst`. `VBCASTn(src,dst)` broadcast the n-th element of each of the lower and upper 128-bit to all the other elements. `VMIX0(src1,src2,dst)` operate data as shown in Figure \[fig:mix\]. `VMIX1(src1,src2,dst)` operate data as shown in Figure \[fig:mix\]. `VADDPS(src1,src2,dst)` add `src1` to `src2`, and store the result to `dst`. `VSUBPS(src1,src2,dst)` subtract `src1` from `src2`, and store the result to `dst`. `VSUBPS_M(mem256, src, dst)` subtract `mem256` from `src`, and store the result to `dst`. `VMULPS(src1,src2,dst)` multiply `src1` by `src2`, and store the result to `dst`. `VRSQRTPS(src,dst)` compute the inverse-square-root of `src`, and store the result to `dst`. `VMINPS(src1,src2,dst)` compare the values in each pair of elements in `src1` and `src2`, and store the not larger ones to `dst`. `VPSRLD(imm,src,dst)` shift each element in the lower 128-bit of `src` left by `imm` bit, and store the result to `dst`. `VPSRRD(imm,src,dst)` shift each element in the lower 128-bit of `src` right by `imm` bits, and set the result to `dst`. `PREFETCH(mem)` prefetch data on `mem` to the cache memory. ![Instructions [MIX0]{} and [MIX1]{}. Each set of four boxes indicates the lower (or upper) 128-bit of a YMM register. Each box contains a single-precision floating-point number.[]{data-label="fig:mix"}](fig02.eps){width="7.5cm"} Furthermore, we define aliases of XMM and YMM registers. Table \[tab:alias1\] and \[tab:alias2\] show the aliases of YMM registers in calculating Newton’s force and an arbitrary shaped central force, respectively. Aliases with suffix “`_X`” indicate the lower 128-bit of the original YMM register which can be used as XMM registers for the SSE instructions. Note that some of aliases are reused for data other than described in Table \[tab:alias1\] and \[tab:alias2\]. Alias ID Description -------- ---------- ------------------------------------------------------------- `XI` `%ymm0` `YI` `%ymm1` $x$, $y$, and $z$-coordinates of $i$-particles `ZI` `%ymm2` ($x_i$, $y_i$, and $z_i$) `EPS2` `%ymm3` square of the gravitational softening length ($\epsilon^2$) `AX` `%ymm4` `AY` `%ymm5` forces of $i$-particles `AZ` `%ymm6` ($a_{x,i}$, $a_{y,i}$, and $a_{z,i}$) `PHI` `%ymm7` gravitational potentials of $i$-particles ($\phi_i$) `XJ` `%ymm8` `YJ` `%ymm9` $x$, $y$, and $z$-coordinates of $j$-particles `ZJ` `%ymm10` ($x_j$, $y_j$, and $z_j$) `MJ` `%ymm11` masses of $j$-particles ($m_j$) `DX` `%ymm12` `DY` `%ymm13` relative coordinates between $i$- and $j$-particles `DZ` `%ymm14` ($x_{ij}$, $y_{ij}$, and $z_{ij}$) : Aliases of YMM registers for calculating Newton’s forces in List \[list:newtonforce\].[]{data-label="tab:alias1"} Alias ID Description --------- ---------- ----------------------------------------------------- `X2` `%ymm0` `Y2` `%ymm1` squared inter-particle distances `Z2` `%ymm2` `TWO` `%ymm3` constant value of `2.0` in single-precision `AX` `%ymm4` `AY` `%ymm5` forces of $i$-particles `AZ` `%ymm6` `R2CUT` `%ymm7` cutoff radius squared `BUF0` `%ymm8` `BUF1` `%ymm9` buffers used to refer a look-up table `BUF2` `%ymm10` `MJ` `%ymm11` masses of $j$-particles `DX` `%ymm12` `DY` `%ymm13` relative coordinates between $i$- and $j$-particles `DZ` `%ymm14` `ZI` `%ymm15` $z$-components of positions of $i$-particles : Aliases of YMM registers for calculating an arbitrary force shape in List \[list:arbitraryforce\].[]{data-label="tab:alias2"} Newton’s force {#sec:methodnewton} -------------- Figure \[fig:newtonforce\] is a schematic illustration of a force loop to compute the Newton’s force on four $i$-particles with AVX instructions. In this figure, we depict only the lower 128-bit of YMM registers just for simplicity, while, in actual computation, the upper 128-bit is used to compute forces on the same four $i$-particle exerted by another $j$-particle. ![image](fig03.eps){width="16cm"} The overall procedures to calculate the force on four $i$-particles using AVX instructions are summarized as follows: 1. Zero out all the YMM registers, and load the $x$, $y$, and $z$ coordinates of four $i$-particles, and squared softening lengths to the lower 128-bit of `XI`, `YI`, `ZI`, and `EPS2` (i.e. `XI_X`, `YI_X`, `ZI_X`, and `EPS2_X`), and copy them to the upper 128-bit of `XI`, `YI`, `ZI`, and `EPS2`, respectively. 2. Load the $x$, $y$, and $z$ coordinates and the masses of two $j$-particles to `XJ`. 3. Broadcast the $x$, $y$, and $z$ coordinates and the masses of two $j$-particles in `XJ` to `XJ`, `YJ`, `ZJ`, and `MJ`, respectively. 4. Subtract `XI`, `YI`, and `ZI` from `XJ`, `YJ`, and `ZJ` respectively. The results ($x_{ij}$, $y_{ij}$, and $z_{ij}$) are stored in `DX`, `DY`, and `DZ`, respectively. 5. Square $x_{ij}$ in `DX`, $y_{ij}$ in `DY`, and $z_{ij}$ in `DZ` and sum them up to compute the squared distance between two $j$-particles and four $i$-particles. The results are stored in the alias `YJ`. The squared softening lengths `EPS2` are also added. Eventually, the softened squared distances ${\hat{r}_{ij}}^2 \equiv r_{ij}^2+\epsilon^2$ between two $j$-particles and four $i$-particles are stored in `YJ`. 6. Calculate inverse-square-root for $\hat{r}_{ij}^2$ in `YJ`, and store the result $1/\hat{r}_{ij}$ in the alias `ZJ`. 7. Multiply $1/\hat{r}_{ij}$ in `ZJ` by $m_j$ in `MJ` to obtain $m_j/\hat{r}_{ij}$, and store the results in `MJ`. 8. Accumulate $m_j/\hat{r}_{ij}$ in `MJ` into $\phi_i$ in `PHI`. 9. Square $1/\hat{r}_{ij}$ in `ZJ`, multiply the result $1/\hat{r}_{ij}^2$ by $m_j/\hat{r}_{ij}$ in `MJ`, and store them $m_j/\hat{r}_{ij}^3$ in `YJ`. 10. Multiply $x_{ij}$ in `DX`, $y_{ij}$ in `DY`, and $z_{ij}$ in `DZ` by $m_j/\hat{r}_{ij}^3$ in `YJ` obtaining the forces ($m_jx_{ij}/\hat{r}_{ij}^3$, $m_jy_{ij}/\hat{r}_{ij}^3$, and $m_jz_{ij}/\hat{r}_{ij}^3$), and accumulate them into `AX`, `AY`, and `AZ`, respectively. 11. Return to step 1 until all the $j$-particles are processed. 12. Operate sum reduction of partial forces and potentials in the lower and upper 128-bits of `AX`, `AY`, `AZ`, and `PHI`, and store the results in the lower 128-bit of `AX`, `AY`, `AZ`, and `PHI`, respectively. 13. Store forces and potentials in the lower 128-bit of `AX`, `AY`, `AZ`, and `PHI` to the structure `Fodata`. The function `GravityKernel` to compute the forces is shown in List \[list:newtonforce\]. The order of instructions in List \[list:newtonforce\] is slightly different from that described above in order to obtain high issue rate of the AVX instructions by optimizing the order of operations so that operands in adjacent instruction calls do not have dependencies as much as possible. Further optimization is given by explicitly unrolling the force loop, which does not appear in the list. void GravityKernel(pIpdata ipdata, pFodata fodata, pJpdata jpdata, int nj) { int j; PREFETCH(jpdata[0]); VZEROALL; VLOADPS(ipdata->x, XI_X); VLOADPS(ipdata->y, YI_X); VLOADPS(ipdata->z, ZI_X); VLOADPS(ipdata->eps2, EPS2_X); VBCASTL128(XI, XI); VBCASTL128(YI, YI); VBCASTL128(ZI, ZI); VBCASTL128(EPS2, EPS2); VLOADPS((jpdata), XJ); jpdata += 2; VBCAST1(XJ, YJ); VBCAST2(XJ, ZJ); VBCAST3(XJ, MJ); VBCAST0(XJ, XJ); for(j = 0 ; j < nj; j += 2) { VSUBPS(YI, YJ, DY); VSUBPS(ZI, ZJ, DZ); VSUBPS(XI, XJ, DX); VMULPS(DZ, DZ, ZJ); VMULPS(DX, DX, XJ); VMULPS(DY, DY, YJ); VADDPS(XJ, ZJ, ZJ); VADDPS(EPS2, YJ, YJ); VADDPS(YJ, ZJ, YJ); VLOADPS((jpdata), XJ); jpdata += 2; VRSQRTPS(YJ, ZJ); VMULPS(ZJ, MJ, MJ); VMULPS(ZJ, ZJ, YJ); VMULPS(MJ, YJ, YJ); VSUBPS(MJ, PHI, PHI); VMULPS(YJ, DX, DX); VMULPS(YJ, DY, DY); VMULPS(YJ, DZ, DZ); VBCAST1(XJ, YJ); VBCAST2(XJ, ZJ); VBCAST3(XJ, MJ); VBCAST0(XJ, XJ); VADDPS(DX, AX, AX); VADDPS(DY, AY, AY); VADDPS(DZ, AZ, AZ); } VCOPYU128TOL128(AX, DX_X); VADDPS(AX, DX, AX); VCOPYU128TOL128(AY, DY_X); VADDPS(AY, DY, AY); VCOPYU128TOL128(AZ, DZ_X); VADDPS(AZ, DZ, AZ); VCOPYU128TOL128(PHI, MJ_X); VADDPS(PHI, MJ, PHI); VSTORPS(AX_X, fodata->ax); VSTORPS(AY_X, fodata->ay); VSTORPS(AZ_X, fodata->az); VSTORPS(PHI_X, fodata->phi); } Central force with an arbitrary shape {#sec:methodarbitrary} ------------------------------------- In this section, we describe how to accelerate the computation of arbitrarily shaped forces $f(r)/r$, using the AVX instructions, where $f(r)$ is a user-specified function in equation (\[eq:arbitrary\_force\]). Note that the inter-particle softening is also expressed in the force shape function $f(r)$, as well as the long range cut-off. Arbitrary shaped softening including the Plummer softening, $S2$ softening, etc. can be set. The function $f(r)$ is assumed to shape; almost constant at $r<\epsilon$, rapidly decreases at larger $r$, and reaches zero at $r=r_{\rm cut}$. Such assumptions are satisfied in the inter-particle force calculations of PPPM or TreePM methods. In order to calculate central forces with an arbitrary shape in equation (\[eq:arbitrary\_force\]), we refer to a pre-calculated look-up table of $f(r)/r$ and use the linear interpolation between the sampling points. In § \[sec:maketable\] and \[sec:usetable\], we describe our scheme to construct the look-up table, and procedure to calculate the force by using the look-up table with the AVX instructions, respectively. ### Construction of an optimized look-up table {#sec:maketable} In terms of numerical accuracy, the look-up table is preferred to have a large number of sampling points between $0\le r\le r_{\rm cut}$. On the other hand, the size of the look-up table should be as small as possible to avoid cache misses for fast calculations. Thus, it is important how to choose sampling points of the look-up table in order to satisfy such exclusive requirements: accuracy and fast calculations of forces. In many previous implementations, sampling points of the look-up table are chosen so that the sampling points have equal intervals in a squared inter-particle distance $r^2$ at $0<r<r_{\rm cut}$. However, the sampling with equal intervals in $r^2$ is not a good choice, because it has coarser intervals at a smaller inter-particle distance, and the force shape at $r\lesssim\epsilon$ is poorly sampled if the number of sampling points is not large enough, while the shape at $r\simeq r_{\rm cut}$ is sampled fairly well, or even redundantly (see the top panel of Figure \[fig:binning\]). Typically speaking, tens of thousand sampling points in the region $0<r<r_{\rm cut}$ are required to assure the sufficient force accuracy if sampling with equal intervals in $r^2$ is adopted. Such look-up tables need several hundred kilobytes in single-precision, and do not fit into a low-level cache memory. The desirable sampling of the force shapes, therefore, should have almost equal intervals in $r$ at short distances $r \lesssim \epsilon$, and intervals proportional to $r$ (or equal intervals in $\ln r$) at long distances. In the following, we realize such a sampling by adopting rather a new binning scheme, with which we can compute the force efficiently. Here, we consider to construct a look-up table of $f(r)/r$ in the range of $0<r<r_{\rm cut}$. In our binning scheme, the indices of the look-up table are calculated by directly extracting the fraction and the exponent bits of the IEEE754 format of squared inter-particle distances. First, the squared distance $r^2$ is affine-transformed to a single-precision floating-point number $s\equiv r^2 (s_{\rm max}-2)/r^2_{\rm cut}+2$ so that $s$ is in the range of $s_{\rm min} < s < s_{\rm max}$, where $s_{\rm min}\equiv 2$ and $s_{\rm max}\equiv 2^{2^{E}}(2-1/2^F)$. Here, $E$ and $F$ are the pre-defined positive integers, and the numbers of exponent and fraction bits extracted in computing the indices of the look-up table, respectively. Binary expressions of $s_{\rm min}$ and $s_{\rm max}$ in the IEEE754 format of single-precision (32-bit) in the case of $E=4$ and $F=6$ are shown in Table \[tab:IEEE754\]. Except that the most significant bit of the exponent part is always 1, all the bits of $s_{\rm min}$ are 0, and as for $s_{\rm max}$, only the lower $E$ bits of the exponent and the higher $F$ bits of the fraction are 1. Next, the indices of the look-up table for the squared distances $r^2$ or $s$ are computed by extracting the lower $E$ bits of the exponent and the higher $F$ bits of the fraction of $s$ (underlined portion of exponent and fraction bits in Table \[tab:IEEE754\]) and reinterpreting it as an integer. This procedure can be done by applying a logical right shift by $23-F$ bits, and a bitwise-logical AND with $2^{E+F}-1$ to $s$. It should be noted that the resulting size of the look-up table is $2^{E+F}$. [c\|\|c\|c\|c\|c]{} $r$ & $s$ & exponent bits & fraction bits & index\ 0&$2$ ($s_{\rm min}$) & 1000 & 00000000000000000 & 0\ $r_{\rm cut}/2$ & $3.2514\times10^4$ & 1000 & 00000001100000000 & 895\ $r_{\rm cut}$ & --------------------- $1.3005\times 10^5$ ($s_{\rm max}$) --------------------- : $s$-values, their exponent and fraction bits in the IEEE754 expressions, and their indices in the table for $r=0$, $r_{\rm cut}/2$ and $r_{\rm cut}$ in the case of $E=4$ and $F=6$ (underlined portion of exponent and fraction bits). []{data-label="tab:IEEE754"} & 1000 & 00000000000000000 & 1023\ An affine-transformed squared distance at a sampling point with an index specified by a lower $E$ exponent bits $b_{\rm E}$ and an upper $F$ fraction bits $b_{\rm F}$ is expressed as $$s_{b_{\rm E},b_{\rm F}} = 2^{b_{\rm E}+1}\left(1+\frac{b_{\rm F}}{2^F}\right) \;\; \left(0 \le b_{\rm E} < 2^E, \; 0 \le b_{\rm F} < 2^F \right).$$ The ratio between inter-particle distances whose affine-transformed squared distances are $s_{(b_{\rm E}+1),b_{\rm F}}$ and $s_{b_{\rm E},b_{\rm F}}$ is given by $$\begin{aligned} \frac{r_{(b_{\rm E}+1),b_{\rm F}}}{r_{b_{\rm E},b_{\rm F}}} &=& \left(\frac{s_{(b_{\rm E}+1),b_{\rm F}}-2}{s_{b_{\rm E},b_{\rm F}}-2}\right)^{1/2} \nonumber \\ &\simeq& 2^{1/2}, \label{eq:comp_e}\end{aligned}$$ where $b_{\rm E} \gg 1$ is assumed for the last approximation. The interval between inter-particle distances whose affine-transformed distances are $s_{b_{\rm E},(b_{\rm F}+1)}$ and $s_{b_{\rm E},b_{\rm F}}$ is calculated as $$\begin{aligned} r_{b_{\rm E},(b_{\rm F}+1)} - r_{b_{\rm E},b_{\rm F}} &=& \left(\frac{s_{b_{\rm E},(b_{\rm F}+1)}-2}{s_{\rm max}-2}\right)^{1/2} - \left(\frac{s_{b_{\rm E},b_{\rm F}}-2}{s_{\rm max}-2}\right)^{1/2} \nonumber \\ &\simeq& \frac{1}{(2^F+b_{\rm F})^{1/2}} \left(\frac{2^{b_{\rm E}+1}/2^{F+2}}{s_{\rm max}-2}\right)^{1/2}, \label{eq:comp_f}\end{aligned}$$ where we also assume $b_{\rm E} \gg 1$ and $F \gg 1$ for the last approximation. Therefore, the sampling points with the same fraction bits are distributed uniformly in logarithmic scale, and those with the same exponent bits are aligned uniformly in linear scale unless the fraction bit is small. As an example, we illustrate how the sampling points of the look-up table depend on the pre-defined integers $E$ and $F$ in Figure \[fig:comp\_ef\]. We first see the cases in which either of $E$ and $F$ is zero, in order to see the roles of the integers $E$ and $F$. As seen in Figure \[fig:comp\_ef\], the intervals of sampling points are roughly uniform in linear scale for the case $E=0$ (the bottom line in the top panel), and uniform in logarithmic scale for the case $F=0$ (the middle line in the bottom panel), unless $r/r_{\rm cut}$ is small. As expected above, the integers $E$ and $F$ control the number of sampling points in logarithmic and linear scales, respectively. By comparing the sampling points with $(E,F)=(4,0)$ and those with $(4,2)$ (see the top panel of Figure \[fig:comp\_ef\]), it can be seen that all intervals of the sampling points with $(E,F)=(4,0)$ (indicated by the vertical dashed lines and double-headed arrows) are divided nearly equally into $2^F=4$ regions by the sampling points with $(E,F)=(4,2)$. Thus, our binning scheme is a hybrid of the linear and logarithmic binning schemes. ![Sampling points of an inter-particle distance for a look-up table in various cases of pre-defined integers $E$ and $F$. The top and bottom panels take horizontal axes in linear and logarithmic scales, respectively.[]{data-label="fig:comp_ef"}](fig04.eps) Figure \[fig:decide\_ef\] shows the comparison of the several binning in which the number of sampling points is fixed to $2^{E+F}=2^6$. One can see that the binning with $(E,F)=(4,2)$ has sufficient sampling points in the range of $10^{-3}\le r/r_{\rm cut}\le 10^0$, whereas the binning with the other sets of $(E,F)$ only samples the region of $10^{-2}\le r/r_{\rm cut}<10^0$. The number of the extracted exponent bit $E$ should be large enough so that the scale of the softening length should be sufficiently resolved. For example, if $\epsilon/r_{\rm cut}\lesssim 10^{-2}$, $E$ should be set to at least equal to or larger than $4$. ![Comparison of binning among the same number of sampling points in various cases of the integers $E$ and $F$.[]{data-label="fig:decide_ef"}](fig05.eps) In List \[list:bit\_binning\], we present routines for constructing the look-up table. In our implementation, the look-up table contains two values: one is the force at a sampling point $r_k$, $$G^0_k = \frac{f(r_k)}{r_k},$$ and the other is its difference from the next sampling point $r_{k+1}$ divided by the interval of the affine-transformed squared distance $$G^1_k = \frac{G^0_{k+1}-G^0_k}{s_{k+1}-s_k}$$ where subscript $k$ indicates indices of the look-up table, and is expressed as $k=2^{F} \times b_{\rm E} + b_{\rm F}$. Using these two values, we can compute the linear interpolation of $f(r)/r$ at a radius $r$ with $r_{k}\le r \le r_{k+1}$ by $G^0_k + (s-s_k)G^1_k$. The $G^0_k$ and $G^1_k$, are stored in a two-dimensional array declared as `Force_table[TBL_SIZE][2]`, where `TBL_SIZE` is the number of the sampling points ($2^{E+F}$) and the values of the $G^0_k$ and $G^1_k$ are stored in the `Force_table[k][0]` and `Force_table[k][1]`, respectively. Since the values of $G^0_k$ and $G^1_k$ are stored in the adjacent memory address, we can avoid the cache misses in computing the linearly interpolated values of $f(r)/r$. #define EXP_BIT (4) #define FRC_BIT (6) #define TBL_SIZE (1 << (EXP_BIT+FRC_BIT)) // 1024 extern float Force_table[TBL_SIZE][2]; // 8 kB union pack32{ float f; unsinged int u; }; void generate_force_table(float rcut) { unsigned int tick; float fmax, r2scale, r2max; union pack32 m32; float force_func(float); tick = (1 << (23-FRC_BIT)); fmax = (1 << (1<<EXP_BIT))(2.0-1.0/(1<<FRC_BIT)); r2max = rcutrcut; r2scale = (fmax-2.0f)/r2max; for(i=0,m32.f=2.0f;i<TBL_SIZE;i++,m32.u+=tick) { float f, r2, r; f=m32.f; r2 = (f-2.0)/r2scale; float r = sqrtf(r2); Force_table[i][0] = force_func(r); } for(i=0,m32.f=2.0f;i<TBL_SIZE-1;i++) { float x0 = m32.f; m32.u += tick; float x1 = m32.f; float y0 = Force_table[i][0]; float y1 = (i==TBL_SIZE-1) ? 0.0 : Force_table[i+1][0]; Force_table[i][1] = (y1-y0)/(x1-x0); } Force_table[i][1] = 0.0f; } In Figure \[fig:binning\], we compare the conventional binning with equal intervals in squared distances to our binning with $E=4$ and $F=2$ (i.e. 64 sampling points), for the $S2$-force shape [@Hockney81] used in the PPPM scheme. Although we adopt $F=5$ in the rest of this paper, we set $F=2$ here just for good visibility of the difference of the two binning schemes. It should be noted that the number of sampling points is the same (64) in both schemes. Compared with the conventional binning scheme in the top panel, our binning scheme can faithfully reproduce the given functional form even at distances smaller than the gravitational softening length. ![Binning of $f(r)/r$ in the conventional scheme with 64 constant intervals in $r^2$ (top panel) and in our scheme with $E=4$ and $F=2$ (bottom panel) between $[0,r_{\rm cut}]$. Although we adopt $F=5$ elsewhere in this paper, we set $F=2$ here for viewability. $R(r,\epsilon)-R(r,r_{\rm cut})$ is assumed as a functional form of $f(r)$, in which $R(r,\eta)$ is the $S2$-profile [@Hockney81] (see equation (\[eq:s2\])). Solid lines indicate the shape of $f(r)/r$. Vertical dashed lines in both panels are the locations of the gravitational softening length $\epsilon$.[]{data-label="fig:binning"}](fig06.eps) ### Procedure of force calculation {#sec:usetable} In calculating the arbitrary central forces, the data of $i$- and $j$-particles are stored in the structures `Ipdata` and `Jpdata`, respectively, in the same manner as described in the case for calculating the Newton’s force, except that the coordinates of $i$- and $j$-particles are scaled as $$\tilde{{{\boldsymbol r }}}_i = \frac{{{\boldsymbol r }}_i}{r_{\rm cut}/\sqrt{s_{\rm max}-2}}, \label{eq:scalingposition}$$ so that we can quickly compute the affine-transformed squared inter-particle distances between $i$- and $j$-particles. As in the case of the Newton’s force, we compute the forces of four $i$-particles exerted by two $j$-particles using the AVX instructions. Using the scaled positions of the particles, the calculation of the forces is performed in the force loop as follows; 1. Calculate an affine-transformed distance between $i$- and $j$-particles, $s$, as $$s = \min \left(\|\tilde{{\boldsymbol r }}_j-\tilde{{\boldsymbol r }}_i\|^2+2, s_{\rm max}\right),$$ where the function “$\min$” returns the minimum value among arguments. 2. Derive an index $k$ of the look-up table from the affine-transformed squared distance, $s$, computed in the previous step by applying a bitwise-logical right shift by $23-F$ bits and reinterpreting the result as an integer. 3. Refer to the look-up table to obtain $G^0_k$ and $G^1_k$. Note that the address of the pointer to `fcut` is decremented by [1<<(30-(23-F))]{} in advance (see line 24 in List \[list:arbitraryforce\]) to correct the effect of the most significant exponential bit of $s$. 4. Derive an affine-transformed distance $s_k$ that corresponds to the $k$-th sampling point $r_k$ by applying a bitwise-logical left shift by $23-F$ bits to $k$ and reinterpreting the result as a single-precision floating-point number. 5. Compute the value of $f(\|{{\boldsymbol r }}_j-{{\boldsymbol r }}_i\|)/\|{{\boldsymbol r }}_j-{{\boldsymbol r }}_i\|$ by the linear interpolation of $G^0_k$ and $G^0_{k+1}$. Using the values of $G^0_k$ and $G^1_k$, the interpolation can be performed as $$\frac{f(\|{{\boldsymbol r }}_j-{{\boldsymbol r }}_i\|)}{\|{{\boldsymbol r }}_j-{{\boldsymbol r }}_i\|} = G^0_k + G^1_k \left(s - s_k\right).$$ 6. Accumulate scaled “forces” on $i$-particles as $$\tilde{{\boldsymbol a }}_i = \sum_j^N m_j \frac{f(\|{{\boldsymbol r }}_j-{{\boldsymbol r }}_i\|)}{\|{{\boldsymbol r }}_j-{{\boldsymbol r }}_i\|}(\tilde{{\boldsymbol r }}_j-\tilde{{\boldsymbol r }}_i)$$ After the force loop, the scaled “forces” are rescaled back as $${{\boldsymbol a }}_i = \frac{r_{\rm cut}}{\sqrt{s_{\rm max}-2}}\tilde{{\boldsymbol a }}_i.$$ The actual code of the force loop for the calculation of the central force with an arbitrary force shape is shown in List \[list:arbitraryforce\]. Note that bitwise-logical shift instructions such as `VPSRLD` and `VPSLLD` can be operated only to XMM registers or the lower 128-bit of YMM registers. In order to operate bitwise-logical shift instructions to data in the upper 128-bit of a YMM register, we have to copy the data to the lower 128-bit of another YMM register. Bitwise-logical shift operations to the upper 128-bit of YMM registers are supposed to be implemented in the future AVX2 instruction set. Also note that we cannot refer to the look-up table in a SIMD manner and have to issue the `VLOADLPS` and `VLOADHPS` instructions one by one (see lines 89–92 and 94–97 in List \[list:arbitraryforce\]). Except for those operations, all the other calculations are performed in a SIMD manner using the AVX instructions. #define FRC_BIT (6) #define ALIGN32 __attribute__ ((aligned(32))) #define ALIGN64 __attribute__ ((aligned(64))) typedef float v4sf __attribute__ ((vector_size(16))); typedef struct ipdata_reg{ float x[8]; float y[8]; } Ipdata_reg, pIpdata_reg; void GravityKernel(pIpdata ipdata, pJpdata jp, pFodata fodata, int nj, float fcut[][2], v4sf r2cut, v4sf accscale) { int j; unsigned long int ALIGN64 idx[8] = {0, 0, 0, 0, 0, 0, 0, 0}; Ipdata_reg ALIGN32 ipdata_reg; static v4sf two = {2.0f, 2.0f, 2.0f, 2.0f}; fcut -= (1<<(30-(23-FRC_BIT))); VZEROALL; VLOADPS(ipdata->x[0], X2_X); VLOADPS(ipdata->y[0], Y2_X); VLOADPS(ipdata->z[0], Z2_X); VLOADPS(r2cut, R2CUT_X); VLOADPS(two, TWO_X); VBCASTL128(X2, X2); VSTORPS(X2, ipdata_reg.x[0]); VBCASTL128(Y2, Y2); VSTORPS(Y2, ipdata_reg.y[0]); VBCASTL128(Z2, ZI); VBCASTL128(R2CUT, R2CUT); VBCASTL128(TWO, TWO); VLOADPS(jp, MJ); jp += 2; VBCAST0(MJ, X2); VBCAST1(MJ, Y2); VBCAST2(MJ, Z2); VSUBPS_M(ipdata_reg.x, X2, DX); VMULPS(DX, DX, X2); VADDPS(TWO, X2, X2); VSUBPS_M(ipdata_reg.y, Y2, DY); VMULPS(DY, DY, Y2); VADDPS(X2, Y2, Y2); VSUBPS(ZI, Z2, DZ); VMULPS(DZ, DZ, Z2); VADDPS(Y2, Z2, Y2); VBCAST3(MJ, MJ); VMULPS(MJ, DX, DX); VMULPS(MJ, DY, DY); VMULPS(MJ, DZ, DZ); VMINPS(R2CUT, Y2, Z2); for(j = 0; j < nj; j += 2){ VLOADPS(jp, MJ); jp += 2; VCOPYU128TOL128(Z2, Y2_X); VPSRLD(23-FRC_BIT, Y2_X, Y2_X); VPSRLD(23-FRC_BIT, Z2_X, X2_X); VSTORPS(X2_X, idx[0]); VSTORPS(Y2_X, idx[4]); VPSLLD(23-FRC_BIT, Y2_X, Y2_X); VPSLLD(23-FRC_BIT, X2_X, X2_X); VGATHERL128(Y2, X2, Y2); VSUBPS(Y2, Z2, Z2); VBCAST0(MJ, X2); VBCAST1(MJ, Y2); VSUBPS_M(ipdata_reg.x, X2, X2); VSUBPS_M(ipdata_reg.y, Y2, X2); VLOADLPS(fcut[idx[4]], BUF0_X); VLOADHPS(fcut[idx[5]], BUF0_X); VLOADLPS(fcut[idx[0]], BUF1_X); VLOADHPS(fcut[idx[1]], BUF1_X); VGATHERL128(BUF0, BUF1, BUF1); VLOADLPS(fcut[idx[6]], BUF2_X); VLOADHPS(fcut[idx[7]], BUF2_X); VLOADLPS(fcut[idx[2]], BUF0_X); VLOADHPS(fcut[idx[3]], BUF0_X); VGATHERL128(BUF2, BUF0, BUF2); VMIX1(BUF1, BUF2, BUF0); VMIX0(BUF1, BUF2, BUF2); VMULPS(Z2, BUF0, BUF0); VBCAST2(MJ, Z2); VBCAST3(MJ, MJ); VSUBPS(ZI, Z2, Z2); VADDPS(BUF0, BUF2, BUF2); VMULPS(BUF2, DX, DX); VMULPS(BUF2, DY, DY); VMULPS(BUF2, DZ, DZ); VADDPS(DX, AX, AX); VADDPS(DY, AY, AY); VADDPS(DZ, AZ, AZ); VCOPYALL(X2, DX); VCOPYALL(Y2, DY); VCOPYALL(Z2, DZ); VMULPS(X2, X2, X2); VMULPS(Y2, Y2, Y2); VMULPS(Z2, Z2, Z2); VADDPS(TWO, X2, X2); VADDPS(Z2, Y2, Y2); VADDPS(X2, Y2, Y2); VMULPS(MJ, DX, DX); VMULPS(MJ, DY, DY); VMULPS(MJ, DZ, DZ); VMINPS(R2CUT, Y2, Z2); } VCOPYU128TOL128(AX, X2_X); VADDPS(AX, X2, AX); VCOPYU128TOL128(AY, Y2_X); VADDPS(AY, Y2, AY); VCOPYU128TOL128(AZ, Z2_X); VADDPS(AZ, Z2, AZ); VMULPS_M(accscale, AX_X, AX_X); VMULPS_M(accscale, AY_X, AY_X); VMULPS_M(accscale, AZ_X, AZ_X); VSTORPS(AX_X, fodata->ax); VSTORPS(AY_X, fodata->ay); VSTORPS(AZ_X, fodata->az); } Although the AVX instruction set takes the non-destructive 3-operand form, the copy instruction between registers appeared in the code above, which was intended to avoid the inter-register dependencies. Parallelization on multi-core processors ---------------------------------------- On multi-core processors, we can parallelize the calculations of the forces of $i$-particles for both of the Newton’s force and arbitrary central forces using the [OpenMP]{} programming interface by assigning a different set of four $i$-particles onto each processor core. List \[list:parallel\] shows a code fragment for the parallelization of the computations of the Newton’s force. The calculation of an arbitrary force can be parallelized similarly to that of Newton’s force. #define ISIMD 4 extern Ipdata ipos[NI_MEMMAX / ISIMD]; extern Jpdata jpos[NJ_MEMMAX]; extern Fodata iacc[NI_MEMMAX / ISIMD]; int nig = ni / ISIMD + (ni % ISIMD ? 1 : 0) #pragma omp parallel for for(i = 0; i < nig; i++) GravityKernel(&ipos[i], &iacc[i], jpos, nj); Application programming interfaces ---------------------------------- With the implementations of the force calculation accelerated with the AVX instructions described above, we develop a set of application programming interfaces (APIs) for $N$-body simulations, which is compatible to GRAPE-5 library[^4], except that our library do not support functions to search for neighbours of a given particle. The APIs are shown in List \[list:api\]. `g5_set_xmj` sends the data of $j$-particles to the array of the structure `Jpdata`. `g5_calculate_force_on_x` sends the data of $i$-particles to the array of the structure `Ipdata`, and computes the forces and potentials of $i$-particles and returns them into the arrays `ai` and `pi`, respectively. In the function `g5_open`, we derive statistical bias of the fast approximation of inverse-square-root, `VRSQRTPS` instruction. As [@Nitadori06] reported, the results of this instruction contains a bias which is implementation-dependent. We statistically correct this bias in the same way as [@Nitadori06]. Softening length and the number of $j$-particles are set by the functions `g5_set_eps_to_all` and `g5_set_n`, respectively. `g5_close` does nothing and is just for compatibility with the GRAPE-5 library. List \[list:sample\] shows a code fragment to perform an $N$-body simulation, using this APIs. void g5_open(void); void g5_close(void); void g5_set_eps_to_all(double eps); void g5_set_n(int nj); void g5_set_xmj(int adr, int nj, double (xj)[3], double mj); void g5_calculate_force_on_x(double (xi)[3], double (ai)[3], double pi, int ni); int n; // The number of particles double m[NMAX]; // Mass double x[NMAX][3]; // Position double v[NMAX][3]; // Velocity double a[NMAX][3]; // Force double p[NMAX]; // Potential double t; // Time double tend; // Time at the finish time double dt; // Timestep void time_integrator(int, double ()[3], double ()[3], double (*)[3] double); // Function for time integration g5_open(); g5_set_eps_to_all(eps); g5_set_n(n); while(t < tend){ g5_set_xmj(0,n,x,m); g5_calculate_force_on_x(x,a,p,n); time_integrator(n,x,v,a,dt); t += dt; } g5_close(); For the version of arbitrary force shape, we provide a new API call to set the force-table through a function pointer, which is not compatible to the GRAPE-5 API. Accuracy {#sec:accuracy} ======== Newton’s force {#newtons-force} -------------- We investigate accuracy of forces and potentials obtained by our implementation for Newton’s force. For this purpose, we compute the forces and potentials of particles in the Plummer models using our implementations and compare them with those computed fully in double-precision floating-point numbers without any explicit use of the AVX instructions. For the calculations of the forces and the potentials, we adopt the direct particle-particle method and the softening length of $4r_{\rm v}/N$, where $r_{\rm v}$ is a virial radius of the Plummer model and $N$ is the number of particles. Figure \[fig:newton\_error\] shows the cumulative distribution of relative errors in the forces and the potentials of particles, $$\frac{\|{{\boldsymbol a }}_{\rm AVX}-{{\boldsymbol a }}_{\rm DP}\|}{\|{{\boldsymbol a }}_{\rm DP}\|},$$ and $$\frac{\|\phi_{\rm AVX}-\phi_{\rm DP}\|}{\|\phi_{\rm DP}\|},$$ where ${{\boldsymbol a }}_{\rm AVX}$ and $\phi_{\rm AVX}$ are the force and the potential calculated using our implementation, and ${{\boldsymbol a }}_{\rm DP}$ and $\phi_{\rm DP}$ are those computed fully in double-precision. It can be seen that most of the particles have errors less than $10^{-4}$. These errors primarily come from the approximate inverse-square-root instruction `VRSQRTPS`, whose accuracy is about 12-bit, and consistent with the typical errors of $\simeq 10^{-4}$. While the errors of the forces are distributed down to less than $10^{-7}$, the errors of the potentials are mostly larger than $\simeq 3 \times 10^{-5}$. It can be ascribed to the way of excluding the contribution of self-interaction to the potentials. In computing a potential of the $i$-th particle, we accumulate the contribution from particle pairs between the $i$-th particle and all the particles including itself, and then subtract the contribution of the potential between the $i$-th particle and itself, $-m_i/\epsilon$ to finally obtain the correct potential of the $i$-th particle. Note that the potential between the $i$-th particle and itself is largest among the potentials between the $i$-th particle and all the particles, since the separation between $i$-particle and itself is zero. Thus, the subtraction of the “potential” due to the self-interaction causes the cancellation of the significant digits, and consequently degrades the accuracy of the potentials. A remedy for such degradation of the accuracy is to avoid the self-interaction in the force loop. In fact, we do so in calculating the potentials in double-precision ($\phi_{\rm DP}$) in Figure \[fig:newton\_error\]. However, such treatment requires conditional bifurcation inside the force loop, and significantly reduces the computational performance. The potentials of particles are usually necessary only for checking the total energy conservation, and the accuracy obtained in our implementation is sufficient for that purpose. For these reasons, we choose the original way to compute the potentials of particles in our implementation. ![image](fig07.eps){width="16cm"} Central force with an arbitrary shape {#central-force-with-an-arbitrary-shape} ------------------------------------- In order to see accuracies of central forces with an arbitrary shape obtained in our implementation, we choose a force shape which is frequently adopted in cosmological $N$-body simulations using PPPM or TreePM methods. Such methods are comprised of the particle–mesh (PM) and the particle–particle (PP) parts which compute long- and short-range components of inter-particle forces, respectively. Our implementation of the calculation of arbitrarily-shaped central forces can accelerate the calculation of the PP part, in which the force shape is different from the Newton’s force and is expressed as $$f(r) = R(r,\epsilon) - R(r,r_{\rm cut}), \label{eq:s2pp}$$ where $R(r,a)$ is the so-called $S2$-profile with a softening length of $a$ [@Hockney81] given by $$\displaystyle R(r,a) = \left\{ \begin{array}{l} \bigl(224\xi-224\xi^3+70\xi^4+48\xi^5-21\xi^6\bigr)/35a^2\\ \;\;\; \mbox{for ($0 \le \xi < 1$)} \\ \bigl(12/\xi^2-224+896\xi-840\xi^2+224\xi^3+70\xi^4\\ \;\;\; -48\xi^5+7\xi^6\bigr)/35a^3 \; \mbox{for ($1 \le \xi < 2$)} \\ \frac{1}{r^2} \; \mbox{for ($2 \le \xi$)} \\ \end{array} \right. . \label{eq:s2}$$ We calculate forces exerted between $4$K particle pairs with various separations uniformly distributed in $\ln(r)$ in the range of $5\times 10^{-3} < r/r_{\rm cut} < 1$ using our implementation described in section \[sec:methodarbitrary\], where $1$K is equal to $1024$. We set $\epsilon$ and $r_{\rm cut}$ to $3.125 \times 10^{-3}$ and $4.6875 \times 10^{-2}$, and masses to unity. In creating the look-up table of the force shape, we set $E=4$ and $F=5$. Figure \[fig:s2\_error\] shows a functional form of $R(r,\epsilon)$ (solid curve) and $f(r)$ (dashed curve) in the top panel and relative errors of forces including both PP and PM parts, i.e. $R(r,\epsilon)$, in the bottom panel as a function of $r/r_{\rm cut}$. In Figure \[fig:s2\_error\], we can see that the relative errors are less than $10^{-3}$, which are sufficiently accurate for cosmological $N$-body simulations. ![Shape of $R(r,\epsilon)$ and $f(r)$ (upper panel) and the relative errors of forces of particle pairs with a separation $r$ (bottom panel) as a function of $r/r_{\rm cut}$, where the forces include both PP and PM parts. Here, $R_{\rm AVX}$ and $R_{\rm DP}$ are, respectively, an absolute force calculated with our implementation and that obtained by performing all the calculations in double-precision without referring to the look-up table. The separations of particle pairs are distributed uniformly in $\ln(r)$ in the range of $5 \times 10^{-3} < r/r_{\rm cut} < 1$.[]{data-label="fig:s2_error"}](fig08.eps) Performance {#sec:performance} =========== In this section, we present the performance of our implementation of the collisionless $N$-body simulation using the AVX instructions (hereafter AVX-accelerated implementation). For the measurement of the performance, we use an Intel Core i7–2600 processor with 8MB cache memory and a frequency of $3.40$ GHz, which contains four processor cores. In measuring the performance, Intel Turbo Boost Technology is disabled, and Intel Hyper-Threading Technology (HTT) is enabled. A compiler which we adopt is [GCC 4.4.5]{}, with options [-O3 -ffast-math -funroll-loops]{}. To see the advantage of the AVX instructions relative to the SSE instructions, we also develop the implementations with the SSE instructions rather than the AVX instructions both for Newton’s force and arbitrarily-shaped force (SSE-accelerated implementation). Newton’s force {#newtons-force-1} -------------- First, we show the performance of our implementation for Newton’s force. The performance is measured by executing the direct particle-particle calculation of the Plummer model with the number of particles from 0.5K to 32K. The left panel of Figure \[fig:newton\_pfm\] depicts the performances of the AVX- and SSE-accelerated implementations. For comparison, we also show the performance of an implementation without any explicit use of SIMD instructions (labeled as “w/o SIMD” in the left panel of Figure \[fig:newton\_pfm\]). The numbers of interactions per second are $2 \times 10^9$ in the case of the AVX-accelerated implementation with a single thread, which corresponds to $75$ GFLOPS, where the number of floating-point operations for the computation of force and potential for one pair of particles is counted to be $38$. The performances of the SSE- and AVX-accelerated implementations with a single thread are higher than those without SIMD instructions by $10$ and $20$ times, respectively, and higher than those expected from the degree of concurrency of the SSE and AVX instructions for single-precision floating-point number (4 and 8, respectively). This is because a very fast instruction of approximate inverse-square-root is not used in the “w/o SIMD” implementation. On the other hand, the performance with the AVX-accelerated implementation is higher than that of the SSE-accelerated implementation roughly by a factor of two as expected. Furthermore, in the left panel of Figure \[fig:newton\_pfm\], we show the performance of a GPU-accelerated $N$-body code based on the direct particle-particle method implemented using the CUDA language, where the GPU board is NVIDIA GeForce GTX 580 connected through the PCI-Express Gen2 x16 link. The GPU-accelerated $N$-body code computes the forces and potentials of the particles using GPUs, and integrate the equations of the motion of the particles on a CPU. Thus, the communication of the particle data between the main memory of the host machine and the device memory on the GPU boards is required, and can hamper the total efficiency of the code. Of course, if all the calculations are performed on GPUs, we might not suffer from such overhead. However, the performance of such implementation cannot be fairly compared with those of the AVX- and SSE-accelerated implementations, because the communication of the particle data is inevitable when we perform $N$-body simulations with multiple GPUs or with multiple nodes equipped with GPUs, regardless of the $N$-body solvers such as Tree and TreePM methods. The performances of the AVX- and SSE-accelerated implementations are almost independent of the total number of particles, $N$. On the other hand, the performance of the GPU-accelerated implementation strongly depends on the number of particles $N$, due to the non-negligible overhead caused by the particle data communication. For $N=0.5$K, the performance of the GPU-accelerated implementation is only 5% of that for $N=32$K. Thus, for small $N$ ($0.5$K and $1$K), the performance of the AVX-accelerated implementation with four threads is higher than that with GPU-accelerated implementation, although, for large $N$ ($4{\rm K}$–$32{\rm K}$), the performance of the GPU-accelerated implementation is higher than that of the AVX-accelerated implementation. These features can be explained by the communication overhead in the GPU-accelerated implementation. So far, we see the performance of our code in the case that both the numbers of $i$- and $j$-particles ($N_i$ and $N_j$, respectively) are the same and equal to $N$. However, in actual computations of forces in collisionless $N$-body simulations based on various $N$-body solvers such as PPPM, Tree, and TreePM methods, the numbers of $i$- and $j$-particles $N_i$ and $N_j$ are much smaller than the total number of particles $N$. In the Tree method modified for the effective force with external hardwares or softwares as described in @Makino91, for example, $N_i$ is the number of particles, for which a tree traverse is performed simultaneously and the resultant interaction list (size $N_j$) is shared, and typically around $10$–$1000$. Furthermore, if one adopts the individual timestep algorithm, the number of $i$-particles $N_i$ gets even smaller. The number of $j$-particles $N_j$ is also decreased in Tree and TreePM methods. Therefore, we show the performance for typical $N_i$ and $N_j$ in the realistic situations of typical collisionless $N$-body simulations. The right panel of Figure \[fig:newton\_pfm\] shows the performance of the AVX-accelerated implementation using four threads with four processor cores (black lines) and that of the GPU-accelerated one (red lines) for various set of $N_i$ and $N_j$. It can be seen that the obtained performance gets lower for the smaller $N_i$ and $N_j$, regardless of the implementations. For the AVX- and SSE-accelerated implementations, this feature is due to the overhead of storing the particle data into the structures `Ipdata` and `Jpdata` shown in List \[list:structures\]. The amount of the overhead of storing $i$- and $j$-particles are proportional to $N_i$ and $N_j$, respectively, and the computational cost is proportional to $N_i N_j$. Keeping this in mind the low performance with $N_i=16$ compared with those with $N_i \ge 64$ can be ascribed to the overhead of storing $j$-particles to the structure `Jpdata`. For the GPU-accelerated implementation, the overhead originates from the transfer of the particle data to the memory on GPUs. It can be seen that the performance of the AVX-accelerated implementation has rather mild dependence on $N_i$ and $N_j$, while that of the GPU-accelerated one relatively strongly depends on $N_j$. Such difference reflects the fact that the bandwidths and latency of the communication between GPUs and CPUs are rather poor compared with those of memory access between CPUs and main memory. Thus, the performance of the GPU-accelerated implementation is apparently superior to the AVX-accelerated one only when both of $N_i$ and $N_j$ are sufficiently large (say, $N_i>1$K and $N_j>4$K). ![image](fig09.eps) At the end of this section, we apply our AVX-accelerated implementation to Barnes-Hut Tree method [@Barnes86], and measure its performance. Our tree code is based on the PP part of TreePM code implemented by [@Yoshikawa05] and [@Fukushige05], in which they accelerated the calculations of the gravitational forces of the $S2$-profile using GRAPE-5 and GRAPE-6A systems under the periodic boundary condition. We modify the tree code such that it can compute the Newton’s force under the vacuum boundary condition. Since both of GRAPE-6A systems and Phantom-GRAPE library support the same APIs, we can easily utilize the capability of Phantom-GRAPE by simply exchanging the software library. Using the tree code described above, we calculate gravitational forces and potentials of all the particles in a Plummer model and a King model with the dimensionless central potential depth $W_0=9$. We measure the performance on an Intel Core i7–2600 processor. For the comparison with other codes, we also measure the performance of the same code but without any explicit use of SIMD instructions, and the publicly available code [bonsai]{} [@Bedorf12], which is a GPU-accelerated $N$-body code based on the tree method. The performance of the [bonsai]{} code is measured on a system with NVIDIA GeForce GTX 580. Since the [bonsai]{} code utilizes the quadrupole moments of the particle distribution in each tree node as well as the monopole moments in the force calculations, for a fair comparison of the performance with the [bonsai]{} code, we give our tree code a capability to use the quadrupole moments in each tree node, although the original code uses only the monopole moments. We represent these multipole moments as pseudo-particles, using pseudo-particle multipole method [@Kawai01]. Figure \[fig:treeNW\] shows the wall clock time to compute gravitational forces and potentials for each tree code. We show the both results with the code which uses the quadrupole moments (lower panels) and the one which uses only the monopole moments (upper panels). Note that the wall clock time includes the time for tree construction, tree traverse and calculations of forces and potentials but we exclude the time to integrate orbital motion of particles. As expected, the wall clock time with the AVX-accelerated implementation is roughly 10 times shorter than those without any explicit use of SIMD instructions, owing to parallelism to calculate forces and potentials. The wall clock time with the AVX-accelerated implementation is about only three times longer than those with [ bonsai]{}, despite that theoretical peak performance of Intel Core i7–2600 ($220$ GFLOPS) is lower than that of NVIDIA GeForce GTX 580 ($1600$ GFLOPS) by a factor of 7.3 in single-precision. We expect that the performance of our AVX-accelerated implementation could be close to that of the [bonsai]{} in the following situations. When we adopt individual timestep algorithm, the number of $i$-particles is effectively decreased, and a part of GPU cores becomes inactive. Thus, the performance of GPU-accelerated implementation would be degraded more rapidly than that of our AVX-accelerated implementation. Furthermore, when we use GPU-accelerated implementation on massively parallel environments, the communication between CPUs and GPUs is inevitable, which also degrades the performance of GPU-accelerated implementation. ![image](fig10.eps) Force with an arbitrary shape ----------------------------- The left panel of Figure \[fig:s2\_pfm\] shows the performance of our implementation to calculate forces with an arbitrary force shape accelerated with the AVX and SSE instructions. For the comparison, we also plot the performance of an implementation without any explicit use of the SIMD instructions. The numbers of exponent and fraction bits used to referring the look-up table are set to $E=4$ and $F=6$, respectively. The performance of the AVX-accelerated implementation with a single thread is $2$ and $6$ times higher than that of the SSE-accelerated one and the one without any SIMD instructions, respectively. These forces with the use of the AVX instructions are lower than those expected from the degree of concurrency of their SIMD operations, $8$, mainly because the reference of a look-up table is not carried out in a SIMD manner. The performance with multi-thread parallelization is almost proportional to the number of threads up to four threads. If the HTT is activated, the performance with eight threads is higher than that with four threads by a few percent. The right panel of Figure \[fig:s2\_pfm\] shows the performance of the AVX-accelerated implementation with eight threads for a various set of $N_i$ and $N_j$. For $N_i \ge 64$, the performance is almost independent of $N_i$ and $N_j$, and for $N_i = 16$ it is about half the performance with $N_i \ge 64$. This is again due to the overhead of copying $j$-particle data to the structure `Jpdata`, as is the case in the calculation of Newton’s force. Such weak dependence of the performance on $N_i$ and $N_j$ are also preferable for the calculations of the forces in the PPPM and TreePM methods especially with the individual timestep scheme. ![image](fig11.eps) Summary {#sec:summary} ======= Using the AVX instructions, the new SIMD instructions of x86 processors, we develop a numerical library to accelerate the calculations of Newton’s forces and arbitrarily shaped forces for $N$-body simulations. We implement the library by means of inline-assembly embedded in C-language with GCC extensions, which enables us to manually control the assignment of the YMM registers to computational data, and extract the full capability of a CPU core. In computing arbitrarily shaped forces, we refer to a look-up table, which is constructed with a novel scheme so that the binning is optimized to ensure good numerical accuracy of the computed forces while its size is kept small enough to avoid cache misses. The performance of the version for Newton’s forces reaches $2 \times 10^9$ interactions per second with a single thread, which is about $2$ times and $20$ times higher than those of the implementation with the SSE instructions and without any explicit use of SIMD instructions, respectively. The use of the fast inverse-square-root instruction is a key ingredient of the improvement of the performance in the implementation with the SSE and AVX instructions. The performance of the version for arbitrarily shaped forces is $2$ and $6$ times higher than those implemented with the SSE instructions and without any explicit use of the SIMD instructions. Furthermore, our implementation supports the thread parallelization on a multi-core processor with the [OpenMP]{} programming interface, and has a good scalability regardless of the number of particles. While the performance of our implementation using the AVX instructions is moderate compared with that of the GPU-accelerated implementation, the most remarkable advantage of our implementation is the fact that the performance has much weaker dependence on the numbers of $i$- and $j$-particles than that of the GPU-accelerated implementation. This feature is also the case for the calculation of the arbitrarily shaped force, and can be explained by the relatively large overhead of the data transfer between GPUs and main memory of their host computers. In actual calculations of forces with popular $N$-body solvers such as the Tree-method and the TreePM-method combined with the individual timestep scheme, the numbers of $i$- and $j$-particles cannot be always large enough to extract the full capability of GPUs. In that sense, our implementation is more suitable in accelerating the calculations of forces using the Tree- and TreePM-methods. Another advantage of our implementation is its portability. With this library, we can carry out collisionless $N$-body simulations with a good performance even on supercomputer systems without any GPU-based accelerators. Note that massively parallel systems with GPU-based accelerators, at least currently, are not ubiquitous. Even on processors other than the x86 architecture, most of them supports similar SIMD instruction sets (e.g. Vector Multimedia Extension on IBM Power series, and HPC-ACE on SPARC64 VIIIfx, etc.) Our library can be ported to these processors with some acceptable efforts. Finally let us mention the possible future improvement of our implementation. Fused Multiply-Add (FMA) instructions which have already been implemented in the “Bulldozer” CPU family by AMD Corporation, and is scheduled to be introduced in the “Haswell” processor by Intel Corporation in 2013. The use of the FMA instructions will improve the performance and accuracy of the calculations of forces to some extent. The numerical library “Phantom-GRAPE” developed in this work is publicly available at [http://code.google.com/p/phantom-grape/]{}. Acknowledgment {#acknowledgment .unnumbered} ============== We thank Dr. Takayuki Saitoh for valuable comments on this work. A. Tanikawa thanks Yohei Miki and Go Ogiya for fruitful discussion on GPU. Numerical simulations have been performed with computational facilities at the Center for Computational Sciences in University of Tsukuba. This work was supported by Scientific Research for Challenging Exploratory Research (21654026), Grant-in-Aid for Young Scientists (start-up: 21840015), the FIRST project based on the Grants-in-Aid for Specially Promoted Research by MEXT (16002003), and Grant-in-Aid for Scientific Research (S) by JSPS (20224002). K. Nitadori and T. Okamoto acknowledge financial support by MEXT HPCI STRATEGIC PROGRAM. 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[^1]: http://www.intel.com/content/dam/doc/manual/ [^2]: http://support.amd.com/us/Processor\_TechDocs/ [^3]: http://software.intel.com/en-us/avx/ [^4]: http://www.kfcr.jp/downloads/g7pkg2.2.1/g5user.pdf
--- abstract: 'We constrain the rest-frame FUV (1546Å), NUV (2345Å) and U-band (3690Å) luminosity functions (LFs) and luminosity densities (LDs) with unprecedented precision from $z\sim0.2$ to $z\sim3$ (FUV, NUV) and $z\sim2$ (U-band). Our sample of over 4.3 million galaxies, selected from the CFHT Large Area $U$-band Deep Survey (CLAUDS) and HyperSuprime-Cam Subaru Strategic Program (HSC-SSP) data lets us probe the very faint regime (down to $M_\mathrm{FUV},M_\mathrm{NUV},M_\mathrm{U} \simeq -15$ at low redshift) while simultaneously detecting very rare galaxies at the bright end down to comoving densities $<10^{-5}$ Mpc$^{-3}$. Our FUV and NUV LFs are well fitted by single Schechter functions, with faint-end slopes that are very stable up to $z\sim2$. We confirm, but self-consistently and with much better precision than previous studies, that the LDs at all three wavelengths increase rapidly with lookback time to $z\sim1$, and then much more slowly at $1<z<2$–$3$. Evolution of the FUV and NUV LFs and LDs at $z<1$ is driven almost entirely by the fading of the characteristic magnitude, $M^\star_{UV}$, while at $z>1$ it is due to the evolution of both $M^\star_{UV}$ and the characteristic number density $\phi^\star_{UV}$. In contrast, the U-band LF has an excess of faint galaxies and is fitted with a double-Schechter form; $M^\star_\mathrm{U}$, both $\phi^\star_\mathrm{U}$ components, and the bright-end slope evolve throughout $0.2<z<2$, while the faint-end slope is constant over at least the measurable $0.05<z<0.6$. We present tables of our Schechter parameters and LD measurements that can be used for testing theoretical galaxy evolution models and forecasting future observations.' author: - \| Thibaud Moutard,$^{1}$[^1] Marcin Sawicki,$^{1,2}$[^2] Stéphane Arnouts,$^{3}$ Anneya Golob,$^{1}$ Jean Coupon,$^{4}$ Olivier Ilbert,$^{3}$ Xiaohu Yang,$^{5}$ Stephen Gwyn$^{2}$\ $^{1}$Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax,\ Nova Scotia, B3H 3C3, Canada\ $^{2}$NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada\ $^{3}$Aix Marseille Université, CNRS, LAM - Laboratoire d’Astrophysique de Marseille, 38 rue F. Joliot-Curie, F-13388, Marseille, France\ $^{4}$Astronomical Observatory of the University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland\ $^{5}$Department of Astronomy, Shanghai Jiao Tong University, Dongchuan RD 800, 200240 Shanghai, China bibliography: - 'Moutard2019\_CLAUDS\_LF.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'UV & U-band luminosity functions from CLAUDS and HSC-SSP – I. Using four million galaxies to simultaneously constrain the very faint and bright regimes to z $\sim$ 3 ' --- \[firstpage\] galaxies: statistics – galaxies: luminosity function, mass function – ultraviolet: galaxies – galaxies: evolution – galaxies: star formation Introduction ============ The galaxy luminosity function (LF) and its redshift evolution is one of the most fundamental ways to characterize the galaxy population. It provides a direct probe of the hierarchical framework of galaxy formation. Defined by $\phi(L)\ dL$ as the comoving number density of galaxies with luminosity between $L$ and $L + dL$, the LF is a wavelength-dependent measurement that gives a direct test on the modelling of the baryonic physics such as star formation activity, dust attenuation and feedback processes. The present paper is concerned with galaxy LFs at rest-frame ultra-violet (UV: $\lambda = 1000-3000$ Å) and u ($\lambda = 3000-4000$ Å) wavelengths. In this wavelength regime, light in star-forming galaxies is thought to be primarily produced by short-lived massive stars. For this reason, the evolution of the UV LF has historically been used as a probe of the evolution of star-forming activity in the galaxy population. Similarly, the UV luminosity density ($\rho_{UV}$) – which is the luminosity-weighted integral of the LF, $\int L \times \phi(L)\ dL$ – is a direct measurement of the unobscured cosmic star formation density (SFRD, $\rho_{SFR})$ and its evolution with redshift, giving us a sketch of the cosmic star formation history. At $0\lesssim z\lesssim1$, this was first done by @Lilly1996, with UV LFs measurements from spectroscopic samples with optically selected sources [@Lilly1995]. At higher redshift, a lower limit on $\rho_{SFR}$ based on Lyman-break galaxies (LBGs) was determined by @Madau1996 who summed up the UV light from $U$- and $B$-band dropouts detected in the Hubble Deep Field [HDF, @Williams1996], while @Sawicki1997 presented the first measurement of $\rho_{SFR}$ between $z = 1$ and $z\sim3.5$ by making use of photometric redshifts. The realization that significant fractions of UV photons are prevented from escaping from high-$z$ star-forming galaxies by interstellar dust [e.g., @Meurer1997; @Sawicki1998] forced dust corrections to be subsequently applied to this $\rho_{UV} \rightarrow \rho_{SFR}$ conversion method. Subsequently, the GALEX satellite [@Martin2005b] allowed first measurements of the unobscured UV LF at $z\sim 0$ [@Wyder2005; @Budavari2005] and out to $z\sim1.2$ [@Arnouts2005]. The later provided the first UV LF measurements over the entire redshift range $0\le z\le 3.5$ by combining spectroscopically selected GALEX sources and photometric redshifts of optically selected sources at high redshift in the HDFs. @Schiminovich2005 used those UV-LFs to estimate the evolution of the UV luminosity density ($\rho_{UV}$) and of the cosmic star formation rate density ($\rho_{SFR}$) after accounting for typical UV attenuation due to interstellar dust in star-forming galaxies. SFRD measurements made at infra-red or sub-millimetre wavelengths – which measure the stellar energy re-radiated by interstellar dust and thus obviate the needs for dust corrections – can provide a complementary picture to that gleaned from the UV. Although such measurements have been possible for some time for high-$z$ galaxies [e.g., @Hughes1998; @Chapman2005; @Magnelli2013; @Gruppioni2013; @Goto2019], they do not yet provide significant insights at very high redshifts ($z \ga 6$), nor for low-mass galaxies which have low SFRs and low dust content [e.g., @Bouwens2009; @Bouwens2012; @Sawicki2012]. Consequently, UV LF measurements allow us the only self-consistent way to study the evolution of the galaxy population and of the SFRD [at a constant rest-frame wavelenght]{} across the entire redshift range over which galaxies are currently known to exist, $z=0\sim10$. Similarly, UV measurements let us reach galaxies that are too faint to be observed by infra-red and sub-mm surveys. This explains why UV LFs have continuously been used for estimating the evolution of the cosmic star formation rate density over the last two decades [e.g., @Steidel999; @Ouchi2004; @SawickiThompson2006b; @Dahlen2007; @Iwata2007; @Reddy2009; @vanderBurg2010; @Cucciati2012; @Sawicki2012; @McLure2013; @MadauDickinson2014; @Bouwens2015a; @Bouwens2016; @Ono2018; @Khusanova2019 and many others]. The advent of multi-wavelength datasets that contain flux measurements at a great many wavelengths (sometime as many as several dozen — e.g., @Laigle2016) allow the estimation of physical quantities for each galaxy, such as its stellar mass, and the construction of related global descriptors, such as the galaxy stellar mass functions (SMFs) and stellar mass densities (SMDs, $\rho_{M_\star}$) – e.g., [@Ilbert2013; @Muzzin2013; @Moutard2016b; @Davidzon2017]. Such “physical” measurements are an extremely powerful tool to help us understand galaxy evolution, but they suffer from some important limitations: they rely heavily on the assumptions that underpin stellar population synthesis models [e.g., @BC2003; @Maraston2005], and the spectral energy distribution (SED) -fitting technique that’s used for physical parameter estimation (e.g., @Sawicki1998 [@Papovich2001]; see @Conroy2013 for a review). Consequently, the fidelity of the physical parameter estimates continues to be challenged by studies that show that biases may exist in commonly-used approaches: for example, different galaxy star formation histories [e.g., @Leja2019], the assumed stellar initial mass function [IMF; e.g., @Salpeter1955; @Chabrier2003], the common assumption that dust acts as a uniform foreground screen [see, e.g., @Mitchell2013], or the treatment of individual galaxies as consisting of spatially-homogeneous stellar populations [e.g., @Sorba2015; @Sorba2018], can influence the inferred stellar masses and – consequently – SMFs and SMDs. While such “physical” measurements are a powerful tool to help us understand galaxy evolution, model-independent measurements, such as LFs, are therefore an essential complement. One example of direct applications of the LFs is to calibrate or validate galaxy formation models [e.g., @KitzbichlerWhite2007; @Lacey2011; @Somerville2012; @Henriques2013; @Lacey2016; @Sharma2016], since in using the directly-measured quantity (i.e., the LF), the modeller has full control over the comparison process, rather than relying on assumptions made by the observational papers. A related use of UV LFs is in the forecasting of future observations [e.g., @Williams2018; @Maseda2019]. Finally, because UV LFs probe the galaxy population at wavelengths close to those which ionize hydrogen, UV LFs are used in work that aims to assess the contribution of different types of objects to reionizing the Universe, or to maintaining it in its ionized state [e.g., @Inoue2006; @SawickiThompson2006b; @Bouwens2015b; @Ishigaki2018; @Iwata2019]. For these reasons it is important that we have the best possible measurements of the UV LFs over a wide redshift range of cosmic history. Although the situation has improved dramatically from the early days of the Hubble Deep Field, even the largest studies to date are still based on relatively small fields, such as the COSMOS field [@Scoville2007], and are thus susceptible to cosmic variance and poor statistics, particularly at the bright end. With new data that we now have in hand, we can do better. In this paper we therefore set out to provide a state-of-the-art measurement of the rest-frame FUV (1546 Å), NUV (2345 Å), and U-band (3690 Å) luminosity functions using two overlapping and complementary cutting-edge surveys: the recently-completed Canada-France-Hawaii Telescope (CFHT) Large Area U-band Deep Survey [CLAUDS, @Sawicki2019] and the ongoing HyperSuprime-Cam Subaru Strategic Program [HSC-SSP, @Aihara2018overview]. Together, these two surveys probe the Universe to an unprecedented combination of area and depth, as described in Sec. \[sect\_data\] and allow us to produce the most statistically-significant measurements of the UV LFs that are also essentially free of cosmic variance. This paper focuses on providing reference measurements of the rest-frame FUV, NUV, and U-band LFs based on these state-of-the-art surveys, notably to serve as a basis for making observational forecasts and validating theoretical models. We postpone more physically-motivated interpretations to future work (see companion paper, T. Moutard et al. in prep.). Throughout this paper, we use the standard cosmology ($\Omega_m~=~0.3$, $\Omega_\Lambda~=~0.7$ with $H_{\rm0}~=~70$ km s$^{-1}$ Mpc$^{-1}$). Magnitudes are given in the AB system [@Oke1974]. Galaxy sample {#sect_gal_sample} ============= Data {#sect_data} ---- This study uses the $U+grizy$ data from the Canada-France-Hawaii Telescope (CFHT) Large Area U-band Deep Survey (CLAUDS) and the HyperSuprime-Cam Subary Strategic Program (HSC-SSP). These surveys are described in detail in @Sawicki2019 [CLAUDS] and in @Aihara2018overview [and references therein; HSC-SSP], and the procedures for merging the datasets are described in [@Sawicki2019]. Consequently, here we give only a summary of the key details. The CLAUDS and HSC-SSP imaging data overlap over four well-studied fields, namely E-COSMOS, ELAIS-N1, DEEP2-3, and XMM-LSS, each spanning $\sim$4–6 [deg$^{2}$]{}. The $U$-band data cover 18.60 [deg$^{2}$]{} to a depth of $U_{AB}$=27.1 (5$\sigma$ in 2apertures), with selected ultra-deep sub-areas within the E-COSMOS and XMM-LSS fields that cover 1.36 [deg$^{2}$]{} to a depth of $U$=27.7 (5$\sigma$ in 2apertures). CLAUDS $U$-band data were obtained in two somewhat different CFHT/MegaCam filters: data in the ELAIS-N2 and DEEP2-3 fields were taken with the new [$u$]{} filter, while those in XMM-LSS were taken with the older [$u^$]{} filter. The E-COSMOS field contains data in the [$u$]{} filter except in the central region where both [$u$]{} and [$u^$]{} data overlap. The [$u$]{} and [$u^$]{} data are kept separate, even in areas where they overlap. The image quality of the CLAUDS data is excellent, with median seeing of 0.92. For the details of CLAUDS data see [@Sawicki2019]. The HSC-SSP project [@Aihara2018overview] provides deep Subary/HSC imaging in the $grizy$ wavebands in the same fields imaged by CLAUDS. Here we use images from the S16A internal HSC-SSP data release that are deeper than the HSC-SSP public data release 1 [PDR1, @Aihara2018dr1] with depths of $g_{AB} \sim 26.6$, $r_{AB} \sim 26.1$, $i_{AB} \sim 25.7$, $z_{AB} \sim 25.1$ and $y_{AB} \sim 24.2$ (5$\sigma$ in 2apertures), though not as deep as those from the very recent PDR2 [@Aihara2019] . Seeing in the HSC-SSP varies from band to band, with the $i$-band providing the sharpest images ($\sim$0.62); in all bands, the seeing in the HSC images is even better than the (excellent) seeing in the CLAUDS $U$-band data. Figure \[fig\_footprint\] shows the overlap of the CLAUDS (black) and HSC-SSP (green) footprints. The footprints of the the deep HSC observations are somewhat larger than those of the CLAUDS data, so the area of overlap is dictated by the extent of the CLAUDS data, i.e., 18.29 [deg$^{2}$]{} after the masking of areas around bright stars. Our survey contains two layers of different depths: - the Deep layer covers a total area of 18.29 deg$^2$ with $U\geq26.8$, $g\geq26.5$, $r\geq26.1$, $i\geq25.7$, $z\geq25.1$ and $y\geq24.2$, respectively; - while the Ultra-Deep layer covers an area of 1.54 deg$^2$ with $U\geq27.5$, $g\geq27.1$, $r\geq26.9$, $i\geq26.6$, $z\geq26.3$ and $y\geq25.0$, respectively. We use the [[SExtractor]{}]{}-based multi-band catalog described in [@Sawicki2019]. For object detection, this uses the signal-to-noise image, $\Sigma\rm SNR$ constructed from all the available $uu^grizy$ images as $$\Sigma{\mathrm{SNR}}=\sum_{i=1}^{N}\left(\frac{f_{i}-\mu_{i}}{\sigma_{i}}\right),\label{eq:combUgrizy}$$ where $f_i$ is the flux in each pixel, $\sigma_i$ is the RMS width of the background sky distribution, and $\mu_i$ is its mean. Here the index $i$ runs over MegaCam bands [$u$]{} or [$u^$]{} (or both, where available — i.e., in the central area of E-COSMOS) as well as the HSC bands $grizy$. Once the [[SExtractor]{}]{} software [@Bertin1996] has detected objects in the $\Sigma{\mathrm{SNR}}$ image, the multiband catalog is then created by running [[SExtractor]{}]{} in dual image mode, with various measurements recorded for each object, including positions, fluxes (in Kron, isophotal, and fixed-radius circular apertures), fiducial radii, ellipticities, position angles, and central surface brightnesses. For more details see [@Sawicki2019] and A. Golob et al. (submitted to MNRAS). Note that the CLAUDS $U$-band images are as deep or deeper than the HSC-SSP S16A images we used and consequently our catalog is not expected to be biased against $U$-faint objects. Small apertures are known to provide less noisy colours and therefore an improved photometric redshift accuracy than total Kron-like [@Kron1980] apertures [@Sawicki1997; @Hildebrandt2012; @Moutard2016a; @Moutard2016b]. At the same time, total fluxes are needed for deriving galaxy physical properties. Following the approach of @Moutard2016a, the final magnitudes $m_\textsc{final}$ of each source are produced by rescaling isophotal magnitudes $m_\textsc{iso}$ to the Kron-like magnitudes $m_\textsc{auto}$. To preserve the colours based on isophotal apertures, a mean rescaling factor $\delta m$ is applied in each filter $f$: $$m_{\textsc{final},f} = m_{\textsc{iso},f} + \delta m$$ with $\delta m$ defined as $$\delta m = \frac{ \sum_{f} (m_{\textsc{auto},f}- m_{\textsc{iso},f}) \times w_f}{ \sum_{f} w_f}$$ for $f = u,u^,g,r,i,z,y$ and where the weights $w_f$ are simply defined from $\sigma_\textsc{iso}$ and $\sigma_\textsc{auto}$, the photometric uncertainties on $m_\textsc{iso}$ and $m_\textsc{auto}$, with $w_f = 1 /(\sigma_{\textsc{auto},f}^2+\sigma_{\textsc{iso},f}^2)$. To properly constrain the FUV and NUV luminosities at low redshift, we complemented our photometric dataset with FUV (135-175 nm) and NUV (170-275 nm) observations from the GALEX satellite [@Martin2005b]. Both in the XMM-LSS and E-COSMOS fields, the GALEX observations we used were reduced with the [EMphot]{} code [@Guillaume2006; @Conseil2011] dedicated to extract UV photometry by using the CFHTLS (T0007) $u^$-band detections as a priors down to [$u^$]{}$\sim$25. Consequently, the astrometry of the resulting GALEX photometry is that of the CFHTLS, which enabled a straightforward position matching with our photometric dataset (with 0.5 tolerance). Galaxy identification and photometric redshift estimation {#sect_photoz} --------------------------------------------------------- ![Comparison of our photometric redshifts with spectroscopic redshifts from @bra13 [@com15; @lef13; @kri15; @lil07; @mas17; @mas19; @mcl13; @sco18; @sil15; @tas17]. The red diagonal line shows equality (perfect match) and the dashed blue lines define outliers. The total number of galaxy spectroscopic redshifts and the usual photo-z accuracy estimators (outlier rate $\eta$, scatter $\sigma_z$ and bias $b_z$) are reported in the lower-right corner, while corresponding i-band weighted estimators are reported in the upper-left corner or the figure. \[fig\_photoz\] ](figures_ok/ZpZs.pdf){width="\hsize"} To identify and remove foreground Galactic stars we use the machine-learning method and results of A. Golob et al. (submitted to MNRAS). This method uses both photometric and morphological information to classify objects as stars or galaxies. In more detail, we use HST morphological object classification in the COSMOS field from [@Leauthaud2007] to train a gradient boosted tree (GBT) machine classifier to classify objects based on their CLAUDS+HSC-SSP $Ugrizy$ magnitudes, colours, central surface brightnesses, and effective radii. Because the method uses photometric information, it does well even for faint objects where morphologies from ground-based imaging are ambiguous. Having trained the GBT machine classifier, we use it to remove from our sample all objects for which the classifier returned a value greater than 0.89. Doing so, we discarded $\sim$7.4% of the sources as stars. See A. Golob et al. (submitted to MNRAS) for details of the method and its application to our CLAUDS+HSC-SSP dataset. Our photometric redshifts are computed using a hybrid approach that combines a nearest-neighbours machine-learning method (hereafter kNN; A. Golob, in preparation; see also @Sawicki2019) with the template-fitting code [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} [@Arnouts2002; @Ilbert2006]. The kNN method uses the 30-band COSMOS photometric redshifts from [@Laigle2016] as a training set. For each object in our catalog it identifies 50 nearest neighbours in colour space and then fits a weighted Gaussian kernel density estimator (KDE), with each neighbour’s redshift weighted by $(d_\mathrm{NN}\times \Delta z)^{-1}$; here $d_\mathrm{NN}$ is the Euclidean distance in colour space to the object under consideration, and $\Delta z$ is the width of the 68% confidence interval of the neighbour’s redshift in the [@Laigle2016] catalog. We find that this method gives very good results on average (low scatter, $\sigma_z$, and bias, $b_z$) but suffers from more outliers than we would wish. Following @Moutard2016a, [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} photometric redshifts were computed by making use of the template library of @Coupon2015, while considering four extinction laws with a reddening excess E(B-V) $\leq$ 0.3, as described in @Ilbert2009. In addition, as described in @Ilbert2006, [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} tracked down and corrected for any systematic difference between the photometry and the predicted magnitudes in each band, while using the known $N(z)$ at given apparent magnitude as a prior to avoid catastrophic failures. [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} is thereby naturally well suited to take care of any fluctuation of the absolute calibration from field to field and to deal with the confusion between spectrum breaks in the absence of near-infrared observations. Our hybrid photometric redshift method combines the outputs from the kNN method and [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} as follows. We flag outliers in the kNN photo-$z$ catalogue and then replace their photometric redshift values with those from the [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} template-fitting code. Outliers are identified and flagged by comparison of the kNN redshift, $z_{KDE}$, with the [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{} redshift, $z_{LPh}$. Specifically, when the threshold of $\Delta z_{phot} = \| z_{KDE} - z_{LPh} \| / \sqrt{2} = 0.15 \times (1+\overline{z})$ is exceeded, with $\overline{z} = (z_{KDE} + z_{LPh}) /2$, we adopt $z_{LPh}$; otherwise we use $z_{KDE}$. Doing so, we notably reduced by half the number of photo-z outliers that are due to the confusion between the Lyman and Balmer breaks. Figure \[fig\_photoz\] shows the comparison of our hybrid photometric redshifts with a large sample of spectroscopic redshifts compiled from the literature. Overall, the hybrid photo-z quality is found to be very good within the ranges of redshift and magnitude we explore in our analysis, namely, up to $z = 3.5$ and for observed magnitudes $17.0<i_\mathrm{AB}<26.5$, with a scatter[^3] of $\sigma_z = 0.0213$, a median bias[^4] of $b_z = -0.0049$, and an outlier rate[^5] of $\eta = 2.443\%$. While the spectroscopic sample we assembled combines many surveys, which makes it as representative as possible, it is much brighter (and bluer) than our photometric sample. In order to account for this effect, we followed the approach of @Moutard2016b and weighted the photo-z accuracy estimators with respect to the i-band distribution of the photometric sample. Using this approach, we found weighted scatter of $\sigma^w_z = 0.024$, weighted median bias of $b^w_z = -0.0085$ and weighted outlier rate of $\eta^w = 6.023\%$ at $17.0 < i_\mathrm{AB} < 26.5$. These measurements confirmed the reliability of our hybrid photometric redshifts, which we use for the rest of the analysis that follows. Galaxy physical parameters {#sect_phys_param} -------------------------- ### Physical parameters and absolute magnitudes Our procedure for estimating galaxy rest-frame FUV, NUV, and U-band magnitudes interpolates (or, in some cases – extrapolates) from the observed photometry using spectral models fitted to the photometry. We therefore describe these models (which also yield some physical parameters for our galaxies, such as their stellar masses) before moving on to describe the estimation of rest-frame magnitudes. Absolute magnitudes and other physical parameters (stellar mass, star formation rate, etc.) were derived with the template-fitting code [L<span style="font-variant:small-caps;">e</span> P<span style="font-variant:small-caps;">hare</span>]{}, after fixing the redshift to its best estimate (i.e., our hybrid photometric redshifts – see Sec. \[sect\_photoz\]). Following @Moutard2016b, we made use of the stellar population synthesis models of @BC2003 and considered two metallicities, exponentially declining star formation histories that follow $\tau^{-1} e^{-t/\tau}$ (as described in @Ilbert2013), and three extinction laws with a maximum dust reddening of E(B-V) $=$ 0.5. Finally, we imposed a low extinction for low-SFR galaxies and the emission-line contribution was taken into account [for more details see @Moutard2016b]. We computed FUV, NUV and U-band absolute magnitudes by adopting the procedure followed by @Ilbert2005 to minimise the dependence of the absolute magnitudes to the template library. Specifically, to minimize the k-correction term, the absolute magnitude in a given passband centered on $\lambda^0$ was derived from the observed magnitude in the filter passband that was the closest from $\lambda^0 \ \times \ (1+z)$, except – to avoid measurements that are too noisy – when the apparent magnitude had an error above 0.3 mag. Moreover, all rest-frame magnitudes were derived with two different template libraries [@BC2003; @Coupon2015], which allowed us to verify that no significant systematic uncertainties were introduced by the choice of template library. ### Absolute magnitude error budget {#sect_absmag_err} Of particular importance for our analysis are the uncertainties affecting the absolute magnitudes for which we measure the luminosity function. The first source of uncertainty is the fitting error, $\sigma_{fit}$, which comes from the propagation of the photon noise. The fitting error contribution is directly estimated from the 1$\sigma$ dispersion of absolute magnitudes derived from observed photometry perturbed with associated errors. The second source of uncertainty on the magnitude, $\sigma_{M,z}$, comes from the photometric redshift uncertainty. One way to estimate its effect is to compare the absolute magnitudes derived with photometric and spectroscopic redshifts. While limited by the completeness of the spectroscopic sample, it is the most comprehensive estimate of the photo-z error contribution we have access to. The last source of uncertainty we considered, $\sigma_\mathit{SED}$, comes from the choice of template library used to derive absolute magnitudes. The $\sigma_\mathit{SED}$ uncertainties are expected to be negligible when the k-correction is small, which we ensured by limiting our analysis to a redshift range where the rest-frame emission is observed in one of our filters. To estimate $\sigma_\mathit{SED}$, we compared the absolute magnitudes derived from the empirical SED library [@Coupon2015], $M^\textsc{emp}$, and from the stellar population synthesis models library [@BC2003], $M^\textsc{sps}$, and take $\sigma_\mathit{SED} = \| M^\textsc{emp}-M^\textsc{sps} \| / \sqrt{2}$. The total absolute magnitude error is then given by $$\sigma_{M} = \sqrt{ ~\sigma_{fit}^2 + \sigma_{M,z}^2~ + ~\sigma_\mathit{SED}^2} ~. \label{eq_err_MABS}$$ Results {#sect_results} ======= FUV, NUV and U-band luminosity functions ---------------------------------------- ### Completeness limits and wedding cake approach {#sect_complim} Following an approach similar to that in @Pozzetti2010, we based our estimate of the luminosity (or absolute magnitude) completeness limit on the distribution of the faintest luminosity (or absolute magnitude) at which a galaxy could have been detected at its redshift, $L_\mathrm{faint}$ (or $M_\mathrm{faint}$). In practice, if the sample is limited by the observed magnitude, $m$, down to the limiting depth $m \leq m_{\mathrm{lim}}$, then $$\log(L_\mathrm{faint}) = \log(L) + 0.4 \ (m - m_{\mathrm{lim}})$$ which, in terms of absolute magnitude, gives us $$\label{eq_Mabs_lim} M_\mathrm{faint} = M - (m - m_{\mathrm{lim}}) ~.$$ In each redshift bin, we conservatively considered the 20% highest redshift galaxies (i.e., those that are closest to the upper limit of the redshift bin). The corresponding absolute magnitude completeness limit, $M_\mathrm{lim}$, was then defined by the absolute magnitude for which 90% of that upper-limit population had an absolute magnitude $M < M_\mathrm{faint}$. Given that our detection images combine all the CLAUDS and HSC-SSP passbands (i.e., $u,u^,g,r,i,z,y$), every band contributes to the completeness limit. Assuming that a source is detected as long as it is bright enough in at least one of the bands, we derived the effective absolute magnitude completeness limit of our sample, $M_\mathrm{lim}$, as the faintest absolute magnitude completeness limit computed in all the bands, i.e., $$\label{eq_Mabs_lim_eff} M_\mathrm{lim} = \max_b (M^b_\mathrm{lim}), ~~ \mathrm{for} \ b=u,u^,g,r,i,z,y, \,$$ where $M^b_\mathrm{lim}$ is the absolute magnitude completeness limit derived from the limiting depth of the passband $b$, following Equation \[eq\_Mabs\_lim\]. As detailed in Sect. \[sect\_data\], our survey contains two layers of different depths: Deep and Ultra-Deep. The advantage of such structure was twofold. 1) The different depths of the Deep and Ultra-Deep layers allowed us to fine-tune our method of measuring the completeness limit by cross-matching the results from the two layers; with this, we ensure that we did not miss more than 10 percent of galaxies in the faintest magnitude bin. 2) In order to take the best advantage of our survey, we adopted a wedding cake approach where the bright end of the LF comes from the Deep layer, down to the corresponding completeness limit, below which the very faint end relies on the Ultra-Deep layer. ### LF measurement Given the depth of the two layers of our survey, we decided to adopt highly conservative absolute magnitude completeness limits, as discussed in the previous section, which allowed us to measure the FUV, NUV and U-band LFs without incompleteness correction at $M < M_{lim}$. However, aiming to validate our method, we also measured the LFs with the tool ALF [@Ilbert2005], using two different LF estimators: the $V_{max}$ [@Schmidt1968] and SWML [the step-wise maximum likelihood; @Efstathiou1988]. We verified that these two estimators were in good agreement with our uncorrected estimation of the LF down to our adopted completeness limit, which de facto confirmed our estimation of the completeness limit [$V_{max}$ and SWML estimators are known to diverge below the completeness limit; @Ilbert2005; @Moutard2016b]. [l\[5]{}[c]{}]{}\ \ \ \[-3mm\] Redshift bin & Deep $^{(a)}$ & Ultra-Deep $^{(b)}$ & Overlap $^{(c)}$ & Total used $^{(d)}$\ \ $0.05 < z < 0.3$ & 201,617 & (18,073) & —– & 201,617\ $0.3 < z < 0.45$ & 296,940 & (30,598) & —– & 296,940\ $0.45 < z < 0.6$ & 331,144 & (31,291) & —– & 331,144\ $0.6 < z < 0.9$ & 735,345 & 84,059 & 63,473 & 755,931\ $0.9 < z < 1.3$ & 1,142,045 & 143,771 & 93,381 & 1,192,435\ $1.3 < z < 1.8$ & 830,481 & 134,570 & 68,037 & 897,014\ $1.8 < z < 2.5$ & 566,007 & 102,226 & 58,784 & 609,449\ $2.5 < z < 3.5$ & (277,050) & 54,977 & —– & 54,977\ \ & (277,050) & (79,962) & &\ $0.05 < z < 3.5$ & 4,103,579 & 519,603 & 283,675 & 4,339,507\ \ \ \ \ \ \ By definition, the luminosity function, $\phi(L)dL$, is defined as the comoving number density of galaxies with luminosity between $L$ and $L+dL$, or in term of absolute magnitude $M$, $\phi(M) dM =\phi(L)d(-L)$. To compute the FUV, NUV and U-band luminosity functions, we first selected a sample of 4,380,629 galaxies with $z < 3.5$ in the Deep layer, which covers an effective area (i.e., after masking) of 17.02 deg$^2$ down to $$\begin{aligned} (U\le26.9) \ \cup \ (g\le26.3) \ \cup\ (r\le25.9) \ \cup \nonumber\\ \ (i\le25.5) \ \cup \ (z\le24.9) \ \cup \ (y\le24.0) \label{eq_deep_sel}\end{aligned}$$ and 599,565 galaxies with $z < 3.5$ in the Ultra-Deep layer, which covers an effective area of 1.45 deg$^2$ down to $$\begin{aligned} (U\le27.4) \ \cup \ (g\le26.9) \ \cup\ (r\le26.7) \ \cup \nonumber\\ \ (i\le26.4) \ \cup \ (z\le25.9) \ \cup \ (y\le24.8) ~. \label{eq_udeep_sel}\end{aligned}$$ As discussed in Sect. \[sect\_phys\_param\], we restricted our analysis to the redshift ranges $0.05\le z \le 3.5$ in UV and $0.05\le z \le 2.5$ in U-band, where both photometric redshifts and absolute magnitudes are well constrained. We defined eight contiguous redshift bins which were chosen by considering the observed bands used to derive the absolute magnitudes: $0.05 < z \leq 0.3$, $0.3 < z \leq 0.45$, $0.45 < z \leq 0.6$, $0.6 < z \leq 0.9$, $0.9 < z \leq 1.3$, $1.3 < z \leq 1.8$, $1.8 < z \leq 2.5$, and $2.5 < z \leq 3.5$. Table \[table\_demo\] summarises the corresponding numbers of galaxies available in the two layers of our survey, as well as the numbers of galaxies we finally considered to measure the LFs after combining the two layers. Note that we only used the Deep layer to measure the LFs at $0.05 < z \leq 0.6$, given the cosmic variance affecting the Ultra-Deep layer at low redshift due to the limited volume it probes. On the other hand, concerning the last redshift bins we considered for the UV and U-band LFs (namely, $2.5 < z \leq 3.5$ and $1.8 < z \leq 2.5$, respectively), we only used the Ultra-Deep layer, given the limited depth of our $g,r,i,z,y$ data in the Deep layer. In total, we thereby made use of 4,339,507 galaxies to measure the FUV, NUV and U-band LFs Figure \[fig\_FUV\_LFs\] shows the FUV LF we measured in the eight redshift bins we defined from $z = 0.05$ to $z = 3.5$. For each redshift bin, we specified the observed passband in which the FUV absolute magnitude was generally derived. At lower redshifts, our CLAUDS+HSC-SSP measurements involve an extrapolation blueward of the observed U-band, and we verify that this extrapolation is reasonable using GALEX data as follows. In the four lowest redshift bins, we compare the LF measured from observed GALEX FUV and NUV (which minimizes the k-correction) with the LF measured from CLAUDS U-band observations. As one can see, the two LF measurements are in very good agreement down to $M_\mathrm{FUV} \simeq -17, -18, -18$ and $-19$ at $0.05 < z \leq 0.3$, $0.3 < z \leq 0.45$, $0.45 < z \leq 0.6$, $0.6 < z \leq 0.9$, respectively, where we reach the depth of the GALEX observations.[^6] This agreement suggests that the FUV absolute magnitude we derived from extrapolation of U-band observations is reliable. Similarly, Fig. \[fig\_NUV\_LFs\] shows the NUV LF in the same redshift bins. In the lowest redshift bin, we compare the LF measured from GALEX NUV and from CLAUDS U-band, and one can see that both LF measurements are in very good agreement down to $M_\mathrm{NUV} \simeq -17$, the depth of the GALEX observations. As with the FUV measurements, this agreement suggests that the NUV absolute magnitude we derived from the extrapolation of U-band observations is well constrained. Finally, in Fig. \[fig\_U\_LFs\], we show the U-band LF we measured in the seven redshift bins from $z = 0.05$ to $z = 2.5$ In Figs. \[fig\_FUV\_LFs\], \[fig\_NUV\_LFs\] and \[fig\_U\_LFs\] we showed the LFs we measured in the Deep (squares) and Ultra-Deep (circles) layers. We adopted the wedding cake approach presented in the previous section when the comoving volume of the redshift bin was large enough to be characterized by an average density close to that of the Universe at that redshift (i.e., when the faint end of the LF is not dominated by the so-called cosmic variance that we discuss in the next section). One can see how at $z > 0.6$, the faint end of the LF is based on the Ultra-Deep layer down to the associated completeness limit, while the rest of the LF is derived from the Deep layer. On the other hand, our LF measurements in the last redshift bins we considered for the UV ($2.5 < z \leq 3.5$) and U-band ($1.8 < z \leq 2.5$) were entirely based on the Ultra-Deep layer, given the very small contribution of the Deep layer at those redshifts (because of its fairly bright $g,r,i,z,y$ limits). ### LF uncertainties In addition to the Poissonian error ($\sigma_{Poi}$) usually taken into account, LF measurements suffers from two addtional main sources of uncertainty: the error on the luminosity or absolute magnitude ($\sigma_M$), as described in Sect. \[sect\_absmag\_err\], and the so-called cosmic variance $(\sigma_{cv})$, which is due to large-scale inhomogeneities in the spatial distribution of galaxies in the Universe. These additional sources of uncertainty can have an important contribution to the total error budget and therefore need to be accounted for. For instance, cosmic variance has been shown to represent a fractional error of $\sigma_{cv} = 10-15\%$ for massive ($M_\ge 10^{11} M_{\odot}$) galaxies with number densities of $\phi < 10^{-3}$ Mpc$^{-3}$ in a 2-deg$^2$ survey, against $\sigma_{cv} \sim 6-8\%$ in a 20-deg$^2$ survey. At the same time, the cosmic variance contribution to the error budget is small compared to the Poissonian error for very massive –i.e., rare– galaxies, while it dominates the error budget for lower mass –i.e., more abundant– galaxies [see @Moutard2016b for a discussion of these issues]. We may expect a similar effect on the luminosity function, where the very bright end suffers from large cosmic variance and suffers from an even larger Poissonian error, while at fainter magnitudes a modest cosmic variance dominates a very small Poissonian error. Aiming to estimate the contribution of the cosmic variance affecting our LF measurements, we adopt the procedure followed by @Moutard2016b, which is based on a method introduced by @Coupon2015. In brief, at given area $a$, we derived cosmic variance from Jackknife resampling of N patches with area $a$, for patch areas ranging from $a=0.2$ to $1.6$ deg$^2$. The cosmic variance measured using subareas of our survey is then extrapolated to the total area, namely, $a=18.29$ deg$^2$ and $a=1.54$ deg$^2$ in the Deep and Ultra-Deep layers, respectively [for more details on the method, please refer to @Coupon2015; @Moutard2016b]. The last source of uncertainty that we need to consider comes from the error on the absolute magnitude, $\sigma_{M}$, as defined in Sect. \[sect\_absmag\_err\]. To convert $\sigma_{M}$ into an error on the number density, $\sigma_{\phi, M}$, we generated 200 mock catalogues with absolute magnitudes perturbed according to $\sigma_{M}$ (cf. Equation \[eq\_err\_MABS\]) and measured the 1$\sigma$ dispersion of the perturbed LFs. The total uncertainty, $\sigma_{\phi}$, affecting the luminosity function at each magnitude bin is then calculated by combining the three sources of error in quadrature, $$\sigma_{\phi} = \sqrt{ ~\sigma_{Poi}^2 + \sigma_{cv}^2~ + ~\sigma_\mathit{\phi, M}^2} ~, \label{eq_err_LF}$$ and plotted in Figs. \[fig\_FUV\_LFs\],\[fig\_NUV\_LFs\] and \[fig\_U\_LFs\]. Note that although $\sigma_{\phi, M}$ is a good estimation of the contribution of the absolute magnitude error in the LF error budget, it cannot take into account the so-called Eddington bias, whose effects we treat as discussed in Sect. \[sect\_fit\_Edd\_treat\]. Redshift evolution of the luminosity functions {#sect_LFmeasurements} ---------------------------------------------- ### Fitting method and Eddington bias treatment {#sect_fit_Edd_treat} Eddington bias [@Eddington1913] affects the observed slope of the bright end of the luminosity function by converting the statistical error on the luminosities of the more abundant (usually fainter) galaxies into a systematic boost of the number of less abundant (usually brighter) galaxies. The result of this effect is that the observed slope of the luminosity function is shallower than the underlying reality. The same effect affects the observed stellar mass functions, where the dominant effect is that of the scattering of lower-mass galaxies into the higher-mass population. Several authors have addressed the Eddington bias over the past few years, especially regarding the high-mass end of the stellar mass function (SMF). These studies have implicated the effect as biasing the rather mild evolution of the high-mass end of the SMF at $z < 1$ [e.g., @Matsuoka2010; @Ilbert2013; @Moutard2016b]; at $z>4$, where the SMF evolution is stronger, stellar mass uncertainties are very large and so still have to be taken into account [e.g., @Caputi2011; @Grazian2015; @Davidzon2017]. Although less discussed in the literature, the LF bright end measurements suffer from a similar effect that needs to be accounted for. In the present study, we accounted for the effects of Eddington bias by following a procedure similar to that described in @Ilbert2013, as we fitted our LF measurement through $\chi^2$ minimization. In this, we only consider the statistical uncertainties (Poisson and cosmic variance) in the $\chi^2$ calculation during the fitting process, while the absolute magnitude uncertainty $\sigma_M$ is taken into account through convolution with the fitted Schechter parametric form(s), which is thereby corrected for the Eddington bias. Adapting the approach of @Moutard2016b, we consider an estimate of $\sigma_M$ that varies with absolute magnitude and redshift, $\sigma_M(M, z)$, in order to avoid over-correction of the Eddington bias. Finally, one may notice from Figs. \[fig\_FUV\_LFs\], \[fig\_NUV\_LFs\] and \[fig\_U\_LFs\] that the extremely bright ends of our LF measurements, typically for comoving densities $< 10^{-5}$ Mpc$^{-3}$, suffer from uncertainties that are significantly larger than what one could expect from purely Poissonian errors. As discussed in Appendixes \[app\_LF\_perfield\] and \[app\_LF\_sourcetype\], this is most probably due to the contamination by stars and QSOs, the identification and cleaning of which depends on the depth of our observations that varies across the survey. A very small number of interlopers is indeed sufficient to affect the actual number of extremely bright and rare galaxies. In any event, we verified that this contamination of the extremely-bright ends had not a significant impact on the fitting of the LFs that is discussed in the following.[^7] ### FUV, NUV & U-band LF fitting {#sect_fit_forms} As can be seen in Figs. \[fig\_FUV\_LFs\] and \[fig\_NUV\_LFs\], the Schechter parametric form [@Schechter1976] appears to be well suited to the fitting of the FUV and NUV LFs down to the completeness limits of our survey and, at least, between $z = 0.05$ and $z = 3.5$. We therefore fitted the FUV and NUV LFs with the classical Schechter function defined as $$\phi(L) \ dL = e^{-\frac{L}{{L}^\star}} \ \phi^\star \left(\frac{L}{{L}^\star} \right) ^{\alpha} \frac{dL}{{L}^\star} ~, \label{eq_Sch}$$ which, in term of absolute magnitude, can be written as $$\phi(M) \ dM = \frac{\ln 10}{2.5} \ \phi^{\star} \left( 10^{ 0.4 \Delta M} \right)^{ \alpha + 1} \exp \left( -10^{ 0.4 \Delta M } \right) \ dM ~, \label{eq_singleSch_mag}$$ with $\Delta M = M^\star - M$. While all three Schechter parameters are well constrained at $0.05 < z \leq 1.8$, the slope $\alpha$ and normalisation $\phi^\star$ start being poorly constrained at $1.8 < z \leq 2.5$ and are no longer constrained at $z > 2.5$. One of the strengths of our dataset is its ability to probe the bright end of the LF, thanks to the large area covered, the location of which is well traced by the characteristic absolute magnitude $M^{\star}$. Given the stability of $\alpha$ at $0.05 < z \leq 1.8$, we can help constraint $M^{\star}$ at $z > 1.8$ by setting $\alpha$ at $z>1.8$ to its average value at $z \leq 1.8$: $\alpha_{z > 1.8} = \mathrm{const.} = \overline{\alpha}_{z<1.8}$. Our Schechter functional fits are plotted in Figs. \[fig\_FUV\_LFs\] and \[fig\_NUV\_LFs\] with solid lines at $z < 1.8$ and dashed lines at $z > 1.8$, and the values of the Schechter parameters are listed in Table \[table\_param\]. While the FUV and NUV LFs are well described by the Schechter function (Eq. \[eq\_singleSch\_mag\]), this is not the case for the U-band LF (Fig. \[fig\_U\_LFs\]). Here, the LF shape deviates from the classical Schechter form at the faint end, where a clear upturn can be seen around $M_U \sim -17$, at least at low redshift (where our completeness limit is the faintest). Where needed, we therefore adopt a double-Schechter to fit the U-band LF, as defined by $$\phi(L) \ dL = e^{-\frac{L}{{L}^\star}} \ \left[ \phi^\star_1 \left(\frac{L}{L^\star} \right) ^{\alpha_1} + \phi^\star_2 \left(\frac{L}{L^\star} \right) ^{\alpha_2} \right] \ \frac{dL}{L^\star} ~, \label{eq_doubleSch}$$ or, in term of absolute magnitude, $$\begin{aligned} \phi(M) \ dM = \frac{\ln 10}{2.5} ~ \left[ \ \phi_1^{\star} \left( 10^{ 0.4 \Delta M } \right)^{ \alpha_1 + 1} + ~ \phi_2^{\star} \left( 10^{ 0.4 \Delta M } \right)^{ \alpha_2 + 1} \ \right] \nonumber\\ ~~ \times~~ \exp \left( -10^{ 0.4 \Delta M } \right) \ dM ~~ \label{eq_doubleSch_mag}\end{aligned}$$ with $\Delta M = M^\star - M$ and, in our case, $\alpha_1 < \alpha_2$. For further detail about the relevance of fitting the U-band LF with a double-Schechter function, please refer to Appendix \[app\_U\_LF\_fitting\]. While all the double-Schechter parameters were well constrained at $z \leq 0.5$ and could be fitted simultaneously, we had to constrain the parameters of the faint end before fitting the LF at higher redshift (notably due to the difference between the uncertainties affecting the Deep and Ultra-Deep LF contributions, which tend to drastically reduce the Ultra-Deep layer contribution at the faint end of the LF in the $\chi^2$ fitting). To constrain the fitting at $z > 0.5$, we set the faint-end slope $\alpha_1$ to the its average value at $z \leq 0.45$: $\alpha_1 (z > 0.45) = \mathrm{const.} = \overline{\alpha_1}_{z<0.45}$. On the other hand, at $1.3 < z \leq 2.5$, the completeness limit prevented us from observing a second Schechter component at the faint end of the LF. At these high redshifts we only fitted the LF with a single Schechter function. Finally, we also showed the parametric form we would obtain by fitting the U-band LF with a single Schechter (dash-dotted lines in Fig. \[fig\_U\_LFs\]), for comparison. ### Redshift evolution of the FUV, NUV and U-band LFs Figure \[fig\_LF\_evol\] shows the redshift evolution of the fitted LF in UV and in the U-band at $0.05 < z \leq 3.5$ and $0.05 < z \leq 2.5$, respectively. The first noteworthy feature is the evolution of the bright end of the LF, which fades continuously with cosmic time (i.e., with decreasing redshift). The second remarkable thing is the stability of the faint-end slopes in the FUV and NUV, which is clear up to $z \sim 1$. In other words, the populations of FUV and NUV bright galaxies have been continuously decreasing since $z \sim 1$ while the populations of faint galaxies in these bands have remained stable. The same trends apply to the U-band LF evolution, although the faint end is noisier. Additionally, it is interesting that the location of the upturn in the faint end slope of the U-band LF is preserved with cosmic time, in spite of the simultaneous recession of the bright end. We can take this analysis further by considering how the values of the Schechter parameters change with redshift. Figure \[fig\_LFparam\_evol\] shows the redshift evolution of $\alpha$ and $\phi^\star$ as a function of $M^\star$, corresponding to the fitted LFs shown in Fig. \[fig\_LF\_evol\]. The values of the slope $\alpha$ confirm the stability of the faint end seen in Fig. \[fig\_LF\_evol\] for the FUV and NUV LFs, with $-1.42 \leq \alpha \leq -1.31$ and $-1.53 \leq \alpha \leq -1.28$ at $0.05 < z \leq 1.3$, respectively. The normalisation follows a similar trend with $\phi^\star = 4.4$–$6.0 \times 10^{-3}$ Mpc$^{-3}$ and $\phi^\star = 4.9$–$6.6 \times 10^{-3}$ Mpc$^{-3}$ at $0.05 < z \leq 1.3$ for the FUV and NUV LFs, respectively. The characteristic absolute magnitude is characterized by a clear fading of $\sim$2.3 mag and $\sim$2 mag for $M^\star_\mathrm{FUV}$ and $M^\star_\mathrm{NUV}$, respectively, between $z \sim 2.2$ and $z \sim 0.15$. Regarding the U-band Schechter parameters, it is relevant to consider both the single and the double Schechter parametric forms, since our data are deep enough to allow us to observe a double-Schechter profile which has not been documented and discussed in the literature so far. When considering the single-Schechter parameters, the slope $\alpha$ and normalisation $\phi^\star$ appear particularly stable across cosmic time since $z \sim 1.6$, with $-1.4 \leq \alpha \leq -1.3$ and $\phi^\star = 3.0$–$5.0 \times 10^{-3}$ Mpc$^{-3}$; concurrently, $M^\star_{\mathrm{U},\ {{\mathcal{S}ing}}}$ fades by $\sim$1.3 mag. However, a single Schechter may not be appropriate for the U-band LF, since – as we discussed in the previous section – the double Schechter appears to better fit the U-band LF measured at $0.05 < z \leq 1.8$. When considering a double-Schechter, the dispersion observed in the faint end slope of the U-band LF is slightly larger, with $-1.70 \leq \alpha_1 \leq -1.44$ at $0.05 < z \leq 0.6$ (where we let $\alpha_1$ vary without any constraint except $\alpha_1 < \alpha_2$) and $-1.80 \leq \alpha_1 \leq -1.44$ at $0.05 < z \leq 1.8$. At the same time, the normalisation of the faint end appears stable, increasing slightly from $\phi^\star_1 = 1.0 \times 10^{-3}$ to $3.0 \times 10^{-3}$ Mpc$^{-3}$. At the bright end, $-0.60 \leq \alpha_2 \leq -0.20$ at $0.05 < z \leq 0.6$ and $-0.70 \leq \alpha_2 \leq -0.20$ at $0.05 < z \leq 1.8$, while $\phi^\star_2$ increases from $4.1 \times 10^{-3}$ to $7.3 \times 10^{-3}$ Mpc$^{-3}$ in the same redshift interval and the characteristic absolute magnitude $M^\star_{\mathrm{U},\ {{\mathcal{D}oub}}}$ fades by $\sim$1.7 mag. While one may note that, on average, the double-Schechter parameters evolve in the same direction as the single-Schechter parameters, we note that the characteristic absolute magnitude depends substantially on the parametric form we adopted. In particular, the difference $\|M^\star_{\mathrm{U},\ {{\mathcal{D}oub}}}-M^\star_{\mathrm{U},\ {{\mathcal{S}ing}}}\|$ reaches $\sim0.8$ mag in our lowest redshift bin, i.e., where the faint-end upturn in the U-band LF is best probed. Considering a double-Schechter fit of the U-band LF is therefore imperative to compare our estimation of $M^\star_\mathrm{U}$ with other estimates based on shallower surveys. [l\[9]{}[c]{}]{}\ \ \ \[-3mm\] & & & & & &\ & Deep & Ultra-Deep & Deep & Ultra-Deep & & & &\ \ $0.05 < z < 0.3$ & -14.21 & —– & 111,819 & —– & -18.269$\pm0.054$ & 4.85$\pm0.35$ & -1.405$\pm0.019$ & 25.719$^{+0.009}_{-0.012}$\ $0.3 < z < 0.45$ & -15.17 & —– & 150,738 & —– & -18.572$\pm0.038$ & 5.22$\pm0.25$ & -1.369$\pm0.017$ & 25.873$^{+0.005}_{-0.006}$\ $0.45 < z < 0.6$ & -15.79 & —– & 168,370 & —– & -18.797$\pm0.073$ & 4.13$\pm0.50$ & -1.408$\pm0.053$ & 25.885$^{+0.012}_{-0.014}$\ $0.6 < z < 0.9$ & -17.16 & -16.34 & 265,051 & 17,888 & -19.113$\pm0.041$ & 4.40$\pm0.30$ & -1.402$\pm0.038$ & 26.048$^{+0.009}_{-0.011}$\ $0.9 < z < 1.3$ & -18.02 & -17.27 & 422,362 & 28,890 & -19.554$\pm0.065$ & 4.97$\pm0.57$ & -1.432$\pm0.068$ & 26.304$^{+0.018}_{-0.022}$\ $1.3 < z < 1.8$ & -18.79 & -17.80 & 305,245 & 36,289 & -20.016$\pm0.074$ & 3.20$\pm0.38$ & -1.446$\pm0.074$ & 26.317$^{+0.022}_{-0.025}$\ $1.8 < z < 2.5$ & -19.67 & -18.91 & 180,033 & 22,979 & -20.261$\pm0.042$ & 2.82$\pm0.11$ & -1.43 & 26.355$^{+0.005}_{-0.006}$\ $2.5 < z < 3.5$ & —– & -19.73 & —– & 54,089 & -20.841$\pm0.046$ & 1.69$\pm0.10$ & -1.43 & 26.373$^{+0.011}_{-0.013}$\ \ [l\[9]{}[c]{}]{}\ \ \ \[-3mm\] & & & & & &\ & Deep & Ultra-Deep & Deep & Ultra-Deep & & & &\ \ $0.05 < z < 0.3$ & -14.38 & —– & 117,326 & —– & -18.514$\pm0.025$ & 5.08$\pm0.2$ & -1.399$\pm0.011$ & 25.847$^{+0.006}_{-0.006}$\ $0.3 < z < 0.45$ & -15.37 & —– & 157,999 & —– & -18.798$\pm0.03$ & 6.01$\pm0.26$ & -1.308$\pm0.014$ & 26.009$^{+0.007}_{-0.008}$\ $0.45 < z < 0.6$ & -16.02 & —– & 174,053 & —– & -19.026$\pm0.062$ & 4.56$\pm0.47$ & -1.364$\pm0.043$ & 26.009$^{+0.009}_{-0.013}$\ $0.6 < z < 0.9$ & -17.27 & -16.49 & 297,674 & 17,545 & -19.416$\pm0.053$ & 4.24$\pm0.4$ & -1.396$\pm0.044$ & 26.159$^{+0.008}_{-0.010}$\ $0.9 < z < 1.3$ & -18.25 & -17.49 & 404,623 & 28,462 & -19.859$\pm0.052$ & 4.7$\pm0.43$ & -1.385$\pm0.046$ & 26.385$^{+0.009}_{-0.01}$\ $1.3 < z < 1.8$ & -19.13 & -18.14 & 293,618 & 34,610 & -20.367$\pm0.045$ & 3.13$\pm0.23$ & -1.391$\pm0.038$ & 26.422$^{+0.006}_{-0.008}$\ $1.8 < z < 2.5$ & -20.05 & -19.24 & 173,206 & 23,007 & -20.622$\pm0.034$ & 2.72$\pm0.08$ & -1.4 & 26.472$^{+0.004}_{-0.004}$\ $2.5 < z < 3.5$ & —– & -20.15 & —– & 23,598 & -21.152$\pm0.038$ & 1.71$\pm0.11$ & -1.4 & 26.489$^{+0.015}_{-0.017}$\ \ [l\[9]{}[c]{}]{}\ \ \ \[-3mm\] & & & & & &\ & Deep & Ultra-Deep & Deep & Ultra-Deep & & & &\ \ $0.05 < z < 0.3$ & -15.17 & —– & 121,413 & —– & -19.865$\pm0.030$ & 3.60$\pm0.12$ & -1.424$\pm0.007$ & 26.291$^{+0.004}_{-0.005}$\ $0.3 < z < 0.45$ & -16.20 & —– & 155,008 & —– & -20.042$\pm0.020$ & 5.62$\pm0.13$ & -1.22$\pm0.008$ & 26.466$^{+0.003}_{-0.003}$\ $0.45 < z < 0.6$ & -16.86 & —– & 166,133 & —– & -20.125$\pm0.017$ & 5.05$\pm0.11$ & -1.178$\pm0.009$ & 26.439$^{+0.002}_{-0.002}$\ $0.6 < z < 0.9$ & -17.91 & -17.22 & 356,389 & 13,958 & -20.435$\pm0.012$ & 5.28$\pm0.08$ & -1.154$\pm0.008$ & 26.576$^{+0.002}_{-0.002}$\ $0.9 < z < 1.3$ & -18.92 & -18.21 & 461,661 & 24,246 & -20.841$\pm0.014$ & 4.66$\pm0.09$ & -1.251$\pm0.01$ & 26.723$^{+0.002}_{-0.002}$\ $1.3 < z < 1.8$ & -19.74 & -18.85 & 354,083 & 29,295 & -21.241$\pm0.017$ & 2.96$\pm0.07$ & -1.352$\pm0.013$ & 26.737$^{+0.004}_{-0.004}$\ $1.8 < z < 2.5$ & —– & -20.32 & —– & 30,689 & -21.677$\pm0.039$ & 2.2$\pm0.12$ & -1.394$\pm0.033$ & 26.808$^{+0.011}_{-0.011}$\ \ [l\[7]{}[c]{}]{}\ \ \ \[-3mm\] Redshift & $M^\star_{\mathrm{U},\ {{\mathcal{D}oub}}}$ $^{(a)}$ & $\phi^\star_1$ $^{(b)}$ & $\alpha_1$ & $\phi^\star_2$ $^{(b)}$ & $\alpha_2$ & $\log(\ \rho_{\mathrm{U},\ {{\mathcal{D}oub}}}$ $^{(c)} \ )$\ \ $0.05 < z < 0.3$ & -18.961$\pm0.050$ & 3.4$\pm0.36$ & -1.568$\pm0.024$ & 7.16$\pm0.25$ & -0.213$\pm0.099$ & 26.301$^{+0.003}_{-0.003}$\ $0.3 < z < 0.45$ & -19.500$\pm0.040$ & 2.41$\pm0.47$ & -1.557$\pm0.051$ & 7.91$\pm0.3$ & -0.419$\pm0.092$ & 26.479$^{+0.002}_{-0.003}$\ $0.45 < z < 0.6$ & -19.744$\pm0.027$ & 1.96$\pm0.08$ & -1.56 & 5.59$\pm0.12$ & -0.506$\pm0.043$ & 26.452$^{+0.002}_{-0.002}$\ $0.6 < z < 0.9$ & -20.244$\pm0.023$ & 1.46$\pm0.11$ & -1.56& 5.04$\pm0.09$ & -0.773$\pm0.042$ & 26.588$^{+0.002}_{-0.002}$\ $0.9 < z < 1.3$ & -20.819$\pm0.033$ & 0.51$\pm0.63$ & -1.56 & 4.28$\pm0.47$ & -1.187$\pm0.092$ & 26.727$^{+0.014}_{-0.001}$\ \ \ \ \ \ \ ### Comparison with previous studies Often simply referred to as the UV LF, the FUV LF has been extensively studied up to redshift $z \sim 9$, where rest-frame (not dust-corrected) UV can be constrained from mid-infrared observations. In Fig. \[fig\_FUV\_LFparam\_lit\], we compare our best-fit Schechter parameters for the FUV LF with values from the literature across redshift. As one can see, our results are in overall good agreement with the literature at $0.05 < z \leq 3.5$. Given its unsurpassed combination of depth and area, our homogeneous dataset provides the definitive reference measurement of the rest-frame FUV LF out to $z\sim3$ at this time. It is remarkable how well-behaved the values of the Schechter parameters are with redshift in Fig. \[fig\_FUV\_LFparam\_lit\] over the redshift range we measured them: $M^\star$ increases monotonically with lookback time, while both $\alpha$ and $\phi^\star$ remain essentially constant. The FUV LF slope $\alpha$, which we measured directly from $z \sim 1.6$, is of particular interest as it appears flatter than what is reported in the literature higher redshifts, from $z \sim 9$. This points to the existence of two regimes in the evolution of the FUV LF’s faint end, which flattened from $z \sim 9$ before stabilizing at or before $z \sim 1.6$. Similarly, the evolution of $\phi^\star$ we measure is very stable from $z \sim 1.6$, and seem to be right in the middle of the literature values. At the same time, the continuous fading of the FUV LF’s bright end characteristic absolute magnitude, $M^\star$ we observe from $z \sim 3$ is in line with the literature, though much better constrained with our data. ![image](figures_ok/zMix_FUV_SchParam_LitComp.pdf){width="\hsize"} ![image](figures_ok/zMix_NUV_SchParam_LitComp.pdf){width="0.46\hsize"} ![image](figures_ok/zMix_U_SchParam_LitComp.pdf){width="0.46\hsize"} In contrast to the FUV LF, the NUV and U-band LFs are much less well documented in the literarture, especially at $ z > 1.5$. Figure \[fig\_NUV\_U\_LFparam\_lit\] shows our measurements of the NUV and U-band Schechter parameters compared with those from the literature. For the literature compilation we only considered analyses where the parameters were free to vary over the redshift range covered by the literature, i.e., up to $z \sim 1.5$. Our NUV LF Schechter parameters are in overall good agreement with the literature, but provide measurements that are much less noisy. This is particularly clear for the redshift dependence of $\alpha$ (and to a lower extent for $\phi^\star$ ), for which our measurement is more stable than what is found the literature. Our $\alpha$ values, in particular, show a remarkable stability with redshift. At the same time, the evolution of $M^\star_\mathrm{NUV}$ we measured is in very good agreement with the literature, although even less noisy; with our excellent statistics, it shows a remarkably steady progression with cosmic time. For the U-band LF, the comparison with the literature is different if we consider the single or double Schechter function fit. When considering a single-Schechter (star sybmols in Fig. \[fig\_NUV\_U\_LFparam\_lit\]), the agreement with the literature is particularly good, especially for $\alpha$ and $\phi^\star$, while one may notice a little discrepancy for $M^\star_\mathrm{U}$ at $z < 0.5$. This is expected as the faint-end excess of galaxies in the U-band LF is more pronounced at low redshift, which directly affects our estimation of $M^\star_\mathrm{U}$ due to the well known degeneracy between the Schechter parameters $\alpha$ and $M^\star$ (as observed in Fig. \[fig\_LFparam\_evol\]). Thus, $M^\star_{\mathrm{U}, {{\mathcal{D}oub}}}$ is in overall good agreement with the literature from $z \sim 0$ up to $z \sim 1$ (i.e., over all the redshift range where the comparison is possible), while exhibiting a much less noisy evolution. Redshift evolution of the luminosity densities ---------------------------------------------- ### Luminosity density from the LF {#sect_LDmeasurement} In principle, the luminosity density (LD) is obtained by summing the light from all the galaxies in unit volume. In practice, the LD can be estimated by integrating the LF. The luminosity density of galaxies with luminosity greater than $L$ is defined by $$\rho(L) = \int_{L}^{\infty} L'\ \phi(L') \ dL' ~. \label{eq_LDintegral}$$ If the LF has the (single) Schechter form, this reduces to $$\rho(L) = \phi^\star \ L^\star \ \Gamma(\alpha+2, L/L^\star) \label{eq_LD_oneSchechter}$$ where $\Gamma$ is upper incomplete gamma function. In the case of a double-Schechter LF, Equation \[eq\_LDintegral\] becomes $$\rho(L) = L^\star \ \left[ \ \phi_1^\star \ \Gamma(\alpha_1+2, L/L^\star) + \ \phi_2^\star \ \Gamma(\alpha_2+2, L/L^\star) \ \right] ~. \label{eq_LD_twoSchechter}$$ We derive the rest-frame FUV, NUV, and U-band LDs using the Schechter parameters we obtained in Sec. \[sect\_LFmeasurements\] and Equations \[eq\_LD\_oneSchechter\] or \[eq\_LD\_twoSchechter\], as appropriate. We integrate over luminosity from $\infty$ down to $M_\mathrm{FUV},M_\mathrm{NUV},M_\mathrm{U} = -15$ to avoid heavy extrapolations. This limit is $\geq3$ magnitudes below $M^\star$ for all of our LF measurements, and – given our relatively shallow values of $\alpha$ – it therefore captures the vast bulk of the luminosity that escapes the galaxy population. We present the resulting LD values in the last column of Table \[table\_param\] and discuss the results in the next section. ### Redshift evolution of the FUV, NUV and U-band LDs Figures \[fig\_FUV\_LD\_evol\] and \[fig\_NUV\_U\_LD\_evol\] show the redshift evolution of our FUV, NUV and U-band luminosity densities measured as described in Sec. \[sect\_LDmeasurement\]. For comparison, we show LD values we recalculated from literature LF measurements for the same luminosity limits as those we applied to the CLAUDS+HSC-SSP data. . ![image](figures_ok/zMix_FUV_LDevol.pdf){width="\hsize"} ![image](figures_ok/zMix_NUV_LDevol.pdf){width="0.46\hsize"} ![image](figures_ok/zMix_U_LDevol.pdf){width="0.46\hsize"} In Fig. \[fig\_FUV\_LD\_evol\] one can see how our results support a picture where the FUV luminosity density has continuously decreased from $\rho_\mathrm{FUV} \sim 10^{26.35}$ down to $\sim 10^{25.7}$ erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$ between $z \sim 2$ and $z \sim 0.2$, in good agreement with the literature. At the same time, our results show $\rho_\mathrm{FUV}$ to be stable at $1 \lesssim z \lesssim 2$ (and even at $1 \lesssim z \lesssim 3$ if we assume that the slope of $\alpha=-1.43$ we have set at $z > 1.8$ from lower-$z$ measurements is correct). In that respect, our results appear to be consistent with a picture where the cosmic UV luminosity density experienced a relatively stable phase before decreasing exponentially from redshift $z \sim 1$. In Fig. \[fig\_NUV\_U\_LD\_evol\]a, one can see a similar trend for the redshift evolution of the NUV luminosity density, with a continuous decrease from $\rho_\mathrm{NUV} \sim 10^{26.4}$ down to $\sim 10^{25.85}$ erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$ between $z \sim 1$ and $z \sim 0.2$, after a less pronounced evolution at $1 \lesssim z \lesssim 2$ (and also at $1 \lesssim z \lesssim 3$, assuming a slope of $\alpha=-1.4$ at $z > 1.8$). The evolution of the U-band luminosity density shows a similar trend, at least at $z < 2$, with a continuous decrease of from $\rho_\mathrm{U} \sim 10^{26.8}$ down to $\sim 10^{26.25}$ erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$ between $z \sim 2$ and $z \sim 0.2$ and a more stable evolution at $1 \lesssim z \lesssim 2$, irrespective of whether we consider the double- or single-Schechter fits of the U-band LF, with the difference $\vert \log( \rho_{\mathrm{U},\ {{\mathcal{D}oub}}}) - \log( \rho_{\mathrm{U},\ {{\mathcal{S}ing}}}) \ \vert \lesssim~0.01$ dex. That is in fairly good agreement with the literature shown in Figs. \[fig\_NUV\_U\_LD\_evol\]a and b, although comparison is only possible up to $z \sim 1$ for $\rho_\mathrm{NUV}$ and $\rho_\mathrm{U}$. These results are in broad agreement with the picture first presented by [@Sawicki1997], namely that of a broad plateau at $1 \lesssim z \lesssim 3.5$ followed by a steep decline from $z\sim1$ to $z\sim 0$ [the latter first measured by @Lilly1996].[^8] Our CLAUDS+HSC-SSP measurements (Sec. \[sect\_results\]) show that the evolution of the FUV and NUV LDs out to $z\sim 1$ is primarily driven by changes in $M^\star$ rather than in the faint-end slope, $\alpha$, or the number density of galaxies, $\phi^\star$. At higher redshifts, $z \gtrsim 1$, while $M^\star$ continues to brighten, $\phi^\star$ begins to drop, with the two effects balancing each other to give the much milder, evolution seen at $z \gtrsim 1$ in Figs. \[fig\_FUV\_LD\_evol\] and \[fig\_NUV\_U\_LD\_evol\]a. The interpretation is more complicated in the rest-frame U-band because of the double-Schechter form of the U-band LF. There, we suspect that the build-up of the population of quiescent galaxies may contribute to the LF (bright end) and LD, as we explore in a forthcoming companion paper (T. Moutard et al., in prep.). Summary ======= In this paper we presented our measurements of the $0 < z \lesssim 3$ rest-frame FUV (1546Å), NUV (2345Å), and U-band (3690Å) galaxy luminosity functions and luminosity densities using more than 4.3 million galaxies from the CLAUDS and HSC-SSP surveys. The unprecedented combination of depth (U$\sim$27) and area ($\sim$18deg$^2$) of this dataset allows us to constrain the shape and evolution of these LFs with unmatched statistical precision and essentially free of cosmic variance. The main results of this paper are the LF and LD measurements presented in the Figures and Tables in Section \[sect\_results\]. In addition to these main products, we wish to highlight again the following observations: 1. The rest-frame FUV and NUV luminosity functions are described very well by the classic Schechter form over the full redshift range studied. The evolution of the Schechter parameters is very smooth with redshift: In particular, the values of $M^\star$ for both the FUV and NUV increase monotonically with increasing redshift, while the faint-end slopes are very stable up to $z \sim 2$, with slope values conservatively within $-1.42 \leq \alpha_{FUV} \leq -1.31$ and $-1.53 \leq \alpha_{NUV} \leq -1.28$ over $0.05 < z \leq 1.3$. 2. In contrast to the FUV and NUV LFs, the rest-frame U-band luminosity functions are best described by a double Schechter model, $M^\star_{\mathrm{U},\ {{\mathcal{D}oub}}}$, $\phi^\star_\mathrm{U,1}$, $\phi^\star_\mathrm{U,2}$, and $\alpha_\mathrm{U,1}$ evolving continuously through $0.2<z<2$, assuming that $\alpha_\mathrm{U,2}$ is simultaneously stable with redshift, which is confirmed to at least $z\sim0.5$ (we are unable to measure it independently beyond this redshift). We speculate that the second Schechter component in the rest-frame U-band LF is due to the population of quiescent galaxies – a topic we are currently investigating in a companion paper (T. Moutard, in prep.). 3. We measured the rest-frame FUV, NUV, and U-band luminosity densities by integrating the corresponding LFs down to $M=-15$ at $z \sim 0.2$. At all three wavelengths we confirm previous results but with much better statistical precision afforded by our wide-and-deep CLAUDS+HSC-SSP dataset: at all three rest wavelengths the luminosity density increases monotonically and rapidly with lookback time from $z \sim 0.2$ to $z \sim 1$ and then flattens to a much gentler slope at $z > 1$. 4. The very shallow evolution of the FUV and NUV LDs from $ z \sim 3$ to $z \sim 1$ is driven by two competing effects acting within the LFs: the fading of the characteristic magnitude $M^\star$, which is balanced by the increase in the number of objects, $\phi^\star$ to produce the essentially flat LDs we observe over this wavelength range. At $z<1$ the rapid evolution of the luminosity densities is essentially due to the continuing fading of $M^\star$ only as both $\phi^\star$ and $\alpha$ remain essentially constant from $z \sim 1$ to $z \sim 0.2$. We hope that the $0<z<3$ LF and LD measurements we presented in this paper will serve as a useful reference to the community for making observational forecasts and validating theoretical models. In the future, we plan to extend the range of our LF and LD measurements to higher redshifts (0<z<7) by incorporating Lyman Break Galaxy luminisity functions that we plan to do in a consistent way across this redshift range. Acknowledgements {#acknowledgements .unnumbered} ================ We gratefully acknowledge the anonymous reviewer, whose insightful comments helped in improving the clarity of the paper. We thank the CFHT observatory staff for their hard work in obtaining these data. The observations presented here were performed with care and respect from the summit of Maunakea which is a significant cultural and historic site. We thank Guillaume Desprez and Chengze Liu for helpful suggestions. This work is based on observations obtained with MegaPrime/ MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This research uses data obtained through the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories, Chinese Academy of Sciences, and the Special Fund for Astronomy from the Ministry of Finance. This work uses data products from TERAPIX and the Canadian Astronomy Data Centre. It was carried out using resources from Compute Canada and Canadian Advanced Network For Astrophysical Research (CANFAR) infrastructure. These data were obtained and processed as part of CLAUDS, which is a collaboration between astronomers from Canada, France, and China described in [@Sawicki2019]. This work is also based in part on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by the Subaru Telescope and Astronomy Data Center at National Astronomical Observatory of Japan. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at http://dm.lsst.org. This work was financially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada, by the Programme National Cosmology et Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, and by the Centre National d’Etudes Spatiales (CNES). Fitting the U-band luminosity function {#app_U_LF_fitting} ====================================== As observed in Figs. \[fig\_U\_LFs\] and \[fig\_field\_U\_LFs\], the U-band LF exhibits an upturn at the faint end, which results in a deviation from the shape of a pure Schechter function and argues for fitting the U-band LF with a double-Schechter function. In order to assess whether a double-Schechter function is quantitatively better adapted to the U-band LF, we need to compare the the goodness-of-fit of two non-linear models with different numbers of parameters. While one might be inclined to compare the associated reduced $\chi^2$, defined as $\chi^2_\nu = \chi^2 / \nu$ (where $\nu$ is the number of degrees of freedom), $\nu$ is generally not of the commonly assumed form $\nu = N - M$ for non-linear models [@Andrae2010]. Consequently, the best way to compare the goodness-of-fit of single- and double-Schechter functions is actually to return to the distribution of the fit residuals. To better appreciate the significance of the residual between the observed LF, $\phi$, and its parametric form, $\phi_{mod}$, it is relevant to consider the weighted residual, $\xi$, defined by $$\label{eq_w_res} \xi = \frac{\phi-\phi_{mod}}{\sigma_{\phi}} ~,$$ where the residual $\phi-\phi_{mod}$ is normalized by the LF uncertainty $\sigma_{\phi}$. Thereby, at given absolute magnitude, good agreement between the model and the data is met when the residual is smaller than the statistical uncertainty, i.e., when $-1 \lesssim \xi \lesssim 1$ (or when $\vert \ \xi \ \vert \lesssim 1$). Then, aiming at characterising the distribution of the residuals, it may be convenient to define $\Xi_{mod}$ as the normalized median absolute deviation of the model $\phi_{mod}$: $$\begin{aligned} \label{eq_dev_mod} \Xi_{mod} = 1.48 \times \mathrm{median}\left( \ \left\vert \frac{\phi-\phi_{mod}}{\sigma_{\phi}} \right\vert \ \right) \nonumber\\ = 1.48 \times \mathrm{median}( \ \vert \ \xi \ \vert \ ) ~, ~~~~~~~~~~\end{aligned}$$ $\Xi_{mod}$ being thereby a measure of the typical deviation of the model around the data relative to the statistical uncertainty on the data. In other words, there is overall good agreement of the model with the data when $\Xi_{mod} \lesssim 1$ and the better agreement, the smaller $\Xi_{mod}$ is. In Fig. \[fig\_U\_LF\_fit\], we plotted the weighted residual as a function of the U-band absolute magnitude, $M_U$, and we compare the residuals we obtained when fitting the LF with single- and double-Schechter functions. The typical deviation for the single- and double- Schechter fits, $\Xi_{{\mathcal{S}ing}}$ and $\Xi_{{\mathcal{D}oub}}$, are reported in the lower-left corner of each sub-panel in Fig. \[fig\_U\_LF\_fit\], while the corresponding reduced $\chi^2$ are reported in the lower-right corners of the sub-panels, for information. As one can see, the advantage of using a double-Schechter function to fit the U-band LF is clear up to $z = 0.9$. Associated residuals are indeed smaller than the statistical uncertainty from the faint end (notably around $M_U \sim -17$ where the upturn is observed; see Fig. \[fig\_U\_LFs\]) to the bright end, before the disagreement start increasing around $M_U \sim -21.5$ due to Eddington bias (see Sect. \[sect\_fit\_Edd\_treat\]) and contamination by stars and quasars (see App. \[app\_LF\_sourcetype\]). One may notice that this translates into $\Xi_{{\mathcal{D}oub}}< 0.75$, which traces a pretty good agreement between the data and the best-fit solution with a double-Schechter function, while the best single-Schechter solution is clearly worse, with $\Xi_{{\mathcal{S}ing}}> 1.4$ (i.e., $\Xi_{{\mathcal{S}ing}}\simeq$ 2–4 $ \times \ \Xi_{{\mathcal{D}oub}}$). At $0.9 < z < 1.3$, although single- and double-Schechter functions appear to provide similar results, the typical deviations of the two models tend to confirm that a double-Schechter profile better fits the U-band LF, with $\Xi_{{\mathcal{D}oub}}< 1 < \Xi_{{\mathcal{S}ing}}$. At higher redshifts, the completeness limits of our data ($M_{U,\ lim} = -18.85$ and $-20.32$ at $1.3 < z < 1.8$ and $1.8 < z < 2.5$, respectively) prevent us from detecting any excess of galaxies at the faint end (the excess is typically visible for $M_U \gtrsim -17$ at lower redshift, as recalled above). No definitive conclusion can therefore be drawn about the relevance of fitting the U-band LF with a double-Schechter function at $z > 1.3$, where a simple Schechter function fits well the LF (for $M_U < M_{U,\ lim}$), with $\Xi_{{\mathcal{S}ing}}< 1$ . Variation of the luminosity functions from field to field {#app_LF_perfield} ========================================================= Figures \[fig\_field\_FUV\_LFs\], \[fig\_field\_NUV\_LFs\] and \[fig\_field\_U\_LFs\] show, respectively, the FUV, NUV and U-band raw LFs we measured in our eight redshift bins, for each of the four fields of our survey: DEEP2-3, ELAIS-N1, XMM-LSS and E-COSMOS. In these figures, the Deep and Ultra-Deep layers are not separated, which explains why XMM-LSS and E-COSMOS, which contain the Ultra-Deep layer, appear deeper than DEEP2-3 and ELAIS-N1. The deviation between the LFs measured in each field illustrates the cosmic variance affecting the LF measurement in each field. One can see how the cosmic variance depends on the cosmic volume probed for a given redshift bin and a given effective area: it is thereby not surprising to observe the largest deviation between field LFs in our lowest redshift bin, $0.05 < z \leq 0.3$. On the other hand, the deviation between the LFs one can observe from field to field at the extremely bright end, for very small comoving densities $<10^{-5}$ Mpc$^{-3}$, is likely to be due contamination by stars and QSOs that could not be discarded by the procedure described in Sect. \[sect\_photoz\]. Indeed, the photometric identification of stars and QSOs depends on the SNR of the sources (i.e., the depth of the data), which is different in our different fields. In that respect, what can be seen in Fig. \[fig\_field\_U\_LFs\] at $0.9 < z \leq 1.3$ (where most of the U-band absolute magnitudes are derived from observed $i$-band) is particularly striking but not surprising: XMM-LSS and E-COSMOS host indeed our Ultra-Deep layer and include much deeper $g,r,i,z,y$ observations than DEEP2-3 and ELAIS-N1. Luminosity function per type of source {#app_LF_sourcetype} ====================================== Figures \[fig\_type\_FUV\_LFs\], \[fig\_type\_NUV\_LFs\] and \[fig\_type\_U\_LFs\] show, respectively, the FUV, NUV and U-band LFs we could measure in our eight redshift bins, depending on the type of sources we identified with the procedure described in Sect. \[sect\_photoz\]: galaxies, quasars and stars. Stars with $z > 0$ are obviously not real, and the redshifts of QSOs are most probably wrong, but the exercise allows us to see how and where these two populations may contaminate our LF measurements. Indeed, as one can see in all FUV, NUV and U-band LFs, sources classified as stars and QSOs are completely dominated by the galaxy population at low luminosities (faint absolute magnitudes) down to the completeness limit, but their incidence increases with increasing luminosity to become as numerous as galaxies at the very bright end of the LFs. This seems to confirm that the extremely bright end of our LF measurements may significantly suffer from contamination by stars and QSOs, typically for comoving densities $< 10^{-5}$ Mpc$^{-3}$. None the less, we verified that this very limited population did not affect our analysis, as described in Sect. \[sect\_fit\_Edd\_treat\]. \[lastpage\] [^1]: thibaud.moutard@lilo.org [^2]: Canada Research Chair [^3]: Using the NMAD (normalized median absolute deviation) to define the scatter, $\sigma_z = 1.48 \times \mathrm{median}\left(~\frac{\|z_{phot}-z_{spec}\|}{1+z_{spec}}~\right)$. [^4]: We simply define the photo-z bias as $b_z = \frac{z_{phot}-z_{spec}}{1+z_{spec}}$. [^5]: $\eta$ is defined as the percentage of galaxies with $\frac{\|z_{phot}-z_{spec}\|}{1+z_{spec}}>0.15$. [^6]: Note that GALEX fluxes measured with EMphot only use u band priors down to [$u^*$]{}$\sim$25 (see Sect. \[sect\_data\]). [^7]: In practice, we found that the difference between the best-fitting parameters obtained by considering or excluding the LF points with comoving densities $< 10^{-5}$ Mpc$^{-3}$ was smaller than the typical error on those parameters. [^8]: The [@Sawicki1997] measurements were performed at rest-frame UV wavelengths but then were extrapolated to rest-frame 3000Å (roughly mid-way between the NUV and U-band of the present study) for homogeneity with the $z \lesssim 1$ measurements of [@Lilly1996].
--- abstract: 'It is well-known that r-mode oscillations of rotating neutron stars may be unstable with respect to the gravitational wave emission. It is highly unlikely to observe a neutron star with the parameters within the instability window, a domain where this instability is not suppressed. But if one adopts the ‘minimal’ (nucleonic) composition of the stellar interior, a lot of observed stars appear to be within the r-mode instability window. One of the possible solutions to this problem is to account for hyperons in the neutron star core. The presence of hyperons allows for a set of powerful (lepton-free) non-equilibrium weak processes, which increase the bulk viscosity, and thus suppress the r-mode instability. Existing calculations of the instability windows for hyperon NSs generally use reaction rates calculated for the $\Sigma^-\Lambda$ hyperonic composition via the contact $W$ boson exchange interaction. In contrast, here we employ hyperonic equations of state where the $\Lambda$ and $\Xi^-$ are the first hyperons to appear (the $\Sigma^-$’s, if they are present, appear at much larger densities), and consider the meson exchange channel, which is more effective for the lepton-free weak processes. We calculate the bulk viscosity for the non-paired $npe\mu\Lambda\Xi^-$ matter using the meson exchange weak interaction. A number of viscosity-generating non-equilibrium processes is considered (some of them for the first time in the neutron-star context). The calculated reaction rates and bulk viscosity are approximated by simple analytic formulas, easy-to-use in applications. Applying our results to calculation of the instability window, we argue that accounting for hyperons may be a viable solution to the r-mode problem.' author: - 'D. D. Ofengeim$^1$[^1], M. E. Gusakov$^1$, P. Haensel$^2$, M. Fortin$^2$' title: Bulk viscosity in neutron stars with hyperon cores --- Introduction {#sec:intro} ============ There are two well-known types of viscosities in a fluid. The shear viscosity $\eta$ comes from the momentum diffusion between fluid layers moving with different velocities. The bulk viscosity $\zeta$ appears due to non-equilibrium reactions in the compressing and decompressing fluid [@LL-VI]. Both these viscosities are important in numerous studies of neutron stars (NSs) [@GlamGual2018], in particular, for damping of their r-mode oscillations [@Haskell2015]. The Rossby (or simply r-) modes are a subclass of the inertial oscillation modes, restoring force of which is the Coriolis force in a rotating star. The r-modes appear to be unstable to the gravitational wave emission due to the Chandrasekhar-Friedman-Schutz instability [@Chandra1970; @FriSch1978]. It is damped by the shear and bulk viscosities at low and high temperatures, respectively. The domain in the $\nu,T$ plot ($\nu$ is the rotation frequency and $T$ is the internal temperature of the NS) where the star is unstable is called the r-mode instability window. It is highly unlikely to observe a NS with $\nu$ and $T$ within it. See the reviews [@AK2001; @Haskell2015]. However, one meets a paradox [@Haskell2015]: a lot of observed NSs in low-mass X-ray binaries (LMXBs) have their $\nu$ and $T$ in the unstable domain for NSs with the nucleonic ($npe\mu$) core composition. Namely, their typical temperatures are too hot to damp the instability by $\eta$ and too low to do it via $\zeta$. A lot of possible solutions to this paradox were proposed, mainly to introduce an additional damping mechanism. Some of them are reviewed in [@Haskell2015]. Here we focus on the option to modify the bulk viscosity $\zeta$ by the presence of hyperons in the NS core. In a nucleonic core $\zeta$ is mainly provided by the modified Urca process, e.g. $n + n \to n + p + e + \tilde{\nu}_e$ and the inverse. In the most massive nucleonic NSs the direct Urca, $n \to p + e + \tilde{\nu}_e$ and the inverse, can operate. These non-equilibrium processes have the rates $\propto T^6\Delta\mu$ and $\propto T^4\Delta\mu$, respectively ($\Delta\mu$ is the chemical equilibrium distortions due to the fluid motions) [@Yak2001; @HLY2000; @HLY2001]. This means that at low temperatures these rates are strongly suppressed by a factor of $\sim (kT/\mu)^{4-6}$ ($\mu$ is a typical baryon chemical potential). The bulk viscosity due to these processes can damp the r-mode instability only at $T \sim 10^9 - 10^{10}\,$K, while NSs in LMXBs typically have $T \sim (0.3-1)\times 10^8\,$K. The suppression of the reaction rates due to nucleon pairing even worsens the problem [@HLY2000; @HLY2001]. However, there are numerous models of the NS core equation of state (EoS) predicting the presence of hyperons (baryons with at least one strange quark) in deep layers of the core [@HPY2007; @Vidana2015]. The most-widely used ones are the relativistic mean field (RMF) models due to their relative simplicity [@Glend2000]. The presence of hyperons dramatically changes the bulk viscosity. At low temperatures the main contribution to $\zeta$ comes from weak non-leptonic processes, e.g., $\Sigma^- + p \leftrightarrow n +n$ or $\Lambda + p \leftrightarrow n + p$. At $T < 10^9\,$K their typical rate $\propto T^2\Delta\mu$ is much larger than the Urca process rates. There were numerous calculations of the reaction rates of these processes and the corresponding bulk viscosity [@LO2002; @HLY2002; @vDD2004; @NO2006; @GK2008] in both normal and paired matter. Existing calculations of the r-mode instability windows for hyperonic NSs [@LO2002; @ReisBon2003; @NO2006] yield that the hyperonic enhancement of $\zeta$ is generally not enough to solve the r-mode paradox (except, maybe, for the most massive stars $\sim 2\,$[M$_\odot$]{}, central regions of which may be free of baryon pairing). In the recent reviews [@Haskell2015; @Vidana2015] it is argued that the hyperonic bulk viscosity is unable to close the instability window for the observed NSs. However, previous calculations of the instability window for hyperonic NSs should be revisited. First, they used the $\Sigma^-\Lambda$ hyperonic composition of the NS core. Various modern EoS models [@GHK2014; @RadSedWeb2018; @TolosCool2018], in particular those, calibrated to the up-to-date hypernuclear data [@Fortin+2017; @Prov+2018] predict that $\Lambda$ and $\Xi^-$ are likely the first hyperons that appear with growing density ($\Sigma^-$-hyperons either appear at higher densities or do not appear at all in NSs). Second, calculations of [@LO2002; @ReisBon2003; @NO2006] employed reaction rates for non-leptonic weak processes derived using the contact exchange by the $W$ boson of two baryon currents. Still, it is well-known (see, e.g., Ref. [@GalReview2016]), that the most effective channel for a weak inelastic collision between a hyperon and another baryon is the meson (e.g $\pi$-meson) exchange. However, this channel was analyzed only once in Ref. [@vDD2004] to calculate $\zeta$ in the NS hyperonic core. To the best of our knowledge, the results of Ref. [@vDD2004] have never been used to compute the r-mode instability window. In the present work we revisit the bulk viscosity in a non-superfluid hyperonic NS core. We consider RMF EoS models (Sec. \[sec:EoSs\]), for which the $\Lambda$ and $\Xi^-$ hyperons appear first ($\Sigma^-$ hyperons are also present in some of our EoSs, but we focus on the $\Lambda\Xi^-$ composition for simplicity). We derive relations between $\zeta$ and the rates of the weak non-leptonic processes for an arbitrary EoS (Sec. \[sec:zeta-lambda\]). Then, adopting the one meson exchange weak interaction model, we calculate the rates for all weak non-leptonic processes operating in the $npe\mu\Lambda\Xi^-$ matter and responsible for the bulk viscosity (Sec. \[sec:lambdas\]). Simple analytic approximations are proposed for $\zeta$ and the reaction rates. We continue by applying our results to calculate the r-mode instability windows for hyperonic NSs (Sec. \[sec:windows\]). Our results indicate that the hyperonic solution to the r-mode paradox is likely more viable than it was thought before. Conclusions and some discussion are given in Sec. \[sec:conc\]. Modern equations of state {#sec:EoSs} ========================= --------------------- ------------------ --------------- ---------------------------- ------------------- -------- $n_b$ $\rho$ $M$ $R$ \[fm$^{-3}$\] \[$10^{14}\,$g cm$^{-3}$\] \[[M$_\odot$]{}\] \[km\] GM1A typical NS 0.332 5.92 1.40 13.72 $\Lambda$ onset 0.348 6.25 1.48 13.71 $\Xi^-$ onset 0.408 7.49 1.67 13.64 max mass 0.926 20.10 1.992 11.94 $\Xi^0$ onset 0.988 21.85 — — TM1C typical NS 0.315 5.63 1.40 14.31 $\Lambda$ onset 0.347 6.28 1.55 14.23 $\Xi^-$ onset 0.463 8.76 1.85 13.87 max mass 0.852 18.42 2.054 12.48 $\Xi^0$ onset 0.936 20.76 — — [NL3$\omega\rho$]{} typical NS 0.293 5.16 1.40 13.73 $\Lambda$ onset 0.352 6.39 1.95 14.03 $\Xi^-$ onset 0.474 9.29 2.50 13.86 $\Sigma^-$ onset 0.500 9.97 2.56 13.77 max mass 0.699 16.04 2.707 12.94 FSU2H $\Lambda$ onset 0.328 5.82 1.38 13.30 typical NS 0.331 5.87 1.40 13.31 $\Xi^-$ onset 0.421 7.73 1.69 13.35 $\Sigma^-$ onset 0.592 11.52 1.91 12.95 max mass 0.901 19.32 1.993 11.98 --------------------- ------------------ --------------- ---------------------------- ------------------- -------- : \[tab:astro\] Parameters of key-point NS models for the used EoS models: the central baryon density, $n_b$, and energy density, $\rho$, mass $M$ and radius $R$. ![\[fig:M-R\] Mass — radius relations for the chosen EoS models.](lgP-lgrho.eps){width="\textwidth"} \ ![\[fig:M-R\] Mass — radius relations for the chosen EoS models.](M-R.eps){width="\textwidth"} Four RMF models for the core EoS are employed in this work: GM1A and TM1C from Ref. [@GHK2014], [NL3$\omega\rho$]{} from Ref. [@Horowitz01], and FSU2H from Ref. [@Prov+2018]. The [two last]{} EoSs are calibrated to the up-to-date (hyper)nuclear data [following the approach presented in Ref. [@Fortin+2017]]{}, the former two are not. [For the FSU2H in particular we use [a]{} $\Sigma^-$ potential in the symmetric nuclear matter of $40$ MeV [so that $\Sigma^-$ appear at large enough densities and masses: $M>1.9$ [M$_\odot$]{} (see also the discussion in Ref. [@Prov+2018])]{}.]{} [In each case, the crust EoS is calculated consistently to the core one, similarly as it was done in [@Prov+2018; @Fortin16]]{}. The main astrophysical parameters for [the four]{} models are listed in Table \[tab:astro\]. [Fig. \[fig:P-rho\] shows the pressure $P$ as a function of the density and Fig. \[fig:M-R\] the associated relations between the mass $M$ and the radius $R$ of NSs as obtained when solving the Tolman-Oppenheimer-Volkov equations]{} (e.g. [@Lind1992]) for these EoSs. One can see that for [the models considered here]{} $\Lambda$ appears first, $\Xi^-$ comes after, and then other hyperon species emerge at rather high densities and NS masses. [This]{} allows us to diminish the number of reactions responsible for the bulk viscosity we have to consider. In particular, within this EoS set we can limit ourselves to the properties of $npe\mu\Lambda\Xi^-$ composition up to $M\leqslant 1.9$ [M$_\odot$]{}. [All models we consider are consistent with the existence of the most massive NSs with a precisely measured mass: PSR J$1614-2230$ [@Demorest2010; @Arzoumanian18] and PSR J$0348+0432$ [@Antoniadis2013] with NL3$\omega\rho$ giving the largest maximum mass of all models: $\sim 2.7$ [M$_\odot$]{} compared to $\sim 2$ [M$_\odot$]{} for the three other paramterizations. However only NL3$\omega\rho$ and FSU2H have values of the symmetry energy and its slope consistent with modern experimental constraints (see the discussion in e.g. [@Fortin16; @Oertel17]). Of all models, FSU2H gives the lowest radii $R\sim 13$ km of NSs with the canonical mass $1.4$ [M$_\odot$]{}. Note that for the hyperonic FSU2H EoS hyperons [are present]{} in NSs with a mass larger than $1.38$ [M$_\odot$]{}.]{} Figure \[fig:yi\] shows [that the four models]{} have significantly different composition, and we thus [expect them to give]{} different properties for [the]{} bulk viscosity. ![image](lgyi-nb_inline.eps){width="95.00000%"} With the method presented in Ref. [@GHK2014] we have calculated [the]{} Landau effective masses ${m^_{\text{L} j}}$ and Landau parameters $F_0^{jk}$ and $F_1^{jk}$ ($j$ and $k$ for all baryon species presented for a given EoS). [The quantities]{} ${m^_{\text{L} j}}$ and $F_0^{jk}$ are necessary for bulk viscosity calculations. We would like to stress that baryon Fermi velocities $v_{\text{F}j} = {p_{\text{F} j}}/{m^_{\text{L} j}}$ are close to the unity [(i.e. to the speed of light)]{} in a wide range of densities for all EoSs considered, see Figure \[fig:relativity\] for details. In other [words]{}, baryons (particularly nucleons) are essentially relativistic even at densities [typical]{} of a moderately heavy NS, $M \sim 1.5-1.9$ [M$_\odot$]{}. Thus one has to work in the [relativistic]{} framework like, e.g., in Refs. [@LO2002; @vDD2004; @NO2006], rather than in the nonrelativistic one (as, e.g., in Ref. [@HLY2002]), while calculating reaction rates for the bulk viscosity. ![image](vFermi-nb_inline.eps){width="95.00000%"} Bulk viscosity in a non-superfluid matter and reaction rates {#sec:zeta-lambda} ============================================================ [Bulk viscosity is generated due to non-equilibrium reactions.]{} In [the case]{} of the nucleon $npe\mu$ matter the main reactions are the Urca processes [@HLY2000; @HLY2001]. When [the]{} hyperons appear, the non-leptonic weak processes become the main source for the bulk viscosity (see, e.g., [@LO2002; @HLY2002]), since they are much more intensive [at typical NS temperatures]{}. There are a lot of such processes. If $\Lambda$ is the only hyperon species in the matter, the reactions are \[eq:weakReact\] $$\begin{aligned} n + p &\leftrightarrow \Lambda + p, \label{eq:Lpnp} \\ n + n &\leftrightarrow \Lambda + n, \label{eq:Lnnn} \\ n + \Lambda &\leftrightarrow \Lambda + \Lambda. \label{eq:LLnL}\end{aligned}$$ When $\Xi^-$-hyperons appear, [we have two more reactions]{} $$\begin{aligned} n + \Xi^- &\leftrightarrow \Lambda + \Xi^-, \label{eq:LXnX} \\ \Lambda + n &\leftrightarrow \Xi^- + p. \label{eq:XpLn}\end{aligned}$$ [The]{} appearance of any [additional]{} hyperon species [increases]{} [the]{} number of [the relevant]{} processes significantly. Notice also that we consider only those reactions which change the strangeness by unity, [$\vert\Delta S\vert = 1$]{}. Non-equilibrium rates of these processes, $\Delta\Gamma_\alpha$, $\alpha = (a)$, $(b)$, $(c)$, $(d)$, $(e)$, depend on [the]{} chemical equilibrium perturbations $\Delta\mu_\alpha$, where, e.g., $\Delta\mu_{(a)} = \mu_n - \mu_\Lambda$, $\Delta\mu_{(e)} = \mu_\Lambda + \mu_n - \mu_{\Xi^-} - \mu_p$, etc. In the subthermal regime, $\Delta\mu_\alpha \ll kT$ ($k$ is the Boltzmann constant), the reaction rates can be written as $$\label{eq:Gamma-lambda} \Delta\Gamma_\alpha = \lambda_\alpha \Delta\mu_\alpha.$$ [In what followsthe quantities $\lambda_\alpha$ and $\Delta\Gamma_\alpha$ will be both referred to as “the reaction rates”]{} There are also strong hyperon reactions in the NS core. In the absence of pairing they are $\sim 14-16$ orders [of magnitude]{} faster than the weak non-leptonic ones. For NS oscillations of interest, with frequency $\sim 10^2 - 10^4$ Hz, the core matter can be considered as equilibrated with respect to them. [In spite of that, strong processes are also important for the bulk viscosity calculation (see below).]{} There [are]{} no strong [hyperon]{} reactions in [the]{} $npe\mu\Lambda$ matter. If we add $\Xi^-$, the only strong process is \[eq:strongReact\] $$\label{eq:XpLL} \Xi^- + p \leftrightarrow \Lambda + \Lambda.$$ If we add $\Sigma^-$, the [strong]{} process $$\label{eq:SpLn} \Sigma^- + p \leftrightarrow \Lambda + n$$ [becomes available]{}. Adding $\Xi^0$ we switch on the process $$\label{eq:XnLL} \Xi^0 + n \leftrightarrow \Lambda + \Lambda.$$ Linear combinations of these reactions are also possible. [The complete set of reactions for the full baryon octet can be found in appendix C of [@GHK2014]]{}. We follow Ref. [@GK2008] in [describing the recipe to derive the bulk viscosity in a form convenient for studying dissipation during NS oscillations]{}. \(i) Let us consider a small harmonic perturbation of the fluid with the velocity ${\pmb{u}}$. [It is assumed that the perturbation depends on time $t$ as $\propto \exp(i\omega t)$, where $\omega$ is the frequency of the perturbation.]{} [The unperturbed background is taken to be in full hydrostatic and thermodynamic equilibrium.]{} \(ii) The fluid motion causes small departures $\delta n_j \propto \exp(i\omega t)$ from the equilibrium values [of baryon number densities,]{} $n_j$. Perturbations of chemical potentials and pressure can then be presented as $$\label{eq:dP+dmu} \delta \mu_j = \sum_k {\frac{\partial \mu_j}{\partial n_k}} \delta n_k, \quad \delta P = \sum_j n_j \delta\mu_j,$$ where $\partial\mu_j/\partial n_k$ should be calculated near equilibrium. These derivatives are related to the Landau effective masses and Landau parameters $F_0^{jk}$ (see, e.g, equation D1 in Ref. [@GHK2014]). \(iii) The bulk viscosity $\zeta$ is defined as [@GK2008] $$\label{eq:zetaDef} \delta P - \delta P_\text{eq} = - \zeta \operatorname{div}{\pmb{u}}.$$ Here $\delta P_\text{eq}$ is the pressure perturbation derived [assuming that weak processes (\[eq:weakReact\]) are prohibited]{}.[^2] [Notice that since]{} we use complex exponents, one [has]{} to calculate $\mathrm{Re}\zeta$ [when]{} considering dissipation. \(iv) The relation between the reaction rates and $\operatorname{div}{\pmb{u}}$ is provided by the continuity equations $$\label{eq:cont-Start} {\frac{\partial n_j}{\partial t}} + \operatorname{div}n_j {\pmb{u}} = \Delta\Gamma_j,$$ where $\Delta\Gamma_j$ is the total [number of particles of the $j$ species produced in unit volume per unit time (reaction rate)]{} due to both weak and strong[^3] reactions. These equations should be linearized with respect to $\delta n_j$ and ${\pmb{u}}$. To calculate $\zeta$, one can neglect spatial variations of unperturbed $n_j$ (the result is applicable to both uniform and non-uniform matter, e.g. [@GYG2005]). Density variations $\delta n_j$ are [linearly]{} dependent, because they are related by the electric neutrality condition $$\label{eq:chargeless} \sum_j e_j \delta n_j = 0$$ ($e_j$ is the electric charge of the particle species $j$) and equilibrium conditions with respect to strong reactions \[e.g., the reactions in Eqs (\[eq:strongReact\])\]: \[eq:strongEquil\] $$\begin{aligned} \delta\mu_{\Xi^-} + \delta\mu_p &= 2\delta\mu_\Lambda, \label{eq:strongEquil-XpLL} \\ \delta\mu_{\Sigma^-} + \delta\mu_p &= \delta\mu_\Lambda + \delta\mu_n, \label{eq:strongEquil-SpLn} \\ \delta\mu_{\Xi^0} + \delta\mu_n &= 2\delta\mu_\Lambda, \label{eq:strongEquil-XnLL}\end{aligned}$$ etc., supplemented with Eq. (\[eq:dP+dmu\]) for $\delta\mu_j$. Therefore, for any number of particle species, only four of density perturbations $\delta n_j$ are independent. Another important consequence of Eqs. (\[eq:strongEquil\]) is that for all non-leptonic weak processes we have $$\label{eq:unifiedDmu} \Delta\mu_\alpha = \Delta\mu_{(a)} = \delta\mu_n - \delta\mu_\Lambda = \Delta\mu.$$ This is, [in particular]{}, true for reactions that are listed in Eqs. (\[eq:weakReact\]). The most convenient choice of four independent thermodynamic parameters is: the baryon number density $n_b$ ([conserved]{} in all reactions), the electron and muon fractions $y_{e,\mu} = n_{e,\mu}/n_b$ ([conserved since we restrict ourselves to non-leptonic reactions]{}), and the strangeness fraction $y_s = \sum_j S_j n_j/n_b$, [where]{} $S_j$ is the strangeness of the species $j$. [Only weak processes contribute to the strangeness production]{} since [it is conserved]{} in strong reactions. As we consider weak non-leptonic reactions with $\Delta S = 1$ only, the total strangeness production [rate]{} $\Delta\Gamma_S$ is just the sum of all [partial]{} rates $\Delta\Gamma_\alpha$. Employing Eq. (\[eq:unifiedDmu\]) and bearing in mind that $S_j<0$, we have $$\label{eq:lambTotDef} \Delta\Gamma_S = - \lambda \Delta\mu, \quad \lambda = \sum_\alpha \lambda_\alpha,$$ where $\lambda$ is the total reaction rate of all non-leptonic weak processes. The continuity Eqs. (\[eq:cont-Start\]) [lead]{} to \[eq:cont-Fin\] $$\begin{aligned} \delta n_b &= \frac{i}{\omega} n_b \operatorname{div}{\pmb{u}}, \label{eq:cont-Fin:nb} \\ \delta y_e &= \delta y_\mu = 0, \label{eq:cont-Fin:ylep} \\ \delta y_s &= -\frac{i\Delta\Gamma_S}{\omega n_b} = \frac{i\lambda}{\omega n_b}\Delta\mu. \label{eq:cont-Fin:ys}\end{aligned}$$ Considering all thermodynamic quantities as functions of $n_b$ and $y_{e,\mu,s}$ and accounting for Eq. (\[eq:cont-Fin:ylep\]), we get \[eq:dPDmu-dnbdy\] $$\begin{aligned} \delta P &= {\frac{\partial P}{\partial n_b}}\delta n_b + {\frac{\partial P}{\partial y_s}}\delta y_s, \label{eq:dP-dnbdy} \\ \Delta\mu &= {\frac{\partial \Delta\mu}{\partial n_b}}\delta n_b + {\frac{\partial \Delta\mu}{\partial y_s}}\delta y_s \label{eq:Dmu-dnbdy}\end{aligned}$$ with $\partial \Delta\mu/\partial X = \partial \mu_n/\partial X - \partial \mu_\Lambda/\partial X$ [stemming from Eq. (\[eq:unifiedDmu\])]{}. Near-equilibrium derivatives with respect to $n_b$ and $y_s$ can be derived from Eqs. (\[eq:dP+dmu\]), (\[eq:chargeless\]), and (\[eq:strongEquil\]). The quantity $\delta P_\text{eq}$ should be calculated with Eq. (\[eq:dP-dnbdy\]) [assuming that]{} all reactions are switched off, i.e. $\delta y_s = 0$ as well as [$\delta y_e=0$ and $\delta y_\mu=0$]{}. Combining Eqs. (\[eq:zetaDef\]), (\[eq:cont-Fin\]), and (\[eq:dPDmu-dnbdy\]) we have [(cf. the formulas (22) in [@GK2008] and (17) in [@HLY2002])]{} $$\label{eq:zetaFin} \mathrm{Re} \zeta = {\zeta_\mathrm{max}}\frac{2 \lambda/{\lambda_\mathrm{max}}}{1 + (\lambda/{\lambda_\mathrm{max}})^2},$$ where \[eq:zlmax\] $$\begin{aligned} {\zeta_\mathrm{max}}&= \frac{n_b}{2\omega} {\frac{\partial P}{\partial y_s}} {\frac{\partial \Delta\mu}{\partial n_b}} \left( {\frac{\partial \Delta\mu}{\partial y_s}} \right)^{-1}, \label{eq:zlmax:z} \\ {\lambda_\mathrm{max}}&= n_b \omega \left( {\frac{\partial \Delta\mu}{\partial y_s}} \right)^{-1}. \label{eq:zlmax:l}\end{aligned}$$ Eq. (\[eq:zetaFin\]) shows a well-known feature of the hyperon bulk viscosity [@LO2002; @HLY2002; @vDD2004; @NO2006; @Haskell2015]: it has a maximum with respect to the rate of non-equilibrium processes $\lambda$. Consequently, it has a maximum with respect to temperature since [$\lambda$]{} grows with it. [Apart from]{} $\lambda$ the bulk viscosity depends on two [parameters i.e., ${\zeta_\mathrm{max}}$ which is the maximum possible bulk viscosity, and ${\lambda_\mathrm{max}}$ which is the optimal total reaction rate for a given oscillation frequency $\omega$]{}. They are [determined]{} by [the]{} thermodynamic properties of [the]{} EoS only, [and]{} not by reactions operating in the matter. ![\[fig:zlmax\] The maximum bulk viscosity $\zeta_\text{max30} = {\zeta_\mathrm{max}}/(10^{30}\,\text{g}\,\text{cm}^{-1}\,\text{s}^{-1})$ and the optimum total reaction rate $\lambda_\text{max45} = {\lambda_\mathrm{max}}/(10^{45}\,\text{erg}^{-1}\,\text{cm}^{-3}\,\text{s}^{-1})$ at $\omega=10^4\,$s$^{-1}$ as functions of density $\rho_{14} = \rho/(10^{14}\,\text{g}\,\text{cm}^{-3})$ for different EoS models. Squares, diamonds, and circles mark the points of $\Xi^-$, $\Sigma^-$, and $\Xi^0$ onsets. Crosses show the state in the center of the maximum mass NS. The thicker grey lines are for the fit by Eq. (\[eq:zlmaxFit\]). The thinner grey lines show [60%]{} (${\zeta_\mathrm{max}}$) and 20% (${\lambda_\mathrm{max}}$) deviations from the fit \[i.e. ${\zeta_\mathrm{max}}^\text{appr}\times (1\pm 0.6)$ and ${\lambda_\mathrm{max}}^\text{appr}\times(1\pm 0.2)$\].](viscPars3045-rho14_Full_edges.eps){width="\columnwidth"} Figure \[fig:zlmax\] shows ${\zeta_\mathrm{max}}$ and ${\lambda_\mathrm{max}}$ as [functions]{} of energy density $\rho$. All the curves start from zero at the points of $\Lambda$ onset. The appearance of a new hyperon [causes]{} a rapid increment of the optimum rate ${\lambda_\mathrm{max}}$, however, without discontinuity. The maximum viscosity ${\zeta_\mathrm{max}}$ increases when each of cascade hyperons appears, and decreases when $\Sigma^-$ [appears]{}. But the main feature of plots in Figure \[fig:zlmax\] is that both ${\zeta_\mathrm{max}}$ and ${\lambda_\mathrm{max}}$ are strongly sensitive to the EoS model. However, at not too high densities, $\rho \lesssim 3\rho_0$, for all EoSs considered ${\lambda_\mathrm{max}}(\rho)$ has similar behaviour and values. When only $\Lambda$ and $\Xi^-$ hyperons are [present]{} in the core, the [averaged]{} behavior of the curves in Fig. \[fig:zlmax\] is roughly reproduced by formula $$\label{eq:zlmaxFit} \begin{pmatrix} {\zeta_\mathrm{max}}^\text{appr} \\ {\lambda_\mathrm{max}}^\text{appr} \end{pmatrix} = \begin{pmatrix} \zeta_0/\omega_4 \\ \lambda_0 \omega_4 \end{pmatrix} \left( \frac{x}{1+s x} \right)^t, \quad x = \frac{\rho-\rho_\Lambda}{\rho_0},$$ where $\omega_4 = \omega/(10^4\,\text{s}^{-1})$ and $\rho_\Lambda$ is the density of $\Lambda$ hyperon onset (see Table \[tab:astro\]). The fitting parameters are $\zeta_0 = 6.5\times 10^{30}\,$g$\,$cm$^{-1}\,$s$^{-1}$, $\lambda_0 = 8.0\times 10^{45}\,$erg$^{-1}\,$cm$^{-3}\,$s$^{-1}$, $t = 0.34$, and $s = 1.0$ for ${\zeta_\mathrm{max}}$ ([maximum]{} error [$\sim 60\%$]{}) and $s = 1.5$ for ${\lambda_\mathrm{max}}$ ([maximum]{} error $\sim 20\%$) respectively. We emphasize that the power $t$ describing the behavior at $\rho \to \rho_\Lambda$ is the same for both these quantities. The thicker grey curves in Fig. \[fig:zlmax\] show how this fit works, and the [thinner ones visualize]{} [60%]{} and 20% [uncertainties]{} for ${\zeta_\mathrm{max}}$ and ${\lambda_\mathrm{max}}$, correspondingly. Of course, Eq. (\[eq:zlmaxFit\]) does not reproduce kinks at [the]{} $\Xi^-$ onset points and it [does not describe behavior of the curves after appearance of $\Sigma^-$ or $\Xi^0$ hyperon]{}. However, the four EoSs we use here are significantly different, and we can hope that, for the $npe\mu\Lambda\Xi^-$ matter, [any other RMF model would give ${\zeta_\mathrm{max}}$ and ${\lambda_\mathrm{max}}$ within the range of uncertainties predicted by our fit (\[eq:zlmaxFit\]).]{} When plotting r-mode instability windows, the averaged fit for ${\lambda_\mathrm{max}}^\text{appr}$ appears to be rather accurate, but the fit for ${\zeta_\mathrm{max}}^\text{appr}$, without additional corrections, fails to reproduce the r-mode instability window for some specific EoS. See the end of Sec. \[sec:windows\] and the caption to Fig. \[fig:windowCheck\] for a description of how one should use Eq. (\[eq:zlmaxFit\]) to solve this problem. Now, the question is how close [the “real”]{} reaction rate of weak non-leptonic reactions $\lambda$ [can]{} be to the optimum rate. Nonleptonic weak processes {#sec:lambdas} ========================== General formalism {#sec:lambdas-gen} ----------------- [The formalism of reaction rate calculation that we use follows]{} [@HLY2002; @vDD2004]. In general, we [consider a process in which a pair of baryons]{}[^4] transforms into another one, $$\label{eq:processGen} 1 + 2 \leftrightarrow 3 + 4,$$ where for [baryon strangenesses]{} the rule $\|S_1 + S_2 - S_3 - S_4\|=1$ holds. If the baryon composition is $np\Lambda\Xi^-$, then we [are left with]{} only the five processes listed in Eq. (\[eq:weakReact\]). An inelastic collision $1+2\to 3+4$ is described by a matrix element $\mathcal{M}_{12\to 34}$. Hereafter we assume that during its calculation the particle wavefunctions are normalized to one particle per unit volume. Then, [setting]{} $\hbar=c=1$ and treating particles [as]{} non-polarized, the [expression for the rate]{} of a direct reaction $1+2 \rightarrow 3+4$ is $$\begin{gathered} \label{eq:Gdir} \Gamma_{\to} = \int \prod_{j=1}^4 \frac{{\mathrm{d}}^3 {\pmb{p}}_j}{(2\pi)^3 2{m^_{\text{L} j}}} (2\pi)^4 \delta(p_1 + p_2 - p_3 - p_4) \times \\ \frac{1}{s}\sum_\text{spins} \bigl\| \mathcal{M}_{12\to 34} \bigr\|^2 f_1 f_2 (1-f_3) (1-f_4),\end{gathered}$$ where $p_j = (\epsilon_j,{\pmb{p}}_j)$ is a $j$’th quasiparticle 4-momentum, $s$ is the symmetry factor, which is equal to 2 for the reactions (\[eq:Lnnn\]) and (\[eq:LLnL\]), otherwise $s=1$, and $$\label{eq:fjDef} f_j = f(z_j), \quad z_j = \frac{\epsilon_j - \mu_j}{kT}, \quad f(z) = \frac{1}{1+e^z}$$ is the Fermi [distribution]{} function. Since the [fermions in the NS core matter are]{} strongly degenerate, one can perform the phase space decomposition [@ShapTeuk1983] in (\[eq:Gdir\]): $$\label{eq:GdirDecomp} \Gamma_{\to} = \frac{\prod_j {p_{\text{F} j}}}{4(2\pi)^8 s} (kT)^3 \mathcal{I}\left( \frac{\Delta\mu}{kT} \right) \mathcal{A}\mathcal{J},$$ where $\Delta\mu = \mu_1 + \mu_2 - \mu_3 - \mu_4$ ([recall that]{} Eq. \[eq:unifiedDmu\] [states]{} that all [$\Delta\mu_\alpha$]{} are equal in our problem). For the factors $\mathcal{I}$, $\mathcal{A}$, and $\mathcal{J}$ we have [@HLY2002] \[eq:IAJ\] $$\begin{aligned} \label{eq:IAJ-I} \mathcal{I}(\xi) =& \int \prod_j \left[ {\mathrm{d}}z_j f(z_j) \right] \delta\left(\sum_j z_j-\xi\right) = \frac{4\pi^2\xi + \xi^3}{6(1-e^{-\xi})}, \\ \nonumber \mathcal{A} =& \int \prod_j{\mathrm{d}}\Omega_j \delta({\pmb{p}}_1 + {\pmb{p}}_2 - {\pmb{p}}_3 - {\pmb{p}}_4) \\ \label{eq:IAJ-A} &= \frac{2(2\pi)^3}{\prod_j {p_{\text{F} j}}}(q_\text{max} - q_\text{min}) \Theta(q_\text{max} - q_\text{min}), \\ \label{eq:IAJ-J} \mathcal{J} =& \frac{1}{\mathcal{A}} \int \prod_j {\mathrm{d}}\Omega_j \delta({\pmb{p}}_1 + {\pmb{p}}_2 - {\pmb{p}}_3 - {\pmb{p}}_4) \left\langle \bigl\| \mathcal{M}_{12\to 34} \bigr\|^2 \right\rangle,\end{aligned}$$ where $\langle \rangle$ means summation over the final spin states and averaging over the initial ones, $\Theta(x)$ is the Heaviside function, and \[eq:qMinMax\] $$\begin{aligned} \label{eq:qMinMax-min} q_\text{min} =& \max \left\{ \|{p_{\text{F} 1}}-{p_{\text{F} 3}}\|, \|{p_{\text{F} 2}}-{p_{\text{F} 4}}\| \right\}, \\ \label{eq:qMinMax-max} q_\text{max} =& \min \left\{ {p_{\text{F} 1}}+{p_{\text{F} 3}}, {p_{\text{F} 2}}+{p_{\text{F} 4}} \right\}, \end{aligned}$$ are the minimum and maximum momentum transfers. An inverse reaction $3+4\rightarrow 1+2$ has the rate $\Gamma_{\leftarrow} = \Gamma_\to (\Delta\mu \to -\Delta\mu)$, so the total process rate is $$\label{eq:DeltaGamma-IAJ} \Delta\Gamma_{12\leftrightarrow 34} = \frac{\prod_j {p_{\text{F} j}}}{4(2\pi)^8 s} (kT)^3 \Delta\mathcal{I}\left( \frac{\Delta\mu}{kT} \right) \mathcal{A}\mathcal{J},$$ where $$\label{eq:DeltaIdef} \Delta\mathcal{I}(\xi) = \mathcal{I}(\xi) - \mathcal{I}(-\xi) = \frac{2\pi^2}{3}\xi \left( 1 + \frac{\xi^2}{4\pi^2} \right).$$ In the subthermal limit, $\Delta\mu \ll kT$, Eq. (\[eq:DeltaGamma-IAJ\]) takes the already mentioned form of Eq. (\[eq:Gamma-lambda\]). [The next tasks consist in (i) deriving an expression for $\langle\|\mathcal{M}\|^2\rangle$ and then (ii) averaging it via the angular integrations, yielding in this way the formula for $\mathcal{J}$, Eq. (\[eq:IAJ-J\])]{}. Matrix element {#sec:lambdas-Mfi} -------------- ![\[fig:diagr\] The lightest meson exchange Feynman diagrams for the inelastic scatterings in Eqs. (\[eq:weakReact\]). Open and filled circles mark weak and strong vertices, respectively.](Diagrams.eps){width="\columnwidth"} A non-leptonic weak reaction can go via two channels. The first one is a direct $W$-boson exchange [between]{} two baryons, the weak contact interaction. The second channel is a virtual meson exchange, when a $W$-[boson]{}, emitted by one of the quarks confined in a baryon, decays into a [pair of quark and antiquark]{} that participate in further formation of an intermediate meson and an outgoing baryon. The $W$ exchange in the weak non-leptonic reactions is well-studied in context of the bulk viscosity in NS cores, e.g. [@LO2002; @HLY2002; @vDD2004; @NO2006]. The meson-exchange channel is commonly used in studies of non-leptonic hyperon decays in laboratory, see e.g. [@GalReview2016] for a review. [In particular, the nucleon-induced $\Lambda$ decay and formation, [$np\leftrightarrow\Lambda p$]{} and [$nn\leftrightarrow\Lambda n$]{}, is explored in hypernuclear physics [@PRB1997; @IM2010; @BauGarPer2017] and in nucleon-nucleon scatterings [@Parreno+1999]. These processes are studied, e.g., within the one meson exchange (OME) approach, including the full pseudoscalar and vector meson octets [@PRB1997], as well as with one-loop corrections [@PerOb+2013] and account for decay of the virtual meson into a couple of others [@IM2010]]{}. The process [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} is studied in the hyperon-induced $\Lambda$ decay in double-strange hypernuclei [@PRB2002; @BauGarPer2015] within the OME approach. [To the best of our knowledge]{}, weak processes with $\Xi^-$, like [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{} and [$\Lambda n\leftrightarrow\Xi^-p$]{}, are not studied neither experimentally nor theoretically, since the strong reactions $\Xi^-p\to\Lambda\Lambda$ and $\Xi^-n\to\Lambda\Sigma^-$ operate much more effectively. In general, the $W$ exchange channel for the non-leptonic hyperon decay is less effective than the meson-exchange channel. Moreover, some of the processes [have no $W$]{} exchange contribution due to the absence of a weak $sd$ quark current [@GrKl1990]. For instance, in the set of processes (\[eq:weakReact\]) only [$np\leftrightarrow\Lambda p$]{} and [$\Lambda n\leftrightarrow\Xi^-p$]{} can operate with the [$W$]{} exchange[^5]. However, only once [@vDD2004] the OME channel was used for [calculating]{} the bulk viscosity in the NS core. Three reactions were considered in [that]{} work, $n n\leftrightarrow\Sigma^-p$, [$np\leftrightarrow\Lambda p$]{}, and [$nn\leftrightarrow\Lambda n$]{}, using both OME and $W$ exchanges. In particular, it was inferred that OME is $\sim 10$ times more intensive for [$np\leftrightarrow\Lambda p$]{}. But no handy formulae were given to make [results of [@vDD2004] convenient for applying]{} in further calculations involving the bulk viscosity. In [the present]{} work we try to [reproduce]{} the results of [@vDD2004] and adopt them to the modern hyperon [compositions]{} [of the NS core]{}. Considering OME, we take into account the lightest meson exchange only, the $K^0$/$\bar{K}^0$ mesons for [$n\Lambda\leftrightarrow\Lambda\Lambda$]{}, and the $\pi$ mesons for the other reactions. [All]{} these mesons are pseudoscalar. Corresponding diagrams are shown in Fig. \[fig:diagr\] for each of five processes considered. [An important deficiency of our approach is that]{} we do not account for any other mesons, [e.g. the $\rho$ one]{}. Commonly, [their effect is to decrease]{} the reaction rate up to $3-4$ times [which]{} is not crucial for our purposes, see the discussion in Sec. \[sec:conc\]. Vertex Strong $g$ Weak $A$ Weak $B$ Reference ------------------- ---------------- ---------- ---------- ---------------------------- -- -- pp$\pi$ $13.3$ — — [@PRB1997], tab. III np$\pi$ $13.3\sqrt{2}$ — — [@PRB1997], tab. III nn$\pi$ $-13.3$ — — [@PRB1997], tab. III $\Lambda$n$\pi$ — $-1.07$ $-7.19$ [@vDD2004], sec. V $\Lambda$p$\pi$ — $1.46$ $9.95$ [@vDD2004], sec. V $\Lambda$nK $-14.1$ — — [@PRB1997], tab. III $\Lambda\Lambda$K — $0.67$ $-12.72$ [@PRB2002]$^a$, tab. IV $\Xi^-\Lambda\pi$ — $2.04$ $-7.5$ [@Okun], ch. 30.3.1 $\Xi^-\Xi^-\pi$ $-5.4$ — — [@RSY1999]$^b$, eq. (2.14) : \[tab:constants\] Phenomenological interaction constants in vertices in Fig. \[fig:diagr\]. $^a$ They use the opposite sign for $\gamma^5$.\ $^b$ Their strong $f$ couplings are related to $g$ couplings as $g = f (m_2 + m_4)/m_\pi$. There is one weak (marked by $\circ$) and one strong (marked by $\bullet$) vertex for the baryon-meson interaction in each diagram. Both weak and strong vertices are phenomenological. For the pseudoscalar meson exchange they correspond to, [respectively,]{} $$\label{eq:vertices} \circ = G_\text{F} m_\pi^2 (A + B\gamma^5), \quad \bullet = g \gamma^5,$$ where $G_\text{F} = 1.436\times 10^{-49}\,\text{erg}\,\text{cm}^3$ is the Fermi coupling constant, $m_\pi$ is the charged pion mass, and $\gamma^5 = -i \gamma^0 \gamma^1 \gamma^2 \gamma^3$. The phenomenological constants $g$, $A$, and $B$ for the vertices [in the diagrams]{} in Fig. \[fig:diagr\] are listed in Tab. \[tab:constants\]. Some of [these constants]{} are measured in laboratory, [while some are evaluated]{} theoretically. The meson propagator $D_M(q)$, where $q$ is the 4-momentum transfer, is discussed in Sec. \[sec:lambdas-gator\]. Wavefunctions of the ingoing and outgoing quasiparticles are considered within the RMF approach, [i.e.]{}, they [have the form of]{} relativistic bispinors, $$\label{eq:psi} \psi_j = C_j u_j e^{i p_j^\mu x_{j\mu}}.$$ For strongly degenerate baryons in the NS core one can use the approximation $\|{\pmb{p}}_j\|={p_{\text{F} j}}$. Further, for the bispinor $u_j$ one should use ${m^_{\text{L} j}}$ instead of $\epsilon_j$ and the Dirac effective mass ${m^_{\text{D} j}}$ instead of the rest mass $m_j$. The Landau and Dirac effective masses are related by the formula [@Glend2000] $$\label{eq:mL-mD} {m_{\text{L} j}^{* 2}} = {p_{\text{F} j}}^2 + {m_{\text{D} j}^{* 2}}.$$ Then for the normalization [constants]{} $C_j$ (one particle per unit volume) and the bispinor $u_j$ one [obtains]{} \[eq:CandU\] $$\begin{aligned} \label{eq:CandU-C} &C_j = \frac{1}{\sqrt{2{m^_{\text{L} j}}}}, \\ \label{eq:CandU-uu-S} &\bar{u}_j u_j = 2 {m^_{\text{D} j}}, \\ \label{eq:CandU-uu-M} &\sum_\text{spins} u_j \bar{u}_j = \gamma^0 {m^_{\text{L} j}} - {\pmb{\gamma}}\cdot{\pmb{p}}_j + {m^_{\text{D} j}}.\end{aligned}$$ [Let us notice that]{} a quasiparticle dispersion relation $p_j^0 = \epsilon_j({\pmb{p}}_j)$ is more complex than the free particle [one]{}, in particular $\epsilon_j({p_{\text{F} j}}) = \mu_j \neq {m^_{\text{L} j}}$. [The [$np\leftrightarrow\Lambda p$]{}, [$nn\leftrightarrow\Lambda n$]{}, and [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} processes involve direct and exchange diagrams.]{} [However, the [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{} and [$\Lambda n\leftrightarrow\Xi^-p$]{} processes do not involve exchange diagrams due to, for example, the rule $\|\Delta S\|=1$ which holds in each [weak]{} vertex]{}[^6]. In what follows, for a process in the general form (\[eq:processGen\]) we consider the direct and exchange diagrams [that]{} differ by $1\leftrightarrow 2$ permutation, with weak vertices $1,3$ and $2,3$. For the direct diagram one has $$\label{eq:MfiDir} \mathcal{M}_{12\to 34}^\text{dir} = G_\text{F} m_\pi^2 \bar{u}_3 (A_{13} + B_{13}\gamma^5)u_1 D_M(q) \bar{u}_4 g_{24}\gamma^5 u_2$$ The exchange diagram corresponds to $\mathcal{M}_{12\to 34}^\text{exch} = \mathcal{M}_{12\to 34}^\text{dir}\bigr\|_{1\leftrightarrow 2}$, and the total matrix element is $\mathcal{M}_{12\to 34} = \mathcal{M}_{12\to 34}^\text{dir} - \mathcal{M}_{12\to 34}^\text{exch}$. If there is no exchange diagram for the process considered, one should (artificially) set $A_{23} = B_{23} = g_{14} = 0$. After [averaging]{} over the initial and summing over the final spin states of the squared $\mathcal{M}_{12\to 34}$ we get $$\begin{gathered} \label{eq:MfiSqAvSum} \left\langle \left\| \mathcal{M}_{12\to 34} \right\|^2 \right\rangle = G_\text{F}^2 m_\pi^4 \left[ X(q) D_M^2(q) \right. \\ \left. + X'(q') D_M^2(q') + Y(q,q') D_M(q)D_M(q') \right],\end{gathered}$$ where $$\label{eq:qqP} q = p_3 - p_1, \quad q' = p_3 - p_2,$$ and \[eq:XYZ\] $$\begin{aligned} \label{eq:XYZ-X} X(q) = X(\|{\pmb{q}}\|^2) = m_M^4 &X_0 + m_M^2 X_1 \|{\pmb{q}}\|^2 + X_2 \|{\pmb{q}}\|^4, \\ \nonumber Y(q,q') = Y(\|{\pmb{q}}\|^2 ,\|{\pmb{q}}'\|^2) = m_M^4 &Y_0 + m_M^2 Y_1 \|{\pmb{q}}\|^2 \\ \label{eq:XYZ-Y} +&\ m_M^2 Y_2 \|{\pmb{q}}'\|^2 + Y_3 \|{\pmb{q}}\|^2 \|{\pmb{q}}'\|^2, \\ \label{eq:XYZ-Z} X'(q) = X'(\|{\pmb{q}}\|^2) = m_M^4 &X'_0 + m_M^2 X'_1 \|{\pmb{q}}'\|^2 + X'_2 \|{\pmb{q}}'\|^4,\end{aligned}$$ with dimensionless $X_k$, $X'_k$, and $Y_k$ being functions of ${p_{\text{F} 1...4}}$ listed in Appendix \[app:Mfi\]. The last issue [to be resolved]{} before we can evaluate Eq. (\[eq:IAJ-J\]) is to define meson propagators $D_M$. Meson propagators {#sec:lambdas-gator} ----------------- In general, the meson propagator is $$\label{eq:DM-Gen} D_M^{-1}(\omega,{\pmb{q}}) = \omega^2 - {\pmb{q}}^2 - m_M^2 - \Pi_M(\omega,{\pmb{q}}),$$ where $\omega$ and ${\pmb{q}}$ are the energy and momentum transferred by the virtual meson, $m_M$ is the bare (vacuum) meson mass ($m_\pi = 139\,$MeV and $m_K = 494\,$MeV)[^7], and $\Pi_M$ is the meson polarisation operator. Within a widely used free meson approach [@FM1979; @Maxwell1987; @vDD2004] the polarisation operator is $\Pi_M = 0$ and $\omega^2$ is omitted due to some reasons. In the almost beta-equilibrated matter of the NS core we indeed have $\omega = 0$ for neutral mesons, but [for]{} the charged pions in the diagrams for the processes [$np\leftrightarrow\Lambda p$]{} (Fig. \[fig:diagr\]a) and [$\Lambda n\leftrightarrow\Xi^-p$]{} (Fig. \[fig:diagr\]e) [we]{} have $\omega = \mu_e \ne 0$. Thus the approach by [@vDD2004] to the meson propagator has to be revisited. If we substitute $\omega = \mu_e$ into the free pion propagator, we [get into trouble as soon as]{} $\mu_e > m_\pi$ at $n_b \gtrsim 0.2\,$fm$^{-3}$, and the pion propagator can be positive at some [real values of momentum transfer]{}. This means [that the real]{} pions appear in the matter, but it is inconsistent with our EoS models, which (artificially) prohibit pionization. [This troubling feature]{} appears not only for all four EoSs [that we are using]{} (see Sec. \[sec:EoSs\]), but also for a number of other realistic nucleon EoS models like APR [@APR1998] and BSk21 [@BSK2013]. Therefore we [are forced]{} to account for the polarisation operator $\Pi_{\pi^-}$ of negative pions [hoping]{} that at $\omega=\mu_e$ it is large enough to make $D_{\pi^-}<0$ for all densities. [We find it convenient]{} to introduce the “effective” virtual pion mass, $$\label{eq:mPiEff} \tilde{m}_{\pi-} = \sqrt{m_\pi^2 - \mu_e^2 + \Pi_{\pi^-}(\mu_e,{\pmb{q}})}.$$ Then the propagator takes a simple form $$\label{eq:DpiNonrel} D_{\pi^-}^{-1} = - {\pmb{q}}^2 - \tilde{m}_{\pi^-}^2({\pmb{q}}).$$ Notice that $\mu_e$ varies with density, so $\tilde{m}_{\pi^-}$ technically depends not only on the momentum transfer ${\pmb{q}}$ but also on $n_b$. Obviously, $\tilde{m}_{\pi^-}$ should be strictly real when the [appearance of real pions (pionization)]{} is prohibited. In [nuclear matter characteristic of atomic nuclei]{} we have [@KKW2003] $\Pi_{\pi^-} = \Pi_S + \Delta\Pi_S + \Pi_P$, where $\Pi_S$ comes from the s-wave $n\pi$-scattering, $\Delta\Pi_S$ comes from the s-wave absorption and $\Pi_P$ is the p-wave contribution. Only $\Pi_S$ is positive, so we focus on it [in order to get]{} an upper estimate of $\Pi_{\pi^-}$. The [leading-order]{} contribution to $\Pi_S$ in the nucleon-hyperon NS core comes from the terms [@Kolom-Private] $$\label{eq:PiS-hypCore} \Pi_S(\omega) = \frac{\omega}{f_\pi^2}\sum_j (-I_{3j}) n_j + \frac{\sigma_N}{f_\pi^2}\left( \frac{\omega^2}{m_\pi^2} - 1 \right) n_b,$$ where $j$ is the baryon index, $I_{3j}$ is the isospin projection of the $j^\text{th}$ baryon, $f_\pi = 92.4\,$MeV and $\sigma_N \approx 45\,$MeV. In the nucleonic matter Eq. (\[eq:PiS-hypCore\]) [coincides]{} with equation (11) of [@KKW2003]. ![\[fig:mPiEff\] Thick lines show the upper estimate of the ‘effective’ pion mass for the EoS models employed. Thin lines show what happens if we do not account for the polarization operator in Eq. (\[eq:mPiEff\]).](mPiEffToBare-nb.eps){width="\columnwidth"} Thick curves in Fig. \[fig:mPiEff\] show the ratio $\tilde{m}_{\pi^-}/m_\pi$ with $\Pi_{\pi^-} = \Pi_S$ for the EoS models we use in this work. Notice that in this case, according to Eq. (\[eq:mPiEff\]), $\tilde{m}_{\pi^-}$ technically depends on $n_b$ only. Thin curves are for $\tilde{m}_{\pi^-}$ with $\Pi_{\pi^-} = 0$. They prove what was claimed in the beginning of this section: $\mu_e$ [exceeds]{} the bare pion mass at $n_b \sim 0.2\,$fm$^{-3}$, so we have to account for the polarization operator to avoid a [pionization instability]{}. The s-wave part is only an upper estimate of $\Pi_{\pi^-}$, so actual values of $\tilde{m}_{\pi^-}/m_\pi$ are located below the thick lines in Fig. \[fig:mPiEff\]. For densities between the hyperon onset point and the maximum mass point the upper limit for $\tilde{m}_{\pi^-}$ varies in the range $(0.7...1.6)m_\pi$. Thus $m_\pi$ is a rough upper limit for $\tilde{m}_{\pi^-}$. Consequently, $1/D_{\pi^-} = -{\pmb{q}}^2 - m_\pi^2$ is a rough lower estimate for the propagator modulus. It can be used for making a lower estimate of the reaction rates. An account for the variation of the $\tilde{m}_{\pi^-}$ upper limit mentioned above can affect a rate value not more than by a factor of order 2, which is [acceptable]{} for our purposes. Of course, accounting for other terms in $\Pi_{\pi^-}$ may dramatically [change $D_{\pi^-}$ compared to the prediction from simple expression (\[eq:DpiNonrel\])]{} with $\tilde{m}_{\pi^-} = m_\pi$. Then “the effective pion mass” should be [replaced]{} by the effective pion gap [@Migdal+1990], which can be much less than $m_\pi$. Correspondingly, the pion propagator [would increase]{}. However, these effects are [model-dependent]{}, [so we prefer to use Eq. (\[eq:DpiNonrel\]) with $\tilde{m}_{\pi^-} = m_\pi$ in what follows, [similarly to how it was done in]{} [@FM1979; @Maxwell1987; @vDD2004].]{} What should we do with propagators of neutral mesons, [$\bar{K}^0$]{} and $\pi^0$? The former one is [a quite]{} heavy meson, and [it is harder to affect its propagator essentially]{}. Thus [$\bar{K}^0$]{} can be safely described by a free-[particle]{} propagator. The latter [meson]{}, $\pi^0$, requires more careful discussion, but one can artificially set the free-particle propagator [for]{} it [within]{} the same range of reliability as for $\pi^-$. All in all, for each meson propagator we use $$\label{eq:DMforUse} D_M^{-1} = -{\pmb{q}}^2 - m_M^2.$$ This can [lead to underestimating the reaction rates]{}. But this effect will be (partially) compensated by neglecting [the contribution due to the vector mesons]{}, see Sec. \[sec:conc\] for a more detailed discussion. Reaction rates {#sec:lambdas-rates} -------------- Taking $\langle\left\| \mathcal{M}_{12\to 34} \right\|^2\rangle$ from Eq. (\[eq:MfiSqAvSum\]), $D_M$ from Eq. (\[eq:DMforUse\]), and substituting them into Eq. (\[eq:IAJ-J\]), we can [calculate]{} $\mathcal{J}$ (see Appendix \[app:Iang\] for details) and, consequently, [get]{} the reaction rate $\Delta\Gamma_{12\leftrightarrow 34}$ from Eq. (\[eq:DeltaGamma-IAJ\]). In the subthermal regime, $\Delta\mu\ll kT$, it can be expressed in terms of $\lambda_{12\leftrightarrow 34}$ (see Eq. (\[eq:Gamma-lambda\])) $$\label{eq:lambda1234} \lambda_{12\leftrightarrow 34} = \lambda_0^{12\leftrightarrow 34} \mathcal{W}_{12\leftrightarrow 34},$$ where, restoring natural units, \[eq:lambda0+Wdef\] $$\begin{gathered} \label{eq:lambda0} \lambda_0^{12\leftrightarrow 34} = \frac{G_\text{F}^2 m_N^4}{6\pi^3 \hbar^{10}} \left( q_\text{max} - q_\text{min} \right) (kT)^2 \Theta_{12\leftrightarrow 34} \\ \approx \frac{1.7\times 10^{45}}{\text{erg cm$^3$ s}} \times \frac{q_\text{max} - q_\text{min}}{\hbar \left( 3\pi^2 n_0 \right)^{1/3}} T_8^2 \Theta_{12\leftrightarrow 34},\end{gathered}$$ with the nucleon mass[^8] $m_N = 939\,$MeV, $T_8 = T/(10^8\,\text{K})$, $\Theta_{12\leftrightarrow 34} = \Theta(q_\text{max} - q_\text{min})$, and $$\begin{gathered} \label{eq:Wdef} \mathcal{W}_{12\leftrightarrow 34} = \frac{1}{s}\left( \frac{m_\pi}{2m_N} \right)^4 \left( X_0 J_0 + X_1 J_1 + X_2 J_2 \right. \\ \left. + X'_0 J'_0 + X'_1 J'_1 + X'_2 J'_2 \right. \\ \left. + Y_0 J_3 + Y_1 J_4 + Y_2 J'_4 + Y_3 J_5, \right)\end{gathered}$$ is a dimensionless function of ${p_{\text{F} 1}},{p_{\text{F} 2}},{p_{\text{F} 3}},{p_{\text{F} 4}}$, with $X_k$, $X'_k$, and $Y_k$ defined in Appendix \[app:Mfi\], and $J_k$ and $J'_k$ defined in Appendix \[app:Iang\]. Actually, $\mathcal{W}$ is related to $\mathcal{J}$ in a simple way: $$\label{eq:J-W} \mathcal{J} = 16 s G_\text{F}^2 m_N^4 \mathcal{W}.$$ In the suprathermal regime, $\Delta\mu\gtrsim kT$, one has to use $$\label{eq:DeltaGamma1234} \Delta\Gamma_{12\leftrightarrow 34} = \lambda_{12\leftrightarrow 34}\Delta\mu \left[ 1 + \left( \frac{\Delta\mu}{2\pi kT} \right)^2 \right].$$ ![image](W-rho14_edges.eps){width="95.00000%"} Process $W_0$ $a$ $b$ $p$ error --------------------------------------------- ------- ------ ------ ----- ------- -- [$np\leftrightarrow\Lambda p$]{} 1.1 — — — 30% [$\Lambda n\leftrightarrow\Xi^-p$]{} 0.9 — — — 50% [$nn\leftrightarrow\Lambda n$]{} 0.48 — — — 20% [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} 0.38 0.37 0.87 2 30% [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{} 0.068 — — — 30% : \[tab:Wapprox\] Fitting parameters in Eq. (\[eq:Wapprox\]) we recommend for using in practice. The $\mathcal{W}$ function incorporates all specific properties of the process $12 \leftrightarrow 34$ (recall that $X_k$, $Y_k$, etc. depend on weak and strong coupling constants that are different for different processes). Fig. \[fig:W-rho\] shows how it depends [on the (energy) density $\rho$]{} for each kind of processes in Eq. (\[eq:weakReact\]) for all EoSs we use. It appears to be strongly model-dependent: $\mathcal{W}$ varies up to a factor of $3$ from one EoS to another. Fortunately, it [appears]{} to be a slow function of $\rho$. Since the main [aim]{} of our calculations is [application]{} in [the]{} r-mode physics, it is enough to provide a simple (even if not too precise) approximation of the reaction rate. For [$np\leftrightarrow\Lambda p$]{}, [$nn\leftrightarrow\Lambda n$]{}, [$\Lambda n\leftrightarrow\Xi^-p$]{}, and [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{} processes we can reliably treat $\mathcal{W}$ as a constant, while for [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} it is safer [to]{} account [that it grows]{} with $\rho$. The approximation [that]{} we recommend is $$\label{eq:Wapprox} \mathcal{W}_\text{appr} = W_0 \left( \frac{x+a}{x+b} \right)^p, \quad x = \frac{\rho - \rho_\text{start}}{\rho_0},$$ where $\rho_\text{start}$ is the density where the process $12\leftrightarrow 34$ [switches]{} on, and $\rho_0 = 2.8\times 10^{14}\,\text{g}\,\text{cm}^{-3}$ is the nuclear matter saturation density. Note that $\rho_\text{start}$ may not coincide with the density of $\Lambda$ or $\Xi^-$ onset, and should be derived as a lowest density where $\Theta_{12\leftrightarrow 34}>0$. Parameters $W_0$, $a$, $b$, and $p$ represent a very rough fit of what we have in Fig. \[fig:W-rho\]. The latter three are required for [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} only, other processes can be described with a single constant $W_0$. In Table \[tab:Wapprox\] we give the parameters of this fit for each process. The thicker grey lines in Fig. \[fig:W-rho\] show how these fits work. The ‘error’ column in Table \[tab:Wapprox\] represents ‘ranges of deviations’, $\|\mathcal{W}-\mathcal{W}_\text{appr}\|/\mathcal{W}_\text{appr}$. Most of $\mathcal{W}$ curves lie within these ranges (we stress that [it is more]{} important to reproduce $\mathcal{W}$ behavior far from $\rho_\text{start}$ than close to it). In Fig. \[fig:W-rho\] the [thinner]{} grey lines display [boundaries of these error ranges.]{} Process $l_0$ $c$ $q$ error --------------------------------------------- ------- ------ ------ ------- -- [$np\leftrightarrow\Lambda p$]{} 1.7 0.06 0.36 20% [$\Lambda n\leftrightarrow\Xi^-p$]{} 1.5 0.00 0.36 30% [$nn\leftrightarrow\Lambda n$]{} 2.9 0.3 0.4 20% [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} 3.5 0.8 1.0 30% [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{} 1.6 0.5 1.0 40% : \[tab:lambdaApprox\] Fitting parameters in Eq. (\[eq:lambdaApprox\]) we recommend to use. ![image](lambda-rho14_edges.eps){width="95.00000%"} Thus, [in order to quickly estimate]{} reaction rates for an arbitrary EoS, one can take $\mathcal{W}$ from Eq. (\[eq:Wapprox\]) and substitute it into Eq. (\[eq:lambda1234\]) to obtain $\lambda$ for the process considered. The [quantity]{} $\lambda_0$ can be easily calculated for each process when [the number density]{} $n_j$ of each particle species is known. However, one may desire an approximate formula that does not require knowledge of particle fractions, e.g. to explore some phenomenological $P(\rho)$ models, supplemented with [an arbitrarily]{} chosen $\rho_\text{start}$. [For that purpose,]{} we provide an approximate expression for $\lambda_0$ that depends on $\rho$ and $\rho_\text{start}$ only, $$\label{eq:lambdaApprox} \lambda_{0\,\text{appr}} = l_0 \left( \frac{x}{1 + c x} \right)^q T_8^2, \;\;\; \lambda_\text{appr} = \lambda_{0\,\text{appr}} \mathcal{W}_\text{appr},$$ with the same $x$ as in Eq. (\[eq:Wapprox\]). Recommended values of $c$, $q$, and $l_0$ and maximum relative deviations for each process are given in Table \[tab:lambdaApprox\]. Fig. \[fig:lambda-rho\] shows [the density dependence $\lambda(\rho)$]{} for all five processes [that]{} we consider for EoS models from Sec. \[sec:EoSs\] at $T = 10^8\,$K. Grey lines show $\lambda_\text{appr}$ (thicker [lines]{}) and [boundaries of its uncertainty (thinner lines)]{} due to both $\mathcal{W}$ and $\lambda_0$ approximation errors. For instance, for [$np\leftrightarrow\Lambda p$]{} the thinner lines correspond to $\lambda_{0\,\text{appr}}^\text{{$np\leftrightarrow\Lambda p$}}\times(1\pm 0.3)\times \mathcal{W}_\text{appr}^\text{{$np\leftrightarrow\Lambda p$}}\times(1\pm 0.2)$. The reaction rates are also model-dependent, [similarly to the $\mathcal{W}$ functions]{}. There is an explicit hierarchy[^9] of $\lambda$ typical values. The [processes]{} [$np\leftrightarrow\Lambda p$]{} and [$\Lambda n\leftrightarrow\Xi^-p$]{} [turn out]{} to be the most effective. The next are [$nn\leftrightarrow\Lambda n$]{} and [$n\Lambda\leftrightarrow\Lambda\Lambda$]{}. The latter one has [stronger]{} $\rho$ dependence since it is more sensitive to the $\Lambda$ [fraction]{}. The least intensive is [the [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{} process]{}. There are two reasons for this. First, it is most sensitive to low $\Xi^-$ density. Second, it has the lowest $B$ and $g$ coupling constants (see Tab. \[tab:constants\]), and it has no exchange term contribution in our approximation. The same hierarchy of reaction rates can be seen in Fig. \[fig:W-rho\] for the $\mathcal{W}$ functions. Notice that $\rho_\text{start}$ points (where $\lambda$’s rise up from zero in Fig. \[fig:lambda-rho\]) differ from $\Lambda$ onset densities for [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} and from $\Xi^-$ onset densities for [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{}, since the conditions $\Theta_{n\Lambda\leftrightarrow \Lambda\Lambda}>0$ and $\Theta_{n\Xi\leftrightarrow \Lambda\Xi}>0$ can be satisfied only for high enough $n_\Lambda$ and $n_\Xi$. OME vs $W$ exchange {#sec:lambdas-OMEvsW} ------------------- Let us compare the reaction rates derived using the OME interaction to what one has for the contact $W$ exchange interaction. Only two processes among the [considered ones]{} go via $W$ exchange, [$np\leftrightarrow\Lambda p$]{} and [$\Lambda n\leftrightarrow\Xi^-p$]{}. Here we focus on the former one. For [simplicity]{} we use the non-relativistic matrix element [@LO2002; @vDD2004; @NO2006] $$\label{eq:Mfi-contact} \left\langle \left\| \mathcal{M}_\text{{$np\leftrightarrow\Lambda p$}}^W \right\|^2 \right\rangle = 2 G_\text{F}^2 \sin^2 2\theta_\text{C} m_n m_p^2 m_\Lambda \chi_\text{{$np\leftrightarrow\Lambda p$}},$$ where $\sin\theta_\text{C} = 0.231$, $\theta_\text{C}$ is the Cabibbo angle, and $\chi_\text{{$np\leftrightarrow\Lambda p$}} = 1 + 3\|c_A^{np}\|^2 \|c_A^{p\Lambda}\|^2 \approx 3.47$ with the axial coupling constants $c_A^{np} = -1.26$ and $c_A^{p\Lambda} = -0.72$ [@LO2002; @vDD2004].[^10] We use here the bare baryon masses, [as in]{} [@LO2002; @vDD2004; @NO2006]. The matrix element in Eq.(\[eq:Mfi-contact\]) does not depend on angles between the reacting particles momenta, so Eq. (\[eq:IAJ-J\]) yields $\mathcal{J} = \langle\|\mathcal{M}_\text{{$np\leftrightarrow\Lambda p$}}\|^2\rangle$. The reaction rate in [the case]{} of $W$ exchange can be expressed in the same form as for [the OME]{} interaction (Eq. \[eq:lambda1234\]). Using Eq. (\[eq:J-W\]), one finds that $\lambda_\text{{$np\leftrightarrow\Lambda p$}}$ [obtained via]{} the $W$ exchange is given by Eq. (\[eq:lambda1234\]) with $$\begin{gathered} \label{eq:WlpnpContact} \mathcal{W}_\text{{$np\leftrightarrow\Lambda p$}}^W = \\ \frac{\sin^2 2\theta_\text{C}}{8 s_\text{{$np\leftrightarrow\Lambda p$}}} \frac{m_n}{m_N} \frac{m_\Lambda}{m_N} \left( \frac{m_p}{m_N} \right)^2 \chi_\text{{$np\leftrightarrow\Lambda p$}} \approx 0.10.\end{gathered}$$ This is $7-15$ times less than for [$np\leftrightarrow\Lambda p$]{} using the OME interaction, [in accordance]{} with the results [obtained in]{} [@vDD2004]. ![\[fig:OMEvsW\] Equilibrium reaction rates $\Gamma_0$ [for]{} the [$np\leftrightarrow\Lambda p$]{} process. Thick lines are for the OME channel, thin lines are for the contact $W$ exchange channel multiplied by $10$.](G039-nb_vDD04.eps){width="\columnwidth"} To compare our [results]{} with [@vDD2004], we calculate [the equilibrium rate]{} of reactions for the [$np\leftrightarrow\Lambda p$]{} process, $\Gamma_{(0)}^\text{{$np\leftrightarrow\Lambda p$}}$, which is related to the subthermal reaction rate $\lambda_\text{{$np\leftrightarrow\Lambda p$}}$ according to $$\label{eq:G0-lambda} \Gamma_{(0)}^{12\leftrightarrow 34} = \frac{3 k T}{2\pi^2} \lambda_{12\leftrightarrow 34}.$$ We plot these rates for each EoS model from Sec. \[sec:EoSs\] in Fig. \[fig:OMEvsW\]. This figure is similar to figure 7 from [@vDD2004]: our thick lines correspond to their solid line ($\Gamma_{(0)}^\text{{$np\leftrightarrow\Lambda p$}}$ using OME), and our thin lines correspond to their dotted line ($10\times \Gamma_{(0)}^\text{{$np\leftrightarrow\Lambda p$}}$ using contact $W$ exchange). As expected, the OME interaction yields the equilibrium rate $\sim 10$ times greater than the $W$ exchange. But, surprisingly, our [calculations]{} give $\Gamma_{(0)}$ systematically $\gtrsim 4$ times lower than in [@vDD2004], both for [the OME and the $W$]{} exchange channels. Comparison of the reaction rates and ${\lambda_\mathrm{max}}$ {#sec:lambdas-Topt} ------------------------------------------------------------- ![\[fig:Topt\] Optimum temperatures for the bulk viscosity at $\omega=2\pi\times (400\,$Hz) [assuming]{}: nonsuperfluid and nonsuperconducting matter (top; see Eq. \[eq:Topt-tot\]); strong superfluidity of charged baryons and nonsuperfluid neutral ones (middle; see Eq. \[eq:Topt-ntrl\]), and optimum temperature for the case when only the reaction [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} operates (bottom; see Eq. \[eq:Topt-nLLL\]). In the top panel diamonds and circles mark the $\Sigma^-$ and $\Xi^0$ onsets, correspondingly, where the set of reactions included in the total $\lambda$ becomes incomplete. In each case the curves are plotted at $\rho\geqslant 1.01\rho_\text{start}$ to avoid discontinuities.](Toptimum.eps){width="\columnwidth"} Now we are able to answer the question from the end of the previous section, namely, how close the total rate $\lambda$ (the sum of all $\lambda_{12\leftrightarrow 34}$, see Eq. \[eq:lambTotDef\]) can be to the optimum rate ${\lambda_\mathrm{max}}$. [To answer it,]{} we need to calculate “the optimum temperature”, at which the bulk viscosity reaches its maximum, $$\label{eq:Topt-tot} T_\text{opt}^\text{(tot)} = 10^8\,\text{K} \times\sqrt{\omega_4} \sqrt{\frac{{\lambda_\mathrm{max}}\bigr\|_{\omega_4=1}}{\lambda\bigr\|_{T_8=1}}},$$ and check whether [such a temperature can exist]{} in the NSs [we are interested in]{}. The upper panel in Fig. \[fig:Topt\] [shows]{} $T_\text{opt}^\text{(tot)}$ at $\omega=2\pi\times (400 \, {\rm Hz})$ as a [function of density]{}. The chosen frequency is typical for those NSs in LMXBs, which could be subject to the r-mode instability [@Haskell2015]. We plot the curves up to the points of $\Sigma^-$ or $\Xi^0$ onset, where the set of considered reactions [becomes]{} incomplete. A typical optimum temperature value is within the range of $(0.5-1)\times 10^8\,$K, that [might be]{} close to the typical internal temperature of NSs in LMXBs. Thus application of our hyperon bulk viscosity to the problem of r-mode stability has some chances for success. [Up to this point we were considering only a non-superfluid (non-paired) nucleon-hyperon matter]{}. Baryon pairing is known to suppress reaction rates dramatically [@HLY2002] and affects substantially hydrodynamics of NS matter, in particular, the relation between the bulk viscosity(-ies) and the reaction rates [@GK2008]. Anyway, here we do not account for the latter effect, and use non-superfluid ${\lambda_\mathrm{max}}$ to compare [it]{} with suppressed reaction rates. As is widely accepted [@PageReview2013; @SedrClark2018], neutral baryons in the NS cores have lower pairing critical temperatures than the charged ones. Thus, the first step will be to suppress processes involving $p$, $\Xi^-$, etc. A conservative way to do that is to switch off completely all the processes involving charged baryons (in our case [$np\leftrightarrow\Lambda p$]{}, [$\Lambda n\leftrightarrow\Xi^-p$]{}, and [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{}). Then one can introduce the optimum temperature [for only reactions with neutral particles]{} $$\label{eq:Topt-ntrl} T_\text{opt}^\text{(ntrl)} = 10^8\,\text{K} \times \sqrt{\omega_4} \sqrt{\frac{{\lambda_\mathrm{max}}\bigr\|_{\omega_4=1}}{(\lambda_\text{{$nn\leftrightarrow\Lambda n$}} + \lambda_\text{{$n\Lambda\leftrightarrow\Lambda\Lambda$}})\bigr\|_{T_8=1}}}.$$ It is plotted in the middle panel of Fig. \[fig:Topt\]. It appears to be about $1.5$ times higher than in the unpaired case, $T_\text{opt}^\text{(ntrl)} \sim (0.8-1.5)\times 10^8\,$K. [One can go further and suggest]{} that the critical temperature of $\Lambda$’s is significantly lower than the neutron critical temperature [@Takatsuka2006] since the $\Lambda\Lambda$ interaction is known to be weak [@NagaraEvent2001]. A way to partially account for pairing of neutral baryons is to switch off the [$nn\leftrightarrow\Lambda n$]{} process, since it is more sensitive to the neutron superfluidity (since more neutrons are involved in the process), and consider [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} only. Introducing the optimum temperature for this case, $$\label{eq:Topt-nLLL} T_\text{opt}^\text{\text{{$n\Lambda\leftrightarrow\Lambda\Lambda$}}} = 10^8\,\text{K} \times \sqrt{\omega_4} \sqrt{\frac{{\lambda_\mathrm{max}}\bigr\|_{\omega_4=1}}{ \lambda_\text{{$n\Lambda\leftrightarrow\Lambda\Lambda$}}\bigr\|_{T_8=1}}},$$ we get the bottom panel of Fig. \[fig:Topt\]. The optimum temperature is significantly higher in this case, especially at densities close to the threshold of the [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} process[^11]. A typical hyperon NS [core]{} with the central density $\sim 3\rho_0$ should be rather hot, $\sim (2-5)\times 10^8\,$K, to achieve the most effective viscous damping in its interiors. However, even if the regime $\zeta = {\zeta_\mathrm{max}}$ is not reached in the NS core, the calculated bulk viscosity can significantly affect the r-mode stability, as it is demonstrated in the next section. R-mode instability windows {#sec:windows} ========================== Considering the r-mode instability windows, we follow the approach [of]{} [@NO2006]. Namely, we focus on the quadruple $l = m = 2$ r-mode, [which is treated]{} within the non-superfluid non-relativistic hydrodynamics (cf. Sec. \[sec:zeta-lambda\]), but with radial density profiles $\rho(r)$, $n_j(r)$, etc., taken from the numerical solution to the Tolman-Oppenheimer-Volkoff equations [@OppVol1939; @Tolman1939]. The stability criterion [for the]{} r-mode is $$\label{eq:stabCrit} \frac{1}{\tau_\text{GW}(\nu)} + \frac{1}{\tau_\zeta(\nu,{\widetilde{T}})} + \frac{1}{\tau_\eta({\widetilde{T}})} > 0,$$ where $\tau_\text{GW} < 0$ is the driving timescale of the instability due to [the gravitational]{} wave emission (Chandrasekhar-Friedman-Schutz instability [@Chandra1970; @FriSch1978]), $\tau_\zeta > 0$ is the damping timescale due to the bulk viscosity, and $\tau_\eta > 0$ describes damping due to the shear viscosity. These timescales depend on the rotation frequency $\nu$ and the [redshifted internal]{} temperature ${\widetilde{T}}$ (assumed to be constant over the NS core). The $\nu({\widetilde{T}})$ dependence, [for which]{} the inequality (\[eq:stabCrit\]) [becomes]{} an equality, [corresponds to]{} the critical frequency curve in the $\nu-{\widetilde{T}}$ plane. The region of $\nu$ and ${\widetilde{T}}$, where the condition (\[eq:stabCrit\]) is violated (above the critical $\nu$ curve) is the r-mode instability window for a NS. [Observing]{} NSs with frequency and temperature in this domain is highly unlikely [@Haskell2015]. [The]{} necessary formulas for $\tau_\text{GW}$ and $\tau_\zeta$ can be found in [@NO2006]. For the latter timescale we use $\zeta$ obtained in the two previous Sections (Eqs. \[eq:zetaFin\], \[eq:zlmax\], supplemented with Eqs. \[eq:lambda1234\], \[eq:lambda0+Wdef\] for required processes). [The]{} derivation of $\tau_\eta$ is given in [@LOM1998]. The main contribution to the shear viscosity $\eta$ comes from leptons, $e$ and $\mu$, independently of whether baryons are in the normal or in the superfluid state [@SchSht2018]. Moreover, if protons are superconducting, lepton shear viscosity $\eta$ is enhanced [@SchSht2018; @Sht2018]. Since the shear viscous damping is mostly important at low temperatures, where protons are paired, we have to use the “superconducting” expression for $\eta$. Luckily, there is an upper estimate for $\eta$ which is independent of pairing properties (the “London limit”, $T_{cp} \gg 10^9\,$K; see [@Sht2018] for details and the analytic expression). ![image](InstWindow-6.eps){width="90.00000%"} Fig. \[fig:windows\] [shows]{} the instability windows for various NS models. The top two panels are for the bulk viscosity unaffected by baryon pairing (all five processes in Eq. \[eq:weakReact\] [operate]{}). We restrict ourselves [to NS]{} with $M\leqslant 1.9\,$[M$_\odot$]{} to avoid the appearance of $\Sigma^-$ hyperons. Similarly to Sec. \[sec:lambdas-Topt\], we consider the $p$ and $\Xi^-$ pairing effects [excluding]{} all reactions involving these particles (two middle panels in Fig. \[fig:windows\]), and [simulating]{} $n$ pairing effects by excluding the reaction [$nn\leftrightarrow\Lambda n$]{} (bottom panels in Fig. \[fig:windows\]). However, in all plots we use the expressions (\[eq:zetaFin\]), (\[eq:zlmax\]) for a relation between the reaction rates and the bulk viscosity, i.e. we ignore influence of pairing effects on hydrodynamics of the core matter (similar to Sec. \[sec:lambdas-Topt\]). Figure \[fig:windows\] presents the instability windows for FSU2H and TM1C EoSs only. [Plots for GM1A EoS are similar to those for FSU2H EoS.]{} In turn, [NL3$\omega\rho$]{} critical frequency curves resemble the ones for TM1C, except for the substantially greater $\Lambda$ onset mass (see Table \[tab:astro\]) and a slower growth with increasing $M$. For instance, [NL3$\omega\rho$]{} NS with $M=2.55\,$[M$_\odot$]{} and TM1C one with $M=1.9\,$[M$_\odot$]{} have almost the same stable $\nu,{\widetilde{T}}$-regions. The latter difference is due to the fact that [NL3$\omega\rho$]{} has a [smaller]{} hyperon fraction than the other three EoSs that we use. Three main conclusions can be made from [inspecting]{} Fig. \[fig:windows\]. First (obvious), is that different EoS models yield different instability windows for the same $M$. However, the shape of the critical frequency curve is similar in all cases. Second, [the]{} top of the critical curve is reached at a temperature of the order of the corresponding optimum temperature $T_\text{opt}$: ${\widetilde{T}}\sim T_\text{opt}$ (see Sec. \[sec:lambdas-Topt\]). Thus, $T_\text{opt}$ appears to be a good estimate of a NS internal temperature at which r-modes are [the most stable]{}. Finally, the third [conclusion is]{} that for [all]{} EoSs considered above a high enough mass can close the instability window in [most]{} of the area shown in the Figure (except for the right bottom plot). This area is important since it contains the observed sources (LMXBs) that are [difficult to reconcile with current models of r-mode oscillations of NSs]{} (see e.g. [@Haskell2015; @GCK2014]). They are shown in Fig. \[fig:windows\] by blue data points.[^12] All these sources appear to be inside the stability regions for high enough NS masses even if $p$ and $\Xi^-$ are “frozen” due to the superfluid gaps. In particular, [for the FSU2H EoS]{} almost all data points [lie]{} within the contour defined by NSs with a mass of $1.7\,$[M$_\odot$]{}and below with strongly paired charged particles. This is in contrast to Ref. [@NO2006], approach to the weak non-leptonic reactions of which [requires]{} at least partially non-suppressed processes with charged particles. At variance with Ref. [@NO2006] we however account for the [$nn\leftrightarrow\Lambda n$]{} process, not considered by [@NO2006], which appears to be the main contributor to the bulk viscosity in the case of “frozen” charged particles. Another difference [with respect to]{} [@NO2006] is that in [that]{} paper the maximum of the stability curves occurs at $T \gtrsim 10^9\,$K, while we have the maximum of the critical frequency at $T \sim 10^8\,$K (except, maybe, in the case when only [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} is operating). This is [a consequence of the fact that]{} we use the OME interaction to calculate the reaction rates, while [@NO2006] used the contact one. Of course, leaving [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} [as]{} the only operating process is not a good way to study effects of $n$ pairing. When the neutron [superfluidity gap rises]{}, both [$nn\leftrightarrow\Lambda n$]{} and [$n\Lambda\leftrightarrow\Lambda\Lambda$]{} reaction rates decrease dramatically (the latter one does [it more slowly]{} than the former one), and none of them is affected in the regions of the NS core where neutrons are not paired yet. A careful consideration of this phenomenon is beyond the scope of the present paper. ![\[fig:windowCheck\] Comparison of the critical frequency curves calculated using the exact bulk viscosity (solid lines) and fitting Eqs. (\[eq:zlmaxFit\]), (\[eq:Wapprox\]), (\[eq:lambdaApprox\]) (dashed lines). The hyperon onset density $\rho_\Lambda$ is adjusted for each EoS. Using Eq. (\[eq:zlmaxFit\]), $\zeta_\text{max}$ is multiplied by $1.4$ for FSU2H and by $0.8$ for TM1C. All processes in the set (\[eq:weakReact\]) are switched on.](InstWindow-checkApprox.eps){width="\columnwidth"} In Secs. \[sec:zeta-lambda\] and \[sec:lambdas\] we [provided]{} the simple approximate expressions for the bulk viscosity. One should [substitute]{} $\zeta_\text{max}$ and $\lambda_\text{max}$ from Eq. (\[eq:zlmaxFit\]) and the reaction rates from combining Eqs. (\[eq:lambda1234\]), (\[eq:Wapprox\]), and (\[eq:lambdaApprox\]), into Eq. (\[eq:zetaFin\]) for the bulk viscosity. The resulting approximation depends on $T$, $\rho$, $\rho_\Lambda$ (the density of the hyperons onset), and various $\rho_\text{start}$ — the densities of the reaction thresholds (for [$np\leftrightarrow\Lambda p$]{} and [$nn\leftrightarrow\Lambda n$]{}, $\rho_\text{start}\approx \rho_\Lambda$). The value of $\rho_\Lambda$ is fixed for a given EoS but $\rho_\text{start}$ should be accurately adjusted for each EoS model in order to obtain a fit that reproduces the instability windows for this EoS. Strictly speaking, the parameter $\zeta_0$ in the fitting expression (\[eq:zlmaxFit\]) for the maximum bulk viscosity is also very important. While we provided the value $\zeta_0 = 6.5\times 10^{30}\,\text{g}\,\text{cm}^{-1}\,\text{s}^{-1}$ averaged over the four EoSs we use here, its actual value should be adjusted for a given EoS. For instance, FSU2H requires $\zeta_0 \approx 1.4\times$the averaged value, and for GM1A, TM1C, and [NL3$\omega\rho$]{} one needs, respectively, correcting factors $1.45$, $0.8$, and $0.55$. With these comments [taken into account]{}, the described fit of the bulk viscosity reproduces the critical frequency curves from Fig. \[fig:windows\] rather [accurately]{}, as shown in Fig. \[fig:windowCheck\]. Higher accuracy can be achieved if one also adjusts the parameter $s$ in Eq. (\[eq:zlmaxFit\]). Conclusion {#sec:conc} ========== Let us summarize [the scope of]{} the present article. First, we calculated the bulk viscosity $\zeta$ for a set [of hyperonic]{} EoSs. We considered models for which the core is composed of $npe\mu\Lambda\Xi^-$ matter, in contrast to most of the previous works [@LO2002; @HLY2002; @vDD2004; @NO2006] (see, however [@ChatBand2006]). We consider the full set of [weak]{} non-leptonic processes (Eq. \[eq:weakReact\]), operating in such NS cores and [generating]{} $\zeta$. Three of them, [$n\Lambda\leftrightarrow\Lambda\Lambda$]{}, [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{}, and [$\Lambda n\leftrightarrow\Xi^-p$]{}, are considered for the first time. The rates $\lambda_{12\leftrightarrow 34}$ for these processes are calculated using the relativistic OME interaction, [as in]{} Ref. [@vDD2004] (see Eqs. (\[eq:lambda1234\]), (\[eq:lambda0+Wdef\]), and Appendices \[app:Mfi\], \[app:Iang\]). [Expressions for]{} $\zeta$ and $\lambda$’s [are]{} derived within the non-superfluid hydrodynamics (Eqs. \[eq:zetaFin\] and \[eq:zlmax\], which [are]{} appropriate for an arbitrary hyperon composition). Second, we calculated the r-mode instability windows following the approach of [@NO2006]. We show that [the]{} positions of the critical frequency curve [maxima]{} are shifted to [lower]{} temperatures compared to [previous]{} calculations (cf. Fig. \[fig:windows\] and, e.g., Ref. [@NO2006]), even if we assume strong pairing of charged baryons and moderate pairing of neutral particles in the core. This [is due to the fact that]{} we calculated the reaction rates using OME interaction instead of the contact $W$ exchange, as Ref. [@NO2006] did. Third, we derived simple approximations for $\zeta$ and $\lambda$’s as a function of $\rho$. [Namely,]{} for each $\lambda_{12\leftrightarrow 34}$ one may use Eqs. (\[eq:lambda1234\]), (\[eq:Wapprox\]), (\[eq:lambdaApprox\]) together with the parameters from Tables \[tab:Wapprox\], \[tab:lambdaApprox\] \[or Eqs. (\[eq:lambda1234\]), (\[eq:lambda0\]), and (\[eq:Wapprox\]) if one wants to specify all particle fractions\]. [In turn, to calculate]{} $\zeta$ one may use Eqs. (\[eq:zetaFin\]) and (\[eq:zlmaxFit\]) together with the approximations for $\lambda$’s. However, [this approximation should be used with caution: if]{} one wants to reproduce [the]{} r-mode critical curve for some specific hyperonic EoS, [one]{} has to adjust the parameters $\zeta_0$ and $\rho_\Lambda$ to this EoS accurately; see the end of Sec. \[sec:windows\] and the caption to Fig. \[fig:windowCheck\] for an illustration. The value of $\zeta_0$ given in Sec. \[sec:zeta-lambda\] is just a rough averaging, appropriate for phenomenological NS models without the detailed hyperon microphysics. [We would like to point out four limitations of the work presented here]{}: (i) simplified calculation of the reaction rates; (ii) restricted hyperonic composition; (iii) almost no account for baryon pairing; (iv) simplified calculation of r-mode instability windows. \(i) The first deficiency [in the]{} $\lambda_{12\leftrightarrow 34}$ calculation is [that]{} we consider only the lightest meson exchange. In our cases the lightest meson is $\pi$ ($139\,$MeV) for [$np\leftrightarrow\Lambda p$]{}, [$nn\leftrightarrow\Lambda n$]{}, [$n\Xi^-\leftrightarrow\Lambda\Xi^-$]{}, and [$\Lambda n\leftrightarrow\Xi^-p$]{}, and $K$ ($494\,$MeV) for [$n\Lambda\leftrightarrow\Lambda\Lambda$]{}. Both [of]{} them are pseudoscalar mesons responsible for the long-range interaction. On the one hand, the long-range interaction is typically the most important in rough, first-order approximations, and the up-to-date NS physics does not [necessitate]{} [very]{} precise calculations of $\lambda$’s. On the other hand, typical distance between the baryons in the NS core is $\lesssim 1\,$fm, while at such distances the transition potential for weak non-leptonic processes strongly deviates from the OME model (at least in atomic hypernuclei [@IM2010; @PerOb+2013]). So, it is unclear whether the OME interaction model is sufficient for the astrophysical purposes or not. Typically, [accounting]{} for the heavier mesons (first of all, $\rho$ with the mass $770\,$MeV) yields an effect of a factor of few. For decay rates of the hypernuclei, the rates calculated using the $\pi$ exchange only (disregarding the short-range correlations, form factors and final state interactions) are 2–3 times lower than what [is]{} obtained using many [meson approach]{} [@PRB1997; @PRB2002]. In the context of NSs, a comparison of $\pi$ and $\pi+\rho$ exchanges was performed by [Friman and Maxwell]{} [@FM1979] for the neutrino pair bremsstrahlung from $nn$ scattering, $n + n \to n + n + \nu + \tilde{\nu}$. Their result is [that]{} $\pi$ exchange yields the rate 2–5 times greater than in case of $\pi+\rho$ exchange. A similar effect was obtained using the realistic $T$-matrix instead of one $\pi$ exchange (see the review [@SchSht2018] for details). Another deficiency is our simplistic treatment of the in-medium effects on the meson propagator $D_M$, mainly the pion one ($M = \pi$). As described in Sec. \[sec:lambdas-gator\], the expression (\[eq:DMforUse\]) we adopt for the propagators [allows]{} us to account for the s-wave part of the polarization operator $\Pi$ (in a rather simplistic way), but it provides no account for the p-wave part of $\Pi$. This means that we [underestimate]{} $D_M$, and, consequently, [also]{} $\lambda$’s. Different [calculations]{} of the in-medium modified propagators are divergent [@SchSht2018], the most impressive result is [that]{} it can increase the reaction rate up to several orders of magnitude [@Migdal+1990; @Vos2001]. All in all, are our reaction rates [under or over-estimated]{}? If the in-medium effects on $D_M$ are close to results [obtained in]{} [@Vos2001], our $\lambda$’s are surely [underestimated]{}. If the in-medium effects are not so dramatic, [the situation is unclear]{}. However, it seems more likely that the effects of $D_M$ in-medium renormalization are stronger than the influence of heavy mesons, so one can expect [that]{} the reaction rates are higher than [the ones we obtain]{}. \(ii) Throughout our work we have focused on a $\Lambda\Xi^-$ hyperon composition. For a [number]{} of EoS models, $\Sigma^-$ [appears]{} in the core (for instance, in deep layers of massive [NL3$\omega\rho$]{} and FSU2H stars; see also [@Prov+2018; @TolosCool2018; @Fortin+2017]). The relation between $\zeta$ and $\lambda$ inferred in Sec. \[sec:zeta-lambda\] is still true in this case, but the total rate $\lambda$ should include the rates of weak non-leptonic processes involving $\Sigma^-$, and may deviate from the $\Lambda\Xi^-$ case. The expressions for the rate $\lambda_{12\leftrightarrow 34}$, given in Sec. \[sec:lambdas\], are applicable for an arbitrary weak non-[leptonic]{} process $12 \leftrightarrow 34$ operating via the pseudoscalar meson exchange. [However, finding the necessary coupling constants in the literature is not an easy task.]{} \(iii) The main [limitation]{} of our work is [that we do not account for baryon pairing]{}. First of all, it affects the reaction rates. It can be accounted for by introducing [reduction]{} factors $\mathcal{R}$ [@HLY2002]. Some of them are already calculated and analytically approximated, some of them (in particular, $\mathcal R$ for [$nn\leftrightarrow\Lambda n$]{} in the case of $n$ pairing) are [available, but]{} still not published. We emphasize that a rough account for $\mathcal{R}$’s [via excluding processes involving paired baryons is too simplistic and may be misleading]{}. Second, baryon superfluidity affects the relation [between]{} the bulk viscosity and the reaction rates. Moreover, [the number]{} of kinetic coefficients named “the bulk viscosity” increases. These effects were studied in detail by [@GK2008; @KG2009]. Third, superfluidity affects the r-mode hydrodynamics. Several attempts to explore this effect were [made]{} [@LeeYosh2003; @HaskAnd2010; @KG2017; @DGK2019], but it is currently an unsolved problem. \(iv) The previous paragraph partially overlaps with the [last limitation]{} we would like to address, [that is]{} the simplistic calculation of the r-mode critical frequency curves. Besides the fact that the damping and driving timescales (see Eq. \[eq:stabCrit\]) differ in the presence of pairing, the “$\tau$-approach” to the critical $\nu$ curve itself is just an estimate. It is widely accepted as it is rather accurate in the non-paired case, but in the presence of pairing this approach should be revisited [@DGK2019]. Next, we calculate the damping timescale $\tau_\zeta$ due to the bulk viscosity employing the same approach as in Ref. [@NO2006]. In particular, we used their fitting formula for the angle averaged $(\operatorname{div}{\pmb{u}})^2$, which was fitted to [NS models obtained using their specific collection of EoSs]{}. It can be less accurate for our [choice of]{} EoSs. [Finally]{}, we use [non-relativistic hydrodynamics]{}, which is also inaccurate in NSs. [Improving the model presented in this work and overcoming, in particular, the limitations]{} (ii) and (iii), i.e. including more hyperon species and calculating the $\mathcal{R}$-factors that are currently unavailable, [will be the]{} subject of our future work. This work is supported in part by the Foundation for the Advancement of Theoretical Physics and mathematics “BASIS” \[Grant No. 17-12-204-1 (M.E.G.) and 17-15-509-1 (D.D.O.)\] and by RFBR Grant No. 18-32-20170 (M.E.G.). D.D.O. is grateful to N. Copernicus Astronomical Center for hospitality and perfect working conditions. This work was supported in part by the National Science Centre, Poland, grant 2018/29/B/ST9/02013 (P.H.), and grant 2017/26/D/ST9/00591 (M.F.). We thank E.E. Kolomeitsev and P.S. Shternin for valuable discussions. Coefficients in Eqs. (\[eq:XYZ\]) {#app:Mfi} ================================= Let us introduce dimensionless variables $$\label{eq:AlphaBetaX} \alpha_j = \frac{{m^_{\text{L} j}}}{m_M}, \quad \beta_j = \frac{{m^_{\text{D} j}}}{m_M}, \quad x_j = \frac{{p_{\text{F} j}}}{m_M}.$$ In these notations the coefficients in Eq. (\[eq:XYZ-X\]) take the form: $$\begin{gathered} \label{eq:X0} X_0 = g_{24} \left( 2\alpha_2\alpha_4 - 2\beta_{2}\beta_{4} - x_{2}^2 - x_{4}^2 \right) \\ \times \left[ \left( A_{13}^2 + B_{13}^2 \right) \left( 2\alpha_{1}\alpha_{3} - x_{1}^2 - x_{3}^2 \right) \right. \\ \left. + 2\beta_{1}\beta_{3} \left( A_{13}^2 - B_{13}^2 \right) \right],\end{gathered}$$ $$\begin{gathered} \label{eq:X1} X_1 = g_{24}^2 \left( A_{13}^2 + B_{13}^2 \right)\left( 2 \alpha_{1} \alpha_{3} + 2 \alpha_{2} \alpha_{4} \right. \\ \left. - x_{1}^2 - x_{2}^2 - x_{3}^2 - x_{4}^2 - 2 \beta_{2} \beta_{4} \right) \\ + 2 g_{24}^2 \left( A_{13}^2 - B_{13}^2 \right)\beta_{1}\beta_{3},\end{gathered}$$ $$\label{eq:X2} X_2 = g_{24}^2 \left( A_{13}^2 + B_{13}^2 \right).$$ In Eq. (\[eq:XYZ-Y\]) we have: $$\begin{gathered} \label{eq:Y0} Y_0 = g_{14}g_{24}\left( A_{13}A_{23} - B_{13}B_{23} \right) \left[ \beta_2\beta_3 x_1^2 \right. \\ \left. + \beta_1\beta_3 x_2^2 + \beta_1\beta_3 x_4^2 + \beta_2\beta_3 x_4^2 - \beta_3\beta_4 x_3^2 \right. \\ \left. - \beta_3\beta_4 x_4^2 + 2\alpha_1\alpha_2\beta_3\beta_4 - 2\alpha_1\alpha_4\beta_2\beta_3 \right. \\ \left. - 2\alpha_2\alpha_4\beta_1\beta_3 + 2\beta_1\beta_2\beta_3\beta_4 \right] \\ + g_{14}g_{24}\left( A_{13}A_{23} + B_{13}B_{23} \right) \left[ - x_2^2 x_1^2 - x_3^2 x_4^2 \right. \\ \left. - \alpha_1\alpha_2 x_1^2 + \alpha_2\alpha_3 x_1^2 + \alpha_2\alpha_4 x_1^2 - \alpha_1\alpha_2 x_2^2 \right. \\ \left. + \alpha_1\alpha_3 x_2^2 + \alpha_1\alpha_3 x_4^2 + \alpha_2\alpha_3 x_4^2 + \alpha_1\alpha_4 x_2^2 \right. \\ \left. + \alpha_1\alpha_4 x_3^2 + \alpha_2\alpha_4 x_3^2 - \alpha_3\alpha_4 x_3^2 - \alpha_3\alpha_4 x_4^2 \right. \\ \left. + \beta_1\beta_2 x_1^2 - \beta_2\beta_4 x_1^2 + \beta_1\beta_2 x_2^2 - \beta_1\beta_4 x_2^2 \right. \\ \left. - \beta_1\beta_4 x_3^2 - \beta_2\beta_4 x_3^2 + 2\alpha_1\alpha_3\beta_2\beta_4 \right. \\ \left. + 2\alpha_2\alpha_3\beta_1\beta_4 - 2\alpha_3\alpha_4\beta_1\beta_2 - 2\alpha_1\alpha_2\alpha_3\alpha_4 \right],\end{gathered}$$ $$\begin{gathered} \label{eq:Y1} Y_1 = g_{14}g_{24} A_{13}A_{23} \left[ \left( \alpha_2-\alpha_3 \right) \left( \alpha_1-\alpha_4 \right) \right. \\ \left. + \left( \beta_2+\beta_3 \right) \left( \beta_4-\beta_1\right) \right] \\ + g_{14}g_{24} B_{13}B_{23} \left[ \left( \alpha_2-\alpha_3 \right) \left( \alpha_1-\alpha_4 \right) \right. \\ \left. - \left( \beta_2-\beta_3 \right) \left( \beta_1-\beta_4 \right) \right],\end{gathered}$$ $$\label{eq:Y1p} Y_2 = Y_1\Bigl\|_{1\leftrightarrow 2},$$ $$\label{eq:Y2} Y_3 = g_{14}g_{24}\left( A_{13}A_{23} + B_{13}B_{23} \right).$$ In Eq. (\[eq:XYZ-Z\]): $$\label{eq:Z012} X'_{0,1,2} = X_{0,1,2}\Bigl\|_{1\leftrightarrow 2}.$$ Transforming Eq. (\[eq:IAJ-J\]) {#app:Iang} =============================== It is convenient to introduce the dimensionless variables $$\label{eq:xj} {\pmb{x}}_j = \frac{{\pmb{p}}_j}{m_M}, \quad {\pmb{x}} = \frac{{\pmb{q}}}{m_M} = {\pmb{x}}_3 - {\pmb{x}}_1, \quad {\pmb{x}}' = \frac{{\pmb{q}}'}{m_M} = {\pmb{x}}_3 - {\pmb{x}}_2.$$ Similarly, we introduce $x_\text{min,max} = q_\text{min,max}/m_M$ that can be expressed in terms of $x_j = \|{\pmb{x}}_j\| = {p_{\text{F} j}}/m_M$. The non-weighted angular integral Eq. (\[eq:IAJ-A\]) can be written in a dimensionless form $\mathcal{A} = A/m_M^3$ with $$\label{eq:Adimless} A = \frac{2(2\pi)^3}{\prod_j x_j} \left( x_\text{max} - x_\text{min} \right) \Theta\left( x_\text{max} - x_\text{min} \right).$$ Substituting $\langle\left\|\mathcal{M}_{12\to 34}\right\|^2\rangle$ from Eq. (\[eq:MfiSqAvSum\]) and $D_M$ from Eq. (\[eq:DMforUse\]) into Eq. (\[eq:IAJ-J\]), we find $$\begin{gathered} \label{eq:Jfin} \mathcal{J} = G_\text{F}^2 m_\pi^4 \left( X_0 J_0 + X_1 J_1 + X_2 J_2 \right. \\ \left. + X'_0 J'_0 + X'_1 J'_1 + X'_2 J'_2 \right. \\ \left. + Y_0 J_3 + Y_1 J_4 + Y_2 J'_4 + Y_3 J_5 \right).\end{gathered}$$ The dimensionless functions $J_k(x_1, x_2, x_3, x_4)$, $k = 1...5$, are the following: $$\begin{gathered} \label{eq:J012} J_k = \frac{1}{A} \int \prod_j {\mathrm{d}}\Omega_j \frac{x^{2k}}{(x^2+1)^2}\delta\left( {\pmb{x}}_1 + {\pmb{x}}_2 - {\pmb{x}}_3 - {\pmb{x}}_4 \right) \\ = \frac{\Theta(x_\text{max}-x_\text{min})}{x_\text{max}-x_\text{min}} \int_{x_\text{min}}^{x_\text{max}} {\mathrm{d}}x \frac{x^{2k}}{(x^2+1)^2}\end{gathered}$$ for $k = 0,1,2$, $$\begin{gathered} \label{eq:J3} J_3 = \frac{1}{A} \int \prod_j {\mathrm{d}}\Omega_j \frac{\delta\left( {\pmb{x}}_1 + {\pmb{x}}_2 - {\pmb{x}}_3 - {\pmb{x}}_4 \right)}{(x^2+1)(x'^2+1)} \\ = \frac{\Theta(x_\text{max}-x_\text{min})}{x_\text{max}-x_\text{min}} \int_{x_\text{min}}^{x_\text{max}} \frac{{\mathrm{d}}x}{(x^2+1)\sqrt{t_1^2(x) - t_2^2(x)}}\end{gathered}$$ $$\begin{gathered} \label{eq:J4} J_4 = \frac{1}{A} \int \prod_j {\mathrm{d}}\Omega_j \frac{x^2 \delta\left( {\pmb{x}}_1 + {\pmb{x}}_2 - {\pmb{x}}_3 - {\pmb{x}}_4 \right)}{(x^2+1)(x'^2+1)} \\ = \frac{\Theta(x_\text{max}-x_\text{min})}{x_\text{max}-x_\text{min}} \int_{x_\text{min}}^{x_\text{max}} \frac{x^2 {\mathrm{d}}x}{(x^2+1)\sqrt{t_1^2(x) - t_2^2(x)}},\end{gathered}$$ $$\begin{gathered} J_5 = \frac{1}{A} \int \prod_j {\mathrm{d}}\Omega_j \frac{x^2 x'^2 \delta\left( {\pmb{x}}_1 + {\pmb{x}}_2 - {\pmb{x}}_3 - {\pmb{x}}_4 \right)}{(x^2+1)(x'^2+1)} \\ = J_3 + \frac{\Theta(x_\text{max}-x_\text{min})}{x_\text{max}-x_\text{min}} \int_{x_\text{min}}^{x_\text{max}} {\mathrm{d}}x \left( \frac{x^2}{x^2+1} \right. \\ \label{eq:J5} \left. - \frac{1}{\sqrt{t_1^2(x) - t_2^2(x)}} \right),\end{gathered}$$ where we use notation of Ref. [@Maxwell1987]: $$\begin{aligned} \label{eq:t1} t_1 &= x_1^2 + x_2^2 - x^2 +1 - 2 x_3 x_4 \cos\theta_1 \cos\theta_2, \\ \label{eq:t2} t_2 &= 2 x_3 x_4 \sin\theta_1 \sin\theta_2,\end{aligned}$$ with $$\begin{aligned} \label{eq:cos1} \cos\theta_1 &= \frac{x_3^2 - x_1^2 + x^2}{2 x_3 x}, \\ \cos\theta_2 &= \frac{x_2^2 - x_4^2 - x^2}{2 x_4 x}.\end{aligned}$$ For the ‘exchange’ integrals we have $$\label{eq:Jx5prime} J'_k = J_k\bigr\|_{x_1 \leftrightarrow x_2}$$ for $k = 0,1,2,4$, that corresponds to ${\pmb{x}} \to {\pmb{x}}'$ within the integrals ($J'_k = J_k$ for $k = 3,5$). Substituting Eq. (\[eq:Jfin\]) into Eq. (\[eq:DeltaGamma-IAJ\]), we immediately obtain Eq. (\[eq:lambda1234\]). Reduction of multidimensional integrals to their one-dimensional forms is performed according to the standard technique, see, e.g., Refs. [@ShapTeuk1983; @FM1979; @Maxwell1987]. The identities $$\label{eq:helpForJ} 1 = \int {\mathrm{d}}^3{\pmb{x}} \delta\left( {\pmb{x}} + {\pmb{x}}_3 - {\pmb{x}}_1 \right), \quad {\pmb{x}}' = {\pmb{x}} + {\pmb{x}}_1 - {\pmb{x}}_2$$ are helpful [@Maxwell1987]. The one-dimensional integrals in the right-hand sides of Eqs. (\[eq:J012\]) — (\[eq:J5\]) could be simply evaluated, both numerically and analytically. One can find analytic results in Refs. [@FM1979; @Maxwell1987]. 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[^1]: ddofengeim@gmail.com [^2]: [See [@LO2002] for an alternative approach to the definition of $\zeta$. The resulting expression for the coefficient ${\rm Re}(\zeta)$, which is responsible for dissipation, is the same in both approaches (as it should be).]{} [^3]: While chemical disturbance with respect to strong reactions is negligible, [rates of]{} these reactions [are]{} comparable to the rates of weak reactions (\[eq:weakReact\]). See [@Jones2001; @GK2008] for more details. [^4]: Stricly speaking, in the dense nucleon-hyperon matter of NS cores we have to consider [‘the baryon quasiparticles’ instead of ‘baryons’, the latter being appropriate in vacuum or in a few baryon systems. Hereafter by ‘baryon’ or ‘particle’ we will mean ‘the baryon quasiparticle’]{}. [^5]: This [limitation]{} was not so pronounced [when]{} the $\Sigma^-\Lambda$ hyperon composition of the core was considered [@HLY2002; @LO2002; @vDD2004; @NO2006]. [^6]: Strictly speaking, diagrams with permuted particles 1 and 2 would appear if we included the next to the lightest meson. [^7]: We do not discriminate between [masses of different members of isomultiplets]{}, and use values as in [@Glend2000]. [^8]: It is introduced here just [to]{} make $\mathcal{W} \lesssim 1$. [^9]: We emphasize that in the superfluid matter the hierarchy is different. [^10]: We emphasise that here $\langle\|\mathcal{M}_{12\to 34}\|^2\rangle$ is the matrix element, squared, summed over the final spin states, and averaged over the initial spines. Our notation should not be confused with notations used in [@LO2002] and [@vDD2004]. [^11]: In all these three cases $T_\text{opt}$ tends to infinity in the vicinity of the corresponding $\rho_\text{start}$, but in the former two cases this divergence is insensible at $\rho \geqslant 1.01\rho_\text{start}$, where the curves in Fig. \[fig:Topt\] are plotted. [^12]: \[ftnt:LMXBs\] These sources are the same as in [@GCK2014] but with SAX J1810.8–2609 added ($\nu$ from [@Allen+2018], ${\widetilde{T}}$ derived using [@Bilous+2018]). For all the sources ${\widetilde{T}}$ was derived from the effective surface temperature, inferred from observations, assuming $M=1.4\,$[M$_\odot$]{} and $R=10\,$km. See [@GCK2014] for details.
--- abstract: 'We describe a rearrangement of the standard expansion of the symmetry breaking part of the QCD effective Lagrangian that includes into each order additional terms which in the standard chiral perturbation theory ($\chi$PT) are relegated to higher orders. The new expansion represents a systematic and unambiguous generalization of the standard $\chi$PT, and is more likely to converge rapidly. It provides a consistent framework for a measurement of the importance of additional “higher order” terms whose smallness is usually assumed but has never been checked. A method of measuring, among other quantities, the QCD parameters $\hat{m}\langle\bar{q}q\rangle$ and the quark mass ratio $m_s/\hat{m}$ is elaborated in detail. The method is illustrated using various sets of available data. Both of these parameters might be considerably smaller than their respective leading order standard $\chi$PT values. The importance of new, more accurate, experimental information on low-energy $\pi-\pi$ scattering is stressed.' address: - \| Division de Physique Théorique[^1], Institut de Physique Nucléaire,\ F-91406 Orsay Cedex, France - ' Department of Physics, Purdue University, West Lafayette IN 47907\' author: - 'J. Stern and H. Sazdjian' - 'N. H. Fuchs' title: \| What $\pi-\pi$ scattering tells us about\ chiral perturbation theory --- = Introduction ================ Owing to quark confinement, the connection between QCD correlation functions and hadronic observables is far from being straightforward. In the low-energy domain, such a connection is described by chiral perturbation theory ($\chi$PT) [@sw79; @gl84; @gl85]. The latter provides a complete parametrization (in terms of an effective Lagrangian) of low-energy off-shell correlation functions of quark bilinears,which should take into account: (i) the normal and anomalous Ward identities of chiral symmetry, explicitly broken by quark masses; (ii) spontaneous breakdown of chiral symmetry; (iii) analyticity, unitarity and crossing symmetry. On the other hand, such a parametrization (effective Lagrangian) should be sufficiently general, and should not introduce any additional dynamical assumptions beyond those listed above, that could be hard to identify as emerging from QCD. The specificity of QCD then resides in numerical values of low-energy constants which characterize the above parametrization. The theoretical challenge is to calculate these low-energy parameters from the fundamental QCD Lagrangian. While such a calculation is awaited, these parameters can be subjected to experimental investigation. Chiral symmetry guarantees that the same parameters that are introduced through the low-energy expansion of QCD correlation functions also define the low-energy expansion of hadronic observables — pseudoscalar meson masses, transition and scattering amplitudes. In this paper, a new method will be elaborated that allows a detailed measurement of certain low-energy parameters, using the $\pi-\pi$ elastic scattering data [@gl84; @drv]. Instead of concentrating on a particular set of scattering lengths and effective ranges [@gl84] whose extraction from experimental data is neither easy nor accurate, emphasis will be put on a detailed fit of the scattering amplitude in a whole low-energy domain of the Mandelstam plane, including the unphysical region. In this way it is possible to obtain some [experimental]{} insight on the low-energy parameter $2\hat{m}B_0$, where $\hat{m}$ is the average of the up and down quark masses, $B_0$ is the condensate $$B_0 = - \frac{1}{F_0^2} \langle 0\|\bar{u}u\|0\rangle = - \frac{1}{F_0^2} \langle 0\|\bar{d}d\|0\rangle = - \frac{1}{F_0^2} \langle 0\|\bar{s}s\|0\rangle$$ and $\|0\rangle $ and $F_0$ stand for the ground state and pion decay constant respectively at $m_u = m_d = m_s = 0$. It is usually [assumed]{} that the parameter $2\hat{m} B_0$ differs from the pion mass squared by not more than $(1-2)\%$ [@gl85], and the standard chiral perturbation theory could hardly tolerate an important violation of this assumption [@gor-gw]. On the other hand, this assumption has never been confronted with experiment otherwise than indirectly – through the Gell-Mann Okubo formula for pseudoscalar meson masses [@gor-gw]. However, even the latter represents at best a consistency argument rather than a proof: the Gell-Mann Okubo formula can hold quite independently of the relation between $2\hat{m}B_0$ and $M_\pi^2$ [@fss91]. An independent measurement of $2\hat{m}B_0$ is not only possible (as shown in the present work) but, for several reasons, it appears to be desirable: \(i) The effective Lagrangian ${\cal L}_{eff}$ contains, in principle, an infinite number of low-energy constants, which are all related to (gauge invariant) correlation functions of massless QCD. Among them, $B_0$ plays a favored role: The order of magnitude of all low-energy constants other than $B_0$ can be estimated using sum-rule techniques [@sumrules], which naturally bring in the scale $\Lambda \sim 1~GeV$ characteristic of massive bound states. The expected order of magnitude of a low-energy constant related to a connected $N$-point ($N > 1$) function of quark bilinears $\bar{q}\Gamma q$, that is not suppressed by the Zweig rule or by a symmetry, is $F_0^2 \Lambda^{2-N}$ multiplied by a dimensionless constant of order 1. If quarks were not confined [@njl], a similar estimate would relate $B_0$ and the mass of asymptotic fermion states with quark quantum numbers. However, in a confining theory, no similar relation between $B_0$ and the spectrum of massive bound states can be derived: $\bar{q}q$ is an irreducible color singlet and there is no complete set of intermediate states which could be inserted into the matrix element $\langle 0\| \bar{q}q \|0\rangle$. $B_0$ could be as large as $\Lambda \sim 1~GeV$ or as small as the fundamental order parameter of chiral symmetry breaking, $F_0 \sim 90~MeV$. [A priori]{}, there is no way to decide in favor of one of these scales, at least before the non-perturbative sector of QCD is controlled analytically or by reliable numerical methods, using, for instance, sufficiently large lattices. In this paper, we suggest how the question of the scale of $B_0$ can be addressed experimentally. \(ii) To the extent that $2\hat{m}B_0$ and $(m_s + \hat{m})B_0$ are close to $M_\pi^2$ and $M_K^2$, respectively, the ratio of quark masses $$r \equiv m_s/\hat{m}$$ must approach $2 \frac{M_K^2}{M_\pi^2} -1 = 25.9$ [@gor-gw; @w-gl]. There exists an independent measurement of the ratio $r$ in terms of observed deviations from the Goldberger-Treiman relation [@fss90] in non-strange and strange baryon channels. This model-independent measurement indicates a considerably lower value for $r$ than 25.9, unless the pion-nucleon coupling constant turns out to be below the value given by Koch and Pietarinen [@koch-piet] by at least 4–5 standard deviations [@arndt-etal]. \(iii) A reformulation of $\chi$PT which allows $2\hat{m}B_0$ to be considerably lower than $M_\pi^2$ has been given in Ref.[@fss91]. It is as systematic and unambiguous as the standard $\chi$PT itself, and is particularly suitable in the case where $B_0$ is as small as $F_\pi$. It is based on a different expansion of the same effective Lagrangian, with the same infinity of independent terms. To all orders, the two perturbative schemes are identical but, in each finite order, they can (but need not) substantially differ. For each given order, the new scheme contains more parameters than the standard $\chi$PT, the latter being reproduced for special values of these additional parameters. Already at the leading order $O(p^2)$, the new scheme contains one additional free parameter $$\eta = \frac{2\hat{m}B_0}{M_\pi^2}.$$ If $\eta$ is set equal to 1, one recovers the leading $O(p^2)$ order of the standard $\chi$PT. The new expansion can therefore be formally viewed as a [generalization]{} of the standard scheme and – in this sense – it will be referred to as [improved $\chi$PT]{}, since it aims to improve the convergence of the standard perturbation theory. Demonstrating that such an improvement is irrelevant, by measuring, for instance, the ratio (1.3) and finding it close to unity, would be an important experimental argument in favor of the standard $\chi$PT. \(iv) In some cases, the convergence of standard $\chi$PT actually appears to be rather slow. Most of the indications in this direction can be traced back to the fact that the leading $O(p^2)$ order of the standard $\chi$PT underestimates the Goldstone boson interaction and, in particular, the $\pi-\pi$ scattering amplitude. This manifests itself through virtual processes and/or final state interactions, as in $\gamma \gamma \rightarrow \pi^0 \pi^0$ [@gammagamma], $\eta-3\pi$ [@eta3pi], etc. It might even be that although the next order $O(p^4)$ improves the situation, it fails to reach the precision we may rightly expect from it. For example, the $I=0$ $s$ wave $\pi-\pi$ scattering length, which is $a_0^0 = 0.16$ in leading order [@sw66], gets shifted to $a_0^0 = 0.20$ by $O(p^4)$ corrections [@gl84], while the “experimental value” [@yellow; @fp; @nagels] is $a_0^0 = 0.26 \pm 0.05$. (In this paper it will be argued that scattering lengths are not the best quantities to look at. A more detailed amplitude analysis will reveal a possible amplification of the discrepancy, which exceeds one standard deviation.) \(v) $\chi$PT should be merely viewed as a theoretical framework for a precise measurement of low-energy QCD correlation functions. Its predictive power rapidly decreases with increasing order in the chiral expansion : More new parameters enter at each order and more experimental data have to be included to pin them down. For this reason, a slow convergence rate might sometimes lead to a qualitatively wrong conclusion with respect to a measurement based only on the first few orders. This might concern, in particular, the measurement of the ratio $\eta$ (1.3) within the standard $\chi$PT. In the corresponding leading order, $\eta$ is fixed to be 1, independently of any experimental data. This property of standard $\chi$PT could bias the measurement of $\eta$ if $\eta$ turned out to be considerably different from 1: one would presumably have to go to a rather high order and include a large set of data to discover the truth. In this case, the [improved $\chi$PT]{} would be a more suitable framework to measure $\eta$ faithfully. The reason is that in the improved $\chi$PT, $\eta$ is a [free parameter]{} from the start: It defines the [leading order]{} $\pi-\pi$ amplitude. Neglecting, for simplicity, Zweig-rule violation (cf. Ref. [@fss91] and Sec. IV A), the latter reads $$A(s\|tu) = \frac{1}{F_0^2} (s-\eta M_\pi^2).$$ Using in this formula the value of $a_0^0 = 0.26 \pm 0.05$, one concludes that $\eta = 0.4 \pm 0.4$ already at the leading order. The measurement then has more chances to saturate rapidly – say, at the one-loop level – provided $\eta$ is much closer to $0.4 \pm 0.4$ than to 1. The same remark applies to measurements of the quark mass ratio $r = m_s/\hat{m}$, which, incidentally, is closely related to $\eta$ [@fss90]: A slow convergence of the standard $\chi$PT could lower the leading-order result $r = 25.9$ by considerably more than the usually quoted $(10-20)\%$ [@gl85; @donoghuereview]. \(vi) The question of the actual value of $\eta$ and/or of $r = m_s/\hat{m}$ has to be settled experimentally. None of the known properties of QCD, nor the fact that light quark masses are tiny compared with the hadronic scale $\Lambda \sim 1~GeV$, imply that $\eta$ should be close to 1 and that $r$ should be close to 25.9. The proof of this negative statement is provided by the existence of a mathematically consistent generalization of the standard $\chi$PT that does not contradict any known fundamental property of QCD and allows for any value of $\eta$ between 0 and 1 (and for any value of $r$ between 6.3 and 25.9) [@fss91]. Only in the special case of $\eta$ and $r$ close to 1 and 25.9, respectively, can the standard $\chi$PT claim a decent rate of convergence. In Sec. II, the precise mathematical definition of the improved $\chi$PT, in terms of the effective Lagrangian, is briefly summarized. It is not a purpose of this paper to present a full formal development of this theory; incidentally, most of it can be read off from existing calculations [@gl85] after rather minor extensions (which will be presented elsewhere). Here, we will mainly concentrate on phenomenological aspects of the problem in connection with low-energy $\pi-\pi$ scattering. The content of Sec. III is independent of any particular $\chi$PT scheme. In that section, a new low-energy representation of the $\pi-\pi$ scattering amplitude is given that provides the most general solution of analyticity, crossing symmetry and unitarity up to and including the chiral order $O(p^6)$. (Partial wave projections of this representation coincide with a particular truncation of the well-known Roy equations [@roy].) Subsequently, this representation is used both to constrain the experimental data and to perform a comparison with theoretical amplitudes as predicted by the two versions of $\chi$PT. For the case of the improved $\chi$PT, the one-loop amplitude is worked out in Sec. IV. Finally, a method permitting a detailed fit of the experimental amplitude in a whole low-energy domain of the Mandelstam plane is developed in Sec. V. This method is then applied to various sets of existing data. Formulation of improved $\chi$PT ==================================== Following rather closely the off-shell formalism which was elaborated some time ago by Gasser and Leutwyler [@gl85], we consider the generating functional $Z(v^\mu,a^\mu,\chi)$ of connected Green functions made up from $SU(3)\times SU(3)$ vector and axial currents as well as from scalar and pseudoscalar quark densities, as defined in QCD with three massless flavors. The sources $v^\mu,~a^\mu$ and $\chi$ are specified through the Lagrangian $${\cal L} = {\cal L}_{QCD} + \bar{q}(\not \! v + \not \! a \gamma_5)q - \bar{q}_R \chi q_L - \bar{q}_L \chi^{\dag} q_R,$$ which defines the vacuum-to-vacuum amplitude $\exp i Z$. Here, $q_{L,R} = \case{1}{2}(1 \mp \gamma_5)q$ stand for the light quark fields $u,\,d,\,s$ and ${\cal L}_{QCD}$ is invariant under global $SU(3) \times SU(3)$ transformations of $q_L$ and $q_R$. $v^\mu$ and $a^\mu$ are traceless and hermitean, whereas $$\chi = s + ip$$ is a general $3 \times 3$ complex matrix ($s$ and $p$ are hermitean). Explicit chiral symmetry breaking by quark masses is accounted for by expanding $Z$ around the point $$v^\mu = a^\mu = 0,~~~~~\chi = {\cal M}_q \equiv \left( \begin{array}{rrr} m_u &~~ &~~ \\ {}~~ & m_d &~~ \\ {}~~ &~~ & m_s \end{array} \right).$$ The scalar-pseudoscalar source $\chi$ and the quark mass matrix ${\cal M}_q$ are closely tied together by chiral symmetry. (Notice that our source $\chi$ differs from the $\chi$ defined in Ref. [@gl85] by a factor of $2B_0$.) Instead of calculating $Z$, the effective theory parametrizes it by means of an effective Lagrangian which depends on the sources and on eight Goldstone boson fields $$U(x) = \exp \frac{i}{F_0} \sum_{a=1}^8 \lambda^a \varphi_a(x).$$ Leaving aside anomaly contributions described by the Wess-Zumino action, the effective Lagrangian ${\cal L}_{eff}(U,v^\mu,a^\mu,\chi)$ is merely restricted by the usual space-time symmetries and by the requirement of invariance under local chiral transformations \[$\Omega_{L,R} \in SU(3)$\] $$U(x) \rightarrow \Omega_R(x) U(x) \Omega_L^{\dag}(x),~~~\chi(x) \rightarrow \Omega_R(x) \chi(x) \Omega_L^{\dag}(x)$$ compensated by the inhomogeneous transformation of the sources $v^\mu$ and $a^\mu$: $$\begin{aligned} v^\mu + a^\mu &\rightarrow& \Omega_R (v^\mu + a^\mu + i \partial ^\mu) \Omega_R^{\dag} \nonumber \\ v^\mu - a^\mu &\rightarrow& \Omega_L (v^\mu - a^\mu + i \partial ^\mu) \Omega_L^{\dag}.\end{aligned}$$ (This gauge invariance of the nonanomalous part of $Z$ is necessary and sufficient to reproduce all $SU(3) \times SU(3)$ Ward identities.) Otherwise, the effective Lagrangian remains unrestricted. ${\cal L}_{eff}$ can be written as an infinite series of local terms, $${\cal L}_{eff} = \sum_{n,m} \ell ^{nm} {\cal L}_{nm},$$ where ${\cal L}_{nm}$ denotes an invariant under the transformations (2.5) and (2.6) that contains the $n$-th power of the covariant derivatives $D_\mu$ and the $m$-th power of the scalar-pseudoscalar source $\chi$. The sum over independent invariants that belong to the same pair of indices $(n,m)$ is understood. The covariant derivatives are defined as $$D_\mu U = \partial _\mu U - i(v_\mu + a_\mu)U + iU(v_\mu - a_\mu),$$ and likewise for $D_\mu \chi$. The expansion coefficients $\ell ^{nm}$ represent properly subtracted linear combinations of massless QCD correlation functions that involve $n$ vector and/or axial currents and $m$ scalar and/or pseudoscalar densities, all taken at vanishing external momenta. The first two terms in the sum (2.7), for instance, read ($n$ is even) $$\begin{aligned} \ell ^{01} {\cal L}_{01} &=& \frac{1}{2} F_0^2 B_0 \langle U^{\dag}\chi + \chi^{\dag}U \rangle \nonumber \\ \ell ^{20} {\cal L}_{20} &=& \frac{1}{4} F_0^2 \langle D^\mu U^{\dag} D_\mu U \rangle.\end{aligned}$$ Everything said so far is rather general and independent of any particular perturbative scheme. Chiral perturbation theory is an attempt to reorder the infinite sum (2.7) as $${\cal L}_{eff} = \sum_d {\cal L}^{(d)},$$ where ${\cal L}^{(d)}$ collects all terms that in the limit $$p \rightarrow 0,~~~M_\pi \rightarrow 0,~~~ p^2/M_\pi^2 ~~\text{fixed}$$ behave as $O(p^d)$ ($p$ stands for external momenta). In order to relate the expansions (2.10) and (2.7), one needs to know the [effective infrared dimension $d(m_q)$ of the quark mass]{}. The invariant ${\cal L}_{nm}$ then contributes as $O(p^{d_{nm}})$, where $$d_{nm} = n + m\,d(m_q).$$ For infinitesimally small quark masses, one should have $$d(m_q) = 2,~~~m_q \rightarrow 0.$$ This follows from the mathematical fact that in QCD $$\lim_{m_q \rightarrow 0} \frac{(m_i + m_j)B_0}{M_P^2} = 1.$$ (Here, $i,j = u,d,s$ and $M_P$ is the mass of the pseudoscalar meson $\bar{\imath}j,~i\not =j$.) The assumption that in the [real world, i.e.]{}, for physical values of quark masses, the effective dimension of the quark mass is 2, underlies the [standard $\chi$PT]{}. It amounts to the well-known rule which asserts that each insertion of the quark mass matrix and/or of the scalar-pseudoscalar source $\chi$ counts as two powers of external momenta. Equivalently, the standard $\chi$PT can be viewed as an expansion around the limit $$(p,m_q) \rightarrow 0,~~~p^2/M_P^2~~\text{fixed}.$$ Since, by definition, the low-energy constants $\ell^{nm}$ are independent of quark masses, they are $O(1)$ in the limit (2.15). It is easy to see that the convergence of the standard $\chi$PT could be seriously disturbed if $B_0 \ll \Lambda \sim 1~GeV$, say $B_0 \sim 100 ~MeV$ [@fss91; @fss90]. The expansion of $M_P^2$ reads $(i,j = u,d,s;~ i \not = j)$ $$M_P^2 = (m_i + m_j)B_0 + (m_i + m_j)^2 A_0 + \ldots,$$ where the dots stand for non-analytic terms and for higher order terms. $A_0$ can be expressed in terms of two-point functions of scalar and pseudoscalar quark densities divided by $F_0^2$ [@fss90]. It satisfies a superconvergent dispersion relation, whose saturation leads to the order of magnitude estimate $A_0 \sim 1 - 5$. For $B_0$ as small as 100 MeV, the first and second order terms in Eq. (2.16) then become comparable for quark masses as small as (10 – 50) MeV. In order to accommodate this possibility, the [improved $\chi$PT]{} attributes to the quark mass [and]{} to the vacuum condensate parameter $B_0$ the effective dimension 1, $$d(m_q) = d(B_0) = 1,$$ reflecting their smallness compared to the scale $\Lambda$. This does not contradict mathematical statements such as (2.14). It only means that, although for physical values of quark masses the ratio in Eq. (2.14) remains on the order of 1, it is allowed to differ from 1 considerably. To summarize, in the improved $\chi$PT each insertion of the quark mass-matrix ${\cal M}_q$ and/or of the scalar-pseudoscalar source $\chi$ counts as a single power of external momentum (pion mass) [and so does the parameter $B_0$]{}. This leads to a new expansion of the effective Lagrangian $${\cal L}_{eff} = \sum_d \tilde{\cal L}^d,$$ where each $\tilde{\cal L}^d$ contains more terms ${\cal L}_{nm}$ than does the corresponding term ${\cal L}^d$ in the case of the standard counting. The improved $\chi$PT is a simultaneous expansion in $p/\Lambda, m_q/\Lambda$ and $B_0/\Lambda$ around the limit $$(p,m_q,B_0) \rightarrow 0,~~~~ p^2/M_P^2 ~\text{and}~ m_qB_0/M_P^2~~\text{fixed}.$$ This is just another way to realize the chiral limit (2.11). The fact that – in the effective theory – we treat $B_0$ as an arbitrary expansion parameter does not contradict the general belief that, within QCD, this parameter is fixed and – hopefully – calculable. After all, quantum electrodynamics is also based on an expansion in $\alpha$, in spite of the general belief that there might exist a more fundamental theory in which the value of $\alpha$ is fixed and calculable [@edd]. Reconstruction of the low-energy\ $\pi-\pi$ scattering amplitude\ neglecting $O(\lowercase{p}^8)$ effects. ========================================== The analysis of low-energy $\pi-\pi$ scattering, traditionally based on analyticity, crossing symmetry and unitarity [@chew-mandelstam; @yellow; @roy], considerably simplifies if, in addition, one takes into account the Goldstone character of the pion. First, in the chiral limit (2.11), higher $(\ell \geq 2)$ partial waves are suppressed. The reason stems from the fact that in the limit (2.11) the whole amplitude behaves as $O(p^2)$ and furthermore, it does not contain light dipion bound state poles. Unitarity then implies that the scattering amplitude is dominantly real, since its imaginary part behaves as $O(p^4)$. Analyticity then forces the leading $O(p^2)$ part of the amplitude $A(s\|tu)$ to be a polynomial in the Mandelstam variables. Furthermore, higher than first order polynomials are excluded: They would be $O(p^2)$ only provided their coefficients blew up as $M_\pi^2 \rightarrow 0$, which would contradict the finiteness of the S-matrix in the limit $m_q \rightarrow 0$ with the external momenta kept fixed at a non-exceptional value. Finally, crossing symmetry allows one to express the $O(p^2)$ part of the scattering amplitude $A(s\|tu)$ as $$A_{Lead}(s\|tu) = \frac{\alpha}{3F_\pi^2}M_\pi^2 + \frac{\beta}{3F_\pi^2} (3s-4M_\pi^2),$$ where $\alpha$, $\beta$ are two dimensionless constants which are $O(1)$ in the chiral limit. The linear amplitude (3.1) does not contribute to $\ell \geq 2$ partial waves. Consequently, the latter behave in the chiral limit as $O(p^4)$ and, owing to unitarity, the absorptive parts of $\ell \geq 2$ waves are suppressed at least to $O(p^8)$. This conclusion holds independently of more quantitative predictions of $\chi$PT, which in the actual case merely concern the values of the two parameters ${\alpha}$ and ${\beta}$ in Eq. (3.1). The second simplification resides in the suppression of inelasticities arising from intermediate states that consist of more than two Goldstone bosons. The behavior of $n$-pion invariant phase space in the chiral limit (2.11) is given by its dimension: It scales like $p^{2n-4}$. Amplitudes with an arbitrary number of external pion legs are dominantly $O(p^2)$. Consequently, the contribution of multi-pion $(n > 2)$ intermediate states to the absorptive part of the elastic $\pi-\pi$ amplitude is suppressed in the chiral limit at least to $O(p^8)$. The smallness of higher partial waves and of inelasticities are of course well-known phenomenological facts [@yellow]. It is important that these “remarkable accidents" (see page 53 of [@yellow]) can be put under the rigorous control of chiral power counting: The previous discussion suggests that a rather simple amplitude analysis of low-energy $\pi-\pi$ scattering can be performed [up to and including $O(p^6)$ contributions]{}. In the following we confirm and elaborate this expectation in detail. It will be shown in particular that, neglecting $O(p^8)$ contributions, the whole scattering amplitude can be expressed in terms of low-energy $s$ and $p$ wave phase shifts and six (subtraction) constants. (The latter are related to the experimental phase shifts $via$ unitarity.) The resulting expression (3.2) will prove particularly useful both for constraining low-energy experimental data and for providing a basis for a confrontation of chiral perturbation theory up to two loops with experiment. Statement of the theorem ---------------------------- Let $\Lambda$ denote a scale (slightly) below the threshold for production of non-Goldstone particles. The $\pi-\pi$ amplitude can be written as $$\begin{aligned} \frac{3}{32\pi} A(s\|tu) &=& T(s)+ T(t) + T(u) \nonumber \\ &+& \frac{1}{3} [2U(s) - U(t) - U(u)] \nonumber \\ &+& \frac{1}{3}[(s-t) V(u) + (s-u)V(t)] \nonumber \\ &+& R_{\Lambda}(s\|t,u).\end{aligned}$$ The remainder, $R_{\Lambda}$, behaves in the chiral limit as $O(p^8)$ relative to the scale $\Lambda$: up to possible logarithmic terms, $$R_{\Lambda} = O([p/\Lambda]^8),$$ where $p$ stands for external pion momenta. In practice, $\Lambda \lesssim 1~GeV$. The functions $T,U$ and $V$ are analytic for $s < 4M_\pi^2$, whereas for $4M_\pi^2 < s < \Lambda^2$ their discontinuities are given by the three lowest partial wave amplitudes $f_\ell^I(s)$:[^2] $$\begin{aligned} I\!m\,T(s) &=& \frac{1}{3} \{ I\!m\,f_0^0(s) + 2I\!m\,f_0^2(s) \} \nonumber \\ I\!m\,U(s) &=& \frac{1}{2} \{ 2I\!m\,f_0^0(s) - 5I\!m\,f_0^2(s) \} \nonumber \\ I\!m\,V(s) &=& \frac{27}{2} \frac{1}{s-4M_\pi^2}I\!m\,f_1^1(s).\end{aligned}$$ The real parts of the functions $T,U$ and $V$ are defined only up to polynomials $$\begin{aligned} \delta T(s) &=& x(s- \frac{4}{3} M_\pi^2) \nonumber \\ \delta U(s) &=& y_0 + y_1 s+ y_2 s^2 + y_3s^3 \nonumber \\ \delta V(s) &=& - (y_1 + 4M_\pi^2 y_2 + 16M_\pi^4 y_3) + (y_2 + 12M_\pi^2 y_3) s -3 y_3 s^2 ,\end{aligned}$$ where $x$ and the $y$’s are five arbitrary real constants: because of the relation $s+t+u=4M_\pi^2$, the two sets of amplitudes $T,U,V$ and $T+\delta T, U+\delta U, V+\delta V$ lead to the same scattering amplitude A. (It is shown in Appendix B that Eqs. (3.5) actually represent the most general transformation of $T,U,V$ leaving the scattering amplitude invariant.) After conveniently fixing the “gauge freedom” (3.5), the functions $T,U$ and $V$ can be written as $$\begin{aligned} T(s) =& t_0 + t_2 s^2 + t_3 s^3 + & \frac{s^3}{\pi} \int_{4M_\pi^2}^{{\Lambda}^2} \frac{dx}{x^3} \frac{1}{x-s}I\!m\,T(x) \nonumber \\ U(s) =&&\frac{s^3}{\pi} \int_{4M_\pi^2}^{{\Lambda}^2} \frac{dx}{x^3} \frac{1}{x-s} I\!m\,U(x) \nonumber \\ V(s) =& v_1 + v_2s + v_3s^2 + &\frac{s^2}{\pi}\int_{4M_\pi^2}^{{\Lambda}^2} \frac{dx}{x^2} \frac{1}{x-s} I\!m\,V(x),\end{aligned}$$ where the imaginary parts are given by Eqs. (3.4) and the $t$’s and $v$’s are constants. It will be shown shortly that Eq. (3.2) is a rigorous consequence of analyticity and crossing symmetry and of the Goldstone nature of the pion. Unitarity ------------- The low-energy representation (3.2) of the scattering amplitude is exact up to an $O(p^8)$ remainder. In the whole interval $4M_\pi^2 < s < \Lambda^2$, unitarity can be imposed with the same accuracy in terms of partial waves $f_\ell^I$. As already pointed out, deviations from the unitarity condition $$%\begin{equation} I\!m\,f_{\ell}^I(s) = \sqrt{\frac{s-4M_\pi^2}{s}} \|f_\ell^I(s)\|^2 %\end{equation}$$above the inelastic threshold are of the order $O(p^8)$. The amplitude (3.2) contains all partial waves. For $\ell \geq 2$, the partial waves are real. Nevertheless, unitarity automatically is satisfied for $\ell \geq 2$ up to $O(p^8)$ terms, since higher partial waves anyway are $O(p^4)$ or smaller. Consequently, it is sufficient to impose unitarity for the three lowest waves $f_0^0, f_1^1$ and $f_0^2$ (hereafter denoted as $f_a, a = 0,1,2$ according to their isospin). Projections of Eq. (3.2) into the three lowest partial waves read $$\begin{aligned} Re\,f_a(s) &=& P_a(s) + \frac{s^3}{\pi} {\int\hspace{-1em}-}_{4M_\pi^2}^{{\Lambda}^2} \frac{dx}{x^3} \frac{I\!m\,f_a(x)}{x-s} \nonumber \\ &+& \frac{1}{\pi} \int_{4M_\pi^2}^{{\Lambda}^2} \frac{dx}{x} \sum_{b=0}^2 W_{ab}(s,x) I\!m\,f_b(x) + O(p^8)\end{aligned}$$ for the two $s$ waves $(a = 0,2)$, whereas the $p$ wave projection is $$\begin{aligned} Re\,f_1(s) &=& P_1(s) + \frac{s^2(s-4M_\pi^2)}{\pi} {\int\hspace{-1em}-}_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^2(x-4M_\pi^2)} \frac{I\!m\,f_1(x)}{x-s} \nonumber \\ &+& \frac{1}{\pi} \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x} \sum_{b=o}^2 W_{1b} (s,x) I\!m\,f_b(x) + O(p^8)\end{aligned}$$ Here, $P_a(s)$ are third order polynomials whose coefficients are defined in terms of the six constants $t_0, t_2, t_3$ and $v_1, v_2, v_3$ which appear in Eqs. (3.6). These polynomials are tabulated in Appendix C, together with the nine kernels $W_{ab}(s,x)$ which define the left-hand cut contributions to the partial waves. Eqs. (3.7a) and (3.7b) may be viewed as a particular truncation of the infinite system of Roy equations, which slightly differs from the form in which these equations have been used in the past [@bfp]. Here, the truncation in angular momentum and energy is performed under the systematic control of chiral power counting. In particular, Eqs. (3.7a,b) do not require a model-dependent evaluation of “driving-terms” which in the standard treatment behave in the chiral limit as $O(p^4)$, owing to the use of twice-subtracted dispersion relations. The price to pay is the occurrence of six ([a priori]{} unknown) constants in the polynomials $P_a(s)$ instead of only two constants (usually, the two $s$ wave scattering lengths) which characterize the inhomogeneous terms in standard Roy equations [@roy; @bfp]. Eqs. (3.7a) and (3.7b) can be used to fully reconstruct from the data the [whole amplitude $A(s\|tu)$ up to and including accuracy $O(p^6)$]{} in the whole low-energy domain of the Mandelstam plane, including the unphysical region. For this purpose one has to know the absorptive parts of three lowest partial waves for $4M_\pi^2 < s < \Lambda^2$ [and]{} the six constants $t_0, t_2, t_3,v_2,v_2,v_3$. Suppose one knew $I\!m\,f_a(s)$ with associated error bars in the whole interval $4M_\pi^2 < s < \Lambda^2 \lesssim 1~GeV^2$. Then one could calculate the dispersion integrals on the right hand side of Eqs. (3.7a,b). One would then determine the constants $t$ and $v$ from the best fit to the values $Re\,f_a(s)$ determined from the input $I\!m\,f_a(s)$ via the unitarity condition. The $\chi^2$ of this fit may be considered as a measure of the internal consistency of the input data $I\!m\,f_a(s)$. In practice, [experimental]{} information on $I\!m\,f_a(s)$ is only available for $s$ well above the threshold. In this case, a more sophisticated iteration procedure [@iterate]of Eqs. (3.7a,b) has to be used in order (i) to extrapolate the experimental data down to the threshold and, simultaneously, (ii) to determine the six constants $t$ and $v$. In both cases, the resulting amplitude is given by the formula (3.2). Proof of the reconstruction theorem --------------------------------------- Formulae (3.2) and (3.6) can be proven following the original derivation of the Roy equations [@roy]. The proof is based on fixed $t$ dispersion relations for the three $s$-channel isospin amplitudes $F^{(I)}$ $${\bf F}(s,t,u) = \left ( \begin{array}{c} F^{(0)}\\F^{(1)}\\F^{(2)}\\ \end{array} \right ) (s,t,u),$$ combined with the crossing symmetry relations $${\bf F}(s,t,u) = C_{su}{\bf F}(u,t,s) = C_{st}{\bf F}(t,s,u) = C_{ut}{\bf F}(s,u,t).$$ (Properties of the crossing matrices $C_{su},C_{st}$ and $C_{ut}$ are reviewed in Appendix A.) The standard Roy equations are derived from twice-subtracted dispersion relations – cf. the minimal number of subtractions required by the Froissart bound. In this case, however, the high-energy tail of the dispersion integral, which is hard to control in a model independent way, contributes to the $O(p^4)$ part of the amplitude. (In standard Roy equations, this contribution is contained in the so-called driving terms [@bfp].) If, on the other hand, one requires at low energy the precision $O(p^4)$ or higher, then it is more appropriate to stick to less predictive [triply-subtracted]{} fixed-$t$ dispersion relations: $$\begin{aligned} {\bf F}(s,t) &=& C_{st} \{ {\bf a_+}(t) + (s-u){\bf b_-}(t) + (s-u)^2 {\bf c_+}(t) \} \nonumber \\ &+ &\frac{1}{\pi}\int_{4M_\pi^2}^{\infty} \frac{dx}{x^3} \left \{ \frac{s^3}{x-s} + \frac{u^3}{x-u}C_{su} \right \} I\!m\,{\bf F}(x,t).\end{aligned}$$ Here the subscript $\pm$ refers to the eigenvalues $\pm 1$ of the crossing matrix $C_{tu}$. \[Notice that in the $s$-channel isospin basis (3.8), $C_{tu} = diag(+1,-1,+1)$.\] The subtraction term then represents the most general quadratic function in $s$ (for fixed $t$) symmetric under $s-u$ crossing. By construction, the dispersion integral in Eq. (3.10) exhibits $s-u$ crossing symmetry too. The task is now to impose the remaining two crossing relations and to determine the subtraction functions [a]{},[b]{}, and [c]{}. This can be achieved, neglecting in Eq. (3.10) contributions of chiral order $O(p^8)$ and higher. Let $\Lambda$ be a scale set by the threshold of production of non-Goldstone particles. Let us split the dispersion integral in Eq. (3.10) into low energy ($x \leq \Lambda^2$) and high energy ($x > \Lambda^2$) parts. For $4M_\pi^2 < s < \Lambda^2$, the imaginary part can be written as $$I\!m\, {\bf F} = I\!m\, \mbox{\boldmath $\Phi_+$}(s) + \left ( 1 + \frac{2t}{s-4M_\pi^2} \right ) I\!m\, \mbox{\boldmath $\Phi_-$}(s) + {\bf A}_{\ell \geq 2} (s,t),$$ where the first two terms stand for the contributions of $s$ and $p$ waves: $$I\!m\, \mbox{\boldmath $\Phi_+$}(s) = \left ( \begin{array}{c} I\!m\, f_0^0 (s) \\ 0 \\ I\!m\, f_0^2 (s) \end{array} \right ),~~~ I\!m\, \mbox{\boldmath $\Phi_-$}(s) = \left ( \begin{array}{c} 0 \\ 3I\!m\, f_1^1 (s) \\ 0 \end{array} \right ).$$ ${\bf A}_{\ell \geq 2}$ then collects the absorptive parts of all higher partial waves. The reason for this particular splitting resides in the chiral counting mentioned at the beginning of this section: The first two terms in Eq. (3.11) dominantly behave as $O(p^4)$, whereas ${\bf A}_{\ell \geq 2}$ is suppressed to $O(p^8)$. The dispersion integral ${\bf I}(s,t)$ in Eq. (3.10) then splits into three parts, $${\bf I}(s,t) = {\bf I}_{\ell < 2} (s,t) + {\bf I}_{\ell \geq 2} (s,t) + {\bf I}_H (s,t).$$ ${\bf I}_{\ell < 2} $ (${\bf I}_{\ell \geq 2} $) is the contribution of low-energy $\ell < 2$ ($\ell \geq 2$) partial waves, and ${\bf I}_H$ represents the high frequency part in which no partial wave decomposition is performed. Extracting from ${\bf I}_H$ its leading low energy behavior, one can write $${\bf I}_H (s,t) = (s^3 + u^3 C_{su}) {\bf H}_{\Lambda} + {\bf R}_H,$$ where ${\bf H}_{\Lambda}$ are constants which can be expressed as integrals over high-energy $\pi - \pi$ total cross sections, and the remainder behaves at low energies as $${\bf R}_H = O([p/\Lambda]^8).$$ The low-energy high angular momentum part ${\bf I}_{\ell \geq 2}$ is also suppressed to $O(p^8)$, reflecting the leading behavior of the absorptive part ${\bf A}_{\ell \geq 2}$ in the chiral limit and the fact that the corresponding dispersion integral (3.10) extends over a finite interval $x \in [4M_\pi^2,\Lambda^2]$. Hence, it remains to concentrate on the low-energy low angular momentum part ${\bf I}_{\ell < 2}$. Using Eq. (3.11), one easily checks the identity $$\begin{aligned} {\bf I}_{\ell < 2} &=& \mbox{\boldmath $\Phi$}(s,t,u) - C_{st} \left \{ \mbox{\boldmath $\Phi$}_+ (t) + \frac{s-u}{t-4M_\pi^2} \mbox{\boldmath $\Phi$}_- (t) \right \} \nonumber \\ & &+ (4M_\pi^2 - 2t)(s^2 + u^2 C_{su})\frac{1}{\pi}\int_{M_\pi^2}^{\Lambda^2} \frac{dx}{x^3} \frac{I\!m\, \mbox{\boldmath $\Phi$}_- (x)}{x-4M_\pi^2},\end{aligned}$$ where $$\begin{aligned} \mbox{\boldmath $\Phi$}(s,t,u) &=& \left \{ \mbox{\boldmath $\Phi$}_+ (s) + \frac{t-u}{s-4M_\pi^2} \mbox{\boldmath $\Phi$}_- (s) \right \} \nonumber \\ & & +C_{su} \left \{ \mbox{\boldmath $\Phi$}_+ (u) +\frac{t-s}{u-4M_\pi^2} \mbox{\boldmath $\Phi$}_- (u) \right \} \nonumber \\ & & +C_{st} \left \{ \mbox{\boldmath $\Phi$}_+ (t) + \frac{s-u}{t-4M_\pi^2} \mbox{\boldmath $\Phi$}_- (t) \right \}\end{aligned}$$ and denote the following dispersion integrals over the imaginary parts of low-energy $s$ and $p$ waves \[cf. Eq. (3.12)\]: $$\begin{aligned} \mbox{\boldmath $\Phi$}_+ (s) &=& \frac{s^3}{\pi} \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^3} \frac{I\!m\, \mbox{\boldmath $\Phi$}_+ (x)}{x-s} \nonumber \\ \mbox{\boldmath $\Phi$}_- (s) &=& \frac{s^2(s-4M_\pi^2)}{\pi} \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^2(x-4M_\pi^2)} \frac{I\!m\, \mbox{\boldmath $\Phi$}_- (x)}{x-s}.\end{aligned}$$ One observes from Eq. (3.17) that the function $(s,t,u)$ exhibits the full three-channel crossing symmetry. Furthermore, the second and third terms in Eq. (3.16) represent a function that is quadratic in $s$ (at fixed $t$) and symmetric under $s-u$ crossing. These terms can therefore be absorbed into the subtraction polynomial in the dispersion relations (3.10) by a suitable redefinition of (yet unknown) subtraction functions ${\bf a}_+,{\bf b}_-,{\bf c}_+$. Consequently, the whole amplitude ${\bf F}$ can be rewritten as $${\bf F}(s,t) = \mbox{\boldmath $\Phi$}(s,t,u) + {\bf P}(s,t,u) + O([p/\Lambda]^8),$$ where [P]{} is of the form $$\begin{aligned} {\bf P} =& C_{st} \left \{ \mbox{\boldmath $\alpha$}_+ (t) + (s-u) \mbox{\boldmath $\beta$}_- (t) + (s-u)^2 \mbox{\boldmath $\gamma$}_+ (t) \right \} \nonumber \\ &+ (s^3 + u^3 C_{su}) {\bf H}_{\Lambda}.\end{aligned}$$ Notice that the unspecified $O(p^8)$ contributions in Eq. (3.19) originate both from the high-energy remainder ${\bf R}_H$ (3.14) and from the low-energy higher angular momentum part ${\bf I}_{\ell \geq 2}$. Crossing symmetry of the scattering amplitude [F]{} should hold order by order in the chiral expansion. Since the function (3.17) exhibits full crossing symmetry, it remains to impose the latter for the function [P]{} (3.20). Because of the manifest $s-u$ symmetry, it is enough to require $${\bf P}(s,t,u) = C_{st}{\bf P}(t,s,u).$$ Neglecting $O(p^8)$ contributions, this equation represents the necessary and sufficient condition for the complete crossing symmetry of the amplitude [F]{}. Eq. (3.21) can be easily solved. Considering $s$ and $t$ as independent variables, one easily finds that $(t)$, $(t)$, and $(t)$ should be cubic, quadratic and linear functions of $t$ respectively. Hence, [P]{}$(s,t,u)$ is a general crossing symmetric polynomial in the Mandelstam variables of (at most) third order. Such a polynomial contains six independent parameters (see Appendix A). Indeed, after some simple but lengthy algebra, one verifies that Eq. (3.21) leaves a six parameter freedom in the original expression (3.20) for [P]{}. It remains to rewrite the result (3.19) in terms of the single amplitude $$A(s\|tu) = A(s\|ut) = \frac{32\pi}{3} \left \{ F^{(0)}(s,t,u) - F^{(2)}(s,t,u) \right \}.$$ The function gives rise to a contribution of the form (3.2) in which only the dispersion integrals of Eq. (3.6) occur. (One easily checks that $I\!m\, T,~I\!m\, U$ and $I\!m\, V$ are given by Eqs. (3.4).) Furthermore, taking into account the ambiguity (3.5) in the definition of $T,~U$ and $V$, it is clear that a general crossing symmetric polynomial may be conveniently parametrized by the six independent parameters $t_0,t_2,t_3,v_1,v_2,v_3$ as in Eqs. (3.6). Perturbative $\pi-\pi$ amplitude and the\ effective infrared dimension of the quark mass ================================================ We are now in a position to compare the two alternative low-energy expansions of the amplitude $A(s\|tu)$ generated by chiral perturbation theory according to the two possible values of the effective dimension of the quark mass: 2, in the case of the standard $\chi$PT, and 1 in the case which was defined in Sec. II as improved $\chi$PT. Up to and including two loops, the amplitude $A$ should be of the general form (3.2). Consequently, neglecting $O(p^8)$ contributions, one can work with the three functions $T, U$ and $V$ of a single variable and decompose them as $$T(s) = \sum_{n=0}^2 T^{(n)}(s),~ U(s) = \sum_{n=0}^2 U^{(n)}(s),~ V(s) = \sum_{n=0}^2 V^{(n)}(s),$$ where $n$ refers to the number of loops (including tree contributions of the corresponding order). It will be shown that the amplitudes $T, U$ and $V$ start to be sensitive to the effective dimension of the quark mass at leading $(n=0)$, one-loop $(n=1)$ and two-loop levels respectively. Leading $O(p^2)$ order -------------------------- If the dimension of the quark mass is 2, [i.e.]{}, if each power of the scalar pseudoscalar source $\chi$ in ${\cal L}_{eff}$ counts for two powers of pion momentum (mass), then the effective Lagrangian is dominated by the well-known expression $${\cal L}^{(2)} = \frac{1}{4} F_0^2 \{\langle (D^\mu U)^+ (D_\mu U) \rangle + 2B_0 \langle \chi^+ U + U^+ \chi \rangle \}.$$ This formula collects all possible invariants of dimension 2. To leading order, the pion and the kaon masses read $$\begin{aligned} \overcirc{M}_\pi^2 &=& 2\hat{m}B_0 \nonumber \\ \overcirc{M}_K^2 &=& (m_s + \hat{m}) B_0\end{aligned}$$ and the $\pi-\pi$ amplitude takes the well-known form, first given by Weinberg [@sw66] $$A_{lead} (s\|tu) = \frac{1}{F_0^2} (s- 2\hat{m}B_0) = \frac{1}{F_0^2} (s - \overcirc{M}_\pi^2)$$ This represents the standard scenario of chiral perturbation theory. It can hardly be circumvented provided the scale of $B_0$ is large compared to the pion mass, typically, $B_0 \gtrsim 1~GeV$. On the other hand, if $B_0$ turned out to be much smaller than the GeV-scale, [e.g.]{}, comparable to the fundamental order parameter $F_0$ ($\sim 93~ MeV$), then the above way of counting effective infrared dimensions would be modified. Both the quark mass and the condensate $B_0$ should then be considered as quantities comparable to the pion mass. They should both be attributed effective infrared dimension 1 and they should both be viewed as expansion parameters. In this case, every insertion of the source $\chi(x)$ counts as a single power of pion momentum and the formula (4.2) no longer represents the most general expression of dimension 2. Instead, the complete collection of invariants of dimension 2 now reads $$\begin{aligned} \tilde{{\cal L}}^{(2)} &=& \frac{1}{4} F_0^2 \{ \langle D^\mu U D_\mu U^+ \rangle + 2B_0 \langle \chi^+ U + \chi U^+\rangle \nonumber \\ &+& A_0 \langle \chi^+ U \chi^+ U + \chi U^+ \chi U^+ \rangle + Z_0^S \langle \chi^+ U + U^+ \chi \rangle^2 + \nonumber \\ &+& Z_0^P \langle \chi^+ U - \chi U^+ \rangle ^2 + 2H_0 \langle \chi^+ \chi \rangle \} .\end{aligned}$$ where the tilde over the symbol ${\cal L}$ here (and below) indicates the use of the modified chiral power counting. The terms containing two powers of $\chi$ are usually included into the next-to-the-leading part ${\cal L}^{(4)}$ of the effective Lagrangian. Here, they appear of the same dimension and they are expected to be of a comparable size as the standard expression (4.2). The low-energy constants $A_0, Z_0^S$ and $Z_0^P$ represent appropriately subtracted zero-momentum transfer two-point functions of scalar and pseudoscalar quark densities, divided by $F_0^2$. These two-point functions are order parameters of spontaneous chiral symmetry breaking and, consequently, they satisfy superconvergent dispersion relations. A simple saturation of the latter with a few of the lowest massive hadronic states suggests that the dimensionless constants $A_0$ and $Z_0^P$ are of the order $1$, say, $A_0 \sim 1 - 5$. On the other hand, $Z_0^S$ violates the Zweig rule in the $0^{++}$ channel and consequently it is expected to be suppressed. The parameters $Z_0^S, Z_0^P$ and $A_0$ are related to the low-energy constants $L_6, L_7$ and $L_8$ of the standard $d = 4$ Lagrangian ${\cal L}^{(4)}$ [@gl85]. Expanding the latter constants in powers of $B_0$, one gets[^3] $$\begin{aligned} L_6 &=& \left ( \frac{F_0}{4B_0} \right ) ^2 \{ Z_0^S + O(B_0^2)\},\nonumber \\ L_7 &=& \left ( \frac{F_0}{4B_0} \right ) ^2 Z_0^P \nonumber \\ L_8 &=& \left ( \frac{F_0}{4B_0} \right ) ^2 \{A_0 + O(B_0^2) \},\end{aligned}$$ where $O(B_0^2)$ terms represent divergent contributions to the two-point functions defining the divergent parts of the bare constants $L_6, L_8$. (The constants $A_0, Z_0^S$ and $Z_0^P$ do not undergo any infinite renormalization.) The leading order pion and kaon masses (denoted by a tilde) now read $$\begin{aligned} \widetilde{M}_\pi^2 &=& 2\hat{m}(\tilde{B} + 4 \hat{m} Z_0^S) + 4\hat{m}^2 A_0 \nonumber \\ \widetilde{M}_K^2 &=& (m_s + \hat{m}) (\tilde{B} + 4 \hat{m} Z_0^S) + (m_s + \hat{m})^2 A_0 .\end{aligned}$$ Here $\tilde{B}$ stands for the dominant $O(p)$ contribution to the $SU(2) \times SU(2)$ quark-antiquark condensate (divided by $F_0^2$) taken at $m_u = m_d = 0$: $$\langle \bar{u}u \rangle_{m_u=m_d=0} = \langle \bar{d}d \rangle_{m_u=m_d=0} = -F_0^2 \tilde{B} + O(m_s^2).$$ Within the modified chiral power counting, $\tilde{B}$ consists of two terms $$\tilde{B} = B_0 + 2 m_s Z_0^S$$ which are both of the order $O(p)$. In principle, they could be of comparable size, if $Z_0^S$ were not suppressed by the Zweig rule. The leading contribution to the $\pi-\pi$ scattering amplitude calculated from the improved $O(p^2)$ Lagrangian (4.5) turns out to be independent of low-energy parameters $A_0$ and $Z_0^P$, and it can be expressed in term of the quark-antiquark condensate $\tilde{B}$, $$A_{lead} (s\|tu) = \frac{1}{F_0^2} (s - 2\hat{m} \tilde{B}),$$ in complete analogy with the standard result (4.4). Although Eq. (4.10) and Weinberg’s formula (4.4) formally coincide if one neglects Zweig-rule violation, their numerical content is rather different, because of different scales of quark-antiquark condensation in each $\chi$PT alternative. In Eq. (4.4), $2\hat{m}B_0$ is the leading approximation to $M_\pi^2$, whereas in the improved $\chi$PT, the relation between the quark-antiquark condensate and the pion mass is more subtle: Indeed, using first Eq. (4.7), formula (4.10) can be rewritten as $$A_{lead} (s\|tu) = \frac{1}{F_0^2} (s - \widetilde{M}_\pi^2) + \frac{\widetilde{M}_\pi^2}{F_0^2}\,\epsilon \,(1 + 2\zeta),$$ where $$\epsilon = \frac{4\hat{m}^2 A_0}{\widetilde{M}_\pi^2}, ~~~~~~~~~~~ \zeta = \frac{Z_0^S}{A_0}.$$ Whereas in the standard $\chi$PT $\epsilon$ would be a small quantity of the order $O(p^2)$, in the improved $\chi$PT, $\epsilon$ is $O(1)$ and there is no reason for it to be particularly small; hence, the second term in Eq. (4.11) represents a [leading order]{} modification of the Weinberg’s formula (4.4). ($\zeta$ measures the Zweig rule violation in the $0^{++}$ channel and can be expected rather small.) Using Eqs. (4.7) one may easily check that $\epsilon$ can indeed be of order 1 for natural values of $A_0$ (cf. footnote \[A\_0\]) and for reasonably small values of quark masses. Setting — for the sake of illustration — $B_0 = 150~MeV$ and $\hat{m} = 25~MeV$, and neglecting Zweig rule violation, one obtains $\epsilon = 0.62, A_0 = 4.8$ and $m_s \simeq 195~MeV$. The leading order mass formula (4.7) implies a relation between the parameter $\epsilon$ and the quark-mass ratio $ r = m_s/\hat{m}$: $$\epsilon = 2 \frac{r_2-r}{r^2-1} ,~~~ r_2 = 2\frac{\widetilde{M}_K^2}{\widetilde{M}_\pi^2} - 1 \simeq 25.9$$ If $r$ decreases from its canonical leading order value $r = r_2$, then $\epsilon$ increases and reaches 1 for $r = r_1$, $$r_1 = 2 \frac{\widetilde{M}_K}{\widetilde{M}_\pi} - 1 \simeq 6.33.$$ Similarly, the order parameter $B_0$ can be expressed as $$\frac{2\hat{m}B_0}{\widetilde{M}_\pi^2} = 1 - [1 + (r+2)\zeta]\epsilon.$$ This ratio decreases from its canonical value 1 down to zero, as $r$ decreases from $r=r_2$ to $r=r_{crit}(\zeta) \gtrsim r_1$, for which $B_0$ vanishes. Notice that stability of the massless QCD vacuum under perturbation by small quark masses implies $B_0 \geq 0$. Next to the leading $O(p^3)$ contribution --------------------------------------------- In the improved chiral perturbation theory, the leading order Lagrangian $\tilde{{\cal L}}^{(2)}$ is followed by a [dimension 3 term]{} $\tilde{{\cal L}}^{(3)}$, which contributes at the tree level before one-loop contributions of dimension 4 start to appear. $\tilde{{\cal L}}^{(3)}$ reads $$\begin{aligned} \tilde{{\cal L}}^{(3)} &=& \frac{1}{4} F_0^2 \{ \xi \langle D_\mu U^+ D^\mu \chi + D_\mu \chi^+ D^\mu U \rangle \nonumber \\ &+& \rho_1 \langle (\chi^+U)^3 + (\chi U^+)^3 \rangle + \rho_2 \langle \chi ^+ \chi (\chi^+ U + U^+ \chi) \rangle \nonumber \\ &+& \rho_3 \langle (\chi^+ U)^2 - (\chi U^+)^2 \rangle \langle \chi^+ U - \chi U^+ \rangle + \ldots \} .\end{aligned}$$ The dots stand for terms that violate the Zweig rule in a nonanomalous channel. Notice that (4.16) differs in its first term from the expression given for $\tilde{{\cal L}}^{(3)}$ in Ref. [@fss91]. The two forms of $\tilde{{\cal L}}^{(3)}$ are equivalent: they are related by a simple redefinition of the Goldstone boson field $U$. The low energy constants $\xi$ and $\rho_i$ are finite — there are no divergences of dimension 3. $\tilde{{\cal L}}^{(3)}$ induces a shift in the pion mass, $$\delta M_\pi^2 = \epsilon \, \widetilde{M}_\pi^2 \, (9\lambda_1 + \lambda_2),$$ where $$\lambda_i = \frac{\hat{m}\rho_i}{4A_0}$$ are dimensionless parameters of order $O(M_\pi)$. Similarly, the leading $\pi-\pi$ amplitude receives a constant $d = 3$ contribution $$\delta \tilde{A} (s\|tu) = \epsilon \, \frac{\widetilde{M}_\pi^2}{3F_0^2} \, (81 \,\lambda_1 + \lambda_2).$$ Finally, the first term in $\tilde{{\cal L}}^{(3)}$ is responsible for splitting of the decay constants $F_\pi, F_K, F_\eta$. Eliminating the low-energy parameter $\xi$, one obtains, to that order $$\frac{F_\pi^2}{F_0^2} = 1 + \frac{2}{r-1} (\frac{F_K^2}{F_\pi^2} - 1).$$ It is convenient to collect all $d=2$ and $d=3$ contributions, and to express the resulting tree amplitude in the form (3.1): $$A_{tree} (s\|tu) = \frac{1}{3F_\pi^2} [ \alpha M_\pi^2 + \beta (3s-4M_\pi^2)] + \frac{M_\pi^2}{3F_\pi^2} \,\delta \alpha ,$$ where $M_\pi$ and $F_\pi$ denote the $experimental$ (charged) pion mass and decay constant.[^4] The parameters $\alpha$ and $\beta$ read $$\frac{\alpha}{\beta} = 1 + 3 \epsilon \, (1+2\zeta),~~~~\beta = \frac{F_\pi^2}{F_0^2},$$ whereas $\delta \alpha = \delta \alpha_3 + \delta \alpha_4$ describes small $O(p^3)$ and $O(p^4)$ corrections. $\delta \alpha$ arises from the genuine $O(p^3)$ contribution (4.19) of $\tilde{\cal L}^{(3)}$ to the $\pi-\pi$ amplitude and from the introduction of the physical mass $M_\pi$ into the formula (4.21). Using Eqs. (4.17) and (4.19), the $O(p^3)$ constant $M_\pi^2~\delta \alpha_3$ can be expressed in terms of the parameters $\lambda_1$ and $\lambda_2$ of $\tilde{\cal L}^{(3)}$: $$M_\pi^2 ~ \delta \alpha_3 = \epsilon \, \beta \, \widetilde{M}_\pi^2 \, [72 \lambda_1 - (27\lambda_1 + 3 \lambda_2)\, \epsilon \, (1+2\zeta)].$$ The remaining term $M_\pi^2 \, \delta \alpha_4$ accounts for the $O(p^4)$ and higher contributions to $M_\pi^2$. One has $$M_\pi^2 \, \delta \alpha_4 = - \beta \, \Delta \! M_\pi^2 \, [1 + 3\epsilon(1+2\zeta)],$$ where $$\Delta \! M_\pi^2 = M_\pi^2 - \widetilde{M}_\pi^2 - \delta M_\pi^2$$ represents the $O(p^4)$ difference between the physical value and the tree approximation of the pion mass squared. The results of standard $\chi$PT are reproduced by setting $\epsilon = \zeta = 0$ in the previous equations; $i.e.$, $r = r_2 \simeq 25.9$. In this case, $\widetilde{M}_\pi^2$ reduces to $\overcirc{M}_\pi^2$ \[Eq. (4.3)\], and $\alpha = \beta \simeq 1$. The improved $\chi$PT still requires $\beta \simeq 1$, but $\alpha$ is now allowed and expected to be considerably larger, since $\epsilon$ is now an $O(1)$ quantity. In fact, the vacuum stability conditions mentioned above imply that for a given quark mass ratio $r$ \[lying between $r_1$ and $r_2$ – cf. Eqs. (4.13) and (4.14)\], the Zweig rule violating parameter $\zeta = Z_0^S/A_0$ should satisfy $$0 \leq \zeta \leq \zeta_{crit}(r) = \frac{1}{2} \frac{r-r_1}{r_2 - r} \frac{r+r_1+2}{r+2} .$$ Using these bounds in Eq. (4.22), one obtains a rather narrow band of allowed values in the plane defined by the ratio $\alpha/\beta$ and $r$. This band is shown in Fig. 1. It is straightforward to rewrite the above result in terms of the amplitudes $T,U$ and $V$. The tree contribution to these amplitudes simply reads $$T^{(0)} (s) = (\hat{\alpha} + \delta \hat{\alpha}) M_\pi^2,~~ U^{(0)}(s) = 0,~~ V^{(0)}(s) = 9\hat{\beta},$$ where $$\hat{\alpha} \equiv \frac{\alpha}{96\pi}\frac{1}{F_\pi^2}, {}~~~~~~~~~~~~ \hat{\beta} \equiv \frac{\beta}{96\pi}\frac{1}{F_\pi^2}$$ and likewise for $\delta \hat{\alpha}$. Our main task is to use all available experimental information to measure $\alpha,\beta$ and, indirectly, the quark mass ratio $r$. One loop $O(p^4)$ order --------------------------- Let ${\cal L}_{nm}$ denote an invariant entering the effective Lagrangian, that contains $n$ powers of covariant derivatives D and $m$ insertions of the scalar-pseudoscalar source $\chi$. (For simplicity, the expansion coefficients $\ell^{nm}$ of Eq. (2.7) are included in ${\cal L}_{nm}$.) In the standard chiral perturbation theory the $d=4$ part of the effective Lagrangian can be written as $${\cal L}^{(4)} = \sum_{n+2m=4} {\cal L}_{nm}.$$ It contains all counterterms which are needed to renormalize one-loop contributions generated by ${\cal L}^{(2)}$. If the dimension of the quark mass is 1, one-loop renormalization gets modified in two respects: (i) The effective dimension of a term ${\cal L}_{nm}$ is $d = n+m$ instead of $d = n+2m$, and (ii) $B_0$ is now a (small) expansion parameter of dimension 1. It follows, in particular, that renormalization has to be performed order by order in $B_0$. The modified $d=4$ part of ${\cal L}_{eff}$ then reads $$\tilde{{\cal L}}^{(4)} = \sum_{n+m=4} {\cal L}_{nm} + B_0 \, ({\cal L}_{21} + {\cal L}_{03}) + B_0^2 \, {\cal L}_{02}.$$ The last two counterterms are needed to renormalize the $B_0$-dependent part of one-loop divergences generated by $\tilde{{\cal L}}^{(2)}$. Terms which are contained both in ${\cal L}^{(4)}$ and in $\tilde{{\cal L}}^{(4)}$ are merely made with four derivatives [@gl85]: $$\begin{aligned} {\cal L}_{40} &=& L_1 \langle D_\mu U^+ D^\mu U\rangle^2 + L_2 \langle D_\mu U^+ D_\nu U\rangle \langle D^\mu U^+ D^\nu U\rangle \nonumber \\ &+& L_3 \langle D_\mu U^+ D^\mu U D_\nu U^+ D^\nu U \rangle \nonumber \\ &-& iL_9 \langle F_{\mu\nu}^R D^\mu UD^\nu U^+ + F_{\mu\nu}^L D^\mu U^+ D^\nu U\rangle \nonumber \\ &+& L_{10} \langle U^+ F_{\mu\nu} ^R U F^{L,\mu\nu}\rangle + H_1 \langle F_{\mu\nu}^R F^{R,\mu\nu} + F_{\mu\nu}^L F^{L,\mu\nu}\rangle .\end{aligned}$$ The meaning and renormalization of low-energy constants in Eq. (4.31) are independent of the symmetry breaking sector and, in particular, of the infrared dimension of the quark mass. The remaining $B_0$-independent terms in Eq. (4.30), cf. ${\cal L}_{22}$ and ${\cal L}_{04}$, are absent from the expression for ${\cal L}^{(4)}$: with quark mass of dimension 2, these terms would count as $O(p^6)$ and $O(p^8)$ respectively. On the other hand, all terms but ${\cal L}_{40}$ contained in ${\cal L}^{(4)}$ are already included either in $\tilde{{\cal L}}^{(2)}$ or in $\tilde{{\cal L}}^{(3)}$. Consequently, $\tilde{{\cal L}}^{(2)} + \tilde{{\cal L}}^{(3)} + \tilde{{\cal L}}^{(4)}$ not only encompasses all terms of the standard ${\cal L}^{(2)} + {\cal L}^{(4)}$ but, in addition, it contains new terms of the type $\tilde{{\cal L}}^{(3)}, {\cal L}_{22}$ and ${\cal L}_{04}$. This phenomenon is general. Order by order, the improved $\chi$PT contains the standard perturbation theory as a special case: It contains more parameters and it could well fit the experimental data even when the standard $\chi$PT fails. The one-loop contribution to the $\pi-\pi$ amplitude $A(s\|tu)$ has been worked out within the standard chiral perturbation theory in Refs. [@gl84; @gl85]. The result can be expressed in terms of four constants: $\alpha =\beta$ (close to 1), the shift $\delta \alpha_4$ ($\delta \alpha_3 = 0$ in this case) introduced in Eqs. (4.21), and two linear combinations of the renormalized constants $L_1, L_2$ and $L_3$. In the improved $\chi$PT, the one-loop $O(p^4)$ amplitude contains, in addition, two parameters which arise from the new terms ${\cal L}_{22}$ and ${\cal L}_{04}$ in $\tilde{{\cal L}}^{(4)}$. Working with the amplitudes $T,U,V$ (the formula (3.2) is valid up to and including two loops), one may obtain a closed form for the one-loop amplitude which encompasses both alternatives of chiral perturbation theory. Let $\varphi_a^{(d)}(s)$ denote the effective dimension-$d$ contribution to the real part of the partial wave amplitude $f_a(s),~(a=0,1,2)$, introduced in section III B: $$Re\,f_a(s) = \sum_{d \geq 2} \varphi_a^{(d)}.$$ From Eqs. (4.27) one finds $$\begin{aligned} \varphi_0^{(2)}(s) &=& 6\hat{\beta}\, (s + \kappa_0) \nonumber \\ \varphi_1^{(2)}(s) &=& \hat{\beta}\, (s-4M_\pi^2) \nonumber \\ \varphi_2^{(2)}(s) &=& -3\hat{\beta}\,(s + \kappa_2),\end{aligned}$$ where $$\kappa_0 \equiv (\frac{5\alpha}{6\beta} - \frac{4}{3}) M_\pi^2,~~~~~ \kappa_2 \equiv (-\frac{2\alpha}{3\beta} - \frac{4}{3}) M_\pi^2 .$$ Similarly, the real parts at the $O(p^3)$ level are $$\varphi_0^{(3)} = 5 M_\pi^2 \, \delta \alpha_3,~~~\varphi_1^{(3)} = 0,~~~ \varphi_2^{(3)} = -2 M_\pi^2 \, \delta \alpha_3,$$ where $\delta \alpha_3$ is given by Eq. (4.23). For $d > 3$, the real parts are no longer defined by the tree amplitude alone. The $O(p^d)$ contribution to the imaginary part of the partial wave amplitudes $I\!m\, f^{(d)}_a(s)$ can be expressed for $s > 4 M_\pi^2$ through elastic unitarity: $$I\!m \, f^{(d)}_a(s) = \sqrt{\frac{s-4M_\pi^2}{s}} \sum_{d_1 + d_2 = d} \varphi_a^{(d_1)}(s) \varphi_a^{(d_2)}(s).$$ This result is an exact property of $\chi$PT amplitudes for $4 \leq d < 8$. The one-loop level contains $d=4$, $d=5$ and $d=6$ contributions to the scattering amplitude $A(s\|tu)$. In the following, we shall merely concentrate on the leading $O(p^4)$ part. The corresponding components of the functions $T,U,V$[^5] will be denoted as $T_{lead}^{(1)}(s), U_{lead}^{(1)}(s)$, and $V_{lead}^{(1)}(s)$. The discontinuities of these functions are given by the $O(p^4)$ absorptive parts $I\!m\,f_a^{(4)}$, following Eqs. (3.4). Hence, the $O(p^4)$ one-loop amplitudes $T,U,V$ can be written as $$\begin{aligned} T_{lead}^{(1)}(s) &=& \frac{1}{3} \{ [\varphi^{(2)}_0(s)]^2 + 2 [\varphi^{(2)}_2 (s)]^2 \} L(s,\mu^2) + \alpha_4(\mu^2) + \alpha_0(\mu^2)s^2 \nonumber \\ U_{lead}^{(1)}(s) &=& \frac{1}{2} \{2[\varphi^{(2)}_0(s)]^2 - 5 [\varphi^{(2)}_2(s)]^2\} L(s,\mu^2) \nonumber \\ V_{lead}^{(1)}(s) &=& \frac{27}{2} \frac{[\varphi^{(2)}_1(s)]^2}{s-4M_\pi^2} L(s, \mu^2) + \beta_2(\mu^2) + \beta_0 (\mu^2)s,\end{aligned}$$ where $L(s,\mu^2)$ is the loop integral subtracted at the point $s = -\mu^2$: $$L(s,\mu^2) \equiv \frac{s+\mu^2}{\pi} \int_{4M_\pi^2}^\infty \frac{dx}{x+\mu^2} \frac{1}{x-s} \sqrt{\frac{x-4M_\pi^2}{x}}.$$ The constants $\alpha_n(\mu^2)$ and $\beta_n(\mu^2)$ behave in the chiral limit as $M_\pi^n$. They describe the most general polynomial part of $T,~U$ and $V$ which is $ O(p^4)$ and takes into account the freedom (3.5). These constants represent renormalized tree contributions of the $d=4$ part of ${\cal L}_{eff}$. Their dependence on the subtraction point $\mu^2$ can be determined by demanding that the scattering amplitude $A(s\|tu)$ be $\mu^2$-independent. Following Appendix B, this requirement is equivalent to the conditions $$\frac{\partial}{\partial \mu^2} T^{(1)}(s) = \delta T(s), ~~\frac{\partial}{\partial \mu^2} U^{(1)}(s) = \delta U(s),~~\frac{\partial}{\partial \mu^2} V^{(1)}(s) = \delta V(s),$$ where $\delta T,~ \delta U$ and $\delta V$ are of the general form (3.5). Taking into account the $s$-independence of $\frac{\partial}{\partial \mu^2} L (s,\mu^2)$ and $L(0,\mu^2) = -L(-\mu^2,0)$, the solution of Eqs. (4.39) can be easily found: $$\begin{aligned} \alpha_0(\mu^2) &=& \alpha_0 (0) + 18 \hat{\beta}^2 L(-\mu^2) \nonumber \\ \beta_0(\mu^2) &=& \beta_0 (0) \nonumber \\ \alpha_4(\mu^2) &=& \alpha_4(0) + (11 \hat{\alpha}^2 - 32 \hat{\beta}^2) M_\pi^4 L(-\mu^2) \nonumber \\ \beta_2(\mu^2) &=& \beta_2(0) + 18 \hat{\beta} (5\hat{\alpha} - 2\hat{\beta}) M_\pi^2 L(-\mu^2).\end{aligned}$$ In these equations we have denoted $$L(s) \equiv L(s,\mu^2 = 0) = \frac{1}{\pi} \left [ 2 + \sigma \ln (\frac{\sigma - 1}{\sigma +1}) \right ] ,~~~~ \sigma = \sqrt{1 - \frac{4M_\pi^2}{s}}.$$ In the following we shall work at $\mu^2 = 0$. The constants $\alpha_0$ and $\beta_0$ are related to the low-energy parameters $L_1, L_2$ and $L_3$ which occur in the expression (4.31) for ${\cal L}_{40}$. One gets $$\begin{aligned} \alpha_0(0) &=& \frac{1}{4\pi F_0^4} [ L_1^r + L_2^r + \frac{1}{2} L_3 - \frac{1}{4}\nu (\bar{\mu}^2) ] \nonumber \\ \beta_0(0) &=& \frac{3}{8\pi F_0^4} (L_2^r - 2L_1^r -L_3) + \frac{1}{1024\pi^3F_0^4}\end{aligned}$$ where $$\nu (\bar{\mu}^2) = \frac{1}{32\pi^2} \left [ \ln \frac{M_\pi^2}{\bar{\mu}^2} + \frac{1}{8} \ln \frac{M_K^2}{\bar{\mu}^2} + \frac{9}{8} \right ]$$ and $\bar{\mu}^2$ denotes the renormalization scale introduced in Ref.[@gl85]. The renormalized constants $L_1^r, L_2^r$ are $\bar{\mu}^2$-dependent, whereas $L_3$ and $L_2^r - 2\,L_1^r$ are not. Furthermore, the combination $L_2^r - 2\,L_1^r$ should be suppressed by the Zweig rule or in the large $N_c$ limit. Notice that the constant $\beta_0$ is independent both of $\mu^2$ and of $\bar{\mu}^2$. The interpretation of the remaining two constants $\alpha_4$ and $\beta_2$ depends on the effective dimension of the quark mass. In the standard chiral perturbation theory, these constants can be expressed in terms of the shifts of the pion mass and decay constant, as calculated within $SU(2) \times SU(2)$ perturbation theory [@gl84]. In the improved $\chi$PT, $\alpha_4(0)$ and $\beta_2(0)$ are independent parameters which describe respective contributions of new terms ${\cal L}_{04}$ and ${\cal L}_{22}$ in the $O(p^4)$ effective Lagrangian $\tilde{\cal L}^{(4)}$. The explicit relationship between $\alpha_4(0), \beta_2(0)$ and the low-energy parameters of $\tilde{\cal L}^{(4)}$ is of no direct use in the present paper and it will be given elsewhere. Concluding this section, it is worth noting that the low-energy theorem of Sec. III considerably simplifies the calculation of two-loop contributions to $A(s\|tu)$: For $d < 8$, all $O(p^d)$ terms can be obtained by a straightforward combination of Eqs. (3.2) and (3.4) with the unitarity condition (4.36). Up to and including two loops, the $\chi$PT expansion of the $\pi-\pi$ scattering amplitude can be viewed as an iteration of the Roy-type Eqs. (3.7a) and (3.7b). The corresponding polynomials $P_a(s)$ appearing at a given order $O(p^d)$ are then defined in terms of the renormalized low-energy constants of the Lagrangians ${\cal L}^{(d)}$ or $\tilde{\cal L}^{(d)}$, according to the effective dimension of the quark mass being respectively 2 or 1. Determination of parameters of ${\cal L}_{\lowercase {eff}}$\ from $\pi-\pi$ scattering data ====================================================== Suppose one has enough experimental information to perform the program formulated in Sec. III and to reconstruct the low-energy amplitude $A(s\|tu)$. Let us call the result of this reconstruction $A_{exp}(s\|tu)$ and the corresponding $T,U,V$ amplitudes given by Eqs. (3.4) $T_{exp}, U_{exp}$ and $V_{exp}$ respectively. We would like to compare the experimental amplitude $A_{exp}$ with the theoretical amplitude $A_{th}$ given in Sec. IV [in a whole low-energy domain of the s-t-u plane including the unphysical region]{}. Such a comparison should lead to a detailed fit, which in turn should provide a rather precise determination of low energy constants entering $A_{th}$. In particular, we would like to measure the parameter $\alpha$ and, in this way, let Nature tell us whether it prefers a quark mass of effective dimension 1 or 2. The theorem proved in Sec. III considerably simplifies the above task: Neglecting $O(p^8)$ contributions, the equation $$A_{exp} (s\|tu) - A_{th} (s\|tu) = 0,$$ which is supposed to hold in a crossing symmetric domain of the Mandelstam plane, is actually equivalent to a set of three [single-variable]{} equations, $$\begin{aligned} T_{exp} (s) - T_{th} (s) &=& \delta T(s) \nonumber \\ U_{exp} (s) - U_{th} (s) &=& \delta U(s) \nonumber \\ V_{exp} (s) - V_{th} (s) &=& \delta V(s)\end{aligned}$$ valid in an interval of $s$. The functions $\delta T(s), \delta U(s)$ and $\delta V(s)$ are the arbitrary and irrelevant polynomials given by Eq. (3.5). In this section, we will analyze Eqs. (5.2). Hereafter we systematically set $M_\pi^2 = 1$. One-loop precision ---------------------- The functions $T_{exp}, U_{exp}$ and $V_{exp}$ are given by Eqs. (3.6). Up to and including (one-loop) order $O(p^4)$, the theoretical amplitude reads $$T_{th} = T^{(0)} + T_{lead}^{(1)}$$ (and likewise for $U$ and $V$), where the tree and leading one-loop contributions are presented in Eqs. (4.27) and (4.37) respectively. We shall concentrate on real parts of Eqs. (5.2). Let us denote the partial wave integrals appearing in Eqs. (3.6) as $$\begin{aligned} \phi_a(s) &=& \frac{s^3}{\pi} {\int\hspace{-1em}-}_4 ^{\Lambda^2} \frac{dx}{x^3} \frac{I\!m\,f_a(x)}{x-s}, ~~~~~~a = 0,2 \nonumber \\ \phi_1(s) &=& \frac{s^2}{\pi} {\int\hspace{-1em}-}_4^{\Lambda^2} \frac{dx}{x^2} \frac{1}{x-4} \frac{I\!m\, f_1(x)}{x-s} ,\end{aligned}$$ It is convenient to take linear combinations of the Eqs. (5.2) for $T$ and $U$ and isolate the contributions of $I=0$ and $I=2$ $s$ waves. The resulting equations can be written as $(a = 0,2)$ $$\phi_a(s) = \frac{\hat{\beta}^2N_a}{6\pi} (s + \kappa_a)^2 D(s) + p_a (s), \eqnum{5.5a}$$ where $$N_0 = 36,~~~~~~~~~~~~N_2 = 9$$ and ($w \equiv \left\vert 1 - \frac{4}{s}\right\vert ^{1/2}$) $$\begin{aligned} D(s) &\equiv& 6\pi~Re\,L(s), \nonumber \\ D(s) &=& 12 + 6 w \ln \left\vert \frac{1-w}{1+w}\right\vert,~~~~s \leq 0, ~~s \geq 4, \nonumber \\ D(s) &=& 12 - 12 w \arctan w^{-1},~~~~ 0 \leq s \leq 4.\end{aligned}$$ The $p_a(s)$ are two third-order polynomials, whose coefficients are given in terms of (i) three constants $t_i$ \[cf. the first of Eqs. (3.6)\], (ii) the parameters $\alpha, \beta, \alpha_0(0)$ and $\alpha_4(0)$ defined in terms of ${\cal L}_{eff}$, and (iii) the irrelevant five constants that characterize the polynomial ambiguity (3.5). The explicit expression for the coefficients of $p_a(s)$ can be easily read off from Eqs. (3.6), (4.27) and (4.37). Similarly, the $V$-equation (5.2) can be written as $$\phi_1(s) = \frac{\hat{\beta}^2}{6\pi} (s-4) D(s) + q(s), \eqnum{5.5b}$$ where $q(s)$ is now a second order polynomial with coefficients given by linear combinations of three parameters $v_i$ \[cf. the last of Eqs. (3.4)\], the ${\cal L}_{eff}$ parameters $\beta_0(0)$ and $\beta_2(0)$ and the irrelevant constants $y_i$. For small $s$, the function $D(s)$ behaves as $$D(s) = s + \frac{1}{10} s^2 + O(s^3) .$$ On the other hand, the functions $\phi_a(s)$ and $\phi_1(s)$ defined in (5.4) behave as $O(s^3)$ and $O(s^2)$ respectively. The polynomials $p_a(s)$ and $q(s)$ should be such to insure this small s behavior on the right hand sides of Eqs. (5.5a) and (5.5b). Using (5.8), one easily finds $$\begin{aligned} p_a(s) &=& \frac{\hat{\beta}^2 N_a}{6\pi} \left\{ -\kappa_a^2 s - \frac{1}{10} \kappa_a(\kappa_a + 20)s^2 + \tau_a s^3 \right\}~~~~a=0,2 \nonumber \\ q(s) &=& \frac{\hat{\beta}^2}{6\pi}\left\{ -s(s-4) + \tau_1 s^2 \right\},\end{aligned}$$ where $\tau_0,\tau_1,\tau_2$ are three yet undetermined parameters. Eqs. (5.5a) and (5.5b) now take the form $(a=0,2)$ $$\begin{aligned} \phi_a(s) &=& \frac{\hat{\beta}^2 N_a}{6\pi} \left\{ s^2 D(s) + 2s[D(s) -s] \kappa_a + [D(s) - s - \frac{1}{10} s^2]\kappa_a^2 + \tau_a s^3 \right \} \nonumber \\ \phi_1(s) &=& \frac{\hat{\beta}^2}{6\pi} \left\{ (s-4) [D(s) - s] + \tau_1 s^2 \right\} .\end{aligned}$$ Once the experimental phase shifts are known, one can compute the integrals $\phi(s)$ on left-hand side of Eq. (5.10) and fit them with the corresponding right-hand side. The parameters of the fit are $\alpha, \beta, \tau_0, \tau_1, \tau_2$. At this stage, one does not need to know the subtraction constants $t_i$ and $v_i$ in the dispersion relations (3.6). The latter are needed, however, if one wants to measure the four parameters of ${\cal L}^{(4)}$, namely $\alpha_0(0), \beta_0(0), \delta\hat{\alpha}+\alpha_4(0)$ and $\beta_2(0)$. (Remember that the parameters $\alpha_0(0)$ and $\beta_0(0)$ determine the two linear combinations (4.42) of the low-energy constants $L_1, L_2$ and $L_3$ that appear in the ${\cal L}_{40}$-part (4.31) of ${\cal L}_{eff}$.) Indeed, comparing coefficients of polynomials on both sides of Eqs. (5.9), one gets 11 linear relations among the “experimental" constants $t_0, t_2, t_3, v_1, v_2, v_3$, the four parameters of ${\cal L}_{eff}$ mentioned above, and the irrelevant five constants $x,y_0,y_1,y_2,y_3$. Eliminating the latter, one can express the four ${\cal L}_{eff}$ parameters as $$\begin{aligned} \alpha_0(0) &=& t_2 - \frac{\hat{\beta}^2}{10\pi} \left\{ 2\kappa_0(\kappa_0 + 20) + \kappa_2(\kappa_2 + 20) \right\} \nonumber \\ \beta_0(0) &=& v_2 + \frac{9\hat{\beta}^2}{\pi} (1 - 8\tau_0 + 5\tau_2) \nonumber \\ &+& \frac{3\hat{\beta}^2}{40\pi} \left\{ 8\kappa_0 (\kappa_0 + 20) - 5\kappa_2(\kappa_2 + 20)\right\},\end{aligned}$$ and $$\begin{aligned} \delta \hat{\alpha}+ \alpha_4(0) &=& t_0 - \hat{\alpha} - \frac{4\hat{\beta}^2}{3\pi} (2\kappa_0^2 + \kappa_2^2) \nonumber \\ \beta_2(0) &=& v_1 - 9\hat{\beta} + \frac{21\hat{\beta}^2}{20\pi} \left\{ 5\kappa_2^2 - 8 \kappa_0^2 \right\} \nonumber \\ &+& \frac{6\hat{\beta}^2}{\pi} \left\{ 5(\kappa_2 - 2\tau_2) -8 (\kappa_0 - 2\tau_0) \right\} .\end{aligned}$$ The remaining two equations do not involve any parameter of ${\cal L}_{eff}$ to be determined. They read $$\begin{aligned} t_3 &=& - \frac{\hat{\beta}^2}{\pi} (2\tau_0 + \tau_2) \nonumber \\ v_3 &=& \frac{9\hat{\beta}^2}{4\pi} (1 - 8\tau_0 + 5\tau_2 - \tau_1).\end{aligned}$$ The two Eqs. (5.12) should be merely expected to measure the strength of neglected two-loop and $\tilde{\cal L}^{(6)}$ contributions, rather than represent a true constraint on the fit based on Eqs. (5.10). Fits to Roy-type equations (3.7a) and (3.7b) ------------------------------------------------ In order to reconstruct the amplitude $A_{exp}(s\|tu)$, one needs a complete set of pion-pion phase shifts $\delta_a(s),~(a=0,1,2)$. (By complete we mean that they extend in energy from the threshold to $\Lambda \lesssim 1~GeV$ for all three isospins and are dense enough in the interval to allow adequate numerical evaluation of our dispersion integrals.) There exists only one complete set of pion-pion scattering phase shifts (extrapolated from experimental data[^6]) that has been published in numerical form, namely that appearing in the paper of Froggatt and Petersen [@fp]. They provide values for $\delta_a(s)$ — without quoted errors — at 20 MeV energy intervals in $4M_\pi^2 < s < \Lambda^2$, for $a = 0,1,2$. The phase shifts $\delta_a$ come from an analysis following that of Basdevant [et al.]{} [@bfp], which employs a truncated set of twice-subtracted Roy equations, makes a particular choice of parametrization for $f_a$ (fixing the $I=0$ scattering length, $a_0$) and uses a Regge type model for estimating the high energy contributions to the dispersion integrals. Data were taken from the Estabrooks-Martin analysis [@em74] of the CERN-Munich experiment on $\pi N \rightarrow \pi \pi N$ [@cern-munich]. Although Basdevant [et al.]{} [@bfp] present graphical results for several choices of values of $a_0$ in their work, numerical results are only presented in the subsequent paper of Froggatt and Petersen [@fp], and only for the unique choice $a_0 = 0.3$. We first check to what extent the Froggatt-Petersen phases satisfy the version of the Roy equations set forth in Section III B. To this end, we compute the integrals on the right hand side of Eqs. (3.7a,b), using the $\delta_a$ from Froggatt and Petersen. Calling the result $Ref_a^{RHS}(s)$, we then determine the parameters $t_i,v_i$ by minimizing $$\sum_a \sum_i [ Re f_a^{LHS}(s_i) - Re f_a^{RHS} (s_i) ]^2,$$ where $Ref_a^{LHS}$ is the real part of $f_a$ determined directly (via unitarity) from $\delta_a$. This is not a proper $\chi^2$ fit, since no uncertainties can be included; consequently, no uncertainties can be quoted for the resulting constants. We find, however, that the values for experimentally determined constants are stable for reasonable variations in the energy interval used for the fit (see Table \[tuv\]). The fit over the largest range, $4 < s < 25$, is excellent: $Ref_a^{RHS}$ and $Ref_a^{LHS}$ agree to 1% over nearly all the interval, the sum in Eq. (5.13) being $O(10^{-4})$ for 63 data points. We see no need to present the results graphically: $Ref_a^{RHS}$ and $Ref_a^{LHS}$ would be indistinguishable. Instead, the values of $Ref_a^{RHS}$ and $Ref_a^{LHS}$ are compared in Table \[fptable\], for 21 energies included in the sum (5.13). We thus conclude that the Froggatt-Petersen phases indeed give a solution of our set of triply-subtracted Roy equations, for the values of parameters $t_i$ and $v_i$ summarized in Table \[tuv\]. (Notice that the parameters $t_3$ and $v_3$ are poorly determined, but that they are sufficiently small not to affect the analysis at the $O(p^4)$ level.) The corresponding low-energy amplitude $A_{exp}(s\|tu)$ will be confronted with the theoretical prediction $A_{th}$ shortly. $K_{e4}$-decay experiments [@ke4] are consistent with the value $a_0 = 0.30$ for the scattering length, characteristic of Froggatt-Petersen phases, but standard $\chi$PT predicts a lower value, namely $a_0 = 0.20 \pm 0.01$ [@gl84]. It would be desirable to have complete sets of phase shifts that fit both experiment and Roy equations for other values of $a_0 < 0.30.$ These are not available.[^7] For this reason, we must use [ad hoc]{} extrapolations down to threshold of existing data at energies $E > (500-600)~ MeV$ obtained from $\pi N \rightarrow \pi\pi N$ and $\pi N \rightarrow \pi\pi\Delta$ production experiments. One such extrapolation has been recently considered by Schenk [@schenk] using a simple parametrization $$\begin{aligned} \tan \delta_i(s) &=& \sqrt{\frac{s-4}{s}} \left[ a_i + \tilde{b}_i ( \frac{s - 4}{4} ) + c_i ( \frac{s-4}{4})^2 \right] (\frac{4-s_i}{s-s_i} ) \nonumber \\ \tilde{b}_i &=& b_i - a_i \frac{4}{s_0 - 4} + (a_i)^3.\end{aligned}$$ for the two $s$ waves ($i=0.2$) and a similar formula for the $p$ wave. The scattering lengths $a_i$ and the slope parameters $b_i$ are fixed at their values predicted by the standard one-loop $\chi$PT [@gl85]: $$\begin{aligned} a_0 \equiv a_0^0 = 0.20, & a_2 \equiv a_0^2 = -0.042, & a_1 \equiv a_1^1 = 0.037 \nonumber \\ b_0 \equiv b_0^0 = 0.24, & b_2 \equiv b_0^2 = -0.075 & .\end{aligned}$$ The remaining parameters are determined by fitting the data obtained from various analyses of dipion production experiments [@cern-munich]. For the $I=0$ $s$ wave, Schenk uses the Ochs energy-independent analysis[^8] of the CERN-Munich experiment [@em74], covering the energy range 610–910 MeV. For his best fit – called solution B – no $\chi^2$ or error bars are quoted. Instead, two additional sets of parameters $c_0$ and $s_0 = E_0^2$ called “A” and “C” are given that bracket together both the Ochs data and the well-known data by Estabrooks and Martin [@em74]. A similar procedure is adopted for the $I=2$ $s$ wave, whereas the parameters of the $p$ wave are determined from the experimental $\rho$ mass and width. Results of this analysis and more details can be found in Ref. [@schenk]. In this way, the parametrization (5.14) provides a complete set of phases – hereafter referred to as Schenk B – that fits the data at higher energies and uses the threshold parameters (5.15) of the standard $\chi$PT. Using this set, we have performed exactly the same kind of fit to the Roy-type Eqs. (3.7a) and (3.7b) as in the case of Froggatt-Petersen phases. Surprisingly enough, we find this fit at least as good as in the case of the Froggatt-Petersen phases, despite the fact that the Schenk B phases were not obtained using Roy equations or any other crossing-symmetry correlation among the three lowest partial waves.[^9] The resulting parameters $t_i$ and $v_i$ are given in the second half of Table \[tuv\], and the quality of the fit can be appreciated from Table \[schenkb\]. Unfortunately, we do not see any simple way to associate the Schenk B phases, and the corresponding parameters $t_i$ and $v_i$, with a set of errors which would be deduced from statistical errors of the experimental data used at the beginning and which would respect the correlations imposed by the Roy equations. The same remark applies to the set of phases of Froggatt and Petersen. Determination of parameters $\alpha,\beta,L_1,L_2$ and $L_3$\ from a complete set of phase shifts --------------------------------------------------------------- The next step is to confront the empirical amplitude $A_{exp}$ with the amplitude $A_{th}$ computed from chiral perturbation theory. In particular, the two solutions of the Roy-type equations (3.7a) and (3.7b) described above can be used to measure the parameters $\alpha,\beta$ and, through Eqs. (4.42), two linear combinations of the low-energy constants $L_1^r,L_2^r$ and $L_3$, defining the four-derivative terms in ${\cal L}_{eff}$. The measurement is based on Eqs. (5.10). First, one evaluates the three functions $\phi_a(s),~a=0,1,2$, defined in Eqs. (5.4), using the complete sets of phase shifts exhibited in Section V B. The results are represented graphically by continuous lines in Figs. 3a,b,c for the case of Froggatt-Petersen phases and in Figs. 4a,b,c for the Schenk B set. Next, one fits the “experimental” functions $\phi_a(s)$ with the theoretical expression represented on the right-hand side of Eqs. (5.10). The parameters of the fit are $\alpha,\beta$ and $\tau_0,\tau_1,\tau_2$. \[Recall that the $\kappa_a$ are defined in terms of the ratio $\alpha/\beta$ – see Eq. (4.34).\] The range in $s$ in which the fit is performed should not exceed the range in which the $O(p^4)$-order $\chi$PT may actually be expected to apply. On the other hand, this range should be large enough to permit a sensitive determination of parameters. For this reason, it might be misleading to consider exclusively the physical region $s \geq 4$ [@drv; @gm]. In the following, we use the interval $-4 \leq s \leq 8$, which most likely represents a rather conservative choice. From Figs. 3a,b,c and 4a,b,c one observes a large difference in scale of individual $\phi_a$: $\phi_0$ is typically an order of magnitude or more larger than $\phi_2$ and nearly two orders of magnitude larger than $\phi_1$. For this reason, we first fit the function $\phi_0$, determining the three parameters $\alpha,\beta$ and $\tau_0$. Then, using the values of $\alpha$ and $\beta$ obtained in this way, we perform two single-parameter fits to $\phi_2$ and $\phi_1$, determining $\tau_2$ and $\tau_1$ respectively. In the absence of error bars for $\phi_a(s)$, it is impossible to perform a true $\chi^2$ fit. Instead, we minimize the sum of squares of the difference between the left- and right-hand sides of Eqs. (5.10), for 66 equidistant points in the interval $-4 \leq s \leq 8$, giving the same weight to each point. In all cases, the parameter $\beta = F_\pi^2/F_0^2$ should remain close to 1, and the fit should be constrained by this condition. We require $$\beta \leq 1.17,$$ corresponding to the lower bound $F_0 \geq 86~MeV$. This bound is consistent both with existing standard $\chi$PT estimates [@gl85] and with the improved $\chi$PT formula (4.20). Leaving the ratio $\alpha/\beta$ unconstrained in the minimization procedure, one tests – for a given set of data – the relevance of the improved $\chi$PT. The corresponding fits are represented by dashed curves in Figs. 3a,b,c and 4a,b,c. The corresponding best values of the parameters are $$\begin{aligned} \alpha/\beta &=& 4.20, ~~~~~~ \beta = 1.17 \nonumber \\ \tau_0 &=& -0.263, ~~ \tau_1 = 3.75, ~~~ \tau_2 = -0.540\end{aligned}$$ for the set of Froggatt-Petersen phases, and $$\begin{aligned} \alpha/\beta &=& 1.63, ~~~~~~ \beta = 1.17 \nonumber \\ \tau_0 &=& -0.032, ~~ \tau_1 = 3.68, ~~~ \tau_2 = -0.640\end{aligned}$$ for phases of the Schenk B set. On the other hand, in order to test the compatibility of the $O(p^4)$ [standard]{} $\chi$PT with a given set of data, one further restricts the fit by requiring $$\alpha = \beta \leq 1.17.$$ Results of the minimization with this constraint are represented by dot-dashed curves in Figs. 3 and 4. The best values of parameters corresponding to this constrained fit are $$\alpha = \beta = 1.17, ~~~\tau_0 = -0.414,~~~\tau_1 = 3.75,~~~\tau_2 =-0.661$$ and $$\alpha = \beta = 1.17, ~~~\tau_0 = -0.045,~~~\tau_1 = 3.68,~~~\tau_2 = -0.653$$ for the Froggatt-Petersen and Schenk B sets of phases respectively. A few remarks are in order. The Froggatt-Petersen data are considerably better fit in terms of a larger value (5.17a) of the ratio $\alpha/\beta$ than the standard $\chi$PT would permit, although without a true $\chi^2$ fit we cannot be quantitative about this observation. The failure of standard $\chi$PT to describe the Froggatt-Petersen $s$ wave is also apparent in Fig. 3a (dot-dashed curve). Concerning the $p$ wave, the fit is reasonably good for both cases (Fig. 3b), reflecting the fact that the theoretical calculation of $\phi_1(s)$ senses the effective infrared dimension of the quark mass starting only at the two-loop level. It is worth noting that the best value $\alpha/\beta$ = 4.20 is overcritical by 5%. This means that the Froggatt-Petersen I=0 $s$ wave would be compatible with the vanishing of the $\bar{q}q$ condensate $B_0$.[^10] From this point of view, the set of Froggatt-Petersen phases with $a_0^0 = 0.30$ appears as an extreme alternative. The opposite extreme is represented by the Schenk B set of phases. Since the latter incorporates [a priori]{} the values of scattering lengths and effective ranges as predicted by the standard $\chi$PT, it is not surprising that the corresponding best value for $\alpha/\beta$ (5.17b) is considerably closer to 1 than in the Froggatt-Petersen case. Furthermore, Fig. 4a seems to indicate that, although the best value for $\alpha/\beta$ is still as large as 1.63, this fact need not be significant. In the absence of error analysis, it is hard to be too affirmative in the interpretation of the Schenk B fit. It remains to exploit the additional information (values of constants $t$ and $v$ as well as the constants $\tau$ resulting from our fits), in order to measure certain parameters of the dimension-4 component of ${\cal L}_{eff}$. Here, we merely concentrate on the constants $L_1,L_2$ and $L_3$ characteristic of ${\cal L}_{40}$, Eq. (4.31), whose meaning and renormalization do not depend on the effective dimension of the quark mass. For this purpose, we have to determine the constants $\alpha_0(0)$ and $\beta_0(0)$ given by Eqs. (5.11a). Using the central values of the parameters $t_2$ and $v_2$ (the second column of Table \[tuv\]) and the best values for $\alpha/\beta$ and the $\tau$’s, as determined in the previous fits, one gets $\alpha_0(0) = 5.81 \times 10^{-4}, \beta_0(0) = 3.99 \times 10^{-3}$ for the Froggatt-Petersen solution, and $\alpha_0(0) = 5.87 \times 10^{-4}, \beta_0(0) = 2.07 \times 10^{-3}$ for the case of Schenk B phases. These numbers are easily converted into information on the constants $L_{1,2,3}$, using Eqs. (4.42) and (4.43). Assuming the Zweig-rule (or large-$N_c$) relation $L_2^r - 2L_1^r = 0$, and identifying the running scale $\bar{\mu}$ with the $\eta$ mass, as done in Refs. [@gl85; @rigg], one obtains $$L_2^r = 2L_1^r = 1.34 \times 10^{-3},~~~L_3 = -4.50 \times 10^{-3}$$ for the Froggatt-Petersen data, and $$L_2^r = 2L_1^r = 0.56 \times 10^{-3},~~~L_3 = -2.15 \times 10^{-3}$$ for the set of Schenk B phases. It is gratifying to see that these values – especially (5.20a) – compare well with other determinations based on [standard]{} $\chi$PT [@gl85; @rigg]. Indeed, there is no reason why the purely derivative terms in ${\cal L}_{eff}$ should be affected by questions concerning the symmetry breaking sector. Estimates of errors in the direct measurements of $\alpha/\beta$ -------------------------------------------------------------------- The uncertainties in the values of the parameters $\alpha,\beta$ and $\tau_a$ arise from uncertainties in the functions $\phi_a$; these uncertainties, in turn, arise from uncertainties in the phase shifts $\delta_a$ over the range of integration in Eqs. (5.4). As we have noted, there is no set of phase shifts $\delta_a$ which exists, together with corresponding errors, in this energy range. In the present subsection, we extend the extrapolation method of Schenk [@schenk], described above, to construct several sets of I=0 phase shifts $\delta_0$, together with estimated errors, in the necessary energy interval. In this way, we obtain values and estimated errors for the parameters $\alpha/\beta$ and $\tau_0$ for each extrapolated data set. Only $I=0$ phases are considered. In fact, we could treat $I=1$ phases similarly (although the insensitivity of $\phi_1$ to $\alpha$ makes this relatively uninteresting); in any case, the paucity of experimental data on $I=2$ makes the production of a complete set of phase shifts impossible without a more extensive recourse to the use of the Roy equations, as in the analysis of Basdevant [et al.]{} [@bfp]. The two original sets of phase shifts (with corresponding errors) used are that of Ochs and that of Estabrooks and Martin. These were each obtained independently from analysis of the same CERN-Munich experiment. The first step is to extrapolate $\delta_0$ down to threshold, using the Schenk formula (5.14). The Ochs phases are fit over the energy range 610-910 MeV, [i.e.]{}, using all his data for which no inelasticity is suggested. (See Table \[ochsdata\].) The Estabrooks-Martin phases are fit over the energy range 570-910 MeV, [i.e.]{}, using all their points in the elastic scattering region except for their first three lowest-energy points, which appear to be less trustworthy. In performing the extrapolation, the scattering length $a_0$ is fixed and the remaining parameters $b_0,c_0,E_0$ are determined by minimization of $\chi^2$ using the phases and errors given to us. We show in Fig. \[fits\] the results of this fitting procedure for the choices $a_0$ = 0.20 (preferred by standard $\chi$PT) and 0.26 (preferred by $K_{e4}$-decay experiment) for each of the two data sets; the resulting parameters are given in columns 2–5 of Table \[phi0fits\]. The $\chi^2$ for these fits is quite good. (We note in passing that the data of Estabrooks and Martin is not well described by the parameters $a_0=0.20,b_0=0.24$ which characterize the Schenk B solution.) The next step is to estimate the uncertainty in the extrapolated phases $\delta_0$. Since the dominant parameter (after $a_0$, which is fixed) is $b_0$, and in view of the strong correlations among the parameters, we proceed as follows: for fixed values of $b_0$ larger than its value for $\chi^2_{min}$, the minimum-$\chi^2$ value, fit the data by allowing $c_0$ and $E_0$ to vary freely, and find the values of $b_0,c_0,E_0$ which give $\chi^2 = \chi^2_{min} +1$; call this solution “a”, in analogy with Schenk’s notation; repeat this procedure for fixed values of $b_0$ smaller than that for $\chi^2_{min}$; call this solution “c”; the uncertainty in the phase shift $\delta_0(E_i)$, for each value of $E=E_i$, is then estimated by interpreting the variation of $\delta_0(E_i)$ from its solution “a” value to its solution “c” value as $\pm 1$ standard deviation in $\delta_0 (E_i)$. (This is similar to the procedure adopted by Schenk, although he allows much greater variation – leading to much larger uncertainties – in order to bracket both Ochs and Estabrooks-Martin phases at the same time.) Now, for each of the four sets of phase shifts $\delta_0$, obtained by the extrapolation procedure described above, we may make the comparison for $\phi_0$ as done for the Froggatt-Petersen and Schenk B phases in Section IV C. However, we now have the important advantage that a true $\chi^2$ fit is possible, so we can have some idea of the precision with which the resulting parameters are determined. For each set, we make two fits: one, corresponding to standard $\chi$PT, for which we fix $\alpha = \beta \leq 1.17$; the other, corresponding to improved $\chi$PT, for which $\beta \leq 1.17$ but $\alpha$ is allowed to vary freely. The fits are all performed over the same interval $-7 \leq s \leq 9$. Results of the determination of the parameters $\alpha/\beta$ and $\tau_0$ are given in columns 6–9 of Table \[phi0fits\]; the reader can judge the quality of the fits from the plots of $\phi_0$ given in Figs.\[phi0figs\]a,b,c,d. The solid curves represent the parametrization of improved $\chi$PT, while the dashed curves represent that of standard $\chi$PT. It is clear, both from the large $\chi^2$ values tabulated for the standard $\chi$PT fits and from examination of the dashed curves in Figs.\[phi0figs\] that standard $\chi$PT is not compatible with these phase shifts. For this reason, we quote no result for $\tau_0$ for this case. On the other hand, improved $\chi$PT can easily accommodate such data. It is important to note that, for a given set of phase shifts, the parameters $\alpha/\beta$ and $\tau_0$ are very well determined by the improved $\chi$PT fit. As a check on our procedure of estimating errors, we have also used a more “conservative” procedure, [viz.]{}, vary all non-fixed parameters within their one-standard-deviation limits to produce solution “$a_{cons}$”, taking max($b_0$),max($c_0$),min($s_0$), and solution “$c_{cons}$”, taking min($b_0$),min($c_0$),max($s_0$); then compute the “conservative” uncertainties in the phase shifts $\delta_0(E_i)$ using these solutions as we did for solutions “a” and “c” before. Clearly, this method does not take into account the strong correlations in $b_0,c_0,s_0$. Thus, when the consequent phase shift errors are used in the fitting of $\phi_0$, these larger errors result in larger errors in the experimental function $\phi_0$. The result is then a $\chi^2$ roughly half of that previously obtained, and errors in $\alpha/\beta$ and $\tau_0$ roughly 2–5 times larger. Nevertheless, the best fit is the same. Summary and Conclusions =========================== A new framework for testing the convergence rate of chiral perturbation theory is proposed. One first replaces the standard expansion of the effective Lagrangian by a more general expansion that is as systematic and unambiguous as the standard $\chi$PT. In addition to the usual terms, the new expansion involves at each given order new contributions that the standard $\chi$PT relegates to higher orders. The size of these additional contributions can then be tested experimentally, in particular in low-energy $\pi-\pi$ scattering. Unless these contributions turn out to be small, the improved $\chi$PT has, in principle, more chance to produce a rapidly convergent expansion scheme. A new low-energy theorem is presented which provides the general solution of constraints imposed by analyticity, crossing symmetry and unitarity on the $\pi-\pi$ scattering amplitude, neglecting $O(p^8)$ contributions. Applications of this theorem are threefold: i\) First, it considerably simplifies the evaluation of the perturbative $\pi-\pi$ amplitude up to and including two loops. This applies both within the “standard $\chi$PT” and within the more general “improved $\chi$PT” which contains the former as a special case. In both cases, the calculation reduces to the iterative insertion of the unitarity condition (4.32) into the dispersive integral for the functions $T,U$ and $V$ in Eq.(3.2). The improved $\chi$PT one-loop amplitude is worked out in detail in Sec. IV. The two-loop amplitude can be easily calculated along the same lines. The reason why the formula (3.2) no longer holds beyond two loops resides in new $O(p^8)$ effects in the absorptive part: inelasticities and higher partial waves. ii\) Next, the low-energy theorem of Sec. III can be used to constrain the low-energy scattering data and to fully reconstruct the corresponding amplitude. The formula (3.2) implies a particular truncation of the infinite system of Roy equations, under a rigorous control of chiral power counting: Neglected contributions are $O(p^8)$, whereas in the original form of the Roy equations [@roy] the model-dependent “driving terms” are of the same order $O(p^4)$ as the effects we are looking for. A complete set of low-energy phases $\delta_0^0,\delta_0^2$ and $\delta_1^1$, together with the six subtraction constants $t$ and $v$ for which the Roy-type Eqs. (2.7a) and (2.7b) are satisfied to a reasonable accuracy (see Tables \[fptable\] and \[schenkb\]),define up to $O(p^8)$corrections the scattering amplitude $A(s,t,u)$ in a whole low-energy region of the Mandelstam plane including the unphysical region. Two examples of such a complete low-energy amplitude are given, based on phase shifts published by Froggatt and Petersen [@fp] and by Schenk [@schenk] respectively. They are both compatible with existing $\pi N \rightarrow \pi\pi N$ and $K_{e4}$ experimental data. iii\) Finally, the low-energy representation (3.2) simplifies the direct comparison of the perturbative amplitude $A^{th}(s,t,u)$ with the amplitude $A^{exp}(s,t,u)$ reconstructed from the data. In particular, parameters of ${\cal L}_{eff}$ contained in $A^{th}$ can be measured through a detailed fit of the amplitude $A^{exp}(s,t,u)$ over a sufficiently large portion of the Mandelstam plane in which the low-energy expansion can still be taken as valid. The fit is particularly sensitive to the ratio $\alpha/\beta$ which parametrizes the leading $O(p^2)$ amplitude. The improved $\chi$PT requires $1 \leq \alpha/\beta \leq 4$, whereas the special case of the standard $\chi$PT corresponds to $\alpha/\beta = 1$. The ratio $\alpha/\beta$ is related to the value of the QCD parameter $2\hat{m}B_0$ in the units of pion mass squared and, [via]{} the pseudoscalar mass spectrum, to the quark mass ratio $r = m_s/\hat{m}$. Examples of measurement of $\alpha/\beta$ exhibited in this paper illustrate the lack of sufficiently precise experimental information on low-energy $\pi-\pi$ scattering. For a fixed value of the scattering length $a_0^0$, the statistical errors of the production data on $\delta_0^0$ are estimated to show up as errors in the measured values of $\alpha/\beta$ of the order of a few percent. On the other hand, for different values of $a_0^0$ in the experimental range $a_0^0 = 0.26 \pm 0.05$, and for different sets of production data, the resulting values of $\alpha/\beta$ vary between 1.5 and 4.2. The two complete low-energy amplitudes mentioned above correspond to these two extremes. In particular, the Froggatt-Petersen phases (for which $a_0^0$ = 0.30) are compatible with the vanishing of the condensate $B_0$ and with the critical value of the quark mass ratio $r = r_1 \simeq 6.3$. The suspicion that a bad convergence of the standard $\chi$PT might bias the usual conclusions that $r = m_s/\hat{m} \simeq 25.9$ and $2\hat{m}B_0 \simeq M_\pi^2$ is at least well motivated, but clearly it requires confirmation. In order to produce a truly unbiased measurement of these fundamental QCD parameters, the method developed in this paper can prove useful provided that it is supplied with more accurate experimental information on low-energy $\pi-\pi$ phase shifts. The current imprecision, illustrated by error bars as large as in $a_0^0 = 0.26 \pm 0.05$ can hide [all]{} cases of interest, including the intriguing critical case $\langle \bar{q}q \rangle = 0$. Here, one faces a challenge of fundamental high precision low-energy experimental physics. We are indebted to Dr. M. Knecht for various discussions and for pointing out some errors in a preliminary version. Discussions on $\pi-\pi$ data with Dr. J.-L. Basdevant and with Dr. J. Gasser at various stages of this work have been extremely useful. One of us (N.H.F.) would like to thank the Division de Physique Théorique, Orsay, for its generous hospitality and support during his sabbatical leave of absence, and the Department of Physics of Arizona State University for its hospitality as well. Notation and conventions ======================== In the first part of this appendix, we fix the notation and normalization for the scattering amplitude. We then exhibit the main properties of the crossing matrices $C$ \[Eq. (3.9)\]. The $S$-matrix element for the transition $\pi^a + \pi^b \rightarrow \pi^c + \pi^d$, where $a,b,c,d$ are pion isospin indices, is connected to the $T$-matrix element by the relation: $$\langle cd \| S \| ab \rangle = \langle cd \| ab \rangle + i(2\pi)^4\,\delta^4(p_a +p_b - p_c -p_d)\,T_{ab,cd}$$ The $T$-matrix element can be written in terms of isospin invariant amplitudes; taking crossing symmetry into account, the decomposition reads: $$\begin{aligned} T_{ab,cd}(s,t,u) &=& A(s\|tu) \delta_{ab}\delta_{cd} + A(t\|su) \delta_{ac} \delta_{bd} + A(u\|ts) \delta_{ad}\delta_{bc},\end{aligned}$$ where $s,t$ and $u$ are the Mandelstam variables, $$s = (p_a + p_b)^2,~~~t = (p_a - p_c)^2,~~~u = (p_a - p_d)^2.$$ The amplitude $A(s\|tu)$ is symmetric in the variables $t,u.$ (The amplitude $A(t\|su)$ is obtained from $A(s\|tu)$ by the exchange of variables $s,t$ and by subsequent analytic continuation.) The $s$-channel isospin amplitudes $F^{(I)}$ \[Eq. (3.8)\] are related to the amplitude $A$ by $$\left( \begin{array}{c} F^{(0)}\\F^{(1)}\\F^{(2)} \end{array} \right) (s,t,u) = \frac{1}{32\pi}\left( \begin{array}{rrr} 3 &~~ 1 &~~ 1 \\ 0 &~~ 1 &~~ -1 \\ 0 &~~ 1 &~~ 1 \end{array} \right) \left( \begin{array}{c} A(s\|tu)\\A(t\|su)\\A(u\|ts) \end{array} \right).$$ The partial wave expansion is $$F^{(I)} = \sum_\ell (2\ell +1) P_\ell (\cos \theta) f_\ell^I(s),$$ where $s = 4(M_\pi^2 + q^2), t = -2q^2(1-\cos\theta)$. With this normalization, the elastic unitarity condition for $f_\ell^I$ takes the form: $$\begin{aligned} I\!m\, f_\ell^I(s) &=& \sqrt{\frac{s-4M_\pi^2}{s}} \|f_\ell^I(s)\|^2,\\ f_\ell^I(s) &=& \sqrt{\frac{s}{s-4M_\pi^2}} e^{i\delta_\ell^I(s)} \sin \delta_\ell^I(s).\end{aligned}$$ The crossing matrices $C$ \[Eq. (3.9)\] have the following forms: $$C_{st} = \left( \begin{array}{rrr} 1/3 &~~1&~~5/3 \\ 1/3 &~~1/2&~~-5/6 \\ 1/3 &~~-1/2&~~1/6 \end{array} \right), C_{tu} = \left( \begin{array}{rrr} 1 &~~0&~~0 \\ 0 &~~-1&~~0 \\ 0 &~~0 &~~1 \end{array} \right), C_{su} = \left( \begin{array}{rrr} 1/3 &~~-1&~~5/3 \\ -1/3 &~~1/2&~~5/6 \\ 1/3 &~~1/2&~~1/6 \end{array} \right),$$ and satisfy the relations $$C_{tu}^2 =C_{su}^2 =C_{st}^2 =1,$$ $$\begin{aligned} C_{st}C_{su} & = C_{tu}C_{st} & = C_{su}C_{tu},\nonumber \\ C_{su}C_{st} & = C_{tu}C_{su} & = C_{st}C_{tu} .\end{aligned}$$ It is worthwhile to notice that Eqs. (A9) imply that the eigenvectors $A_{\pm}$ with eigenvalues $\pm 1$, respectively, of the matrix $C_{tu}$ satisfy $$C_{su}C_{st}A_{\pm} = \pm C_{st}A_{\pm}.$$ These relations are extensively used throughout the calculation of Sec. III C. The invariance of the amplitude $A(s\|tu)$ under the exchange of the variables $t,u$ permits us to construct rather easily all independent crossing symmetric polynomials of a given degree in $s,t,u$. It is convenient to take the variables $t,u$ as independent. Since there are $$k_n \equiv [n/2] + 1$$ symmetric monomials in $t,u$ that are homogeneous of degree $n$, the number of independent parameters in a general crossing symmetric polynomial of degree $N$ is $$K_N \equiv \sum_{n=0}^N k_n.$$ Hence, the most general polynomial of degree three contains six parameters, as claimed in Sec. III C. Ambiguities of the amplitudes $T,U$, and $V$ ============================================ In this appendix, we determine the general expression for the transformations $T \rightarrow T + \delta T,~U \rightarrow U + \delta U,$ and $V \rightarrow V + \delta V$ that leave invariant the scattering amplitude $A$ \[Eq. (3.2)\]. It follows that the variations $\delta T,~\delta U$ and $\delta V$ must satisfy the equation $$\delta T(s) + \delta T(t) + \delta T(u) + \frac{1}{3}[2\delta U(s) - \delta U(t) - \delta U(u)] + \frac{1}{3}[(s-t)\delta V(u) + (s-u)\delta V(t)] = 0.$$ In order to solve Eq. (B1), one notices that only two out of the three variables $s,t,u$ are independent. By successive differentiation with respect to independent variables, one obtains a set of simpler equations which can be solved easily. To simplify notation, let us define $$f \equiv \delta T,~~g \equiv \delta U,~~h \equiv \delta V.$$ We first consider $s$ and $t$ as independent variables, and differentiate Eq. (B1) first with respect to $s$ and then with respect to $t$. We thus obtain the following two equations, where the primes indicate differentiation with respect to the arguments of the functions: $$f'(s) - f'(u) + \frac{1}{3}(2g'(s) + g'(u)) + \frac{1}{3}(h(u) + 2h(t)) - \frac{1}{3}(s-t)h'(u) = 0,$$ $$f''(u) - \frac{1}{3}g''(u) + \frac{2}{3}h'(t) + \frac{1}{3}(s-t)h''(u) = 0.$$ We now consider $t$ and $u$ as independent variables and differentiate Eq. (B4) with respect to $t$, obtaining the result $$h''(t) - h''(u) = 0,$$ which indicates that $h''$ is a constant, and therefore $h$ is a quadratic polynomial, $$h(t) = \frac{1}{2}at^2 + bt + c,$$ where $a,b,c$ are constants. Using this result for $h$ in Eq. (B4), we find a relation between $f$ and $g$: $$f(u) = \frac{1}{3}g(u) + \frac{1}{18}au^3 -\frac{1}{3}(b+2M_\pi^2a)u^2 +du +e,$$ where $d$ and $e$ are constants. We then return to Eq. (B3), consider $s$ and $u$ as independent variables, and differentiate with respect to $s$. This implies $$f''(s) + \frac{2}{3}g''(s) -\frac{4}{3}b -\frac{2}{3}a(4M_\pi^2 -s) = 0,$$ which becomes, after replacing $f$ in terms of $g$ \[Eq. (B7)\], $$g''(s) + as -2(b+2M_\pi^2 a) = 0,$$ the solution of which is $$g(s) = -\frac{a}{6}s^3 +(b+2M_\pi^2 a)s^2 +ks +\ell,$$ where $k$ and $\ell$ are constants. The expression for $f$ then becomes $$f(s) = (d + \frac{k}{3})s + (e+\frac{\ell}{3}).$$ Finally, upon substituting the expressions for $h$ \[Eq. (B6)\], $g$ \[Eq. (B10)\] and $f$ \[Eq. (B11)\] in the original equation (B1), we obtain two constraints on the constants $$\begin{aligned} (e + \frac{\ell}{3}) &=& -\frac{4M_\pi^2}{3} (d + \frac{k}{3}),\\ c &=& -(k + 4M_\pi^2 b + \frac{16}{3}M_\pi^4 a).\end{aligned}$$ After relabeling the constants as follows, $$y_0 = \ell,~ y_1 = k,~ y_2 = (b+2M_\pi^2 a),~ y_3 = -a/6,~ x = d + k/3,$$ the functions $f,g$ and $h$, and hence $\delta T,~\delta U$ and $\delta V$ \[Eqs. (B2)\], take the forms given in Eqs. (3.5). Polynomials and kernels of the Roy type equations ================================================= In this Appendix we explicitly list the polynomials $P_a(s)$ and the kernels $W_{ab}$ which appear in the Roy-type dispersion relations (3.7). $$\begin{aligned} P_0(s) &=& 5t_0 + \frac{5}{9}t_2 [3s^2 + 2(s-4M_\pi^2)^2] +\frac{5}{6}t_3 [2s^3 - (s-4M_\pi^2)^3] \nonumber \\ & +& \frac{2}{9}v_1 (3s-4M_\pi^2) -\frac{8}{27}v_2(s-M_\pi^2)(s-4M_\pi^2) \nonumber \\ & +& \frac{1}{27}v_3(5s-4M_\pi^2)(s-4M_\pi^2)^2 \\ P_2(s) &=& 2t_0 + \frac{2}{9}t_2 [3s^2 + 2(s-4M_\pi^2)^2] + \frac{1}{3}t_3 [2s^3 - (s-4M_\pi^2)^3] \nonumber \\ & -& \frac{1}{9}v_1 (3s-4M_\pi^2) +\frac{4}{27}v_2(s-M_\pi^2)(s-4M_\pi^2) \nonumber \\ & -& \frac{1}{54}v_3(5s-4M_\pi^2)(s-4M_\pi^2)^2 \\ P_1(s) &=&\frac{1}{9}(s-4M_\pi^2)(v_1 + v_2 s) \nonumber \\ &+& \frac{2}{27}(s-4M_\pi^2)v_3 [s^2 - \frac{1}{20}(s-4M_\pi^2)(11s - 4M_\pi^2)]\end{aligned}$$ $$\begin{aligned} \frac{1}{\pi}\int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x} \sum_{b=0}^2 W_{0b}(s,x) I\!m\, f_b(x) &=&\frac{4}{\pi}(s-4M_\pi^2)^2 (s-2M_\pi^2) \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^3} \frac{I\!m\, f_1(x)}{(x-4M_\pi^2)} \nonumber \\ & - & \frac{1}{6\pi}(s-4M_\pi^2)^3 \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^4} \{ I\!m\, f_0(x) + 5 I\!m\, f_2(x) \nonumber \\ & + & 9\left( 1 + \frac{2s}{x-4M_\pi^2}\right) I\!m\, f_1(x) \} G\left(\frac{s-4M_\pi^2}{x}\right)\end{aligned}$$ $$\begin{aligned} \frac{1}{\pi}\int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x} \sum_{b=0}^2 W_{2b}(s,x) I\!m\, f_b(x) &=& -\frac{2}{\pi}(s-4M_\pi^2)^2 (s-2M_\pi^2) \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^3} \frac{I\!m\, f_1(x)}{(x-4M_\pi^2)} \nonumber \\ & -& \frac{1}{12\pi}(s-4M_\pi^2)^3 \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^4} \{ 2I\!m\, f_0(x) + I\!m\, f_2(x) \nonumber \\ & -& 9\left( 1 + \frac{2s}{x-4M_\pi^2}\right) I\!m\, f_1(x) \} G\left(\frac{s-4M_\pi^2}{x}\right)\end{aligned}$$ $$\begin{aligned} \frac{1}{\pi}\int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x} \sum_{b=0}^2 W_{1b}(s,x) I\!m\, f_b(x) &=& -\frac{1}{\pi}(s-4M_\pi^2)^2 (s-2M_\pi^2) \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^3}\frac{I\!m\, f_1(x)}{(x-4M_\pi^2)} \nonumber \\ &\hspace{-2cm} + &\hspace{-1cm}\frac{1}{12\pi}(s-4M_\pi^2)^2 \int_{4M_\pi^2}^{\Lambda^2} \frac{dx}{x^3} [2 - \left(2 + \frac{s-4M_\pi^2}{x}\right)G\left(\frac{s-4M_\pi^2}{x}\right) ] \nonumber \\ &&\hspace{-2cm}\times \{ 2I\!m\, f_0(x) - 5 I\!m\, f_2(x) + 9\left( 1 + \frac{2s}{x-4M_\pi^2}\right) I\!m\, f_1(x) \}\end{aligned}$$ In Eqs. 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[D43]{}, 127 (1991). [clll]{} &\ & & &\ \ $t_0$ &0.0206 & 0.0207 & 0.0208\ $t_2$ &6.4 $\times 10^{-4}$ & 6.5 $\times 10^{-4}$ & 6.7 $\times 10^{-4}$\ $t_3$ &5.2 $\times 10^{-6}$ & 3.5 $\times 10^{-6}$ & 1.6 $\times 10^{-6}$\ $v_1$ &0.0764 & 0.0760 & 0.0755\ $v_2$ &0.0021 & 0.0020 & 0.0020\ $v_3$ &-1.3$\times 10^{-5}$ & -4.4$\times 10^{-6}$ &+3.5$\times 10^{-6}$\ \ $t_0$ &0.0067 & 0.0065 & 0.0063\ $t_2$ &4.6 $\times 10^{-4}$ & 4.8 $\times 10^{-4}$ & 5.1 $\times 10^{-4}$\ $t_3$ &1.4 $\times 10^{-5}$ & 9.9 $\times 10^{-6}$ & 6.9 $\times 10^{-6}$\ $v_1$ &0.0697 & 0.0695 & 0.0693\ $v_2$ &0.0021 & 0.0020 & 0.0020\ $v_3$ &-8.4$\times 10^{-7}$ & 2.3$\times 10^{-6}$ & 6.5$\times 10^{-6}$\ ------ ------- ------- ------- ------- -------- -------- 300. 0.342 0.344 0.005 0.006 -0.029 -0.029 320. 0.377 0.376 0.014 0.012 -0.043 -0.041 340. 0.414 0.411 0.022 0.019 -0.055 -0.052 360. 0.447 0.445 0.028 0.027 -0.067 -0.064 380. 0.479 0.477 0.039 0.037 -0.080 -0.076 400. 0.509 0.507 0.049 0.047 -0.090 -0.087 420. 0.534 0.534 0.058 0.058 -0.100 -0.099 440. 0.557 0.556 0.072 0.071 -0.113 -0.110 460. 0.573 0.572 0.088 0.086 -0.122 -0.121 480. 0.584 0.585 0.105 0.103 -0.132 -0.132 500. 0.590 0.591 0.123 0.122 -0.142 -0.143 520. 0.590 0.591 0.146 0.144 -0.152 -0.153 540. 0.585 0.586 0.171 0.170 -0.161 -0.163 560. 0.574 0.576 0.199 0.199 -0.171 -0.172 580. 0.558 0.558 0.234 0.233 -0.178 -0.181 600. 0.537 0.538 0.272 0.273 -0.186 -0.189 620. 0.512 0.513 0.318 0.319 -0.194 -0.196 640. 0.483 0.484 0.371 0.371 -0.201 -0.203 660. 0.450 0.450 0.428 0.429 -0.209 -0.209 680. 0.414 0.413 0.486 0.490 -0.214 -0.214 700. 0.375 0.373 0.532 0.531 -0.222 -0.217 ------ ------- ------- ------- ------- -------- -------- : Comparison of the left and right hand sides of Eqs.(3.7a,b), using phase shifts of Froggatt and Petersen[]{data-label="fptable"} ----- ---------------- 610 56.3 $\pm$ 3.2 630 59.5 $\pm$ 2.9 650 65.6 $\pm$ 3.2 670 62.5 $\pm$ 3.5 690 68.8 $\pm$ 3.6 710 74.5 $\pm$ 3.8 730 79.4 $\pm$ 3.6 750 81.2 $\pm$ 5.7 770 79.9 $\pm$ 3.9 790 77.5 $\pm$ 5.7 810 84.1 $\pm$ 3.3 830 84.4 $\pm$ 2.6 850 87.1 $\pm$ 2.5 870 89.2 $\pm$ 2.5 890 93.2 $\pm$ 2.9 910 103.3$\pm$ 3.2 ----- ---------------- : Data from energy-independent analysis of Ochs[]{data-label="ochsdata"} ------ ------- ------- ------- ------- -------- -------- 300. 0.236 0.234 0.006 0.005 -0.053 -0.052 320. 0.274 0.272 0.012 0.011 -0.064 -0.063 340. 0.314 0.312 0.019 0.018 -0.075 -0.074 360. 0.356 0.355 0.027 0.026 -0.087 -0.085 380. 0.398 0.397 0.035 0.034 -0.098 -0.097 400. 0.439 0.440 0.045 0.044 -0.109 -0.108 420. 0.479 0.479 0.056 0.055 -0.121 -0.119 440. 0.515 0.515 0.068 0.067 -0.132 -0.130 460. 0.545 0.546 0.083 0.081 -0.143 -0.142 480. 0.569 0.569 0.099 0.098 -0.153 -0.153 500. 0.584 0.584 0.117 0.116 -0.164 -0.164 520. 0.589 0.590 0.138 0.138 -0.174 -0.174 540. 0.585 0.585 0.163 0.163 -0.184 -0.185 560. 0.571 0.571 0.192 0.192 -0.194 -0.195 580. 0.548 0.547 0.226 0.227 -0.203 -0.204 600. 0.517 0.517 0.267 0.268 -0.212 -0.213 620. 0.480 0.479 0.315 0.316 -0.221 -0.222 640. 0.439 0.439 0.370 0.372 -0.229 -0.230 660. 0.395 0.394 0.433 0.437 -0.237 -0.237 680. 0.350 0.350 0.496 0.495 -0.244 -0.244 700. 0.304 0.305 0.540 0.538 -0.252 -0.249 ------ ------- ------- ------- ------- -------- -------- : Comparison of the left and right hand sides of Eqs.(3.7a,b), using phase shifts of Schenk, solution B[]{data-label="schenkb"} [ccccccccc]{} & & & & & & & &\ & & & &\ \ & & & & & &\ & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & & & &\ & & & & & &\ \ & & & & & &\ & & & & & & & &\ & & & & & &\ & & & & & &\ & & & & & & & &\ & & & & & &\ [^1]: Unité de Recherche des Universités Paris 11 et Paris 6 associée au CNRS [^2]: Notation and normalization are reviewed in Appendix A. [^3]: The order of magnitude estimate $A_0 \sim 1 - 5$ is compatible with the standard $\chi$PT estimates. Taking $A_0 \sim 5$, and using the standard value $B_0 \sim 1.2~GeV$, the $A_0$-contribution to $L_8$ in Eq. (4.6) becomes $1.6 \times 10^{-3}$, which is consistent with the standard $\chi$PT measurement of $L_8$ [@gl85].\[A\_0\] [^4]: In practice, $M_\pi$ = 139.6 MeV and $F_\pi$ = 93.1 MeV will be identified with the corresponding theoretical expressions up to and including the highest order of $\chi$PT considered. [^5]: Notice that $O(p^N)$ terms in $V$ contribute to the scattering amplitude $A$ of Eq. (3.2) as $O(p^{N+2})$. [^6]: For a recent review of experimental $\pi - \pi$ scattering data, see [@ochs-newsletter; @gm]. [^7]: J.L. Basdevant, private communication. [^8]: The data by Ochs can be found in his unpublished thesis [@ochs]. We are indebted to Dr. J. Gasser for communicating these unpublished data to us. For the reader’s convenience, they are reproduced in our Table \[ochsdata\]. [^9]: A. Schenk, private communication. [^10]: This critical case has been considered earlier [@mass-dimension].
--- abstract: '[Pointing is the task of tracking a target with a pointer and confirming the target selection through a click action when the pointer is positioned within the target.]{} Little is known about the mechanism by which users plan and execute the click action in the middle of the target tracking process. The Intermittent Click Planning model proposed in this study describes the process by which users plan and execute optimal click actions, from which the model predicts the pointing error rates. In two studies in which users pointed to a stationary target and a moving target, the model proved to accurately predict the pointing error rates [($R^{2}=0.992$ and $0.985$, respectively).]{} The model has also successfully identified differences in cognitive characteristics among first-person shooter game players.' author: - '\' bibliography: - 'sample.bib' --- <ccs2012> <concept> <concept\_id>10003120.10003121.10003122.10003332</concept\_id> <concept\_desc>Human-centered computing User models</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012> Conclusion ========== The model proposed in this study accurately predicted users’ pointing error rate with a simple algorithm regardless of the target motion ($R^{2}$= 0.985 to 0.992 and MAE=2.56% to 3.04%). In particular, the four free parameters obtained from the data fitting remained similar for different pointing situations (see Table \[table\]). Based on this robust explanatory power, the model revealed significant cognitive differences between gamers and non-gamers. Nonetheless, this study has some limitations. First, it is difficult to apply our model in a situation where the trajectory of the cursor is difficult to track. Secondly, further validation is needed as to whether this model is generally applicable for more complex patterns of target motion. Third, our model did not explain how a user’s internal clock encodes the temporal structure cue when the input is randomly repeated. [Fourth, the ICP model was currently only tested for young users in their 20s. For users from other age groups, further research will be needed to determine whether the ICP model predicts error rates well and extracts meaningful parameters. Fifth, the ICP model only predicts the error rate of the click action and does not predict how the user’s tracking movement will be performed before the click action. We envision that this limitation can be overcome by integrating the ICP model with existing control theoretical [@muller2017control; @bye2008bump; @bye2010bump] pointing models that can simulate the user’s target tracking movements.]{} Acknowledgements ================ This research was funded by the National Research Foundation of Korea (2017R1C1B2002101) and the Korea Creative Content Agency (R2019020010).
--- abstract: 'Hierarchical clustering models of cold dark matter (CDM) predict that about 5% - 10% of a galaxy-sized halo with mass $ \sim 10^{12}$ solar masses ($M_{\odot }$) resides in substructures (CDM subhalos) with masses $\la 10^{8} M_{\odot }$. To directly identify such substructures, we propose to observe radio continuum emission from multiply imaged QSOs using VSOP-2 with a high angular resolution.' author: - 'Shigenori Ohashi, Masashi Chiba, and Kaiki Taro Inoue' title: 'LENS MAPPING OF DARK MATTER SUBSTRUCTURE WITH VSOP-2' --- Introduction ============ The currently standard framework for understanding cosmological structure is based on hierarchical clustering of CDM in the accelerating Universe. Low mass systems collapse and form earlier and later they merge to form more massive gravitationally bound systems. The standard model is successful for explaining a wide variety of astrophysical observations such as the temperature fluctuations in the Cosmic Microwave Background and large-scale distribution of galaxies. Despite the great success of the concordance model, there are several open issues. One of them is so-called “the missing satellite problem”. Some authors have argued that this discrepancy could be resolved by considering some suppressing process for star formation, such as gas heating by an intergalactic UV background. In whatever models relying on the suppression of galaxy formation, a typical galaxy-sized halo should contain numerous dark subhalos. It is thus important to constrain the presence or absence of many invisible satellites around a galaxy like the Milky Way in the form of dark subhalos. To obtain the direct evidence for such dark subhalos, we propose to observe the multiply imaged QSOs with a high angular resolution. Irrespective of the presence or absence of associated luminous components, it is possible to probe these subhalos in a foreground lensing galaxy by means of gravitational lensing. Theory of Gravitational Lensing =============================== The phenomena that light rays are deflected by gravity is one of the consequences of Einstein’s General Theory of Relativity. This is referred to as Gravitational lensing. Gravitational lensing is a powerful tool to determine the mass of lens objects which deflect and distort a source image in the background. In a gravitational lens system, Einstein radius is a useful indicator of the scale of image distortion. In the case of an Singular Isothermal Sphere (SIS) with density distribution of $\rho \propto r^{-2}$, Einstein radius is written as, $$\theta_E \sim 1.5 \left(\frac{M_t}{5\times10^7 M_\odot}\right) \left({\frac{r_t}{1.0\mathrm{kpc}}}\right)^{-1}\left(\frac{D_{ls}/D_{os}}{0.5}\right) [\mathrm{mas}]$$ where $r_t$ is a supposed tidal radius of an SIS lens and $M_t$ is a total mass enclosed within $r_t$. $D_{ls}$ and $D_{os}$ are angular diameter distances between lens and source and between observer and source, respectively. Several observations of lens systems in a galactic scale have revealed that the density distribution of a galaxy lens, being composed of both baryonic and dark matter, is well represented by an SIS profile. However, actual density profiles in subhalos are yet to be settled. According to N-body simulations based on the CDM model, i.e., without taking into account dissipative baryonic matter, all dark halos appear to show a shallower density profile at their central parts, expressed as $\rho \propto r^\gamma$ with $-1.5 < \gamma < -1$ (e.g., Navarro et al. 2004). If this is the case, then a dark halo with this profile has a smaller Einstein radius than an SIS case, so the resulting lensing signal to be observed by VSOP-2 would be biased further in favor of more massive subhalos. How baryonic components, if there are any, affect a dark halo profile is yet uncertain: an adiabatic contraction of a halo as driven by gas cooling yields a steeper density profile, whereas starburst activity and associated heating by supernovae explosion may make a gravitational potential shallower owing to galactic wind. Whatever effects of baryonic components are at work, an SIS profile would provide a possibly upper limit to the lensing signal of subhalos. As is shown in equation (1), the size of an Einstein radius depends on the distance ratio, $D_{ls}/D_{os}$. We plot, in Figure 1, this dependence for a subhalo of $M_t = 5 \times 10^7 M_\odot$ and $r_t = 1.0$ kpc, using known lensing systems taken from CASTLES. Simulated Images ================ In Figure 2, we show simulated images of a gravitational lensing system using the lens parameters of MG0414+0534, where a source image and a lensing galaxy are at redshifts of 2.64 and 0.96, respectively. We assume an SIS profile for the density distribution of a subhalo. The locally distorted feature for an arc-like lensed image in Figure 2 can be caused by not only subhalo lensing but also intrinsic substructure within a source image. To distinguish the effect of subhalo lensing alone, we should select a lens system showing multiple images as a target: intrinsic substructure in a source would be seen in all multiple images, whereas subhalos yield particular features in each image. Thus it is possible to extract the lensing signals of subhalos by subtracting commonly-observed features in lensed images. Inoue K.T. & Chiba M. 2003, ApJ, 591,L83 Inoue K.T. & Chiba M. 2005, ApJ, 633,23 Inoue K.T. & Chiba M. 2005, ApJ, 634,70 Navarro J.F. et al. 2004, MNRAS, 349, 1039
--- abstract: 'Breast cancer is the most frequently reported cancer type among the women around the globe and beyond that it has the second highest female fatality rate among all cancer types. Despite all the progresses made in prevention and early intervention, early prognosis and survival prediction rates are still unsatisfactory. In this paper, we propose a novel type of perceptron called L-Perceptron which outperforms all the previous supervised learning methods by reaching 97.42 % and 98.73 % in terms of accuracy and sensitivity, respectively in Wisconsin Breast Cancer dataset. Experimental results on Haberman’s Breast Cancer Survival dataset, show the superiority of proposed method by reaching 75.18 % and 83.86 % in terms of accuracy and F1 score, respectively. The results are the best reported ones obtained in 10-fold cross validation in absence of any preprocessing or feature selection.' author: - \| Hadi Mansourifar\ Department of Computer Science\ University of Houston\ Houston, Texas\ `hmansourifar@uh.edu`\ Weidong Shi\ Department of Computer Science\ University of Houston\ Houston, Texas\ `wshi3@central.uh.edu`\ title: 'Toward Efficient Breast Cancer Diagnosis and Survival Prediction Using L-Perceptron' --- Introduction ============ Nowadays, cancer is the second leading cause of death in the U.S based on IARC report and it would replace heart disease as the main cause of death [@19]. Breast cancer is considered as one of the deadliest diseases with a high fatality rate among women worldwide. Early prognosis is the most effective way to minimize the physical and psychological side effects of prolonged treatments. Machine Learning methods are effective ways for early diagnosis and survival prediction [@8]. Over the past decades, large amounts of cancer data have been collected and are available to the Machine Learning and Data Mining community. In parallel, a wide range of supervised learning methods have been applied on collected datasets [@16; @13; @7; @2; @12]. However, due to vital impacts of correct diagnosis in cancer treatments, the accurate diagnosis and efficient survival prediction is still one of the most challenging tasks for researchers. In this paper, we propose a novel type of perceptron called L-Perceptron which outperforms all the previous supervised learning methods by reaching 97.42 % and 98.73 % in terms of accuracy and sensitivity, respectively in Wisconsin Breast Cancer dataset. Experimental results on Haberman’s Cancer Survival dataset, show the superiority of proposed method by reaching 75.18 % and 83.86 % in terms of accuracy and F1 score, respectively. L-Perceptron has been devised by combination of least square classifiers and traditional perceptron ideas. In L-perceptron, given $p1$ and $p2$ as $Y$ values, a mathematical function is fitted per feature same as least squares classification. What’s trained during update rule is characteristic of each function per feature. For example, in case of polynomials, the best possible polynomial order is trained per feature. This procedure helps L-Perceptron to be not only non-linear but very flexible to handle overfitting problem. The rest of this paper is organized as follows. Section 2 reviews some of the most important researches in breast cancer diagnosis and survival prediction. In section 3, we propose a novel type of perceptron called L-Perceptron. Section 4 reports experimental results on Wisconsin Breast Cancer and Haberman’s Breast Cancer Survival datasets. Finally, section 5 concludes the paper. Related Work ============ In this section, we provide a review on several studies which have been conducted on breast cancer diagnosis and survival prediction. These studies have focused on different approaches to the given problem and achieved high classification accuracies. [@18] proposed an interactive evaluation - diagnosis computer system based on cytologic features. Vikas Chaurasia and Saurabh Pal [@5] compared the performance criterion of supervised learning classifiers, such as Naive Bayes, SVM-RBF kernel, RBF neural networks, Decision Tree (DT) (J48), and simple classification and regression tree (CART), to find the best classifier in breast cancer datasets. The experimental result shows that SVM-RBF kernel is more accurate than other classifiers since it scores at the accuracy level of 96.84 % in the Wisconsin Breast Cancer (original) dataset. Asri et al. [@1] compared the performance of C4.5, Naive Bayes, Support Vector Machine (SVM) and K- Nearest Neighbor (K-NN) to find the best classifier in Wisconsin Breast Cancer (original) showing that SVM proves to be the most accurate classifier with accuracy of 97.13 %. Vikas Chaurasia and Saurabh Pal [@6] used three popular data mining algorithms (Naive Bayes, RBF Network, J48) to develop the prediction models using the Wisconsin Breast Cancer (original). The obtained results indicated that the Naive Bayes performed the best with a classification accuracy of 97.36 % and RBF Network came out to be the second best with a classification accuracy of 96.77 %, and the J48 came out to be the third with a classification accuracy of 93.41 %. [@11] used an evolutionary artificial neural network (EANN) approach based on the pareto-differential evolution (PDE) algorithm augmented with local search for the prediction of breast cancer. The approach is named Memetic Pareto Artificial Neural Network (MPANN). Since the early dates of the researches in the field of computational biomedicine, the cancer survivability prediction has been a challenging problem for many researchers [@9; @3]. In [@10], artificial neural networks and decision trees along with logistic regression were used to develop the prediction models using 202,932 breast cancer patients records, which then pre-classified into two groups of “survived” (93,273) and “not survived” (109,659). The results of predicting the survivability were in the range of 93 % accuracy.[@15] used a fully complex valued fast learning artificial neural network with GD activation function in the hidden layer. The comparison results showed that, FC-FLC provides a better classification performance comparing to the SRAN, MCFIS and ELM classifiers. Liu et al. [@14] used the under-sampling C5 technique and bagging algorithm to deal with the imbalanced problem predictive models for breast cancer survivability. Proposed Method =============== In this section, we propose a novel type of perceptron called L-Perceptron which despite its simplicity, it can outperform traditional supervised learning methods in breast cancer diagnosis and survival prediction. L-Perceptron ------------ Given a set of $n$ training data in $m$ dimensional space $X= \{x_1, x_2, x_3..., x_n\}$, a set of corresponding labels $Y=\{y_1,y_2,y_3,...,y_n \}$, $p1$ and $p2$ as two hyperparameters, $V=\{v_1,v_2,v_3,...,v_n\}$ is created as follows. $$\begin{cases} p1 & if \quad y_ {j}=class 1 \quad (positive \quad instance) \\ p2 & otherwise \end{cases}$$ Where, $p1$ and $p2$ can be manipulated by the user to get the best possible result. Training phase of a L-Perceptron starts by fitting function $F$ by minimizing $S$, given $V$ per each feature as follows. $$r_i= [v_i -F (x_i, \alpha)]$$ $$S= \sum_{i=1} ^{n} (r_i)^2$$ After finding the best fitting function per each feature test phase is formulized as follows. $$\begin{cases} 1 & if \sum_{j=1} ^{m} F_j(X) > 0 \\ 0 & otherwise \end{cases}$$ Where, $F_j$ is fitting function of $j^{th}$ feature. It means that instead of dot product of weights and features the features are passed to their corresponding function and a summation of functions outputs is passed to a step function or activation function to predict the label of input test data. The diagram of L-Perceptron is shown in figure1.Before starting the training phase, the fitting function type must be defined which can be selected among all possible mathematical functions like logarithmic, exponential, polynomial, etc. What’s trained during the update rule is characteristic of the fitting function. For example, in case of polynomials the best possible polynomial order is trained per feature. This lets the model to meet each feature set complexity separately. ![Schematic of L-Perceptron. \[overflow\]](figure1.jpg){width="90mm"} Update Rule ----------- Suppose polynomials are used as fitting function. In this case, the order of each fitting polynomial is trained during the update rule. What’s happening during the update rule is to assign initial polynomial orders, fit a polynomial per feature given the $p1$ and $p2$, calculate the error and repeat this procedure until the best possible orders are found at the end of update rule. To make the update rule faster the upper bound and lower bound of polynomial orders can be limited to some specific range. The upper bound and lower bound range can be defined as hyperparameters as described in implementation section. This restriction adds more flexibility to the L-Perceptron to avoid overfitting. Update rule of L-Perceptron is as follows. ------------------------------------------------------------------------ \ 1. Initialize all degrees to 1 and $Error=a$\ 2. Repeat until no significant change seen in $Error$\ 3. For all Features\ 4. $degree[i]=degree[i]+1$\ 5. Compute the $NewError$\ 6. if ($NewError< Error$)[ $Error = NewError$]{}\ 7. else[ $degree[i]=degree[i]-1$]{}\ ------------------------------------------------------------------------ \ ![Schematic of L-Perceptron training phase. \[overflow\]](figure2.jpg){width="90mm"} Figure 2 shows the schematic of L-Perceptron, in case of using polynomials as fitting function. Experiments =========== In this section, we compare the accuracy results of proposed method versus a set of traditional supervised learning methods by testing on WBCD and HSD. Datasets -------- In this section, we provide a brief introduction to datasets used in the experiments including Wisconsin Breast Cancer dataset (original) and Haberman’s Breast Cancer Survival dataset. ### Wisconsin Breast Cancer dataset The breast cancer dataset is a classic binary classification dataset. The data is accessible from the UC Irvine Machine Learning repository [@17]. This dataset has 699 instances, two classes (malignant and benign), and 9 integer-valued attributes. It contains 16 instances with missing values, but we didn’t remove them from the dataset. It’s consisted of benign: 458 (65.5%), malignant: 241 (34.5%) instances. Attribute -------- ----------------------------- 1 Sample code number 2 Clump thickness 3 Uniformity of cell size 4 Uniformity of cell shape 5 Marginal adhesion 6 Single epithelial cell size 7 Bare nuclei 8 Bland chromatin 9 Normal nucleoli 10 Mitoses 11 Class ### Haberman’s Breast Cancer Survival dataset The Haberman’s Breast Cancer Survival dataset [@4] contains of cases from a study that was conducted between 1958 and 1970 at the University of Chicago’s Billings Hospital on the survival of patients who had undergone surgery for breast cancer. This dataset has 306 instances, 2 classes and 3 integer-valued attributes. Each data point contains following features. Attribute ------- -------------------------------------------- 1 Age of patient at time of operation 2 Patient’s year of operation (year - 1900) 3 Number of positive axillary nodes detected 4 Class Output attribute is a survival status (class attribute) assigned for the output attribute is as follows: Patient lived for 5 years or longer =1. Patient death within 5 years =2. ### Implementation We have developed a Python function called “lp.py” which is available for the public to test on classification problems [@20]. This function has been originally devised for numerical datasets. However, it can be used on categorical datasets if the features supposed as a set of discrete integer values. The input parameters of this functions are as follows.\ lp(trainx, trainy, testx, testy, p1, p2, dlb, dub, ite, threshold)\ Table1 shows implemented parameters and their descriptions. Parameter Description -------------------- ----------------------------------------------------------------------------------- trainx, trainy A set of training data and their corresponding labels testx, testy A set of test data and their corresponding labels p1, p2 Y values for fitting functions as described in section 3 dlb, dub Degree lower bound and degree upper bounds respectively to limit the degree range ite Number of iterations threshold Threshold of activation function to discriminate between the instances. : Parameters and their description. ### Results and discussion We used 10-fold cross validation method to measure the unbiased estimation for performance comparison purposes. The comparison results of different methods tested on Wisconsin Breast Cancer Dataset (WBCD) are presented in Table 2. Experimental results on Haberman’s Survival Dataset (HSD) have been tabulated on Table 3. Table 2 shows that L-Perceptron outperforms other methods in terms of accuracy and sensitivity based on experiments on WBCD. Naive Bayes is in the second rank by showing better results comparing to the rest of methods. Methods Accuracy (%) Sensitivity (%) Specificity (%) F1 Score (%) -------------- ------------------ --------------------- --------------------- ------------------ L-Perceptron 97.42 98.73 96.2 96.50 Naive Bayes 97.36 97.4 97.9 97.64 RBF Network 96.77 97.07 96.23 96.6 J48 93.41 93.4 90.37 91.86 : Accuracy, sensitivity, specificity and F1 score of different methods tested on Wisconsin Breast Cancer Dataset. ![Comparative graph of different classifiers tested on WBCD. \[overflow\]](figure3.png){width="120mm"} Methods Accuracy (%) Sensitivity (%) Specificity (%) F1 Score (%) ---------------------------------- ------------------ --------------------- --------------------- ------------------ L-Perceptron 75.18 90.04 37.08 83.86 Logistic Regression 74.27 94.77 22.95 82.62 Linear Discriminant Analysis 73.78 95.42 19.67 82.71 KNN 71.03 88.23 34.42 81.57 CART 64.02 74.5 26.22 78.44 Naive Bayes 74.17 94.11 27.86 82.52 SVM 69.77 95.42 3.27 82.71 MLP 66.21 62.74 55.73 72.64 Random Forest 67.27 81.69 22.95 80.38 : Accuracy, sensitivity, specificity and F1 score of different methods tested on Haberman’s Survival Dataset. Table 3 shows that L-Perceptron outperforms other methods in terms of accuracy and F1 score based on experiments on HSD. When it comes to compare the methods based on specificity, L-Perceptron is at the second rank after MLP which shows best specificity result among all tested methods. Table 4 shows the initialized parameters including $p1$, $p2$ and fitting degree for each dataset.\ \ \ \ ![Comparative graph of different classifiers tested on HSD. \[overflow\]](figure4.png){width="170mm"} Parameter WBCD HSD ----------------- ---------- ----------- *p1, p2* -2, 3 -1.3, 2.9 *dlb, dub* 4, 4 1, 1 *ite* 2 0 *threshold* 0.5 0.42 : . Initialized parameters used in experiments. Conclusion ========== In this paper, we proposed a novel type of perceptron called L-Perceptron. The proposed method successfully tested on Wisconsin Breast Cancer Dataset and Haberman’s Breast Cancer Survival Dataset. We used 10-fold cross-validation method to measure the unbiased estimation for performance comparison purposes. The experimental results showed that L-Perceptron has the best performance comparing to previous methods in terms of accuracy and sensitivity based on experiments on WBCD. The proposed method reached 97.42 % of accuracy, 98.73 % of sensitivity which are the best performance results reported in the literature among the reported results without any preprocessing or feature selection. We also tested L-Perceptron on Haberman’s Breast Cancer Survival Dataset. The experimental results showed that L-Perceptron has the best performance comparing to previous methods in terms of accuracy and F1 score. The proposed method reached 75.18 % of accuracy, 83.86 % of F1 score which are the best reported performance results. [1]{} Asri H, Mousannif H, Al Moatassime H, Noel T. Big data in healthcare: Challenges and opportunities. 2015 Int Conf Cloud Technol Appl. 2015:1-7. doi:10.1109/CloudTech.2015.7337020. Ayer T, Alagoz O, Chhatwal J, Shavlik JW, Kahn CE, Burnside ES. Breast cancer risk estimation with artificial neural networks revisited. Cancer 2010;116:3310–21. Bellaachia A and Guven E. Predicting breast cancer survivability using data mining techniques. In: Scientific data mining workshop (in conjunction with the 2006 SIAM conference on data mining), April 20-22, 2006, Bethesda, Maryland. C. Blake and C. Merz, “UCI repository of machine learning databases”, Department of Information and Computer Sciences, University of California, Irvine, \[URL: http://archive.ics.uci.edu/ml/\], 1998. Chaurasia V and Pal S. Data mining techniques: to predict and resolve breast cancer survivability. Int J Comput Sci Mobile Comput 2014; 3: 10–22. Chaurasia, Vikas, Saurabh Pal, and B. B. 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Abbass, An Evolutionary Artificial Neural Networks Approach for Breast Cancer Diagnosis, School of Computer Science, University of New South Wales, Australian Defence Force Academy Campus. Lavanya, Dr.K.Usha Rani,..,” Analysis of feature selection with classification: Breast cancer datasets”,Indian Journal of Computer Science and Engineering (IJCSE),October 2011. Li J, Liu H, Ng S-K, et al. Discovery of significant rules for classifying cancer diagnosis data. Bioinformatics 2003; 19: ii93–ii102. Liu Y-Q, Wang C and Zhang L. Decision tree based predictive models for breast cancer survivability on imbalanced data. In: 3rd international conference on bioinformatics and biomedical engineering, 11-13 June 2009, Beijing, China, 2009. Sivachitra, M., and S. Vijayachitra. “Classification of post operative breast cancer patient information using complex valued neural classifiers.” In Cognitive Computing and Information Processing (CCIP), 2015 International Conference on, pp. 1-4. IEEE, 2015. 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--- abstract: 'Studying the internal structure of exoplanet-host stars compared to that of similar stars without detected planets is particularly important for the understanding of planetary formation. The observed average overmetallicity of stars with planets is an interesting point in that respect. In this framework, asteroseismic studies represent an excellent tool to determine the structural differences between stars with and without detected planets. It also leads to more precise values of the stellar parameters like mass, gravity, effective temperature, than those obtained from spectroscopy alone. Interestingly enough, the detection of stellar oscillations is obtained with the same instruments as used for the discovery of exoplanets, both from the ground and from space. The time scales however are very different, as the oscillations of solar type stars have periods around five to ten minutes, while the exoplanets orbits may go from a few days up to many years. Here I discuss the asteroseismology of exoplanet-host stars, with a few examples.' address: ' Laboratoire d’Astrophysique de Toulouse et Tarbes ; CNRS ; Université de Toulouse ; Institut universitaire de France ' author: - Sylvie Vauclair title: ' What do stars tell us about planets? Asteroseismology of exoplanet-host stars ' --- Why bothering about asteroseismology while studying planets? ============================================================= I can see at least four different reasons for which astrophysicists interested in exoplanets should also bother about the asteroseismology of the central stars of planetary systems. First reason: the observations for stellar oscillations and exoplanet searches are done with the same instruments. In some cases, the same observations, analysed on different time scales, can lead to both planet detection and seismic studies. This was the case for the star $\mu$ Arae, observed with HARPS during eight nights in June 2004: these observations, aimed for asteroseismology, lead to the discovery of the exoplanet $\mu$ Arae d (Santos et al. [@santos04]). Second reason : “Some people’s noise is other people’s signal”. Indeed, when searching for exoplanets, the signal to noise ratio is limited by the stellar oscillations, which appear as a noise for the radial velocity variations induced by the planetary motions, while they represent in fact the stellar oscillations signal. A better knowledge of seismology and a better treatment of this low time scale signal could help determining the planetary parameters. Third reason: obtain precise values of the parameters of exoplanet-hosts stars. Asteroseismic studies, combined with spectroscopic observations, can lead to values of the stellar parameters which are much more precise than from spectroscopy alone. In the case of $\iota$ Hor, for example, the mass determined from spectroscopy and position in the HR diagram was $10\%$ wrong (Vauclair et al. [@vauclair08]). Fourth reason: links between asteroseismology and planets discoveries. In two cases at least, asteroseismic studies lead to exoplanet discoveries. One is the case, already mentioned, of $\mu$ Arae. Studies of seismic period variations (the so-called “time delay method") also lead recently to the spectacular discovery of a planet around the extreme horizontal branch star V391 Peg (Silvotti et al. [@silvotti07]). Generally speaking, the goals of the asteroseismology of exoplanets-host stars are to derive their masses, ages, evolutionary stages, outside and inside metallicities, to compare them with the Sun and with stars without detected planets, to obtain hints about the theories of planetary formation and migration, and to try and model the oscillations well enough to decrease the “noise” for planet detection. Basics of asteroseismology for slowly rotating solar-type stars ================================================================= In solar-type stars, acoustic waves are permanently created by the motions which occur in their outer layers, induced by convection and related processes. The waves are damped, but as other waves always appear, the stellar sphere globally behaves as a resonant cavity, and the oscillations can be treated, with a very good approximation, as standing waves. Each mode can be characterized with three numbers: the number of nodes in the radial direction, $n$, and the two tangential numbers $\ell$ and $m$, which appear in the development of the waves on the spherical harmonics: $$Y_l^m (\theta,\phi) = (-1)^m C_{l,m} P_l^m (cos\theta) exp(im\theta)$$ Several combinations of the oscillation frequencies are used to obtain more precise constraints on the stellar internal structure, like the “large separations", which are defined as the differences between two consecutive modes of the same $\ell$ number, and the “small separations", defined as: $$\delta\nu = \delta \nu_{n,l} - \nu_{n-1,l+2}$$ For acoustic modes in a given star, the large separations are nearly constant, and their average value is about equal to half the inverse of the stellar acoustic time, i.e. the time needed for the $l = 0$ waves to travel along the whole radius. Meanwhile, the small separations present variations which are directly related to the structure of the stellar cores (Roxburgh & Vorontsov [@roxburgh94], Roxburgh [@roxburgh07], Soriano et al. [@soriano07], Soriano & Vauclair [@soriano08]). An interesting tool used to compare stellar models with the observations is the “echelle diagram". This diagram presents the mode frequencies in ordinates, and the same frequencies modulo the large separations in abscissae (see Figure 2). Asteroseismology of exoplanet-host stars ========================================== Aseroseismology of exoplanet-host stars is a useful tool to determine their internal structure and their behaviour with respect to stars without detected planets. Let us recall that, due to the present observation bias, many stars may very well host undetected planets. The presently detected planets have to be close to the stars, and their orbital plane must not be perpendicular to the line of sight. In this framework, the special characteristics of exoplanet-host stars compared to the other ones is more related to the migration process of the planets than to their mere existence. An important particularity of exoplanet-host stars is their overmetallicity, compared to the other stars and to the Sun (e.g. Santos et al. [@santos05]). The accretion origin for this overmetallicity (planet engulfment at the beginning of the stellar system formation) is now ruled out for several reasons. One of these reasons concerns the enormous amount of metallic matter which should be accreted to explain the observed overabundances. Furthermore, the accreted metals would not stay in the outer convective zones of the stars, but would fall down due to thermohaline convection induced by the unstable inverse $\mu$-gradient (Vauclair [@vauclair04]). The seismic analysis of the star $\iota$ Hor (see below) also proves that the overmetallicity was there at the origin, in the cloud out of which the stellar system formed. A crucial unknown parameter, very important for the determination of the stellar characteristics, is the helium abundance. Unfortunately, helium is not directly observable in the spectra of solar-type stars. If the stellar systems form inside an overmetallic interstellar cloud, is this cloud also helium-enriched as predicted from theories of the chemical evolution of galaxies or not? This depends on the stellar mass function and there is no clear answer at the present time. Detailed seismic analysis can solve this question, as has been proved for $\iota$ Hor: in this star, the helium abundance is not larger than solar, it may even be smaller (Vauclair et al. [@vauclair08]). Up to now four exoplanet-host stars have been observed on relatively long periods (8 or 9 consecutive nights) with the HARPS and SOPHIE spectrometers. They are: $\mu$ Arae (HARPS, 2004), $\iota$ Hor (HARPS, 2006), 51 Peg (SOPHIE, 2007) and 94 Cet (HARPS, 2007). Figure 1 presents, as an example, part of the radial velocity oscillation curves obtained for 51 Peg (Soriano et al., in preparation). Several other exoplanet-host solar type stars have been observed for about half an hour: they all oscillate, without any exception. At the present time, complete modelling has been performed for $\mu$ Arae and $\iota$ Hor only. The models have been computed using the TGEC (Toulouse-Geneva stellar evolution code), with the OPAL equation of state and opacities (Rogers & Nayfonov [@rogers02], Iglesias & Rogers [@iglesias96]) and the NACRE nuclear reaction rates (Angulo et al. [@angulo99]). Microscopic diffusion is included in all the models (Paquette et al. [@paquette86], Richard et al. [@richard04]). The treatment of convection is done in the framework of the mixing length theory and the mixing length parameter is adjusted as in the Sun. Adiabatic oscillations frequencies are computed using the adiabatic PULSE code (Brassard [@brassard92]). The only COROT main target hosting an exoplanet, HD52265 has also been modelled in advance before the observations, which should be done during the fall 2008 (Soriano et al. [@soriano07]). I give below some information about $\mu$ Arae and $\iota$ Hor ![image](vauclair-Figure1.eps){width="12cm"} The saga of $\mu$ Arae ======================== The exoplanet-host star $\mu$ Arae (HD160691, HR6585, GJ691) is a G5V star with a visual magnitude V = 5.1, an Hipparcos parallax $\pi$ = 65.5 $\pm$ 0.8 mas, which gives a distance to the Sun of 15.3 pc and a luminosity of $\log L/L_{\odot}~=~$0.28$~\pm~$0.012. This star was observed for seismology in August 2004 with HARPS. At that time, two planets were known. The observations aimed for seismology lead to the discovery of a third planet, $\mu$ Ara d, with period 9.5 days (Santos et al. [@santos04]). More recently, evidence for a fourth planet has been discovered (Pepe et al. [@pepe07]). The HARPS seismic observations allowed to identify 43 oscillation modes of degrees $l~=~0$ to $l~=~3$ (Bouchy et al. 2005). In Figure 2, they are presented in the form of an echelle diagram and compared with a model. From the analysis of the frequencies and comparison with models, the values T$_{eff}$ = 5770 $\pm$ 50 K and \[Fe/H\] = 0.32 $\pm$ 0.05 dex were derived. Spectroscopic observations by various authors gave five different effective temperatures and metallicities (see references in Bazot et al. [@bazot05]). The values obtained from seismology are much more precise than those obtained from spectroscopy alone. ![image](vauclair-Figure2.eps){width="10cm"} The special case of $\iota$ Hor --------------------------------- Among exoplanet-host stars, $\iota$ Hor is a special case for several reasons (see Laymand & Vauclair [@laymand07] and Vauclair et al. [@vauclair08]). Three different groups have given different stellar parameters for this star: Gonzalez et al. [@gonzalez01], Santos et al. [@santos04] and Fischer & Valenti [@fischer05]. Meanwhile, Santos et al. [@santos04] suggested a mass of 1.32 M$_{\odot}$ while Fischer and Valenti [@fischer05] gave 1.17 M$_{\odot}$. Some authors (Chereul et al. [@chereul99], Grenon [@grenon00], Chereul and Grenon [@chereul00], Kalas and Delorn [@kalas06], Montes et al. [@montes01]) pointed out that this star has the same kinematical characteristics as the Hyades: its proper motion points towards the cluster convergent. Two different reasons were possible for this behaviour: either the star formed together with the Hyades, in a region between the Sun and the centre of the Galaxy, which would explain its overmetallicity compared to that of the Sun, or it was dynamically canalized by chance (see Famaey et al. [@famaey07]). Solar-type oscillations of $\iota$ Hor were detected with HARPS in November 2006. Up to 25 oscillation modes could be identified and compared with stellar models. The results lead to the following conclusions for $\iota$ Hor (Vauclair et al. [@vauclair08]): (Fe/H) is between 0.14 and 0.18; the helium abundance Y is small, 0.255 $\pm $ 0.015; the age of the star is 625 $\pm$ 5 Myr; the logarithm of the gravity is 4.40 $\pm$ 0.01 and its mass $1.25 \pm 0.01 M_{\odot}$. The values obtained for the metallicity, helium abundance and age of this star are those characteristic of the Hyades cluster (Lebreton et al. [@lebreton01]). In summary, we have found from seismic analysis that this exoplanet-host star has been formed together with the Hyades cluster. As a consequence, the overmetallicity has been present from the beginning and is not due to accretion. The stellar mass has also been derived with a much better precision than from spectroscopy alone. It lies in between the two different values given by Santos et al. [@santos04] and Fischer & Valenti [@fischer05]. Conclusion ============ From the few examples already available, asteroseismology has proved to be a powerful tool for determining stellar parameters, in particular for exoplanet-host stars. The scientific community is well prepared for future observations with on-going or planned projects. Space missions like COROT, and later on KEPLER, are expected to give a large amount of new data for seismology, besides planet searches. Meanwhile ground based instruments devoted to exoplanets like HARPS or SOPHIE can be used for seismology. All these observations will give the possibility of using the seismic tests with a precision never obtained before. They will lead to better parameters for exoplanet-host stars and help for a better understanding of planetary formation and evolution. [99]{} Angulo C., Arnould M., Rayet M., (NACRE collaboration), 1999, Nuclear Physics A 656, 1, http://pntpm.ulb.ac.be/Nacre/nacre.htm Bazot, M., Vauclair, S., 2004 A&A, 427, 965 Bazot, M., Vauclair, S., Bouchy, F., Santos, N. 2005 A&A, 440, 615 Bouchy, F., Bazot, M., Santos, N., Vauclair, S., Sosnowska, D. 2005, A&A, 440, 609 Brassard, P., 1992, ApJS, 81, 747 Chereul, E., Crézé, M., Bienaymé, O., 1999, A&AS, 135,5 Chereul & Grenon,M. 2000, in “Dynamics of Star Clusters and the Milky Way, ASP conf. Series, Deiters et al. 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--- abstract: 'We report the first experimental observation of extreme multistability in a controlled laboratory investigation. Extreme multistability arises when infinitely many attractors coexist for the same set of system parameters. The behavior was predicted earlier on theoretical grounds, supported by numerical studies of models of two coupled identical or nearly identical systems. We construct and couple two analog circuits based on the Lorenz system and demonstrate the occurrence of extreme multistability through the controlled switching of the coupled system to different attractor states purely through a change in initial conditions for a fixed set of system parameters. Numerical studies of the coupled Lorenz circuits are in agreement with our experimental findings.' author: - 'Mitesh S. Patel, Unnati Patel, Abhijit Sen[^1], Gautam C. Sethia,' - 'Calistus N. Ngonghala, and Kenneth Showalter' title: \| Experimental observation of extreme multistability\ in coupled electronic oscillators --- Multistability is a common occurrence in many nonlinear dynamical systems, corresponding to the coexistence of more than one stable attractor for the same set of system parameters [@feudel08]. A large number of theoretical and experimental studies have explored this phenomenon in a variety of physical [@arecchi82; @masoller02; @kastrup94; @schwarz00], chemical [@ganapathy84; @marmillot91] and biological [@foss96; @huisman01] systems. A curious and novel manifestation of this phenomenon arises when a system can have an infinite number of coexisting attractors, where each attractor is associated with a particular set of initial conditions [@sun99; @wang03; @calistus11]. This extreme multistability has been found in coupled identical or nearly identical chaotic systems, such as the Lorenz system [@Ott] and the three-variable autocatalator model [@Peng1990]. The coupled subsystems exhibit generalized synchronization and, in the case of an experimental system with unavoidable variability in conditions, a sensitivity to initial conditions with a corresponding ‘uncertainty’ in the destination dynamics. Extreme multistability was first demonstrated in a system of two coupled identical Lorenz oscillators by Sun [et al.]{} [@sun99] and has more recently been studied in the three-variable autocatalator model by Ngonghala [et al.]{} [@calistus11]. In both studies, a special coupling was applied between two three-variable chaotic systems to form six-variable coupled systems. Numerical simulations of the coupled systems were carried out for a fixed set of system parameters and only the initial conditions were changed. The systems were found to evolve to different attractor states (fixed points, limit cycles, chaotic states) purely through changes in initial conditions, typically with a change in the initial condition of just one of the state variables. This behavior was explained theoretically [@sun99; @calistus11] by analyzing the dynamics of a reduced set of equations governing the differences (“errors") of the corresponding state variables of the three-variable subsystems. The reduced system yields a constant $c$, a conserved quantity that attains a particular value following the decay of transients. Different initial conditions gave rise to new dynamical behavior and a corresponding different value of $c$. The existence of this conserved quantity leads to a ‘slicing’ or foliation of the state space into manifolds corresponding to the different values of $c$. The conserved quantity can be used to reduce the six-variable coupled system to a three-variable system, in which $c$ serves as a bifurcation parameter. Although the full coupled system has a conserved quantity, the asymptotic states to which the system evolves are true attractors, since there are an infinite number of initial conditions that yield any particular value of the conserved quantity $c$. The set comprising these initial conditions constitutes the basin of attraction for a specific attractor, and trajectories emanating from such initial conditions will converge to the attractor. Extreme multistability might have important consequences in the reproducibility of certain experimental systems. For example, some chemical reactions, such as the chlorite-thiosulfate reaction [@orban82] and the chlorite-iodide reaction [@nagypal86] consistently exhibit irreproducibility: despite great care to ensure reproducibility, these reactions show a random long-term behavior for the same set of experimental conditions. The cause of the irreproducibility is not known; however, extreme multistability offers a possible mechanism for the behavior. Model calculations by Wang [et al.]{} [@wang03] demonstrated a type of extreme multistability arising from the specific features of a chemical reaction. However, to the best of our knowledge, there has yet been no direct experimental verification of this new type of dynamical behavior in a controlled laboratory investigation. In this Letter, we report experimental observations of a coupled electronic circuit system that displays extreme multistability.\ Our experiments are carried out on an analog circuit system based on the original model investigated by Sun [et al.]{} [@sun99], consisting of a set of coupled Lorenz equations, namely, $$\begin{aligned} \dot{X}_1 & = \sigma (Y_1 - X_2) \\ \dot{Y}_1 & = r X_1 - Y_1 - X_1 Z_1\\ \dot{Z}_1 & = X_1 Y_1 - b Z_1\\ \dot{X}_2 & = \sigma (Y_2 - X_2)\\ \dot{Y}_2 & = r X_1 - Y_2 -X_1 Z_2\\ \dot{Z}_2 & = X_1 Y_2 - b Z_2\end{aligned}$$ \[eqn\] where $\sigma$, $r$ and $b$ are constants and $(X_1,Y_1,Z_1)$ and $(X_2,Y_2, Z_2)$ are the state variables of the two identical subsystems. For the circuit implementation of the above system, we have scaled the state variables as $$x_{1,2} = \frac{X_{1,2}}{\sqrt{3r}} \;\;;\;\; y_{1,2} = \frac{Y_{1,2}}{\sqrt{3r}} \;\;;\;\;z_{1,2} = \frac{Z_{1,2}}{3r}$$ in order to restrict the output signal voltage range to within $\pm 10$ volts and to avoid saturation of the circuit. This leads to the following system: $$\begin{aligned} \dot{x}_1 & = \sigma (y_1 - x_2), \\ \dot{y}_1 & = \frac{r}{3}(3 - z_1)x_1 - y_1,\\ \dot{z}_1 & = x_1 y_1 - b z_1,\\ \dot{x}_2 & = \sigma (y_2 - x_2),\\ \dot{y}_2 & = \frac{r}{3}(3 - z_2)x_1 - y_2,\\ \dot{z}_2 & = x_1y_2 - b z_2.\end{aligned}$$ \[eqn\] The detailed circuit diagram of the two coupled systems is shown in Fig. (\[circuit\]), in which we have adopted the basic circuit design for a single Lorenz oscillator recently suggested by N. J. Corron [@corron10]. ![image](ckt.jpg){width="96.00000%"} A regulated power supply of $\pm 15V$ energizes the circuit, and the system parameters $r$, $\sigma$, and $b$ for the individual Lorenz circuits are controlled with three circuit resistors, namely, $$r = \frac{10 k\Omega}{3R_3} = \frac{10 k\Omega}{3R_{13}} \;\;;\;\;\sigma = \frac{100 k\Omega}{R_7} = \frac{100 k\Omega}{R_{17}} \;\;;\;\; b = \frac{100 k\Omega}{R_2} = \frac{100 k\Omega}{R_{12}}$$ As a benchmark exercise, each Lorenz oscillator was separately tested by varying the system parameters to obtain its various attractor states, ranging from periodic states to chaotic dynamics, and the behavior was then compared to numerical simulations of the Lorenz equations. Major care was taken to ensure that the two oscillator systems were as nearly ‘identical’ as possible within practical limits. This entailed careful weaning of all the component elements (resistors, capacitors) to match their values as closely as possible and the removal of any intrinsic drifts or biases within the operational amplifiers and multipliers [@circ]. The system parameters for both of the oscillators were then fixed at $\sigma =10$, $r=400$, $b=8/3$, and they were coupled to each other in the manner shown in Fig. (\[circuit\]). To change the initial conditions of the dynamics of the circuit, we have employed a strategy of imposing short voltage pulses on selective nodes of the operational amplifiers as well as shorting relevant capacitors of the circuit initially to set the values of some of the state variables to zero. To implement this combined strategy in a controlled manner we have developed and attached an additional system of relay circuits to the coupled Lorenz circuits (indicated by the boxed hatched portions in Fig. (\[circuit\])). The details of the relay circuit are shown in Fig. (\[relay\]). When this circuit is energized, the timer portion of the circuit produces a high output for 1 second and the relay is turned on through the relay driver circuit. Due to this, the $y_{2}$ and $y_{2in}$ signals get shorted in the coupled circuit and hence $y_2$ is set to $0$ initially. Similarly $y_1, z_1$ and $z_2$ can also be set to $0$. After 1 second the output of the timer circuit falls to a low value and the relay gets switched off. Due to this, the $y_2$ and $y_{2in}$ signal paths open up and the circuit can run with the applied initial conditions. Note that we can also allow $y_2$ (and or $y_1$) to have a finite voltage initially by not shorting it. In this case the initial value is given by the output of the multiplier and op-amp and can be determined exactly from the circuit parameters. The initial conditions of $‘x_1’$ and $‘x_2’$ can be changed through application of short voltage pulses using the $V1$ and $V4$ voltage sources as shown in the circuit diagram. ![Timer relay circuit for setting initial conditions of some the state variables to zero.[]{data-label="relay"}](relay_ckt.jpg){width="45.00000%"} ![The right panels show snapshots of experimental attractor states of the coupled electronic circuit implementation of the model system, Eqs. (\[eqn\]). The oscilloscope images show $z_2 \;{\it vs}\; x_1$ phase space plots for different initial values. The left panels show corresponding plots from numerical simulations of Eqs. (\[eqn\]) with similar initial conditions.[]{data-label="phase_plots"}](all.jpg){width="45.00000%"} Using the above strategy we have run the coupled circuit for a number of initial conditions without changing the circuit parameters. Some typical results in the form of oscilloscope images of phase plots of $z_2\; {\it vs}\; x_1$ are shown in the right-hand column of Fig. (\[phase\_plots\]) indicating two different periodic attractor states and a chaotic state. The uppermost image corresponds to initial conditions of $(x_1= +2V, y_1=z_1=x_2=y_2=z_2 =0)$, the middle image to $(x_1= -2V, y_1=z_1=x_2= y_2= z_2=0)$ and the lower image has $(x_1= -2V, x_2=-1V, y_1= z_1= z_2=0, y_2=14.2V)$. The plots in the left-hand column show corresponding numerical simulation results of the model Eqs. (2) using the same initial conditions. ![Experimental waveforms of $x_1$, $x_2$ and $x_1 - x_2$, where the left and central panels correspond to the periodic attractor states and the right panel to the chaotic attractor state as shown in Fig. (\[phase\_plots\]).[]{data-label="wave_forms"}](ex1.jpg "fig:"){width="30.00000%"} ![Experimental waveforms of $x_1$, $x_2$ and $x_1 - x_2$, where the left and central panels correspond to the periodic attractor states and the right panel to the chaotic attractor state as shown in Fig. (\[phase\_plots\]).[]{data-label="wave_forms"}](ex2.jpg "fig:"){width="30.00000%"} ![Experimental waveforms of $x_1$, $x_2$ and $x_1 - x_2$, where the left and central panels correspond to the periodic attractor states and the right panel to the chaotic attractor state as shown in Fig. (\[phase\_plots\]).[]{data-label="wave_forms"}](ex3.jpg "fig:"){width="30.00000%"} In Fig. (\[wave\_forms\]), we show the oscilloscope wave forms of the variables $x_1$, $x_2$ and the time variation of their difference ($x_1 - x_2$). In all three cases, the differences attain constant values of different magnitudes corresponding to the three distinct attractor states, which is a signature of extreme multistability. The difference ($x_1 - x_2$) represents the conserved quantity $c$ in the system. The coupled Lorenz circuit studied here, as well as other coupled systems that exhibit extreme multistability, exhibit properties of both dissipative and conservative dynamics. The circuit is characterized by infinitely many attractors, each associated with a particular value of the conserved quantity $c = x_1 - x_2$, where the basin of attraction is made up of all sets of initial conditions that evolve to asymptotically yield the particular value of $c$ associated with the attractor. Similarly, the attractor is stable to perturbations in which the asymptotic value of the conserved quantity is maintained. As pointed out in Ref. [@calistus11], because the manifold of initial conditions associated with any attractor is arbitrarily close to manifolds of initial conditions associated with other attractors, all attractors in our system are weak attractors in the Milnor sense [@Milnor1985]. The conserved quantity gives rise to a direction of neutral stability for the stationary states as well as the orbits (in addition to that associated with the direction along the orbit). Hence, if the system is in a particular periodic orbit, say period-2, a perturbation that does not satisfy the condition of the conserved quantity will give rise to the evolution of the system to a new attractor. The new attractor may differ only quantitatively; for example, a small perturbation might shift the period-2 dynamics to a new period-2 dynamics that differs in amplitude. However, larger perturbations give rise to the evolution of the system to qualitatively new attractors, such as a period-4 or period-8 attractor. This characterization of the effects of perturbations also applies to the effects of different initial conditions. Because the coupled system exhibits period-doubling chaos, perturbations or different initial conditions permit the sampling of any of an infinite number of qualitatively different attractors. Even if the coupled system did not display chaotic dynamics, the same mechanism would give rise to an infinite number of quantitatively different attractors. As yet, however, we have been unable to find the phenomenon of extreme multistability in non-chaotic coupled systems. We have described the first experimental demonstration of extreme multistability. Our experiments have been based on an electronic circuit that models two coupled Lorenz attractors. A restrictive feature of this new type of dynamical system is the requirement that the coupled subsystems be identical or nearly identical [@calistus11]. It is this feature that makes electronic circuit models especially attractive, since, as we have demonstrated, two chaotic circuits can be sufficiently matched to give rise to extreme multistability when appropriately coupled. It is likely that extreme multistability will be a rarity in most physical, chemical and biological systems; however, the combined conservative and dissipative features give rise to dynamics that might find technological uses, such as the ability to easily select qualitatively different dynamical states from an infinite number of possibilities. [19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , **, (). , , , , , (). , , (). , , , , , , , (). , , , , (). , , (). , , , , (). , , , , , (). , , (). , , , , (). , , , , , (). , , , , (). , (, , ), ed. , , , **, (). , , (). , , (). , (), <http://www.ccreweb.org/documents/physics/chaos/chaos.html>. , ****, (), ISSN . [^1]: email:senabhijit@gmail.com
--- abstract: 'We update our former analysis of the Nuclear Modification Factors (NMF) for different hadron species at RHIC and LHC. This update is motivated by the new experimental data from STAR which presents differences with the preliminary data used to fix some of the parameters in our model. The main change is the use of AKK fragmentation functions for the hard part of the spectrum and minor adjustments of the coalescence (soft) contribution. We confirm that observation of the NMF for the $f_0$ meson can shed light on its quark composition. [Keywords]{} Heavy Ion Collisions 12.38.Mh' author: - \| L. Maiani$^a$, A.D. Polosa$^b$, V. Riquer$^b$, C.A. Salgado$^a$\ $^a$Dip. Fisica, Università di Roma “La Sapienza” and INFN, Roma, Italy\ $^b$INFN, Sezione di Roma, Roma, Italy title: Update on Counting Valence Quarks at RHIC --- In a recent paper [@Maiani:2006ia] we have studied the $R_{CP}$ and $R_{AA}$ nuclear modification factors for different hadron species at RHIC and LHC using the coalescence model developed in [@Fries:2003vb]. It turns out that these observables are very sensitive to the number of constituent quarks of the hadron species they are calculated for. A striking evidence of this phenomenon is that the $R_{CP}$ is remarkably different for protons and pions, and for $\Lambda$’s and Kaons, being systematically higher for baryons. Some hadrons, especially the lightest scalar mesons, have controversial interpretations as for their quark structure; see e.g. [@closetorn]. We believe that there are solid reasons to understand the observed sub-GeV scalar meson nonet as four-quark states [@ourpaps]. The same structure may be shared by the first $0^+$ super-GeV multiplet. Because of the affinity of $f_0$ to Kaons, the more conventional alternative for $f_0$ is $f_0=s\bar s$. The point made in [@Maiani:2006ia] is that the $R_{CP}$ and $R_{AA}$ observables could indeed be used to discriminate the nature of the $f_0(980)$ ($s\bar s$ meson or $4q$ state). $f_0(980)$ is a good candidate for such a study since it is one of the mesons that are observable in inclusive reactions at RHIC. The approach used to describe the production of hadrons at RHIC, in the intermediate momentum range 1.5 GeV$\leq p_T\leq $4 GeV, is a combination of (i) a coalescence model for the soft part of the spectrum and (ii) fragmentation for the hard part. Parameters for the coalescence component and the inclusive jet cross sections are fitted to the inclusive production of [standard]{} hadrons, pions, p+$\bar p$, Kaons and $\Lambda$s. Fragmentation functions are taken from deep inelastic processes, typically from Z decays at LEP where large statistics are available. While the coalescence picture of a multiquark $f_0$ is a straightforward extension of the model, the fragmentation of $f_0$ requires some additional hypothesis on the functional structure of the fragmentation functions. In [@Maiani:2006ia] we used the fragmentation function set developed by [@vogel] and the following model: $$\begin{aligned} &&D_q^{f_0(4q)}(z,Q^2) \sim 0.5(1-z)^{1.5}~\frac{D_q^{\Lambda+\bar \Lambda}(z,Q^2)}{2}\nonumber\\ &&D_g^{f_0(4q)}(z,Q^2) \sim 0.1 (1-z)^{5}~\frac{D_q^{\Lambda+\bar \Lambda}(z,Q^2)}{2}\end{aligned}$$ In this note, we update our former analysis to take into account the latest data on $R_{AA}$ for $\Lambda$’s and $\Xi$’s [@Abelev:2007xp]. The new data show qualitative change with respect to the preliminary ones [@Salur:2005nj] used in [@Maiani:2006ia]. In particular, they show that the AKK fragmentation functions describe better the inclusive production of $\Lambda$+$\bar \Lambda$, which is underestimated by the ones in [@vogel]. #### New Fragmentation Functions for $f_0$ and $\Xi$. In the present analysis we have adopted the AKK fragmentation functions for $\Lambda$+$\bar \Lambda$ and changed the fragmentation model of a 4-quark meson, in order to keep the agreement with the LEP and NOMAD data on $f_0$ production. Our new parameterization is: $$\begin{aligned} &&D_q^{f_0(4q)}(z,Q^2) \sim 0.5~\frac{D_q^{\Lambda+\bar \Lambda}(z,Q^2)}{2}\nonumber\\&&D_g^{f_0(4q)}(z,Q^2) \sim 0\end{aligned}$$ The new fragmentation function is reported in Fig. \[fig:ndist\], blue curve, against the LEP data (stars). The red curve shows $zD_q^{f_0(4q)}(z,Q^2)$ compared to NOMAD data (squares). ![Comparison with OPAL and NOMAD data on $f_0(980)$. Data from [@opal; @nomad][]{data-label="fig:ndist"}](nomop.eps) For the $\Xi+\bar \Xi$, the new cross sections are reproduced by fragmentation functions with a suppression factor of $0.5$ with respect to $\Lambda + \bar \Lambda$ but no change in $z\to 1$ behavior: $$\begin{aligned} &&D_q^{\Xi+\bar \Xi}(z,Q^2) \sim 0.5~D_q^{\Lambda+\bar \Lambda}(z,Q^2)\nonumber\\&&D_g^{\Xi+\bar \Xi}(z,Q^2) \sim 0\end{aligned}$$ #### Coalescence parameters. The new data require a tuning of the coalescence parameters, to fit the reference cross sections. We report in Table \[tableold\] below the old parameters, used in [@Maiani:2006ia]. In our update we have adjusted the quark and antiquark fugacities in Au+Au at RHIC as follows: $$\begin{aligned} &&{\rm Au+Au~@~RHIC} \left\{ \begin{array}{c} \gamma_{u,d,s}=\gamma_{\bar u, \bar d}\\ \hspace{-0.6truecm}\gamma_{\bar s}=0.9 \end{array}\right .\end{aligned}$$ The comparison with the new cross sections is summarized in Fig. \[calibration\]. ![Left panel: Cross sections for production of $\Lambda$’s in proton-proton collisions. Data from [@xsects]. Right panel: Cross sections for central production of $\Lambda$’s and $\Xi$’s in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV [@Adams:2006ke]. []{data-label="calibration"}](xsectprotons.eps) ![Left panel: Cross sections for production of $\Lambda$’s in proton-proton collisions. Data from [@xsects]. Right panel: Cross sections for central production of $\Lambda$’s and $\Xi$’s in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV [@Adams:2006ke]. []{data-label="calibration"}](xsectslamxi.eps) #### Nuclear Modification Factors. In Fig. \[fig:raa\] we show the updated results for $R_{AA}$ for different hadron species. ![${\bf R_{AA}}$ ratios. Data from [@Abelev:2007xp].[]{data-label="fig:raa"}](ppbar.eps) ![${\bf R_{AA}}$ ratios. Data from [@Abelev:2007xp].[]{data-label="fig:raa"}](lamlambar.eps) ![${\bf R_{AA}}$ ratios. Data from [@Abelev:2007xp].[]{data-label="fig:raa"}](xixibar.eps) ![${\bf R_{AA}}$ ratios. Data from [@Abelev:2007xp].[]{data-label="fig:raa"}](summary.eps) In the last panel we present our new prediction for the $f_0[4q]$ vs. $f_0[s\bar s]$, compared to the theoretical curves for $\Lambda+\bar \Lambda$ and $\Xi + \bar \Xi$ of Fig. \[fig:raa\]. The new fragmentation function set induces a small variation also in the calculation of $R_{CP}$ with respect to the results shown in [@Maiani:2006ia]. We summarize our results for $R_{CP}$ of $f_0$ in Fig. \[fig:newrcp\]. ![The ${\bf R_{CP}}$ with the new set of fragmentation functions. Data for $\Lambda$+$\bar \Lambda$ from [@long].[]{data-label="fig:newrcp"}](newrcp.eps) \[tableold\] $\gamma_{u,d}$ $\gamma_{\bar u,\bar d}$ $\gamma_{s,\bar s}$ $\gamma_{{\rm periph}}$ $\begin{array}{c} \tau A_T \\ ({\small {\rm fm} ^3)}\end{array}$ v$_\perp$ $\begin{array}{c}\epsilon_0\\({\rm GeV}^{-1/2})\end{array}$ N$_{coll}$ ---------------- ---------------- -------------------------- --------------------- ------------------------- ------------------------------------------------------------------ ------------- ------------------------------------------------------------- ------------ Au+Au @RHIC 1 0.9 0.8 0.7 1.27$\cdot 10^3$ 0.55 0.82 1146 (26) Pb+Pb @LHC 1 1 1 0.7 11.5$\cdot 10^3$ 0.68 2.5 3643(82) p+p @RHIC(LHC) 0.4 0.4 0.12 (0.4) – 13.2 (119) 0.55 (0.68) 0 – : [Parameters of the hadron production model in Au+Au @ RHIC, Pb+Pb @LHC and p+p, from Ref. [@Maiani:2006ia]. Numbers in parentheses in the last column refer to peripheral collisions with $b=12$ fm at RHIC and $b=14$ fm at LHC. ]{} \ [99]{} L. Maiani, A. D. Polosa, V. Riquer and C. A. Salgado, Phys. Lett. B [645]{}, 138 (2007) \[arXiv:hep-ph/0606217\]. R. J. Fries, B. Muller, C. Nonaka and S. A. Bass, Phys. Rev. Lett. [90]{}, 202303 (2003) \[arXiv:nucl-th/0301087\]. F. E. Close and N. A. Tornqvist, J. Phys. G [28]{}, R249 (2002) \[arXiv:hep-ph/0204205\]. L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. Lett. [93]{}, 212002 (2004) \[arXiv:hep-ph/0407017\]; Eur. Phys. J. C [50]{}, 609 (2007) \[arXiv:hep-ph/0604018\]; L. Maiani, A. D. Polosa and V. Riquer, arXiv:hep-ph/0703272. D. de Florian, M. Stratmann and W. Vogelsang, Phys. Rev. D [57]{}, 5811 (1998) \[arXiv:hep-ph/9711387\]. B. I. Abelev [et al.]{} \[STAR Collaboration\], arXiv:0705.2511 \[nucl-ex\]. S. Salur \[STAR Collaboration\], Nucl. Phys. A [774]{}, 657 (2006) \[arXiv:nucl-ex/0509036\]. S. Albino, B. A. Kniehl and G. Kramer, Nucl. Phys. B [725]{}, 181 (2005) \[arXiv:hep-ph/0502188\]; Nucl. Phys. B [734]{}, 50 (2006) \[arXiv:hep-ph/0510173\]. K. Ackerstaff [et al.]{} \[OPAL Collaboration\], Eur. Phys. J. C [4]{}, 19 (1998) \[arXiv:hep-ex/9802013\]. P. Astier [et al.]{} \[NOMAD Collaboration\], Nucl. Phys. B [601]{} (2001) 3 \[arXiv:hep-ex/0103017\]. H. Long \[STAR Collaboration\], J. Phys. G [30]{}, S193 (2004). B. I. Abelev [et al.]{} \[STAR Collaboration\], Phys. Rev. C [75]{}, 064901 (2007) \[arXiv:nucl-ex/0607033\]. J. Adams [et al.]{} \[STAR Collaboration\], Phys. Rev. Lett. [98]{}, 062301 (2007) \[arXiv:nucl-ex/0606014\]. See also [http://www.star.bnl.gov/STAR/all/physicsdatabase/66/data.html]{}
--- abstract: 'The tunable magnetism at graphene edges with lengths of up to 48 unit cells is analyzed by an exact diagonalization technique. For this we use a generalized interacting one-dimensional model which can be tuned continuously from a limit describing graphene zigzag edge states with a ferromagnetic phase, to a limit equivalent to a Hubbard chain, which does not allow ferromagnetism. This analysis sheds light onto the question why the edge states have a ferromagnetic ground state, while a usual one-dimensional metal does not. Essentially we find that there are two important features of edge states: (a) umklapp processes are completely forbidden for edge states; this allows a spin-polarized ground state. (b) the strong momentum dependence of the effective interaction vertex for edge states gives rise to a regime of partial spin-polarization and a second order phase transition between a standard paramagnetic Luttinger liquid and ferromagnetic Luttinger liquid.' author: - 'David J. Luitz' - 'Fakher F. Assaad' - 'Manuel J. Schmidt' bibliography: - '../paper\_bosonization/tem\_bib.bib' - 'emw.bib' title: Exact diagonalization study of the tunable edge magnetism in graphene --- Introduction ============ Edge state models\[model\_section\] =================================== Exact diagonalization\[ed\_section\] ==================================== Discussion\[discussion\] ======================== D.J.L. and F.F.A. acknowledge financial support from the DFG for grant AS120/4-3. M.J.S. acknowledges financial support from the Swiss NSF and from the NCCR QSIT. Variational analysis of the generalized model\[appendix\_mean\_field\] ====================================================================== Exact diagonalization of the direct model\[appendix\_ed\_original\_model\] ==========================================================================
--- abstract: 'One of the key science projects of the Low-Frequency Array (LOFAR) is the detection of the cosmological signal coming from the Epoch of Reionization (EoR). Here we present the LOFAR EoR Diagnostic Database (LEDDB) that is used in the storage, management, processing and analysis of the LOFAR EoR observations. It stores referencing information of the observations and diagnostic parameters extracted from their calibration. This stored data is used to ease the pipeline processing, monitor the performance of the telescope and visualize the diagnostic parameters which facilitates the analysis of the several contamination effects on the signals. It is implemented with PostgreSQL and accessed through the psycopg2 python module. We have developed a very flexible query engine, which is used by a web user interface to access the database, and a very extensive set of tools for the visualization of the diagnostic parameters through all their multiple dimensions.' author: - 'O. Martinez-Rubi$^1$, V. K. Veligatla$^1$, A. G. de Bruyn$^{1,2}$, P. Lampropoulos$^2$, A. R. Offringa$^1$, V. Jelic$^{1,2}$, S. Yatawatta$^2$, L. V. E. Koopmans$^1$, and S. Zaroubi$^1$' bibliography: - 'author.bib' title: 'LEDDB: LOFAR Epoch of Reionization Diagnostic Database' --- Introduction ============ The Low-Frequency Array (LOFAR) is an antenna array that observes at low radio frequencies (10 - 240 MHz). It consists of about 70 stations spread around Europe that combine their signals to form an interferometric aperture synthesis array . The LOFAR Epoch of Reionization (EoR) experiment is one of the key science projects (KSP) of LOFAR. It aims to study the redshifted 21-cm line of neutral hydrogen from the Epoch of Reionization . There are many challenges that need to be overcome in order to meet this goal including strong astrophysical foreground contamination, ionospheric distortions, complex instrumental response and different types of noise. The very faint signals from neutral hydrogen require hundreds of hours of observation thereby accumulating petabytes of data. To diagnose and monitor the various instrumental and ionospheric parameters, as well as manage the data, we have developed the LEDDB (LOFAR EoR Diagnostic Database). Its main tasks and uses are: - To store referencing information of the observations, mainly the locations of the data but also other indexing information. - To store diagnostic parameters of the observations extracted through calibration. - To facilitate efficient data management and pipeline processing. - To monitor the performance of the telescope as a function of date. - To visualize the diagnostic parameters. For example we can observe the complex gain of all the stations as a function of time and frequency to visualize ionospheric distortion affecting large part of the array. Data flow ========= The data from the stations is sent to the Central Processing Facility (CEP) located in Groningen (the Netherlands), where it is correlated among other processing steps. Afterwards, the data is stored in the Long Term Archive (LTA) in Groningen. From the LTA we copy the data to the LOFAR EoR CPU/GPU cluster, also in Groningen, where we process it with the LOFAR EoR pipeline. The LEDDB takes care of storing the locations of the data both in the LTA and the LOFAR EoR cluster. It also stores all the diagnostic data produced by the pipeline. Since we can not keep all the data in the LOFAR EoR cluster, we must archive it in the LTA but thanks to the LEDDB we retain access to all its diagnostic information. Database definition =================== The LEDDB is implemented with PostgreSQL and accessed through a python interface provided by the psycopg2 module. It is part of a research project with still evolving requirements, so one of the key points of the design was to make it flexible enough to meet new requirements such as the addition of new diagnostic parameters. The content of the database is categorized under three different blocks: the referencing information, the diagnostic data and the meta-data. In figure \[fig:leddber\] we show the Entity-Relationship diagram of the database with its blocks, the tables involved and their relationships. ![Entity-Relationship diagram of the LEDDB. Only table names and key columns are shown.[]{data-label="fig:leddber"}](O23_f1.eps) \(1) The referencing information block (“REF” in figure \[fig:leddber\]) contains five primary tables: LOFAR\_DATASET (LDS), LOFAR\_DATASET\_BEAM (LDSB), LOFAR\_DATASET\_BEAM\_PRODUCT (LDSBP), MEASUREMENTSET (MS) and [MEASUREMENTSET\_PRODUCT]{} (MSP). They contain information about the observations: their names, date and time information, the pointed fields and other indexing information. They also store the locations of the data, i.e., the host and cluster the data is in and the path to the files. The rest of tables in this block are the secondary tables which are only used to ease the selection on the primary ones. \(2) The diagnostic data block (“DIAG” in figure \[fig:leddber\]) contains the diagnostic parameters related to the observations. There are four primary tables: the GAIN table and three QUALITY tables. They store the gain solutions of the stations and baseline-based, frequency-based and time-based statistic parameters of the data. There is also a secondary table in this block called QUALITY\_KIND. \(3) Finally the meta-data block (“META” in figure \[fig:leddber\]) stores information regarding the relationships of the referencing section and the diagnostic data. Each one of the referencing tables is joined with each related meta-data table. The LEDDB can generate a RefFile or a DiagFile. A RefFile is a file containing locations of data related to the observations. This file is used in the LOFAR EoR pipeline processing tasks. On the other hand, a DiagFile contains references to diagnostic data in the LEDDB. Diagnostic data analysis ======================== The diagnostic data can have multiple dimensions: Time, frequency, baseline (interferometer), station, polarization correlations and other ones depending on the situation. In general they are complex numbers. We provide plotting and animation tools implemented with matplotlib to analyse such multi-dimensional data. In figure \[fig:gain\] we show an example of one of the produced plots. ![Gain as a function of time of one of the polarization auto-correlations of two different stations at 138 MHz for the observation L60639 (Elais field). Note the phase difference between a core station (CS001HBA0) and a remote station (RS508HBA), mainly caused by the ionosphere.[]{data-label="fig:gain"}](O23_f2.eps) Query engine and User Interface =============================== The query engine is a python API which provides fast and flexible access to the database. We use a python based web server (cherrypy) to interface with the query engine. The client-side user interface (UI) in the web page is implemented with JQueryUI framework. In figure \[fig:webui\] we show a snapshot of the web UI. ![Snapshot of the web UI. Each tab in the UI represents a primary table in the database.[]{data-label="fig:webui"}](O23_f3.eps) We estimate that 10 terabytes of diagnostic data will be stored in the LEDDB for the full LOFAR EoR KSP (currently it is 75 gigabytes). In addition to the size challenge, the number of rows of some of the tables is the most important aspect to be taken into account in the design of the database and its query engine, and it is actually the main bottleneck in the queries. We have managed to provide a fast access thanks to efficient table indexing, the minimization of the number of join operations and the use of persistent connections eased by the session handling provided by the cherrypy framework. The query engine provides functionality to sort, filter by column values and by selection in primary and secondary tables. This is used by the UI to provide a very extensive set of options for accessing the data. The UI allows the user to create both RefFiles and DiagFiles. Besides, this UI can be used to launch pipeline jobs with a RefFile and directly plot diagnostic data with a DiagFile. Future developments =================== We will focus on minimizing the access times while the database is growing and improving the tools to analyse the diagnostic parameters. Possibly new diagnostic parameters will added. There is also a plan to migrate the database to a new server specially designed for its purpose. We are indebted to Eite Tiesinga for his assistance in all the matters related to the LOFAR EoR cluster. We also thank the rest of the LOFAR EoR group core members for their contribution. (www.astro.rug.nl/eor/people/core-members).
--- abstract: 'OEDIPUS is a Monte Carlo simulation program which can be used to determine the small-$x$ evolution of a heavy onium using Mueller’s colour dipole formulation, giving the full distribution of dipoles in rapidity and impact parameter. Routines are also provided which calculate onium-onium scattering amplitudes between individual pairs of onium configurations, making it possible to establish the contribution of multiple pomeron exchange terms to onium-onium scattering (the unitarisation corrections).' --- Cavendish-HEP-95/07\ hep-ph/9601220\ January 1996 OEDIPUS: Onium Evolution, Dipole Interaction and Perturbative Unitarisation Simulation[^1] G.P. Salam\ Cavendish Laboratory, Cambridge University,\ Madingley Road, Cambridge CB3 0HE, UK\ e-mail: salam@hep.phy.cam.ac.uk PROGRAM SUMMARY {#program-summary .unnumbered} =============== Title of program: OEDIPUS Catalogue number: Program obtainable from: [ftp://axpf.hep.phy.cam.ac.uk/pub/theory/oedipus.tar.gz]{} , see also [http://www.hep.phy.cam.ac.uk/theory/software/oedipus.html]{} Licensing provisions: None Computers: Tested on Dec Alpha, Sun Operating system: Unix (OSF3.2, SunOS-4.1.3, Solaris-2.4) Program language used: Fortran-90 Memory required to execute with typical data: 1–10 MB No. of bits in a word: 32/64 No. of lines in distributed program, including test data, etc.: 6186 Keywords: Small-$x$; BFKL; Pomeron; Onium-onium scattering; Dipoles; Monte Carlo; Unitarity. Nature of the physical problem BFKL evolution \[1\] of a heavy (quark-)onium, to give full information on the rapidity and transverse positions of gluons carrying a small fraction $x$ of the onium’s longitudinal momentum. This information can be used to calculate a variety of onium-onium scattering cross sections, including multiple pomeron contributions, which restore unitarity. Method of solution A Monte Carlo simulation of the evolution in rapidity of the gluon structure of the onium, using Mueller’s colour dipole formulation of small-$x$ physics \[2\]. Restrictions on the complexity of the problem The number of dipoles (gluons) in an onium grows as an exponential of the rapidity, and inversely with the value of a cutoff used to regulate an ultra-violet divergence. The time for evolution of an onium is proportional to the number of dipoles, while that for calculation of the onium-onium scattering amplitude goes as the square of the number of dipoles. Memory restrictions also arise from the need to store large configurations of dipoles (very large fluctuations can occur in the number of dipoles present). Typical running time On a DEC Alpha 3000 processor, evolution generates about 10000 gluons per second. At a rapidity $Y=16$ and an ultra-violet cutoff of $0.01$ times the onium size, a day’s running will generate about $10^5$ onium-onium scatterings. References \[1\] Y. Y. Balitskiǐ and L. N. Lipatov, Sov. Phys. JETP 28 (1978) 822;\ E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP 45 (1977) 199;\ L. N. Lipatov, Sov. Phys. JETP 63 (1986) 904. \[2\] A. H. Mueller and B. Patel, Nucl. Phys. B 425 (1994) 471;\ A. H. Mueller, Nucl. Phys. B 437 (1995) 107. [LONG WRITE-UP]{} Introduction ============ There has been much interest recently in BFKL [@BaLi78; @KuLF77; @Lipa86] type processes and their possible observation at HERA and the Tevatron. Calculation of the cross sections for a number of these processes, such as diffractive dissociation (see e.g.[@BaLW95a; @BiPe95b]) or exclusive vector meson production (e.g.[@BFLW96]) requires detailed information about the dominant exchanged “object”, known as the BFKL pomeron; for example one needs an understanding of the triple pomeron vertex and of the transverse distribution of small-$x$ gluons. At present, the only BFKL type process which is fully calculable in perturbative QCD is high energy onium-onium scattering. Mueller’s colour dipole formulation of this process [@Muel94a; @MuPa94; @Muel94b; @Muel95] (for a related approach, see [@NiZa93a; @NiZZ94a]), offers a well defined way of performing the necessary calculations, in the large $N_C$ approximation, where $N_C$ is the number of colours. The colour dipole formulation of small-$x$ evolution and high energy onium-onium scattering is particularly suited to Monte Carlo simulation, since the evolution is probabilistic in nature, and because each branching (of one dipole into two) is independent of all other branchings. This paper is divided into three parts. The first gives a brief overview of the small-$x$ dipole evolution of an onium and of the issues relevant to a Monte Carlo simulation of such an evolution. The second section examines the use of the dipole structures of a pair of evolved onia to calculate onium-onium scattering amplitudes, including the multiple pomeron exchange contributions which restore unitarity. The final section gives a detailed description of the structure and use of the OEDIPUS package. Dipole structure of an onium ============================ Background ---------- One starts with an onium where the transverse separation between the quark and anti-quark is ${\ensuremath{\mathbf{b}}}_{01}$. The probability of generating a gluon at a point ${\ensuremath{\mathbf{b}}}_2$ carrying a small fraction $e^{-y}$ of the light-cone momentum of the onium is [@Muel94a]: $$\frac{{\ensuremath{\mathrm{d}}}P}{{\ensuremath{\mathrm{d}}}y {\ensuremath{\mathrm{d}}}^2 {\ensuremath{\mathbf{b}}}_2} = e^{-y / \lambda} \frac{{\ensuremath{\alpha_S}}N_C}{2\pi^2} \frac{b_{01}^2}{b_{02}^2 b_{12}^2}. \label{eq:trns_rte}$$ $N_C$ is the number of colours ($N_C = 3$ for QCD), ${\ensuremath{\alpha_S}}$ is the strong coupling constant; $\lambda$, the effective lifetime in rapidity ($y$) of the dipole, is related to the virtual corrections (or more intuitively, conservation of probability): $$\lambda^{-1} = \frac{{\ensuremath{\alpha_S}}N_C}{2\pi^2} \int {\ensuremath{\mathrm{d}}}^2{\ensuremath{\mathbf{b}}}_2 \frac{b_{01}^2}{b_{02}^2 b_{12}^2}. \label{eq:virtual}$$ In the limit of large $N_C$, the colour structure means that the two new dipoles which arise (${\ensuremath{\mathbf{b}}}_{02}$ and ${\ensuremath{\mathbf{b}}}_{12}$) can themselves independently emit gluons, while the original colour dipole is effectively destroyed. This branching of dipoles repeats itself until the rapidity of any new gluons which would be produced exceeds the maximum rapidity available. Through this branching process, one can establish the probability of any given configuration of dipoles. Notes on implementation {#sc:dipimp} ----------------------- One of the main aspects of eq. (\[eq:trns\_rte\]) is that it has non-integrable divergences at $b_{02} = 0$ and $b_{12} = 0$. This leads to the integral for $\lambda^{-1}$ being infinite. A solution to this problem is to introduce a cutoff on the dipole size, eliminating any region of ${\ensuremath{\mathbf{b}}}_2$ where $b_{02} < \rho$ or $b_{12} < \rho$. This has to be used in both eq. (\[eq:trns\_rte\]) and eq. (\[eq:virtual\]). The lifetime of a dipole of size $b$ is therefore a function of $b/\rho$. In the limit of small $\rho$ the effects of the cutoff should disappear. However the number of small dipoles is large: the number of dipoles of size $c$ from an onium of size $b$, after evolution through $y$ is approximately: $$n^{(1)}(c,b,y) {\ensuremath{\mathrm{d}}}\log c \simeq \frac {b}{c \sqrt{\pi k y}} \exp({\ensuremath{({\ensuremath{\alpha_{\mathcal{P}}}}-1)}}y - \log(c/b)^2/ky) {\ensuremath{\mathrm{d}}}\log c. \label{eq:n1}$$ This is valid for $\|\log(c/b)\| \ll ky$. The BFKL power is ${\ensuremath{({\ensuremath{\alpha_{\mathcal{P}}}}-1)}}= 4 \log 2 {\ensuremath{\alpha_S}}N_C / \pi$, and $k = 14{\ensuremath{\alpha_S}}N_C \zeta(3)/\pi$, with $\zeta(3) \simeq 1.202$ being the Riemann zeta function. The smallest dipoles will be of size $c\sim \rho$, so as $\rho$ is lowered the number of dipoles rises. In addition, the number of dipoles increases exponentially with rapidity. The time taken to generate an onium is proportional to the number of dipoles. The time to calculate onium-onium interactions is proportional to the product of the number of dipoles in the two onia. A final complication will arise because there are very large fluctuations in the numbers of dipoles [@Sala95] — the probability of obtaining a configuration with $n$ dipoles goes as $\exp[-\pi (\log n)^2/4{\ensuremath{\alpha_S}}N_C y]$, so that one has to allow for configurations which are very much larger than the mean. Since configurations generally need to be stored (for example to investigate interactions between pairs of configurations), this can lead to considerable memory consumption, especially at large rapidities. The facility of imposing an upper limit on dipole sizes during the evolution is also included, to allow a crude investigation into the uncertainties due to infra-red effects. When the implementation of the upper cutoff is turned on, both dipoles produced from a branching are required to be smaller than the upper cutoff. The lifetimes of dipoles are adjusted accordingly. The evolution is carried out with fixed ${\ensuremath{\alpha_S}}$, which is justified in the limit of very heavy onia. There is no known unique way of including a running coupling constant, however it would be possible to modify the program to implement a scheme such as that used by Nikolaev and Zakharov [@NiZa93a]. Onium-onium scattering ====================== Background ---------- One wishes to obtain the amplitude $F({\ensuremath{\mathbf{b}}}, {\ensuremath{\mathbf{b}}}', {\ensuremath{\mathbf{r}}}, Y)$, for elastic scattering between two onia of sizes (and orientations) ${\ensuremath{\mathbf{b}}}$ and ${\ensuremath{\mathbf{b}}}'$, whose centres are separated by a transverse distance ${\ensuremath{\mathbf{r}}}$, with a total rapidity between them of $Y$ ($\simeq\log s$, where $\sqrt{s}$ is the centre of mass energy). Let $\gamma$ be a particular dipole configuration for an onium, which contains dipoles of position and size $({\ensuremath{\mathbf{r}}}_1,{\ensuremath{\mathbf{c}}}_1) \ldots ({\ensuremath{\mathbf{r}}}_{n_\gamma}, {\ensuremath{\mathbf{c}}}_{n_\gamma})$. The interaction between two such onia, moving in opposite directions is [@MuPa94; @Muel94b; @Muel95]: $$f_{\gamma, \gamma'} = \sum_{i = 1}^{n_{\gamma}} \sum_{j = 1}^{n_{\gamma'}} f({\ensuremath{\mathbf{r}}}_{i} - {\ensuremath{\mathbf{r}}}_{j}', {\ensuremath{\mathbf{c}}}_{i}, {\ensuremath{\mathbf{c}}}_{j}'), \label{eq:sumamps}$$ where $f({\ensuremath{\mathbf{r}}}, {\ensuremath{\mathbf{c}}}_{i}, {\ensuremath{\mathbf{c}}}_{j}')$, the interaction between dipoles of size ${\ensuremath{\mathbf{c}}}$ and ${\ensuremath{\mathbf{c}}}'$, whose centres are separated by ${\ensuremath{\mathbf{r}}}$ is [@Muel94b; @Sala95b]: $$f({\ensuremath{\mathbf{r}}}, {\ensuremath{\mathbf{c}}}, {\ensuremath{\mathbf{c}}}') = \frac{{\ensuremath{\alpha_S}}^2}{2} \left[ \log \frac{ \|{\ensuremath{\mathbf{r}}}+ {\ensuremath{\mathbf{c}}}/2 - {\ensuremath{\mathbf{c}}}'/2\| \|{\ensuremath{\mathbf{r}}}- {\ensuremath{\mathbf{c}}}/2 + {\ensuremath{\mathbf{c}}}'/2\| } {\|{\ensuremath{\mathbf{r}}}+ {\ensuremath{\mathbf{c}}}/2 + {\ensuremath{\mathbf{c}}}'/2\| \|{\ensuremath{\mathbf{r}}}- {\ensuremath{\mathbf{c}}}/2 - {\ensuremath{\mathbf{c}}}'/2\| } \right]^2. \label{eq:dipdip}$$ One has to average $f_{\gamma, \gamma'}$ over dipole configurations $\gamma$, $\gamma'$ of the two onia. Therefore at the level of one pomeron exchange, the elastic amplitude is: $$F^{(1)}({\ensuremath{\mathbf{b}}},{\ensuremath{\mathbf{b}}}',{\ensuremath{\mathbf{r}}},Y) = - \sum_{\gamma, \gamma'} P_{\gamma}({\ensuremath{\mathbf{r}}}_0, {\ensuremath{\mathbf{b}}}, y) P_{\gamma'}({\ensuremath{\mathbf{r}}}_0 + {\ensuremath{\mathbf{r}}}, {\ensuremath{\mathbf{b}}}', Y-y) f_{\gamma, \gamma'}, \label{eq:mcunit}$$ where $P_\gamma({\ensuremath{\mathbf{r}}}_0, {\ensuremath{\mathbf{b}}}, y)$ is the probability of obtaining a dipole configuration $\gamma$ after evolution through $y$ of an onium of size ${\ensuremath{\mathbf{b}}}$ centred at ${\ensuremath{\mathbf{r}}}_0$. Note that in this case the result can be shown to be independent of $y$, the division of rapidity between the two onia (or equivalently of the longitudinal frame in which the calculation is performed). One can also calculate contributions to the amplitude involving the exchange of $k$ pomerons [@Muel94b]: $$F^{(k)}({\ensuremath{\mathbf{b}}},{\ensuremath{\mathbf{b}}}',{\ensuremath{\mathbf{r}}},Y) = \frac{1}{k!} \sum_{\gamma, \gamma'} P_{\gamma}({\ensuremath{\mathbf{r}}}_0, {\ensuremath{\mathbf{b}}}, y) P_{\gamma'}({\ensuremath{\mathbf{r}}}_0 + {\ensuremath{\mathbf{r}}}, {\ensuremath{\mathbf{b}}}', Y-y) (-f_{\gamma, \gamma'})^k.$$ These multi-pomeron terms, though formally non-leading in $1/N_C$, are enhanced at high rapidities by the large numbers of dipoles in each onium. All numbers of pomeron exchange can be resummed to give an amplitude $F$ which explicitly satisfies the unitarity bound: $$F^{(k)}({\ensuremath{\mathbf{b}}},{\ensuremath{\mathbf{b}}}',{\ensuremath{\mathbf{r}}},Y) = - \frac{1}{k!} \sum_{\gamma, \gamma'} P_{\gamma}({\ensuremath{\mathbf{r}}}_0, {\ensuremath{\mathbf{b}}}, y) P_{\gamma'}({\ensuremath{\mathbf{r}}}_0 + {\ensuremath{\mathbf{r}}}, {\ensuremath{\mathbf{b}}}', Y-y) (1 - {\ensuremath{\mathrm{e}}}^{-f_{\gamma, \gamma'}}).$$ These last two equations are approximations based on there being a large number of dipoles interacting. Multiple pomeron contributions should be calculated in the frame $y = Y/2$, where corrections that would arise from wave-function saturation, which is not being calculated, are expected to be smallest [@Muel94b] (see also [@MuSa96]). Implementation considerations {#sc:ooimp} ----------------------------- One of the main difficulties in calculating onium-onium scattering amplitudes is the large numbers of interactions which have to be worked out: in principle the product of the numbers of dipoles in each onium, multiplied by the number of points at which one needs to know the interaction. The first way to ease the problem is to note that the dipole-dipole interaction, eq. (\[eq:dipdip\]), is relatively local: at large distances it dies off as $1/r^4$ where $r$ is the separation between the centres of the dipoles. Therefore it should be safe to neglect the interaction between dipoles if their separation is sufficiently large, say greater than twice the sum of their sizes. This is found to reproduce the total dipole-dipole interaction to better than one percent[^2]. One generally wants to know the amplitude at many different impact parameters (i.e. onium-onium separations), for example to work out the total cross section (with the normalisation used above, just twice the amplitude, integrated over all impact parameters). One way to do this is to work out the interaction on a grid. Since there are large fluctuations in the spatial extent of the dipole configurations, the grid size has to be variable — it should completely cover the area where dipole-dipole interactions are significant. By summing the interactions at each point of the grid one obtains an approximation to the total amplitude for that configuration pair (the finer the grid, the better the approximation). This approach is necessary if one wants to find a good approximation to the total amplitude for each configuration pair, however it is slow and does not offer any easy way to store the results so that they can be retrieved for later analysis (say if one only wanted to look at 1 and 2 pomeron exchange, but after the run, decided that 3 pomeron exchange would also be interesting) — if one uses a grid which is $100\times 100$, then one requires 40 kilobytes per pair of configurations, for a single value of rapidity. To gain adequate accuracy one needs several tens of thousands of events; with, say, 5 values of rapidity this gives several gigabytes of data. One reason why it is important to sample a large number of configuration pairs, is that rare, large configurations contribute significantly to the total cross section [@Sala95b]. Multiple pomeron contributions are dominated by rare, dense configurations. For a given pair of dipole configurations, however, the variation of the amplitude (at those points where it is significant) with impact parameter is not so large. So it can be advantageous to evaluate fewer points in impact parameter (i.e. making a worse approximation to the amplitudes of individual configuration pairs) in exchange for a larger sample of configurations. The limit of this is to evaluate the onium-onium interaction, not on a grid, but along a radial line. In addition, because there are fewer points in impact parameter it becomes tractable to store the results. In fact the individual amplitudes are not stored event by event, rather a probability distribution for the amplitude is stored at consecutive bins in radial position. This information can then be retrieved and used to determine whichever quantity one is interested in. Structure and use of OEDIPUS ============================ The program comes in many different files in various directories: `basic_src/` contains all the common subroutines which are involved with data input and output, initialisation and evolution of onia, as well as routines for determining onium-onium interaction amplitudes. It does not contain any code for analysing the results from the evolution or interaction: this must be provided as a “driver” program by the user. A number of examples are to be found in the directory [samples/]{}, including drivers to examine the spatial distribution of dipoles in the onium wave function and to determine the total and elastic scattering cross sections. Storage of gluons and dipoles and evolution ------------------------------------------- Gluons and dipoles each have a specific data type: type gluon complex(kind(1d0)) :: psn ! stores the transverse position double precision :: rapidity end type gluon Complex variables are used throughout the program to deal with transverse positions. The dipole type is type dipole type(gluon), pointer :: lo_y_gluon, hi_y_gluon double precision :: size type(dipole), pointer :: child_1, child_2 integer :: index end type dipole It is useful to separate the gluon content of the dipole into high rapidity and low rapidity, mainly for ease of coding. In the case of the initial onium, the two quarks are represented by [gluon]{} types. The [child\_1]{} and [child\_2]{} pointers are needed to store the tree structure of the evolved onium. If they are not [associated]{} then the dipole has no descendents (for example, because any descendents would have a rapidity larger than the maximum of the evolution). Each dipole in an onium has a different [index]{} — this can be useful in analysing the branching structure. The following sequence of routines is needed to produce an onium and evolve it, for each loop (or “event”) of the program. First, various internal counters must be reset by calling subroutine reset_counters Then an onium is produced with subroutine init_onium(onium, onm_size, phi) This returns a pointer [onium]{} to a dipole of size [onm\_size]{}, centred at $(0,0)$, with orientation [phi]{}. To produce a randomly oriented dipole, call subroutine init_onium_rnd_phi(onium, onm_size) The onium is evolved up to a rapidity [maxy]{} by calling subroutine evolve_onium(onium, maxy, n_tree) where, on return, [n\_tree]{} is the total number of dipoles in the tree. To be able to analyse the results, one wants only the dipoles associated with a particular rapidity [y]{}, so one must extract them with: subroutine extract_onium_dpls(onium, y, extrctd_dpls, n_extrctd) The array [extrctd\_dpls(:)]{} is of type [dpl\_pntr]{} (arrays of pointers are not permitted in Fortran 90 — the solution to this is to define a type which contains nothing but a pointer), and on return contains pointers to the [n\_extrctd]{} dipoles. The array must be sufficiently large: one can ensure that it is, by dynamically allocating it to be of size [(n\_tree / 2 + 1)]{}. The array of dipole pointers can then be processed by the user. This sequence allows one to first evolve the onium up to the maximum rapidity that one is interested in, and then examine the dipole structure over a range of rapidities. Additional onia (for use simultaneously) can be produced, evolved, and the dipoles extracted, by repeating the same set of calls (omitting the call to [reset\_counters]{}). Onium-onium interaction {#sc:ooprog} ----------------------- Two sets of routines are provided which calculate amplitudes for onium-onium scattering. The first returns a reasonable estimate for the elastic scattering amplitude between a pair of onium configurations, for all relevant impact parameters: subroutine onm_onm_grid(ext_dpls1, n1, ext_dpls2, n2, grid, gh_lo, & & gh_int, gv_lo, gv_int) It takes a pair of dipole sets ([n1]{} dipoles in [ext\_dpls1]{} and [n2]{} dipoles in [ext\_dpls2]{}) and determines the range of relative separations over which they interact. The onium-onium interaction is then determined for each point of a grid covering this area ([grid(:,:)]{} must be a two-dimensional square array; its size determines the resolution of the sampling) — the program returns information relating the array to the physical grid being used: the horizontal and vertical positions of the bottom left hand edge of the grid ([gh\_lo]{} and [gv\_lo]{}), as well as the horizontal and vertical spacing (or interval) of the grid ([gh\_int]{} and [gv\_int]{}). For each dipole pair, it only works out the contribution to those points on the grid where the dipole-dipole separation is not too large (as discussed in section \[sc:ooimp\]). The amplitudes are always worked out at the centres of the grid rectangles. The second approach works out a binned probability distribution for the amplitude in many sections along a radial line. The details of the binning are held in the [binning]{} module which provides a variable [bn]{} of type [bin\_prms]{}: type bin_prms ! binning vars for amplitude prob distributions integer :: n_f_lo, n_f_hi, nf double precision :: f_mid, f_lo_lgint , f_hi_lgint ! binning vars for r integer :: n_r_lo, n_r_hi, nr double precision :: r_mid, r_hi_lgint end type bin_prms The binning of the amplitude probability distribution is divided into two regions: the lower region is for amplitudes below [f\_mid]{} and the bins are indexed 0 to ([n\_f\_lo - 1]{}), logarithmically spaced with an interval [f\_lo\_lgint]{}. It is important to have a bin explicitly for the absence of interaction (otherwise when integrating the interaction over a large area the smallest bin amplitude could contribute significantly) — this is bin $-1$. The lowest non-zero bin must be sufficiently small to accurately reproduce the integral of the dipole-dipole interaction (bearing in mind that because the interaction dies off as $1/r^4$, there are regions where the amplitude can be quite small, but whose overall contribution to the amplitude is non-negligible). It is to allow adequate coverage of this wide range of small amplitudes that logarithmic binning (in the small amplitude region) is used. For amplitudes above [f\_mid]{} the binning is performed logarithmically with an interval [f\_hi\_lgint]{}. The large amplitude region will contribute more significantly to the total amplitude and dividing the binning into two regions allows one to sample it more accurately in the regions which are more important. The reason to have logarithmic binning for large amplitudes is to be able to store the very large amplitudes which can occur at large rapidities. It is sometimes useful to have the index of the highest amplitude bin, which is [nf = n\_f\_lo + n\_f\_hi]{}. The variable [bn]{} is initialised with default values which reproduce total amplitudes to an accuracy of about one percent. The binning in $r$, the distance along the radius, is also divided into two regions, but for different reasons: there are occasional events with very large transverse extents. To be able to include them with linearly spaced bins would require a prohibitively large number of bins. So for large $r$, logarithmic spacing of the bins is useful. But for small $r$, the most efficient sampling is one where each bin corresponds to a region of similar area — linear binning in $r$ is more appropriate there. There are bins numbered 0 to ([n\_r\_lo - 1]{}) linearly spaced from $r = 0$ to [r\_mid]{}. Above this, there are [n\_r\_hi]{} logarithmically spaced bins, with an interval [r\_hi\_lgint]{}. To help bin and “unbin” the data, the following functions are provided: function bin_of_f(f) returns the bin number associated with an amplitude [f]{}. If the amplitude falls into a bin higher than [nf]{} then it is put into bin [nf]{} — this serves as a form of overflow procedure: if the highest bin contains any entries then the program should have been run with a larger number of amplitude bins. The inverse function function f_of_bin(bin) returns the value of the amplitude corresponding to the (logarithmic) centre of the bin (on average, the optimal value to use in reconstructing amplitudes). For a given dipole-dipole pair, it is useful to know to which radial bins their interaction will contribute. The function function irad(r) aids this by returning the bin corresponding to a given radius (the interaction region for a pair of dipoles is defined by two radii, which are converted into the range of bins which must be sampled). For each radial bin, it is useful to sample at a random radius within the bin. The random value of $r$ returned by function rad_rnd(i) has a probability distribution which increases linearly with $r$ within the bin [i]{}, replicating the distribution of radii that would be obtained by choosing a random position in the corresponding ring. Finally it can be necessary to know the lower radius of a given bin (for example when analysing binned results), which is obtained by calling rad_lo(i) Given a pair of dipoles sets from evolved onia ([n1]{} dipoles in [ext\_dpls1]{} and [n2]{} dipoles in [ext\_dpls2]{}), the following routine subroutine add_f_to_bins(ext_dpls1, n1, ext_dpls2, n2, fbins_single) adds $1$ to the appropriate amplitude bin of fbins_single(-1:bn%nf, 0:bn%nr) for a radius chosen randomly with [rad\_rnd]{} in each radial bin. Data input and output, and initialisation ----------------------------------------- A wide variety of information can be extracted from the evolved onium and the program needs to have the flexibility to input and output whatever information the user wants. In addition, it must be possible to restart a Monte Carlo run to add to previously determined data — this should be as automatic as possible. A number of routines are provided to make this simpler. read_params_fdat(onm_size, maxy, n_prev_events, n_new_events) This routine first parses the command line. Defining [argn]{} to be the $n^{th}$ command line argument, it opens [arg1.dat]{} (a file containing formatted data from the evolution) and [arg1.prm]{} (a file containing the parameters of the evolution). The number of evolutions (or events) to be performed in this run of the program ([n\_new\_events]{}) is the value of [arg2]{}. It then looks to see if [arg3]{} is present: if not (or if it is a dash) then the evolution is a continuation of a previous one and the contents of [arg1.prm]{} are read in, in the following format: jseed(1) jseed(2) r_cut_lo r_cut_hi onm_size maxy data_io n_prev_events with the following definitions: [jseed(1:2)]{} : the pair of integers used as a seed for the random number generator [HWRGEN]{}, which uses l’Ecuyer’s method, described in [@Jame90], as implemented in the HERWIG [@MWAK92] event generator; [r\_cut\_lo]{} : the lower cutoff on dipole sizes; [r\_cut\_hi]{} : the upper cutoff (if it is negative, then no upper cutoff is implemented); [onm\_size]{} : the onium size; [maxy]{} : the maximum rapidity up to which onia are evolved; [data\_io]{} : a variable provided by the user in the module [data\_prms]{} holding any additional information needed by the user’s program for treating the data (e.g. a range of dipole sizes being studied). It can be of any type (including user defined type); [n\_prev\_events]{} : the number of “events” (i.e. Monte Carlo evolutions) processed so far. The routine also performs the initialisation which is necessary before evolution can be carried out: a call to [init\_lives]{} which sets up an array tabulating dipole lifetimes as a function of their size, and a call to [nullify\_child\_set]{} which nullifies pointers to child dipoles (since Fortran 90 has no mechanism for specifying the initial status of a pointer). It is the user’s responsibility to read in the data from unit [idat]{} (since the format and nature of the data will vary from program to program). If [arg3]{} is present then the file [starters/arg3]{} (path from top directory of the distribution) is read in. The format is the same as the first three lines of [arg1.prm]{}. If either value of [jseed(1:2)]{} is zero then the default seed is used; [n\_prev\_events]{} is set to zero. It is then the user’s responsibility to initialise any data variables and provide default values for [data\_io]{}. This mechanism makes it much easier to run the same program with different values of cutoffs and rapidity (the most commonly varied parameter) by having a single set of starter files in the the [starters/]{} directory — effectively by a command line argument, rather than by having to edit a new parameter file each time. A number of starter files are provided with the distribution and are described in the file [starters/README]{}. write_params_fdat(onm_size, maxy, n_events, param_arg, dat_arg) This routine is called to output the updated parameters of the evolution (usually only the seed and the number of evolutions so far, [n\_events]{}, will have changed). It [rewind]{}s both the parameter and data devices. Normally the parameter and data devices ([param\_arg]{} and [dat\_arg]{}) will be the same as those used for input ([iprm]{} and [idat]{}) but the arguments are available (though optional) because there is occasionally a need to use different output devices (e.g. if the user wants to retain the original data). It is the user’s responsibility to output the data. The advantage of using formatted data is that it is easy to examine or plot. But it is very inefficient for storing large amounts of data. Two routines are provided for dealing with unformatted data (the parameters file is still formatted): read_params_ufdt(onm_size, maxy, n_prev_events, n_new_events) write_params_ufdt(onm_size, maxy, n_events, param_dev, ufdt_dev) They are very similar to the previous routines. The main difference is that the data device is now [iufdt]{}, which is opened for unformatted output. Since the most common use of unformatted input and output is for the binned probability distribution of the amplitude (see section \[sc:ooprog\]) the parameter file contains the additional set of parameters: [bn]{} (a variable in the [binning]{} module, of type [bin\_prms]{}) which defines the spacing of the bins. With the unformatted data files it will often be necessary to read them in again to analyse them. Because the various parameters may be needed to know how to read in and process the data a routine read_params_old_ufdt(prnt_size, maxy, n_events) is provided. It looks at [arg1]{} to determine the files to use and opens them (with the [OLD]{} specifier, to prevent them being accidentally altered). It does not perform any initialisation for the evolution. For binned amplitude data, the format is fairly constant, so routines are also provided for reading and writing the data: subroutine allc_read_fbins(n_y, n_prev_events) This allocates an array fbins(-1:bn%nf, 0:bn%nr, n_y) in accord with the format for binned data discussed in section \[sc:ooprog\] reads it in from device [iufdt]{}. The extra dimension comes in because one might want to store results for several ([n\_y]{}) values of rapidity from a single run. The reason for the argument [n\_prev\_events]{} is to produce the correct normalisation for the data: on disk, it is stored so that each bin contains the probability associated with it. When adding extra data, it is convenient to use a normalisation where each bin contains the number of events contributing to it. If [n\_prev\_events]{} is 0 then the bins are not read in, but instead initialised to zero. If the argument is not present then the normalisation of the bins is not changed. To write the data, use the routine subroutine write_fbins(n_events, iufdt_arg) It is sometimes useful to be able to specify a different device, [iufdt\_arg]{} (optional), for output than for input, for example when writing out only a portion of the data. The number of events needs to be provided so that the stored normalisation on output is consistent with each bin containing a probability. The data is stored as [real]{} rather than [double precision]{} to reduce the use of disk space. Some internal details --------------------- The gluons and dipoles which are generated by evolution have to be stored, for example so that one can then work out onium-onium interactions. One of the main problems is that one doesn’t know beforehand how many dipoles will be produced — the mean number of dipoles can be estimated, however the fluctuations above this are very large (see section \[sc:dipimp\]). A solution is to dynamically allocate memory for gluons and dipoles as the evolution proceeds. This slows down the evolution (by nearly a factor of two on some systems) because of the time needed for allocation and deallocation. The alternative is to provide fixed size arrays for the dipoles and gluons, where the size is chosen beforehand — essentially by trial and error. For cases where the largest configurations will be using a significant fraction of a machine’s resources, this has the disadvantage that by having previously allocated the memory, it is taken up for the whole duration of the job (even if there is nothing stored there), whereas with dynamic allocation, the memory is used up only when it is needed. Limits have been coded into the [gluon]{} module for the maximum number of gluons: the variable [max\_gluons]{}. For the situation with dynamic allocation, this is the maximum number of gluons in each onium. It is present simply to allow the program to handle cases which might otherwise cause it to crash because they used all the available data store for that process. In the case of the fixed arrays, the quantity represents the maximum number of gluons in total (i.e. from all evolved onia). Apart from that, when using the subroutines described above for initialising onia, evolving them, etc., the functionality should not depend at all on the method of gluon storage. There are ways round the limitations on the maximum number of dipoles. One could store very large configurations on disk (the computer would normally be doing this anyway with the swap space — but if the program is responsible then the swap space is still left for other processes). Alternatively, there are situations where the dipole structure need not be stored at all. The occasions when a stored dipole structure is needed is when looking at some kind of correlation between dipole pairs (such as onium-onium interaction), because for each dipole, one needs access to all the other dipoles. But in these situations, the time per configuration is normally the limiting factor, since this is proportional to the square of the number of dipoles. In cases where the time taken for some analysis is linear in the number of dipoles, often, it is not necessary to have knowledge of the whole configuration. So an evolution routine could be written which called an analysis routine each time it reached the “tip” of a “branch” of a dipole “tree”, and it would never have to store more than a single path from the base of the tree to the furthest “tip”. Compilation ----------- The program is provided in the form of a number of separate files, the contents of which are documented in [README]{} files. To aid in the compilation process, two shell scripts are provided which select the correct set of source files. Detailed information on their functioning is provided as part of the distribution, however a summary is given here: mkf driver [ dyn \| sml \| lrg ] This is for compiling a driver routine which uses formatted data i/o. The name of the driver should be provided without the [.f90]{} extension. The second command line argument indicates the kind of gluon and dipole storage used — the default is to use dynamic allocation (corresponding to the option [dyn]{}). Two fixed size array storage options are also provided: [sml]{} (up to 5000 gluons) and [lrg]{} (up to 400000 gluons, which with a lower cutoff of $0.01b$, should be adequate for evolution up to rapidities of $y\simeq 15$). The executable is named [driver\_dyn]{} (or [sml]{} or [lrg]{} as appropriate) to allow multiple copies of the executable, with different gluon storage, to coexist. mku driver [ dyn \| sml \| lrg \| noev ] This is for compilation of drivers which use unformatted data i/o. It also compiles in the routines and modules associated with binning of the amplitude. The command line arguments are the same as for [mkf]{}, except for the extra option available as the second argument: [noev]{}. This is to be used for drivers which analyse data from a previous evolution — no evolution routines are compiled, and the data and parameter files are opened with the [’OLD’]{} specifier, meaning that they cannot be modified. Naming of the executable follows the same convention as for [mkf]{} except when the [noev]{} option is used, in which case the executable is named [driver]{} (since there should be no need to have multiple copies compiled with different gluon storage methods). Test run -------- The run described here tests the evolution of the onia, determination and binning of the interaction amplitude, and subsequent analysis of data for onium-onium interactions. It is for moderate rapidities and very small statistics to ensure that it runs quickly (a few seconds) on most systems. It is necessary to first compile the evolution routine (to be found in the [sample/]{} directory of the distribution) with > mku onm-onm-bnd dyn which will include in dynamic allocation of memory for gluons and dipoles. The first command starts an evolution (with 10 events) where the maximum rapidity for each onium is 4, giving a total rapidity for the collisions of 8. The lower cutoff used is $0.1$ times the onium size. The binned, unformatted data are stored in [test\_y8.ufdt]{} (beware: this file is over 1MB in size) and the parameters in [test\_y8.prm]{}. The format of the variable [data\_io]{} (together with the defaults used here) is data_io%y_wvfn = 0d0 data_io%y_int = 0.5d0 data_io%n_y = 5 which specifies that the onium-onium interaction is determined for 5 values of rapidity, each separated by $0.5$ (note that this is for each onium — so the intervals in total rapidity are $1.0$). The maximum rapidity at which dipoles are extracted from each onium is [maxy]{}, but each onium is evolved to [data\_io%y\_wvfn]{} (or [maxy]{}, if this is larger) allowing different runs to effectively use the same dipole configurations but to examine the interactions at different values of rapidity. Various messages are output at the start of the program, as different parts of the startup procedure are accomplished. The line dealing with backups may be preceded by other messages on some systems (e.g. those using the NAG compiler — see section \[sc:prtblty\]), because the compiler does not offer the facility of running a system command. The next command causes the evolution to be continued for a further 5 “events” (simply to allow a test of the continuation function). One then wants to analyse the results of the evolution. In this test we will look at the total amplitudes (the onium-onium amplitude integrated over all impact parameters, which is equal to half the onium-onium total cross-section). The program to do this is compiled with mku ftot noev The option [noev]{} ensures that unnecessary routines (such as those required for evolution) are not compiled in. The output includes various messages at the beginning indicating the initialisation procedure (these messages are sent to standard error, while the numerical output is sent to standard output, to allow it to be redirected to a file). The format of the numerical output from [noev]{} is one line for each value of rapidity, where each line has: (total rapidity) (unitarised amplitude) (1 pomeron amplitude) ... (n_pom pomeron amplitude) (all on one line), and [n\_pom = 4]{} is a parameter in the program. It is the absolute values of the amplitudes which are output. A second test run, which tests the formatted data output is provided and documented within the distribution. In addition, some other driver and analysis routines are provided. These can all be found in the [samples/]{} directory. Examples of results produced with OEDIPUS can be found in [@Sala95; @Sala95b]. Typical rapidities for which adequate statistics are accessible with a day’s running are $Y \le 20$, using a lower cutoff of $0.01b$. Distribution ------------ The distribution (provided as a compressed tarred file) consists of a number of directories. Details of their contents are documented in extensive [README]{} files. A shell script is provided to help automate the installation procedure. Instructions are also included for installation by hand. Portability {#sc:prtblty} ----------- The main issues of portability relate to accessing command line arguments and causing a shell script (which performs backups of data and parameter files) to be executed. Interfaces to the system routines to perform these functions are provided as routines whose names start with the [lcl\_]{} prefix, held in a file basic_src/common/lcl_???.f90 where [???]{} may be [dec]{}, [nag]{} or [std]{} (or the user can generate a new file for his/her operating system). The file to be used must be specified as part of the installation procedure. With the NAG compiler (version 2.1), there does not seem to be any way of causing a shell script to be executed (the usual unix [system]{} subroutine seems to be absent). The routines for accessing command line arguments while available, do not seem to be documented. If the Fortran 90 compiler offers no mechanism for accessing the command line arguments then one can use [lcl\_std.f90]{}, which prompts the user for the information held in each of the arguments. It conforms to the standard. Local implementations are also necessary for writing large, unformatted arrays to disk (on some systems, the i/o buffer is not sufficiently large and the array must be written in small sections). Details are provided in the [lcl\_???.f90]{} files. To port to a non unix system (such as VMS), it would in addition be necessary to rewrite the compilation scripts. Global data ----------- All global data is stored in the form of modules. These are summarised in table \[tbl:modules\] (numbers in brackets refer to default values). [\|l\|l\|p[3.5in]{}\|]{} Module name & Variables & Description\ [constants]{} & [alpha\_s]{} & ${\ensuremath{\alpha_S}}$, the strong coupling constant ($0.17777\ldots$)\ & [nc]{} & $N_C$, the number of colours in QCD ($3$)\ & [alpha\_no\_nc]{} & the value of ${\ensuremath{\alpha_S}}$ to be used when it appears without a factor of $N_C$ (=[alpha\_s]{}).\ cuts & [r\_cut\_lo]{} & the lower cutoff\ & [r\_cut\_hi]{} & the upper cutoff\ & [hi\_cut\_implem]{} & true if an upper cutoff is being used\ [iomodule]{} & [iprm]{} & i/o device for parameters (10)\ & [idat]{} & i/o device for formatted output (12)\ & [iufdt]{} & i/o device for unformatted data (14)\ [gluons]{} & [type gluon]{} & the type definition for gluons\ & [type dipole]{} & the type definition for dipoles\ & [type dpl\_pntr]{} & type definition for a pointer to a dipole\ & [max\_gluons]{} & the maximum number of gluons to be generated\ [data\_defs]{} & [data\_io]{} & variable of user chosen type, with details of data i/o. This module must be provided by the user.\ [sub\_defs]{} & & interface blocks for all the main subroutines\ In addition there are modules associated with the binning routines. These are summarised in table \[tbl:binning\]. -------------------- -------------------- ----------------------------------------------------------------------- [binning]{} [type bin\_prms]{} type definition to store all the parameters for binning of amplitudes [bn]{} variable actually containing the information [fbins\_mdl]{} [fbins]{} array containing amplitude bins [bin\_interface]{} interface blocks for the binning routines -------------------- -------------------- ----------------------------------------------------------------------- : Modules associated with routines for binning the amplitude probability distribution[]{data-label="tbl:binning"} Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thanks B.R. Webber and A.H. Mueller for suggesting this work, as well as for many useful comments. In addition, M.H. Seymour’s introduction to Monte Carlo simulation techniques (in [@Seym92]) has been very helpful during the development of this program. [10]{} Y. Y. Balitskiǐ and L. N. Lipatov, Sov. Phys. JETP 28 (1978) 822. E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP 45 (1977) 199. L. N. Lipatov, Sov. Phys. JETP 63 (1986) 904. J. Bartels, H. Lotter, and M. W[ü]{}sthoff, Zeit. f. Phys. C68 (1995) 121. A. Bialas and R. Peschanski, hep-ph/9512427. J. Bartels, J. R. Forshaw, H. Lotter, and M. W[ü]{}sthof, DESY preprint 95-253, hep-ph/9601201. A. H. Mueller, Nucl. Phys. B415 (1994) 373. A. H. Mueller and B. Patel, Nucl. Phys. B425 (1994) 471. A. H. Mueller, Nucl. Phys. B437 (1995) 107. Z. Chen and A. H. Mueller, Nucl. Phys. 451 (1995) 579. N. N. Nikolaev and B. G. Zakharov, Zeit. f. Phys. C64 (1994) 631. N. N. Nikolaev, B. G. Zakharov, and V. R. Zoller, JETP Lett. 59 (1994) 6. G. P. Salam, Nucl. Phys. B449 (1995) 589. G. P. Salam, Cavendish-HEP-95/05, hep-ph/9509353, to appear in Nucl. Phys. B (1996). A. H. Mueller and G. P. Salam, Cavendish-HEP-95/06, in preparation. F. James, Comp. Phys. Comm. 60 (1990) 329. G. Marchesini et al., Comp. Phys. Comm. 67 (1992) 465. M. H. Seymour, , PhD thesis, Cambridge University, 1992. TEST RUN The input and output of the test run are shown below: > onm-onm-bnd_dyn test_y8 10 y4 Starting a new evolution HWRGEN will use its default seed Lifetimes have been initialised Have allocated bins If there are no error messages then the backups have been done Am about to evolve > onm-onm-bnd_dyn test_y8 5 Continuing a previous evolution.... HWRGEN has been initialised Lifetimes have been initialised Have allocated bins If there are no error messages then the backups have been done Am about to evolve > ftot test_y8 Reading from a previous evolution.... HWRGEN has been initialised Have allocated bins 4.0000E+00 1.3183E-01 1.3466E-01 2.9205E-03 9.6577E-05 3.3521E-06 5.0000E+00 1.8955E-01 2.0176E-01 1.4294E-02 2.4481E-03 4.1440E-04 6.0000E+00 2.0235E-01 2.1490E-01 1.4465E-02 2.2114E-03 3.4580E-04 7.0000E+00 4.2093E-01 4.7515E-01 6.3966E-02 1.1441E-02 1.9750E-03 8.0000E+00 4.4738E-01 5.0385E-01 6.5855E-02 1.0901E-02 1.7452E-03 > The starter routine [y4]{} is the following (default seed, lower cutoff of $0.1$, no upper cutoff ($-2.0$), onium size of $1.0$, and maximum rapidity for each onium of $4.0$): 0 0 0.1 -2.0 1.0 4.0 [^1]: Research supported by the UK Particle Physics and Astronomy Research Council [^2]: Note that this may not correctly reproduce the profile of the interaction at very large onium-onium separations, where the density of dipoles is also dying off also as $1/r^4$ (though enhanced by double leading logarithmic factors). However this region is not important for the total interaction.
--- abstract: 'Nanowire double quantum dots occupied by an even number of electrons are investigated in the context of energy level structure revealed by electric dipole spin resonance measurements. We use numerically exact configuration interaction approach up to 6 electrons for systems tuned to Pauli spin blockade regime. We point out the differences between the spectra of systems with two and a greater number of electrons. For two-electrons the unequal length of the dots results in a different effective $g$-factors in the dots as observed by the recent experiments. For an increased number of electrons the $g$-factor difference between the dots appears already for symmetric systems and it is greatly amplified when the dots are of unequal length. We find that the energy splitting defining the resonant electric dipole spin frequency can be quite precisely described by the two electrons involved in the Pauli blockade with the lower-energy occupied states forming a frozen core.' author: - 'M. P. Nowak' - 'B. Szafran' title: \| Single-electron shell occupation and effective $g$-factor\ in few-electron nanowire quantum dots --- Introduction ============ Few-electron gate-defined quantum dots[@hanson] are exploited for single spin manipulation that allows for realization of single-qubit quantum gates.[@loss] While the desired spin rotation involves a single spin as a carrier of quantum information, multielectron systems provide a feasible environment for readout of the controlled spin. Strong spin-orbit (SO) coupling that is present in InSb and InAs nanowires [@fasth; @pfund] allows for electrical spin rotations[@nowack] that are performed by electric dipole spin resonance (EDSR)[@golovach] which excludes the need for introduction of local magnetic field gradients[@laird; @koppens] or usage of hyperfine interaction.[@osika] The readout of the spin is realized via spin to charge conversion that relies on the Pauli spin blockade.[@ono] The single electron current $(1,0)\rightarrow(1,1)\rightarrow(2,0)\rightarrow(1,0)$ \[the numbers in the brackets correspond to the number of electrons in a particular dot\] is blocked at the transition from $(1,1)$ triplet to $(2,0)$ singlet. Rotation of one of the spins of electrons constituting the $(1,1)$ triplet unblocks the current which serves as a proof for the coherent spin control. On the other hand strong SO coupling leads to unavoidable spin relaxation which results in a spontaneous lifting of the Pauli blockade when one of the $(1,1)$ triplets is close in energy to the $(2,0)$ singlet.[@nowak2014] EDSR lifting of the current blockade is observed already for two electrons bound in the double dot, which indeed is the case for many of the experiments.[@nadj-perge; @nadj2012; @schroer; @frolov] However some of the experimentally studied devices consist an even number $N$ of electrons greater than two.[@berg; @petersson; @stehlik] In this case the system is biased such the Pauli blockade is between $(N-1,1)$ and $(N,0)$ states. It is assumed that such configuration is equivalent to the two-electron system.[@rossella] This approximation resembles the well established concept in chemistry, that the valence electrons are responsible for creation of bonds and the rest in the deep levels can be treated as the frozen core.[@fc] This assumption seems questionable for quantum dots in which the single-electron shells are separated by much smaller energies than for the Coulomb potential, nevertheless this problem has not been discussed by a theoretical study. The present work addresses this issue. We find that for a system with $N>2$ all but two electrons form closed singlet shells. This is in accordance with predictions of Hubbard model,[@tasaki] that appear as a consequence of Mattis-Lieb theorem,[@lieb] and which states that the lowest energy states posses the lowest spin (S=0). As a consequence, in general, the low-energy spectra of multielectron double quantum dots in the $(N-1,1)$ configuration resemble the spectra of quantum dots in $(1,1)$ configuration and the states have similar total spins. We find that the $(N-1,1)$ spectra can be well recreated by a configuration interaction calculation in which one excludes the single-electron orbitals that form the singlet shells. This is analogous to the frozen core approximation regardless of the fact that there are no orbital shells in quantum-dots. The main finding of the work is that though a general resemblance of $N>2$ and $N=2$ spectra is found, the occupation of excited single-electron orbitals in the $N>2$ case leads to lifting of the degeneracy of spin-zero states. This in turn is translated to different effective $g$-factors in the dots. Such differences have been observed in recent EDSR experiments on nanowire quantum dots[@nadj-perge; @nadj2012; @schroer; @frolov; @berg; @petersson; @stehlik] and have been related to the differences in the confinement as predicted by the study on self-organized quantum dots.[@pryor] Here we strictly connect the effective $g$-factors with the number of electrons in the system and the length of the dots. We find that unequal effective $g$-factors for $N=2$ appear only for an asymmetric system but for $N>2$ they are observed already for the dots of the same lengths. Theory ====== In the present work we follow the common approach[@oned] that treats the nanowire quantum dots as quasi-one dimensional. The N-electron system is described by the Hamiltonian, $$H=\sum_{i=1}^N h_i + \sum_{i=1,j=i+1}^N \frac{\sqrt{\pi/2}}{4\pi\varepsilon_0\varepsilon \ell}\mathrm{erfcx}\left[\frac{\|x_1-x_2\|}{\sqrt{2}\ell}\right]. \label{hne}$$ The form of the Coulomb interaction term results from the assumption that the electrons are localized in the ground state of the lateral quantization along the nanowire cross section with the wavefunction of a Gaussian shape with $\psi(y,z)=(\pi^{1/2}\ell)^{-1}\exp\{-(y^2+z^2)/(2\ell^2)\}$. The integration of three-dimensional Coulomb interaction term $H_C= \sum_{i=1,j=i+1}^N \frac{e^2}{4 \pi \varepsilon \varepsilon_0} \frac{1}{\|r_i-r_j\|}$ leads[@bednarek] to the operator including $\mathrm{erfcx}(x)=\exp(x^2)\mathrm{erfc}(x)$ which is the exponentially scaled complementary error function.[@book] To obtain N-electron spin-orbitals we diagonalize the Hamiltonian (\[hne\]) in a basis of Slater determinants consisting of single-electron spin-orbitals. $$\Psi(\nu_1,\nu_2,...,\nu_N)=\sum_{i=1}^M c_i \emph{A} \{\psi_{i_1}(\nu_1)\psi_{i_2}(\nu_2)...\psi_{i_N}(\nu_N)\},$$ where $\nu_i=(x_i,\sigma^i)$ corresponds to the orbital and spin coordinates, $\emph{A}$ is the antisymmetrization operator and $c_i$ is obtained by the diagonalization. We use $M=20$ single-electron spin-orbitals which provides accuracy better than $0.1\;\mu$eV. The single-electron orbitals $\psi(\nu)$ are described by the Hamiltonian, $$h = \frac{\hbar^2 k_x^2}{2m^} + V(x) - \alpha \sigma_y k_x + \frac{1}{2}\mu_B gB\sigma_x, \label{1eham}$$ where $H_{SO 1D}=-\alpha\sigma_yk_x$ corresponds to Rashba SO coupling[@rashba] resulting from $H_{SO}=\alpha(\sigma_xk_y-\sigma_yk_x)$ Hamiltonian averaged in the $y$-direction. $V(x)$ describes the potential profile of the double dot, $$V(x) = \left\{ \begin{array}{l l} V_b & \quad x<-w/2\;\textrm{and}\;x>-(l_1+w/2)\\ V_i & \quad \|x\|<w/2\\ 0 & \quad x>w/2\;\textrm{and}\;x<l_2+w/2 \end{array} \right.$$ where $l_1$ and $l_2$ determine the length of each dot, $w$ is the interdot barrier width, $V_i$ is the barrier height and $V_b$ is the bias potential applied to the bottom of the left dot. We assume a $w=20$ nm thin and $V_i=200$ meV high interdot barrier. The computational box ends at the edges of the defined potential and the magnetic field is applied along the nanowire axis. The single-electron eigenstates are obtained by exact diagonalization of Hamiltonian (\[1eham\]) on a mesh of 201 points with $\Delta x = 1.095$ nm. We adopt parameters corresponding to InSb nanowires, i.e. $m^=0.014$, $\varepsilon=16.5$, $g=-51$ and $\alpha=30$ meVnm which corresponds to spin-orbit length $l_{so}=\hbar/(m^\alpha)=182$ nm comparable to the value measured experimentally in Ref. . We take $\ell=20$ nm. Results ======= Two-electron quantum dot ------------------------ Let us first consider a symmetric system of two quantum dots of lengths $l_1=l_2=100$ nm. We set the bottom of the left dot to $V_b=-3.8$ meV. The bias results in the energy level configuration such that $(2,0)$ singlet[@comment] is the ground state and the lowest-energy excited states are $(1,1)$ states with different spin polarizations. This configuration is necessary for observation of spin Pauli blockade. The inset to Fig. \[spectr2e\](a) shows the lowest part of the energy spectrum. The ground state singlet of (2,0) occupation has mean value of $<S^2>$ operator 0.12 $\hbar^2/4$. Figure \[spectr2e\](a) presents energy levels of (1,1) states. The two Zeeman split energy levels correspond to a spin-positive triplet $T_+$ ($<S^2>=1.98\;\hbar^2/4$) with spins oriented approximately along the magnetic field and to a spin-negative triplet $T_-$ ($<S^2>=1.97\;\hbar^2/4$) with spins oriented against the magnetic field. The horizontal curve corresponds to a degenerate energy level of a singlet (S) and a triplet ($T_0$) states with zero spin projection along the direction of the magnetic field. The degeneracy results from the negligible overlap between the adjacent electrons and hence nearly zero exchange interaction. The mean values of $<S^2>$ operator for these states are: 1.04, 1.01 $[\hbar^2/4]$. In EDSR experiments the spin rotations are performed from one of the non-zero spin triplets: $T_+$ or $T_-$.[@nowak2014] When a resonance to a state with zero spin component along the magnetic field occurs the blockade is lifted. The experimentally measured resonances exhibit a linear dependence of the driving frequency on the magnetic field, equal to (considering $T_+$ as the initial state) $\omega=[E(S)-E(T_+)]/\hbar$. The corresponding energy $\omega\hbar$ is plotted in Fig. \[2ediffen\] with the red-dashed curve. Let us now consider the case in which the dots are of unequal length – $l_1=150$ nm and $l_2=50$ nm – but we keep the $g$-factor constant along the structure. Energy levels of states in which electrons occupy adjacent dots are presented in Fig. \[spectr2e\](b). To keep the energy separation between (2,0) singlet and four (1,1) states 1.5 meV at $B=0$ as in the case of symmetric system we set $V_{bias}=5.07$ meV in the left dot. The striking difference between the spectra of Fig. \[spectr2e\](b) as compared to the symmetric case of Fig. \[spectr2e\](a) is that for the non-zero magnetic field the degeneracy of horizontal energy levels is lifted. The spin densities of the states that correspond to these energy levels calculated as $\sigma^j_x(x)=\sum_{i=1}^N\langle\Psi_j(\nu_1,\nu_2,...,\nu_N)\|\sigma^i_{x}\delta_{x_i,x}\|\Psi_j(\nu_1,\nu_2,...,\nu_N)\rangle$ are depicted in the insets to Fig. \[spectr2e\](b). We observe that in the state with lower energy the spin in the left dot is oriented against the magnetic field, while in the right dot it is oriented along the magnetic field. Further on we will address to this state as to $(\downarrow,\uparrow)$. The state $(\uparrow,\downarrow)$ with an increasing energy in $B$ has an opposite spin configuration. The $(\downarrow,\uparrow)$, $(\uparrow,\downarrow)$ states have zero total spins along the $x$-direction therefore the EDSR transitions to this states lift the Pauli blockade and such transition are visible as a resonance lines in EDSR spectra. We calculate corresponding energy differences $\Delta E_1 =\omega_1 \hbar=[E(\downarrow,\uparrow)-E(T_+)]$, $\Delta E_2 =\omega_2 \hbar=[E(\uparrow,\downarrow)-E(T_+)]$ and plot them in Fig. \[2ediffen\] with black solid lines. We note that this arrangement of resonance lines is present in every EDSR map registered experimentally \[see Refs. \] and attributed to different $g$-factors in the dots. Here it is obtained for a constant $g$ along the structure. The slopes of the curves in Fig. \[2ediffen\] are connected to [effective]{} $g$-factor. For symmetric system we calculate $g^=\frac{E(S)-E(T_+)}{\mu_0 B}$ equal to $g^=-49.93$ for $B=100$ mT. In the case of asymmetric dots the effective $g$-factors are $g_1^=-48.70$, $g_2^=-50.71$. To explain the impact of the dots width on the energy spectra and effective $g$-factors let us inspect the single electron spin-orbitals that constitute the two-electron orbitals. Figures \[spectr1e\] (a,e) show the charge densities of the single-electron states. The densities correspond to the ground states of orbital quantization of each dot. The black curves correspond to $s_{l,\uparrow}$ and $s_{l,\downarrow}$ states \[the main letter denotes the orbital excitation, $(l,r)$ denotes the dot in which the electron is localized and the arrows correspond to the average spin polarization direction\]. The red-dashed curve shows the charge densities of higher energy states $s_{r,\uparrow}$ and $s_{r,\downarrow}$ in which the electron occupies the right dot. We extract the squared absolute values of coefficients – $\|c_i\|^2$ – for each of the Slater determinant that is used in the configuration interaction approach. For the symmetric case we get 0.806 for the determinant consisting of $\{s_{l,\uparrow},s_{r,\downarrow}\}$ single-electron orbitals and 0.194 for consisting $\{s_{l,\downarrow},s_{r,\uparrow}\}$ orbitals for one of the states from the degenerate pair of spin zero two-electron states (the coefficients for the second state are reversed). For the asymmetric case we get 0.992 for $\{s_{l,\downarrow},s_{r,\uparrow}\}$ for the $(\downarrow,\uparrow)$ state and 0.996 for $\{s_{l,\uparrow},s_{r,\downarrow}\}$ for the $(\uparrow,\downarrow)$ state. The lack of the admixture of other Slater determinants is due to small size of the dots which results in a considerable kinetic energy separation of the single-electron orbitals. The energy spectra displayed in Fig. \[spectr1e\](b,f) show the Zeeman splittings of the single-electron energy levels. If we overlay the energy levels of the states in which the electron is localized in the left and right dot (solid black and red-dashed curves in Fig. \[spectr1e\](b,f) respectively) we observe that they are exactly the same \[Fig. \[spectr1e\](c)\] when the dots are of identical length but they differ when the dots are of unequal length \[Fig. \[spectr1e\](g)\]. Now we sum the single-electron energies accordingly to the way the states enter the configuration-interaction calculation, i.e. the $(\downarrow,\uparrow)$ state corresponds to the occupation of $s_{l,\downarrow}$, $s_{r,\uparrow}$ single-electron orbitals with energies $E(s_{l,\downarrow})$ and $E(r_{r,\uparrow})$. The $(\uparrow,\downarrow)$ corresponds to the occupation of $s_{l,\uparrow}$ , $s_{r,\downarrow}$ single-electron orbitals with energies $E(s_{l,\uparrow})$ and $E(s_{r,\downarrow})$. The obtained sums are plotted in Fig. \[spectr1e\](d) for the symmetric and in Fig. \[spectr1e\](h) for the asymmetric case. We find that the degeneracy is lifted due to different Zeeman splittings of single-electron energy levels of electrons confined in dots of different length. In a single one-dimensional quantum dot SO interaction impacts the Zeeman splittings according to $E_z=g\mu_B B\lambda_i$ where[@nowaksp] $$\lambda_i=\int\|\Psi_i(x)\|^2\cos(2\alpha m^x/\hbar^2) dx. \label{integral}$$ Due to a high interdot barrier we can effectively treat the considered system as two separate dots. For a quantum dot in a form of an infinite quantum well the term $\lambda_i$ that controls the strength of the Zeeman splitting is $$\lambda_1(l)=\frac{\hbar^6\pi^2\sin(l\alpha m^/\hbar^2)}{\alpha m^ l (\pi^2 \hbar^4-\alpha^2{m^}^2l^2)},$$ for $\Psi_i(x)$ in a form of $s$-like orbital. $\lambda_i$ changes from 1 for narrow quantum dots to 0 in the limit of infinite dot length. Accordingly, the Zeeman splittings are the strongest (as strong as in the absence of SO coupling) for a narrow quantum dot and become weaker if the length of the dot is increased. The $g^$-factors calculated from $\Delta E_1 =\omega_1 \hbar=[E(\downarrow,\uparrow)-E(T_+)]$ and $\Delta E_2 =\omega_2 \hbar=[E(\uparrow,\downarrow)-E(T_+)]$ depend on the Zeeman splitting of the single electron energy levels as follows \[taking $E_0$ as the orbital excitation energy of $(\downarrow,\uparrow)$,$(\uparrow,\downarrow)$ and $T_+$ states\]: $\Delta E_1 = E_0 + E(s_{l,\downarrow})+E(s_{r,\uparrow})-E_0-E(s_{l,\uparrow})-E(s_{r,\uparrow})=E(s_{l,\downarrow})-E(s_{l,\uparrow})=g\mu_BB\lambda_1(l_1)$ and $\Delta E_2 = E_0+ E(s_{l,\uparrow})+E(s_{r,\downarrow})-E_0-E(s_{l,\uparrow})-E(s_{r,\uparrow})=E(s_{r,\downarrow})-E(s_{r,\uparrow})=g\mu_BB\lambda_1(l_2)$. Therefore $g_1^=g\lambda_1(l_1)$ and $g_2^=g\lambda_1(l_2)$. We plot the ratio $g_1^/g_2^=\lambda_1(l_1)/\lambda_1(l_2)$ in Fig. \[g1g2\](a). For $l_1=150$ and $l_2=50$ we obtain $g_1^/g_2^=0.960$ which matches well the value obtained in the exact calculation of Fig. \[2ediffen\], $g_1^/g_2^=0.961$ It should be noted here that the effect of the SO interaction on the strength of the Zeeman splittings is influenced also by the orientation of the magnetic field.[@nowaksp] For the magnetic field vector forming a $\phi$ angle with the nanowire axis the splitting becomes $E_z=g\mu_B B \sqrt{1-(1-\lambda_i^2)\cos^2\phi}$, i.e. the $g^$ values obtained for the magnetic field oriented perpendicular to the nanowire axis approach the bulk $g$-factor value. Four- and six-electron case --------------------------- Figure \[spectr4e\](a) with black solid curves presents energy levels of four-electron symmetric system with $V_{b}=-14.21$ meV. The levels correspond to the states with (3,1) occupation. The plot omits the ground state with (4,0) occupation that is $1.5$ meV lower in energy with respect to presented energy levels for $B=0$. The mean value of $<S^2>$ operator for the following (3,1) states are: 1.97, 1.04, 1.03, 1.97 $[\hbar^2/4]$. These values are close to the ones obtained for the two electron system. The absence of total spins of two electrons shows that two electrons form a singlet state with zero total spin. Let us extract the coefficients for each Slater determinant that is used to create configuration interaction basis. For the subsequent states whose energy levels are depicted in Fig. \[spectr4e\](a) the only non-zero (and nearly equal to unity, the other coefficients are less than 0.006) are coefficients for Slater determinants consisting of following single electron orbitals: $\{s_{l,\uparrow}s_{l,\downarrow}p_{l,\uparrow}s_{r,\uparrow}\}$, $\{s_{l,\uparrow}s_{l,\downarrow}p_{l,\downarrow}s_{r,\uparrow}\}$, $\{s_{l,\uparrow}s_{l,\downarrow}p_{l,\uparrow}s_{r,\downarrow}\}$ and $\{s_{l,\uparrow}s_{l,\downarrow}p_{l,\downarrow}s_{r,\downarrow}\}$ respectively. The corresponding orbitals are depicted in Fig. \[1efor4e\](a,b,c). Let us assume now that two electrons of the four-electron system form a singlet closed-shell that does not impact the spin properties of the two remaining electrons and thus they can be separated away: we exclude from the configuration interaction basis the $s_{l,\uparrow},s_{l,\downarrow}$ orbitals and limit number of electrons in the calculation to two. The obtained energy levels are depicted with the red-dashed curves in Fig. \[spectr4e\](a). Besides the shift between the energy levels obtained in full four-electron and two-electron calculation with restricted basis the spectra perfectly match. The total spins and the Zeeman splittings in the four-electron energy spectra resemble the ones obtained for two electrons. However, the striking feature of spectrum of Fig. \[spectr4e\](a) is that the two horizontal energy levels become separate in non-zero magnetic field already for a symmetric system. Let us invoke the two-electron approximation with the restricted basis to explain this observation. The $s_{l,\uparrow},s_{l,\downarrow}$ orbitals are occupied by two spin-opposite electrons that form the singlet state and are separated from the basis. The two energy levels that slightly split in the magnetic field are constructed from an $p$-like orbital of electron localized in the left dot ($p_{l,\uparrow}, p_{l,\downarrow}$) \[see Fig. \[1efor4e\](b)\] and an $s$-like orbital formed by an electron localized in the right dot ($s_{r,\uparrow},s_{r,\downarrow}$) \[see Fig. \[1efor4e\](a)\]. The single-electron energy levels are depicted in Fig. \[1efor4e\](d). We observe that the Zeeman splittings between energy levels of $s$-states differ from the ones for $p$-orbitals: 0.282 meV compared to 0.294 meV. Here the dots are symmetric so it its the shape of $\psi_i(x)$ that is changed. We integrate Eq. \[integral\] for an $p$-like orbital and obtain, $$\lambda_2(l)=\frac{4 \hbar^6 \pi^2\sin(l\alpha m^/\hbar^2)}{\alpha m^* l (4\pi^2\hbar^4-\alpha^2{m^}^2l^2)}.$$ The effective $g$-factors are obtained from the energy splittings analogically as in the two-electron case: $\Delta E_1=E_0+E(p_{l,\downarrow})+E(s_{r,\uparrow})-E_0-E(p_{l,\uparrow})-E(s_{r,\uparrow})=E(p_{l,\downarrow})-E(p_{l,\uparrow})=g\mu_BB\lambda_2(l_1)$ and $\Delta E_2=E_0+E(p_{l,\uparrow})+E(s_{l,\downarrow})-E_0-E(p_{l,\uparrow})-E(s_{r,\uparrow})=E(s_{r,\downarrow})-E(s_{r,\uparrow})=g\mu_BB\lambda_1(l_2)$. As a result the two states constructed from $\{p_{l,\downarrow},s_{r,\uparrow}\}$ and $\{p_{l,\downarrow},s_{r,\uparrow}\}$ single-electron orbitals have different energies at $B\ne0$ for $l_1=l_2$. Figure \[g1g2\](b) presents $g_1^/g_2^=\lambda_2(l_1)/\lambda_1(l_2)$. The plot suggests that the $g$-factor ratio can be altered significantly as compared to the two-electron case for an asymmetric system. Namely, if one makes the dot that is occupied by three electrons longer one can amplify the ratio of the $g$-factors in the dots greater than elongating the dot with a single electron. The energy spectra for asymmetric system with $l_1=150$ nm and $l_2=50$ nm are presented in Fig. \[spectr4e\](b). The splitting between the central lines is visibly increased as compared to the symmetric case of Fig. \[spectr4e\](a). We calculate $g_1^/g_2^=0.896$ which is close to the value obtained in analytical calculation from Fig. \[g1g2\](b) equal to 0.910. Figure \[spectr6e\](a) presents the energy spectrum of (5,1) states for six-electron double dot system for the bias $V_b=-30.125$ meV. The energy level structure resembles the spectrum of four-electron system depicted in Fig. \[spectr4e\](a). We again obtain the splitting of the central lines already for a symmetric system. For $B=100$ mT is 7.2 $\mu$eV for four electrons while for six electrons we get 8.6$\mu$eV. Also the total spins of (5,1) states are similar: 1.97, 1.04, 1.03, 1.97 $[\hbar^2/4]$. The coefficients for Slater determinants extracted from the configuration interaction calculation show that mainly a single determinant (with the square of absolute value equal to 0.987) describes each of the discussed six-electron state. The single electron states that constitute the determinants are: $\{s_{l,\uparrow},s_{l,\downarrow},p_{l,\uparrow},p_{l,\downarrow},d_{l,\uparrow},s_{r,\uparrow}\}$, $\{s_{l,\uparrow},s_{l,\downarrow},p_{l,\uparrow},p_{l,\downarrow},d_{l,\downarrow},s_{r,\uparrow}\}$, $\{s_{l,\uparrow},s_{l,\downarrow},p_{l,\uparrow},p_{l,\downarrow},d_{l,\uparrow},s_{r,\downarrow}\}$, $\{s_{l,\uparrow},s_{l,\downarrow},p_{l,\uparrow},p_{l,\downarrow},d_{l,\downarrow},s_{r,\downarrow}\}$ for the following (5,1) states from Fig. \[spectr6e\](a). The single-electron energy levels are depicted in Fig. \[spectr6e\](b). The determinants correspond to the occupation of single-electron orbitals in which two pairs of electrons occupy closed singlet shells: two electrons occupy spin opposite $s$-like orbitals and the next pair occupies two spin opposite $p$-like orbitals. The two remaining electrons occupy an $d$-like orbital in the left dot and an $s$-like orbital in the right dot. The calculated spectrum for the basis excluding the four single electron states that form the two singlet shells is presented in Fig. \[spectr6e\](a) with the red-dashed curves. The spectra obtained in the exact calculation and in the restricted basis agree. For six electrons we calculate the ratio $g_1^/g_2^=\lambda_3(l_1)/\lambda_1(l_2)$ where $$\lambda_3(l)=\frac{\hbar^6\sin(l\alpha m^/\hbar^2)(9\pi^2\hbar^4-2\alpha^2{m^}^2l^2)}{\alpha m^ l (9\pi^2 \hbar^4-\alpha^2{m^}^2l^2)},$$ is determined from integration of Eq. \[integral\] with an $d$-like orbital and plot it in Fig. \[g1g2\](c). We see that it is similar to the four-electron case of Fig. \[g1g2\](b) and is strongly altered as compared to the $N=2$ case. Comparison with the experiments ------------------------------- Our work shows that increasing the number of electrons results in an amplification of the difference between effective $g$-factors in the dots. Table I shows $g_1^/g_2^$ values taken from the experimental works. It is clearly seen that the studies that considered $N>2$ electrons indeed measured ratios that deviate more from 1 as compared to the $N=2$ cases. The actual experimental values could be affected by a number of effects omitted in the present modeling: the detailed structure of the confinement potential, or they can be impacted by non-zero exchange interaction. [@nowakharm] Nevertheless the tendency drawn by these data is clear and agrees with the result of the present study. --------------- --------------------- --------------- Reference No. Number of electrons $g_1^/g_2^$ N=2 0.967 N=2 0.923 N=2 0.922 $N>2$ 0.750 $N>2$ 0.760 $N=6$ 0.872 --------------- --------------------- --------------- : $g_1^/g_2^$ ratio given in the experimental works versus the number of electrons. \[t1\] Summary and conclusions ======================= We investigated nanowire double quantum dots occupied by an even number of electrons tuned to the Pauli spin blockade regime. By the exact configuration interaction study we found that in a system with an even, larger than two, number of electrons all but two electrons form closed singlet shells. This allows to obtain the properties of these structures by configuration interaction calculation where the number of electrons is limited to two and where $N-2$ lowest in energy single-electron orbitals forming singlet shells are excluded. Despite the fact that for $N>2$ the properties of the system are controlled by only two electrons the dots with such occupation cannot be treated as an exact equivalent of two-electron systems. We found that the occupation of excited single-electron orbitals by the valence electrons results in different effective $g$-factors in the adjacent dots. For $N>2$ the difference is obtained already for a symmetric system while for two-electrons it results from the dots asymmetry. 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--- abstract: \| If an invertible linear dynamical systems is Li-York chaotic or other chaotic, what’s about it’s inverse dynamics? what’s about it’s adjoint dynamics? With this unresolved but basic problems, this paper will give a criterion for Lebesgue operator on separable Hilbert space. Also we give a criterion for the adjoint multiplier of Cowen-Douglas functions on $2$-th Hardy space. Last we give some chaos about scalars perturbation of operator and some examples of invertible bounded linear operator such that $T$ is chaotic but $T^{-1}$ is not.\ inverse, chaos, Hardy space, rooter function, Cowen-Douglas function, Spectrum, $C^{}$ algebra, Lebesgue operator. author: - 'Luo Lvlin[^1]and Hou Bingzhe[^2]' date: 'January 24, 2015' title: \| $C^{}$ algebra and inverse chaos \ --- [1.Introduction*]{} The ideas of chaos in connection with a map was introduced by Li T.Y.and his teacher Yorke,J.A.[@LiTYYorkeJA1975], after that there are various definitions of what it means for a map to be chaotic and there is a series of papers on Topological Dynamics and Ergodic Theory about chaos, such as [@PeterWalters1982][@IwanikA1989][@JRBrowm1976][@JohnMilnor2006][@MShub1987]. Following Topological Dynamics,Linear Dynamics is also a rapidly evolving branch of functional analysis,which was probably born in 1982 with the Toronto Ph.D.thesis of C.Kitai [@CKitai1982]. It has become rather popular because of the efforts of many mathematicians, for the seminal paper [@GGodefroy2003] by G.Godefroy and J.H.Shapiro,the notes [@JHShapiro2001] by J.H.Shapiro,the authoritative survey [@KGGrosseErdmann1999] by K.-G.Grosse-Erdmann,,and finally for the book [@FBayartEMatheron2009] by F.Bayart and E.Matheron,the book [@KGGrosseErdmannAPerisManguillot] by K.-G.Grosse-Erdmann and A.Peris. For finite-dimension linear space,the authors have made a topologically conjugate classification about Jordan blocks in [@JWRobbin1972][@NHKuiperJWRobbin1973]. So for the eigenvalues $\|\lambda\|\neq1$, the operators of Jordan block are not Li-Yorke chaotic.With [@FBayartEMatheron2009] we know that a Jordan block is not supercyclic when its eigenvalues $\|\lambda\|=1$, and following a easy discussion it is not Li-Yorke chaotic too. So the definition of Li-Yorke chaos should be valid only on infinite-dimension Frechet space or Banach space such that in this paper the Hilbert space is infinite-dimensional. Because a finite-dimensional linear operator could be regard as a compact operator on some Banach spaces or some Hilbert spaces, we can get the same conclusion from [@NilsonCBernardesJrAntonioBonillaVladimirMullerApeiris2012]P12 or the Theorem $7$ of [@HoubingzheTiangengShiluoyi2009]. For a Frechet space $X$,let $\mathcal{L}(X)$ denote the set of all bounded linear operators on $X$. Let $\mathbb{B}$ denote a Banach space and let $\mathbb{H}$ denote a Hilbert space. If $T\in\mathcal{L}(\mathbb{B})$, then define $\sigma(T)=\{\lambda\in\mathbb{C};T-\lambda \text{ is not invertible}\}$ and define $r_{\sigma}(T)=\sup\{\|\lambda\|;\lambda\in\sigma(T)\}$. \[liyorkehundundedingyi1\] Let $T\in\mathcal{L}(\mathbb{B})$,if there exists $x\in\mathbb{B}$ satisfies: (1)$\varlimsup\limits_{n\to\infty}\|T^{n}(x)\\|>0$; and (2)$\varliminf\limits_{n\to\infty}\\|T^{n}(x)\\|=0$. Then we say that $T$ is Li-Yorke chaotic,and named $x$ is a Li-Yorke chaotic point of $T$,where $x\in\mathbb{B},n\in\mathbb{N}$. Define a distributional function $F_{x}^{n}(\tau)=\frac{1}{n}\sharp\{0\leq i\leq n:\\|T^n(x)\\|<\tau\}$, where $T\in\mathcal{L}(\mathbb{B}),x\in\mathbb{B},n\in\mathbb{N}$. And define $F_{x}(\tau)=\liminf\limits_{n\to\infty} F_{x}^{n}(\tau)$; and $F_{x}^{}(\tau)=\limsup\limits_{n\to\infty} F_{x}^{n}(\tau)$. \[fenbuhundundedingyi2\] Let $T\in\mathcal{L}(\mathbb{B})$,if there exists $x\in\mathbb{B}$ and $(1)$ If $F_{x}(\tau)=0,\exists\tau>0$, and $F_{x}^{}(\epsilon)=1,\forall\epsilon>0$, then we say that $T$ is distributional chaotic or $I$-distributionally chaotic . $(2)$ If $F_{x}^{}(\epsilon)>F_{x}(\tau),\forall\tau>0$, and $F_{x}^{}(\epsilon)=1,\forall\epsilon>0$, then we say that $T$ is $II$-distributionally chaotic. $(3)$ If $F_{x}^{}(\epsilon)>F_{x}(\tau),\forall\tau>0$,then we say that $T$ is $III$-distributionally chaotic. \[liyorkechaoscriteriondingyi0\] Let $X$ is an arbitrary infinite-dimensional separable Frechet space, $T\in\mathcal{L}(X)$,If there exists a subset $X_0$ of $X$ satisfies: $(1)$ For any $x\in X_0,\{T^nx\}_{n=1}^{\infty}$ has a subsequence converging to $0$; $(2)$ There is a bounded sequence $\{a_n\}_{n=1}^{\infty}$ in $\overline{span(X_0)}$ such that the sequence $\{T^na_n\}_{n=1}^{\infty}$ is unbounded. Then we say $T$ satisfies the Li-Yorke Chaos Criterion. \[liyorkechaoscriteriondingli0\] Let $X$ is an arbitrary infinite-dimensional separable Frechet space, If $T\in\mathcal{L}(X)$,then the following assertions are equivalent. $(i)$ $T$ is Li-Yorke chaotic; $(ii)$ $T$ satisfies the Li-Yorke Chaos Criterion. \[youxianweikongjianmeichaoxunhuanxing2\] There are no hypercyclic operators on a finite-dimensional space $X\neq0$. \[chaoxunhuanyudanweikaiyuanpanjiaofeikongdeyinyongyinli34\] Let $\phi\in\mathcal{H}^{\infty}(\mathbb{D})$ and let $M_{\phi}:\mathcal{H}^{2}(\mathbb{D})\to\mathcal{H}^{2}(\mathbb{D})$ be the associated multiplication operator. The adjoint multiplier $M_{\phi}^{}$ is hypercyclic if and only if $\phi$ is non-constant and $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$. \[chaoxunhuanxingjiushichuandixing3\] Let $X$ is an arbitrary separable Frechet space, $T\in\mathcal{L}(X)$.The following assertions are equivalent. $(i)$ $T$ is hypercyclic. $(ii)$ $T$ is topologically transitive; that is,for each pair of non-empty open sets $U,V\subseteq X$, there exists $n\in\mathbb{N}$ such that $T^n(U)\bigcap V\neq\emptyset$. \[duijiaoxianyujiaquanyiweizhuanzhihunhedingliyinyong11\] Let $X$ is a topological vector space,$T$ is a bounded linear operator on $X$.Let $$\begin{aligned} \left\{\begin{array}{l} \Lambda_{1}(T)\triangleq span(\bigcup\limits_{\|\lambda\|=1,n\in\mathbb{N}}\ker(T-\lambda)^{N}\bigcap ran(T-\lambda)^{N});\\ \Lambda^{+}(T)\triangleq span(\Lambda_{1}(T)\bigcup\bigcup\limits_{\|\lambda\|>1,n\in\mathbb{N}}\ker(T-\lambda)^{N});\\ \Lambda^{-}(T)\triangleq span(\Lambda_{1}(T)\bigcup\bigcup\limits_{\|\lambda\|<1,n\in\mathbb{N}}\ker(T-\lambda)^{N}). \end{array}\right.\end{aligned}$$ If $\Lambda^{+}(T)$ and $\Lambda^{-}(T)$ are both dense in $X$,then $T$ is mixing. [2.From Polar Decomposition to functional calculus]{} The Polar Decomposition Theorem [@JohnBConway2000]P15 on Hilbert space is a useful theorem, especially for invertible bounded linear operator. We give some properties of $C^{}$ algebra generated by normal operator. Let $\mathbb{H}$ be a separable Hilbert space over $\mathbb{C}$ and let $X$ be a compact subset of $\mathbb{C}$. Let $\mathcal{C}(X)$ denote the linear space of all continuous functions on the compact space $X$, let $\mathcal{P}(x)$ denote the set of all polynomials on $X$ and let $T$ be an invertible bounded linear operator on $\mathbb{H}$. By the Polar Decomposition Theorem [@JohnBConway2000]P15 we get $T=U\|T\|$, where $U$ is an unitary operator and $\|T\|^2=T^{}T$. Let $\mathcal{A}(\|T\|)$ denote the $C^{}$ algebra generated by the positive operator $\|T\|$ and $1$. \[weierstrassnikefenyinli1\] Let $0\notin X$ be a compact subset of $\mathbb{C}$. If $\mathcal{P}(x)$ is dense in $\mathcal{C}(X)$, then $\mathcal{P}(\frac{1}{x})$ is also dense in $\mathcal{C}(X)$. By the property of polynomials we know that $\mathcal{P}(\frac{1}{x})$ is a algebraic closed subalgebra of $\mathcal{C}(X)$ and we get: $(1)$ $1\in\mathcal{P}(\frac{1}{x})$; $(2)$ $\mathcal{P}(\frac{1}{x})$ separate the points of $X$; $(3)$ If $p(\frac{1}{x})\in\mathcal{P}(x)$,then $\bar{p}(\frac{1}{x})\in\mathcal{P}(x)$. By the Stone-Weierstrass Theorem [@JohnBConway1990]P145 we get the conclusion. \[weierstrasspingfangkefenyinli2\] Let $X\subseteq\mathbb{R_+}$. If $\mathcal{P}(\|x\|)$ is dense in $\mathcal{C}(X)$, then $\mathcal{P}(\|x\|^2)$ is also dense in $\mathcal{C}(X)$. For $X\subseteq\mathbb{R_+}$, $x\neq y$ $\Longleftrightarrow x^2\neq y^2$. By Lemma $\ref{weierstrassnikefenyinli1}$ we get the conclusion. By the GNS construction [@JohnBConway1990]P250 for the $C^{}$ algebra $\mathcal{A}(\|T\|)$, we get the following decomposition. \[gnsfenjiedingliyingyongyinli3\] Let $T$ be an invertible bounded linear operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, $\mathcal{A}(\|T\|)$ is the complex $C^{}$ algebra generated by $\|T\|$ and $1$. There is a sequence of nonzero $\mathcal{A}(\|T\|)$-invariant subspace. $\mathbb{H}_1,\mathbb{H}_2,\cdots$ such that: $(1)$ $\mathbb{H}=\mathbb{H}_1\bigoplus\mathbb{H}_2\bigoplus\cdots$; $(2)$ For every $\mathbb{H}_i$, there is a $\mathcal{A}(\|T\|)$-cyclic vector $\xi^i$ such that $\mathbb{H}_i=\overline{\mathcal{A}(\|T\|)\xi^i}$ and $\|T\|\mathbb{H}_i=\mathbb{H}_i=\|T\|^{-1}\mathbb{H}_i$. By [@WilliamArveson2002]P54 we get $(1)$,and $\|T\|\mathbb{H}_i\subseteq\mathbb{H}_i$,that is $\mathbb{H}_i\subseteq\|T\|^{-1}\mathbb{H}_i$; by Lemma $\ref{weierstrassnikefenyinli1}$ we get $\|T\|^{-1}\mathbb{H}_i\subseteq\mathbb{H}_i$. Hence we get $\|T\|\mathbb{H}_i=\mathbb{H}_i=\|T\|^{-1}\mathbb{H}_i$. For $\forall n\in\mathbb{N}$,$T^n$ is invertible when $T$ is invertible. By the Polar Decomposition Theorem [@JohnBConway2000]P15 $T^n=U_n\|T^n\|$, where $U_n$ is unitary operator and $\|T^n\|^2=T^{n}T^{n}$, we get the following conclusion. \[ncignsfenjiedingliyingyongyinli4\] Let $T$ be an invertible bounded linear operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, let $\mathcal{A}(\|T^k\|)$ be the complex $C^{}$ algebra generated by $\|T^k\|$ and $1$ and let $\mathbb{H}_i^{\|T^k\|}=\overline{\mathcal{A}(\|T^k\|)\xi_k^{i}}$ be a sequence of non-zero $\mathcal{A}(\|T^k\|)$-invariant subspace, there is a decomposition $\mathbb{H}=\bigoplus_i\mathbb{H}_i^{\|T^k\|}$, $\xi_k^{i}\in\mathbb{H},i,k\in\mathbb{N}$. Given a proper permutation of $\mathbb{H}_i^{\|T^k\|}$ and $\mathbb{H}_j^{\|T^{(k+1)}\|}$, we get $T^{}\mathbb{H}_i^{\|T^k\|}=\mathbb{H}_i^{\|T^{(k+1)}\|}$ and $T^{-1}\mathbb{H}_i^{\|T^k\|}=\mathbb{H}_i^{\|T^{(k+1)}\|}$. By Lemma $\ref{weierstrasspingfangkefenyinli2}$,it is enough to prove the conclusion on $\overline{\mathcal{P}(\|T^k\|^2)\xi_k^{i}}$. For any given $\xi_k^{i}\in\mathbb{H}=\bigoplus_j\mathbb{H}_j^{\|T^{(k+1)}\|}$, there is a unique $j\in\mathbb{N}$ such that $\xi_k^{i}\in\mathbb{H}_j^{\|T^{(k+1)}\|}$. $(1)$ Because $T$ is invertible, for any given $\xi_{k}^{i}$, there is an unique $\eta_i\in\mathbb{H}_{s}^{\|T^{(k+1)}\|}$ such that $\eta_i=T^{-1}\xi_{k}^{i}$. For $\forall p\in\mathcal{P}(\|x\|^2)$, we get $p(\|T^{(k+1)}\|^2)\eta_i= T^{}p(\|T^{k}\|^2)\xi_{k}^{i}$. Hence we get $\mathbb{H}_{s}^{\|T^{(k+1)}\|}=\overline{\mathcal{P}(\|T^{(k+1)}\|^2)\xi_{k+1}^{s}}\supseteq T^{}\overline{\mathcal{P}(\|T^{k}\|^2)\xi_{k}^{i}}=T^{}\mathbb{H}_{i}^{\|T^{k}\|}$. $(2)$ Similarly,for any given $\xi_{k+1}^{s}$, there is an unique $\eta_r\in\mathbb{H}_{r}^{\|T^{k}\|}$ such that $\eta_r=T\xi_{k+1}^{s}$. For $\forall p\in\mathcal{P}(\|x\|^2)$,we get $p(\|T^{k}\|^2)\eta_r=p(\|T^{k}\|^2)T\xi_{k+1}^{s}=T^{-1}p(\|T^{(k+1)}\|^2)\xi_{k+1}^{s}$. Hence we get $\mathbb{H}_{r}^{\|T^{k}\|}=\overline{\mathcal{P}(\|T^{k}\|^2)\xi_{k}^{r}}\supseteq T^{-1}\overline{\mathcal{P}(\|T^{(k+1)}\|^2)\xi_{k+1}^{s}}=T^{-1}\mathbb{H}_{s}^{\|T^{(k+1)}\|}$. Let $i=r$,by $(1)(2)$ we get $T^{-1}\mathbb{H}_{s}^{\|T^{(k+1)}\|}\subseteq \mathbb{H}_{i}^{\|T^{k}\|}\subseteq T^{-1}\mathbb{H}_{j}^{\|T^{(k+1)}\|}$. Fixed the order of $\mathbb{H}_{i}^{\|T^{k}\|}$, by a proper permutation of $\mathbb{H}_{j}^{\|T^{(k+1)}\|}$ we get $T^{}\mathbb{H}_{i}^{\|T^{k}\|}=\mathbb{H}_{i}^{\|T^{(k+1)}\|}$. By Lemma $\ref{weierstrassnikefenyinli1}$ and $T$ is invertible,we get $(3)$ For any given $\xi_{k}^{i}$, there is an unique $\eta_i\in\mathbb{H}_{i}^{\|T^{(k+1)}\|}$ such that $\eta_i=T^{}\xi_{k}^{i}$. For $\forall p\in\mathcal{P}(\|x\|^{-2})$, we get $p(\|T^{(k+1)}\|^{-2})\eta_i= T^{-1}p(\|T^{k}\|^{-2})\xi_{k}^{i}$. Hence we get $\mathbb{H}_{i}^{\|T^{(k+1)}\|}=\overline{\mathcal{P}(\|T^{(k+1)}\|^{-2})\xi_{k+1}^{i}}\supseteq T^{-1}\overline{\mathcal{P}(\|T^{k}\|^{-2})\xi_{k}^{i}}=T^{-1}\mathbb{H}_{i}^{\|T^{k}\|}$. $(4)$ For any given $\xi_{k+1}^{i}$,there is an unique $\eta_i\in\mathbb{H}_{i}^{\|T^{k}\|}$ such that $\eta_i=T^{-1}\xi_{k+1}^{i}$. For $\forall p\in\mathcal{P}(\|x\|^{-2})$,we get $T^{-1}p(\|T^{k}\|^{-2})\eta_i= T^{-1}p(\|T^{k}\|^{-2})T^{-1}\xi_{k+1}^{i}=p(\|T^{(k+1)}\|^{-2})\xi_{k+1}^{i}$. Hence we get $T^{-1}\mathbb{H}_{i}^{\|T^{k}\|}=T^{-1}\overline{\mathcal{P}(\|T^{k}\|^{-2})\xi_{k}^{i}}\supseteq \overline{\mathcal{P}(\|T^{(k+1)}\|^{-2})\xi_{k+1}^{i}}=\mathbb{H}_{i}^{\|T^{(k+1)}\|}$. By $(3)(4)$ we get $T^{-1}\mathbb{H}_{i}^{\|T^{k}\|}\subseteq \mathbb{H}_{i}^{\|T^{(k+1)}\|}\subseteq T^{-1}\mathbb{H}_{i}^{\|T^{k}\|}$. that is,$T^{-1}\mathbb{H}_{i}^{\|T^{k}\|}=\mathbb{H}_{i}^{\|T^{(k+1)}\|}$. Let $\xi\in\mathbb{H}$ is a $\mathcal{A}(\|T\|)$-cyclic vector such that $\mathcal{A}(\|T\|)\xi$ is dense in $\mathbb{H}$. Because of $\sigma{\|T\|}\neq\emptyset$, on $\mathcal{C}(\sigma(\|T\|))$ define the non-zero linear functional $\rho_{\|T\|}$:$\rho_{\|T\|}(f)=<f(\|T\|)\xi,\xi>,\forall f\in\mathcal{C}(\sigma(\|T\|))$. Then $\rho_{\|T\|}$ is a positive linear functional, by [@WilliamArveson2002]P54 and the Riesz-Markov Theorem, on $\mathcal{C}(\sigma(\|T\|))$ we get that there exists an unique finite positive Borel measure $\mu_{\|T\|}$ such that $$\begin{aligned} \left.\begin{array}{lr} \int\limits_{\sigma(\|T\|)}f(z)\,d\mu_{\|T\|}(z)=<f(\|T\|)\xi,\xi>, & \forall f\in\mathcal{C}(\sigma(\|T\|)). \end{array}\right.\end{aligned}$$ \[hanshuyansuan5\] Let $T$ be an invertible bounded linear operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, there is $\xi\in\mathbb{H}$ such that $\overline{\mathcal{A}(\|T\|)\xi}=\mathbb{H}$. For any given $n\in\mathbb{N}$, let $\mathcal{A}(\|T^n\|)$ be the complex $C^{}$ algebra generated by $\|T^n\|$ and $1$ and let $\xi_n$ be a $\mathcal{A}(\|T^n\|)$-cyclic vector such that $\overline{\mathcal{A}(\|T^n\|)\xi_n}=\mathbb{H}$. Then: $(1)$ For any given $\xi_n$,there is an unique positive linear functional $$\begin{aligned} \left.\begin{array}{lr} \int\limits_{\sigma(\|T^n\|)}{f(z)\,d\mu_{\|T^n\|}(z)}=<f(\|T^n\|)\xi_n,\xi_n>, & \forall f\in\mathcal{L}^{2}(\sigma(\|T^n\|)). \end{array}\right.\end{aligned}$$ $(2)$ For any given $\xi_n$,there is an unique finite positive complete Borel measure $\mu_{\|T^n\|}$ such that $\mathcal{L}^{2}(\sigma(\|T^n\|))$ is isomorphic to $\mathbb{H}$. Because $T$ is invertible, by Lemma $\ref{ncignsfenjiedingliyingyongyinli4}$ we get that if there is a $\mathcal{A}(\|T\|)$-cyclic vector $\xi$, then there is a $\mathcal{A}(\|T^n\|)$-cyclic vector $\xi_n$. $(1)$:For any given $\xi_n$, define the linear functional, $\rho_{\|T^n\|}(f)=<f(\|T^n\|)\xi_n,\xi_n>$, by [@WilliamArveson2002]P54 and the Riesz-Markov Theorem we get that on $\mathcal{C}(\sigma(\|T^n\|))$ there is an unique finite positive Borel measure $\mu_{\|T^n\|}$ such that $$\begin{aligned} \left.\begin{array}{lr} \int\limits_{\sigma(\|T^n\|)}{f(z)\,d\mu_{\|T^n\|}(z)}=<f(\|T^n\|)\xi_n,\xi_n>, & \forall f\in\mathcal{C}(\sigma(\|T^n\|)). \end{array}\right.\end{aligned}$$ Moreover we can complete the Borel measure $\mu_{\|T^n\|}$ on $\sigma(\|T^n\|)$, also using $\mu_{\|T^n\|}$ to denote the complete Borel measure, By [@PaulRHalmos1974] we know that the complete Borel measure is uniquely. For $\forall f\in\mathcal{L}^{2}(\sigma(\|T^n\|))$,because of $$\begin{aligned} \left.\begin{array}{l} \rho_{\|T^n\|}(\|f\|^2)=\rho_{\|T^n\|}(\bar{f}f)=<f(\|T^n\|)^{}f(\|T^n\|)\xi_n,\xi_n>=\\|f(\|T^n\|)\xi_n\\|\geq0. \end{array}\right.\end{aligned}$$ we get that $\rho_{\|T^n\|}$ is a positive linear functional,hence $(1)$ is right. $(2)$ we know that $\mathcal{C}(\sigma(\|T^n\|))$ is a subspace of $\mathcal{L}^{2}(\sigma(\|T^n\|))$ such that $\mathcal{C}(\sigma(\|T^n\|))$ is dense in $\mathcal{L}^{2}(\sigma(\|T^n\|))$. For any $f,g\in\mathcal{C}(\sigma(\|T^n\|))$ we get $$\begin{aligned} \left.\begin{array}{l} <f(\|T^n\|)\xi_n,g(\|T^n\|)\xi_n>_{\mathbb{H}} =<g(\|T^n\|)^{}f(\|T^n\|)\xi_n,\xi_n>\\ =\rho_{\|T^n\|}(\bar{g}f) =\int\limits_{\sigma(\|T^n\|)}{f(z)\bar{g}(z)\,d\mu_{\|T^n\|}(z)} =<f,g>_{\mathcal{L}^{2}(\sigma(\|T^n\|))}. \end{array}\right.\end{aligned}$$ Therefor $U_0:\mathcal{C}(\sigma(\|T^n\|))\to\mathbb{H},f(z)\to f(\|T^n\|)\xi_n$ is a surjection isometry from $\mathcal{C}(\sigma(\|T^n\|))$ to $\mathcal{A}(\|T^n\|)\xi_n$, also $\mathcal{C}(\sigma(\|T^n\|))$ and $\mathcal{A}(\|T^n\|)\xi_n$ is a dense subspace of $\mathcal{L}^{2}(\sigma(\|T^n\|))$ and $\mathbb{H}$,respectively. Because of $U_0$ a closable operator,it closed extension $U:\mathcal{L}^{2}(\sigma(\|T^n\|))\to\mathbb{H},f(z)\to f(\|T^n\|)\xi_n$ is a unitary operator. Hence for any given $\xi_n$, $U$ is the unique unitary operator induced by the unique finite positive complete Borel measure $\mu_{\|T^n\|}$ such that $\mathcal{L}^{2}(\sigma(\|T^n\|))$ is isomorphic to $\mathbb{H}$. By the Polar Decomposition Theorem [@JohnBConway2000]P15, we get $U^{}T^{}TU=TT^{}$ and $U^{}\|T\|^{-2}U=\|T^{-2}\|$ when $T=U\|T\|$. In fact,when $T$ is invertible, we can choose an specially unitary operator such that $\|T\|^{-1}$ and $\|T^{-1}\|$ are unitary equivalent. We give the following unitary equivalent by Theorem $\ref{hanshuyansuan5}$. \[TjueduizhiniyuTnijueduizhideguanxi15\] Let $T$ be an invertible bounded linear operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$ and let $\mathcal{A}(\|T\|)$ be the complex algebra generated by $\|T\|$ and $1$. There is $\sigma(\|T\|^{-1})=\sigma(\|T^{-1}\|)$ and we get that $\|T\|^{-1}$ and $\|T^{-1}\|$ are unitary equivalent by the unitary operator $F_{xx^{}}^{\mathbb{H}}$, more over the unitary operator $F_{xx^{}}^{\mathbb{H}}$ is induced by an almost everywhere non-zero function $\sqrt{\|\phi_{\|T\|}\|}$,where $\sqrt{\|\phi_{\|T\|}\|}\in\mathcal{L}^{\infty}(\sigma(\|T\|),\mu_{\|T\|})$. That is,$d\,\mu_{\|T^{-1}\|}=\|\phi_{\|T\|}\|d\,\mu_{\|T\|^{-1}}$. By Lemma $\ref{gnsfenjiedingliyingyongyinli3}$,lose no generally,let $\xi_{\|T\|}$ is a $\mathcal{A}(\|T\|)$-cyclic vector such that $\mathbb{H}=\overline{\mathcal{A}(\|T\|)\xi_{\|T\|}}$. $(1)$ Define the function $F_{x^{-1}}:\mathcal{P}(x)\rightarrow\mathcal{P}(x^{-1}),F_{x^{-1}}(f(x))=f(x^{-1})$, it is easy to find that $F_{x^{-1}}$ is linear.Because of $\int\limits_{\sigma(\|T\|)}{f(z^{-1})d\mu_{\|T\|}(z)}=<f(\|T\|^{-1})\xi,\xi>$ $=\int\limits_{\sigma(\|T\|^{-1})}{f(z)d\mu_{\|T\|^{-1}}(z)}$. We get $d\mu_{\|T\|^{-1}}(z)=\|z\|^2d\mu_{\|T\|}(z)$.Hence $\\|F_{x^{-1}}(f(x))\\|_{\mathcal{L}^2(\sigma(\|T\|^{-1}),\mu_{\|T\|^{-1}})}$ $=\int\limits_{\sigma(\|T\|^{-1})}{F_{x^{-1}}(f(x))\bar{F}_{x^{-1}}(f(x))d\mu_{\|T\|^{-1}}(x)}$ $=\int\limits_{\sigma(\|T\|^{-1})}{f(x^{-1})\bar{f}(x^{-1})d\mu_{\|T\|^{-1}}(x)}$ $=\int\limits_{\sigma(\|T\|)}{\|z\|^2f(z)\bar{f}(z)d\mu_{\|T\|}(z)}$ $\leq \sup\sigma(\|T\|)^2\int\limits_{\sigma(\|T\|)}{f(z)\bar{f}(z)d\mu_{\|T\|}(x)}$ $\leq\sup\sigma(\|T\|)^2\\|f(z)\\|_{\mathcal{L}^2(\sigma(\|T\|),\mu_{\|T\|})}$. So we get $\\|F_{x^{-1}}\\|\leq\sup\sigma(\|T\|)^2$, by the Banach Inverse Mapping Theorem [@JohnBConway1990]P91 we get that $F_{x^{-1}}$ is an invertible bounded linear operator from $\mathcal{L}^2(\sigma(\|T\|),\mu_{\|T\|})$ to $\mathcal{L}^2(\sigma(\|T\|^{-1}),\mu_{\|T\|^{-1}})$. Define the operator $F_{x^{-1}}^{\mathbb{H}}:\mathcal{A}(\|T\|)\xi\rightarrow\mathcal{A}(\|T\|^{-1})\xi,F(f(\|T\|)\xi)=f(\|T\|^{-1})\xi$. By Lemma $\ref{weierstrassnikefenyinli1}$ and [@WilliamArveson2002]P55, we get that $F_{x^{-1}}^{\mathbb{H}}$ is a bounded linear operator on the Hilbert space $\overline{\mathcal{A}(\|T\|)\xi}=\mathbb{H}$, $\\|F_{x^{-1}}^{\mathbb{H}}\\|\leq\sup\sigma(\|T\|)^2$.Moreover we get $\left. \begin{array}{rcl} \mathbb{H} & \underrightarrow{\qquad \|T\|\qquad } &\mathbb{H}\\ F_{x^{-1}}^{\mathbb{H}}\downarrow & & \downarrow F_{x^{-1}}^{\mathbb{H}}\\ \mathbb{H} & \overrightarrow{\qquad \|T\|^{-1} \qquad} &\mathbb{H} \end{array} \right.$ $(2)$ Define the function $F_{xx^{}}:\mathcal{P}(xx^{})\rightarrow\mathcal{P}(x^{}x),F_{xx^{}}(f(x,x^{}))=f(x^{}x)$. By [@Hualookang1949] we get that $F_{xx^{}}$ is a linear algebraic isomorphic. For any $x\in \sigma(\|T^{-1}\|)$, by Lemma $\ref{weierstrasspingfangkefenyinli2}$ we get that $\mathcal{P}(\|x\|^2)$ is dense in $\mathcal{C}(\|x\|)$, $\mathcal{C}(\|x\|)$ is dense in $\mathcal{L}^2(\sigma(\|T^{-1}\|),\mu_{\|T^{-1}\|})$. Hence $\mathcal{P}(\|x\|^2)$ is dense in $\mathcal{L}^2(\sigma(\|T^{-1}\|),\mu_{\|T^{-1}\|})$. With a similarly discussion, for any $y\in \sigma(\|T\|^{-1})$,we get that $\mathcal{P}(\|y\|^2)$ is dense in $\mathcal{L}^2(\sigma(\|T\|^{-1}),\mu_{\|T\|^{-1}})$. So $F_{xx^{}}$ is an invertible bounded linear operator from $\mathcal{L}^2(\sigma(\|T\|^{-1}),\mu_{\|T\|^{-1}})$ to $\mathcal{L}^2(\sigma(\|T^{-1}\|),\mu_{\|T^{-1}\|})$. Therefor $F_{xx^{}}\circ F_{x^{-1}}$ is an invertible bounded linear operator from $\mathcal{L}^2(\sigma(\|T\|),\mu_{\|T\|})$ to $\mathcal{L}^2(\sigma(\|T^{-1}\|),\mu_{\|T^{-1}\|})$. Because of $\lambda\in\sigma(T^{}T)\Longleftrightarrow\frac{1}{\lambda}\in\sigma(T^{-1}T^{-1})$, we get that $\lambda\in\sigma(\|T\|)\Longleftrightarrow\frac{1}{\lambda}\in\sigma(\|T^{-1}\|)$, that is,$\sigma(\|T^{-1}\|)=\sigma(\|T\|^{-1})$. For any $p_n\in\mathcal{P}(\sigma(\|T\|^{-1}))\subseteq\mathcal{A}(\sigma(\|T\|^{-1}))$, because of $T^{-1}p_n(\|T\|^{-1})=p_n(\|T^{-1}\|)T^{-1}$, by [@JohnBConway2000]P60 we get that $\mathcal{P}(\|T\|^{-1})$ and $\mathcal{P}(\|T^{-1}\|)$ are unitary equivalent. Hence there is an unitary operator $U\in\mathcal{B}(\mathbb{H})$ such that $U\mathcal{P}(\|T\|^{-1})=\mathcal{P}(\|T^{-1}\|)U$, that is,$U\mathcal{A}(\|T\|^{-1})=\mathcal{A}(\|T^{-1}\|)U$ and $U\overline{\mathcal{A}(\|T\|^{-1})}\xi_{\|T\|}=\overline{\mathcal{A}(\|T^{-1}\|)}U\xi_{\|T\|}$. If let $\xi_{\|T^{-1}\|}=U\xi_{\|T\|}$,then $\xi_{\|T^{-1}\|}$ is a $\mathcal{A}(\|T^{-1}\|)$-cyclic vector and $\overline{\mathcal{A}(\|T^{-1}\|)}\xi_{\|T^{-1}\|}=\mathbb{H}$.Because of $\int\limits_{\sigma(\|T\|^{-1})}{f(z)d\mu_{\|T\|^{-1}}(z)}=<f(\|T\|^{-1})\xi_{\|T\|},\xi_{\|T\|}>$ $=\int\limits_{\sigma(\|T\|)}{f(\frac{1}{z})d\mu_{\|T\|}(z)}$. $\int\limits_{\sigma(\|T^{-1}\|)}{f(z)d\mu_{\|T^{-1}\|}(z)}=<f(\|T^{-1}\|)\xi_{\|T^{-1}\|},\xi_{\|T^{-1}\|}>$. We get $[d\mu_{\|T^{-1}\|}]=[d\mu_{\|T\|^{-1}}]$,that is, $d\mu_{\|T^{-1}\|}$ and $d\mu_{\|T\|^{-1}}$ are mutually absolutely continuous, by [@JohnBConway1990]IX.3.6Theorem and $(1)$ we get that $d\mu_{\|T^{-1}\|}=\|\phi_{\|T\|}(\frac{1}{z})\|d\mu_{\|T\|^{-1}}=\|z\|^2\|\phi_{\|T\|}(z)\|d\mu_{\|T\|}$, where $\|\phi_{\|T\|}(z)\|\neq0,a.e.$ and $\|\phi_{\|T\|}(z)\|\in\mathcal{L}^{\infty}(\sigma(\|T\|),\mu_{\|T\|})$. So we get $\\|F_{xx^{}}\circ F_{x^{-1}}(f(x))\\|_{\mathcal{L}^2(\sigma(\|T^{-1}\|),\mu_{\|T^{-1}\|})}$ $=\int\limits_{\sigma(\|T^{-1}\|)}{F_{xx^{}}\circ F_{x^{-1}}(f(x))\overline{F_{xx^{}}\circ F_{x^{-1}}}(f(x))d\mu_{\|T^{-1}\|}(x)}$ $=\int\limits_{\sigma(\|T^{-1}\|)}{f(x^{-1})\bar{f}(x^{-1})d\mu_{\|T^{-1}\|}(x)}$ $=\int\limits_{\sigma(\|T\|^{-1})}{\|\phi_{\|T\|}(\frac{1}{x})\|f(x^{-1})\bar{f}(x^{-1})d\mu_{\|T\|^{-1}}(x)}$ $=\int\limits_{\sigma(\|T\|^{-1})}{F_{x^{-1}}(\sqrt{\|\phi_{\|T\|}(x)\|}f(x))F_{x^{-1}}(\sqrt{\|\phi_{\|T\|}(x)\|}\bar{f}(x))d\mu_{\|T\|^{-1}}(x)}$ $=\\|\sqrt{\|\phi_{\|T\|}(\frac{1}{x})\|}F_{x^{-1}}(f(x))\\|_{\mathcal{L}^2(\sigma(\|T\|^{-1}),\mu_{\|T\|^{-1}})}$ Hence $F_{xx^{}}$ is an unitary operator that is induced by $Uf(\frac{1}{x})=\sqrt{\|\phi_{\|T\|}(\frac{1}{x})\|}f(\frac{1}{x})$. Define $F_{xx^{}}^{\mathbb{H}}$: $\left\{ \begin{array}{l} \mathcal{A}(\|T\|^{-2})\xi_{\|T\|}\rightarrow\mathcal{A}(\|T^{-2}\|)\xi_{\|T^{-1}\|},\\ F_{xx^{}}^{\mathbb{H}}(f(\|T\|^{-2})\xi_{\|T\|})=f(\|T^{-2}\|)\xi_{\|T^{-1}\|}. \end{array} \right.$ Therefor $F_{xx^{}}^{\mathbb{H}}$ is an unitary operator from $\overline{\mathcal{A}(\|T\|^{-1})\xi_{\|T\|}}$ to $\overline{\mathcal{A}(\|T^{-1}\|)\xi_{\|T^{-1}\|}}$. By Lemma $\ref{gnsfenjiedingliyingyongyinli3}$ and [@Hualookang1949] we get $\overline{\mathcal{A}(\|T\|^{-1})\xi}$ $=\mathbb{H}=\overline{\mathcal{A}(\|T^{-1}\|)\xi}$. That is,$F_{xx^{}}^{\mathbb{H}}$ is an unitary operator and we get $\left. \begin{array}{rcl} \mathbb{H} & \underrightarrow{\qquad \|T\|^{-1}\qquad } &\mathbb{H}\\ F_{xx^{}}^{\mathbb{H}}\downarrow & & \downarrow F_{xx^{}}^{\mathbb{H}}\\ \mathbb{H} & \overrightarrow{\qquad \|T^{-1}\| \qquad} &\mathbb{H} \end{array} \right.$ So $\|T\|^{-1}$ and $\|T^{-1}\|$ are unitary equivalent by $F_{xx^{}}^{\mathbb{H}}$, the unitary operator $F_{xx^{}}^{\mathbb{H}}$ is induced by the function $\sqrt{\|\phi_{\|T\|}(\frac{1}{x})\|}$. \[TjueduizhiyuTxingjueduizhideguanxi16\] Let $T$ be an invertible bounded linear operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$ and let $\mathcal{A}(\|T\|)$ be the complex algebra generated by $\|T\|$ and $1$. There is $\sigma(\|T\|)=\sigma(\|T^{}\|)$ and we get that $\|T\|$ and $\|T^{}\|$ are unitary equivalent by the unitary operator $F_{xx^{}}^{\mathbb{H}}$, more over the unitary operator $F_{xx^{}}^{\mathbb{H}}$ is induced by an almost everywhere non-zero function $\sqrt{\|\phi_{\|T\|}\|}$,where $\sqrt{\|\phi_{\|T\|}\|}\in\mathcal{L}^{\infty}(\sigma(\|T\|),\mu_{\|T\|})$. That is,$d\,\mu_{\|T^{}\|}=\|\phi_{\|T\|}\|d\,\mu_{\|T\|}$. [*3.The chaos between $T$ and $T^{-1}$ for Lebesgue operator*]{} For the example of singular integral in mathematical analysis, we know that is independent the convergence or the divergence of the weighted integral between $x$ and $x^{-1}$, however some times that indeed dependent for a special weighted function. For $T$ is an invertible bounded operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, we get $0\notin\sigma(\|T^n\|)\subseteq\mathbb{R}_{+}$. In the view of the singular integral in mathematical analysis and by Theorem $\ref{hanshuyansuan5}$, we get that $T$ and $T^{-1}$ should not be convergence or divergence at the same time for $T$ is an invertible bounded operator. Anyway,they should be convergence or divergence at the same time for some special operators. Therefor we define the Lebesgue operator and prove that $T$ and $T^{-1}$ are Li-Yorke chaotic at the same time for $T$ is a Lebesgue operator. Then we give an example that $T$ is a Lebesgue operator,but not is a normal operator. Let $dx$ be the Lebesgue measure on $\mathcal{L}^{2}(\mathbb{R}_{+})$, by Theorem $\ref{hanshuyansuan5}$ we get that $d\mu_{\|T^n\|}$ is the complete Borel measure and $\mathcal{L}^2(\sigma(\|T^n\|))$ is a Hilbert space. If $\exists N>0$, for $\forall n\geq N,n\in\mathbb{N}$,$d\mu_{\|T^n\|}$ is absolutely continuity with respect to $dx$, by the Radon-Nikodym Theorem [@JohnBConway1990]P380 there is $f_n\in\mathcal{L}^{1}(\mathbb{R}_{+})$ such that $d\mu_{\|T^n\|}=f_n(x)\,dx$. \[lebesguesuanzileidingyi6\] Let $T$ be an invertible bounded linear operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, moreover if $T$ satisfies the following assertions: $(1)$ If $\exists N>0$,for $\forall n\geq N,n\in\mathbb{N}$ $$\begin{aligned} \left\{\begin{array}{lr} d\mu_{\|T^n\|}=f_n(x)\,dx,& f_n\in\mathcal{L}^{1}(\mathbb{R}_{+}).\\ x^{2}f_n(x)=f_{n}(x^{-1}),& 0<x\leq1. \end{array}\right.\end{aligned}$$ $(2)$ If $\exists N>0$, for $\forall n\geq N,n\in\mathbb{N}$,there is a $\mathcal{A}(\|T^n\|)$-cyclic vector $\xi_n$. And for any given $0\neq x\in\mathbb{H}$ and for any given $0\neq g_n(t)\in\mathcal{L}^2(\sigma(\|T^n\|))$, there is an unique $0\neq y\in\mathbb{H}$ such that $y=g_n(\|T^n\|^{-1})\xi_n$ when $x=g_n(\|T^n\|)\xi_n$. Then we say that $T$ is a Lebesgue operator, let $\mathcal{L}_{Leb}(\mathbb{H})$ denote the set of all Lebesgue operators. \[lebesguesuanzihundundeduichengxing7\] Let $T$ be a Lebesgue operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, then $T$ is Li-Yorke chaotic if and only if $T^{-1}$ is. We prove the conclusion by two parts. $(1)$ Let $\mathbb{H}$ be $\mathcal{A}(\|T\|)$-cyclic, that is,there is a vector $\xi$ such that $\overline{\mathcal{A}(\|T\|)\xi}=\mathbb{H}$. By Lemma $\ref{ncignsfenjiedingliyingyongyinli4}$ we get that if there is a $\mathcal{A}(\|T\|)$-cyclic vector $\xi$, then there is also a $\mathcal{A}(\|T^n\|)$-cyclic vector $\xi_n$. Let $x_0$ be a Li-Yorke chaotic point of $T$, by Theorem $\ref{hanshuyansuan5}$ and the Polar Decomposition Theorem [@JohnBConway2000]P15 and by the define of Lebesgue operator,we get that for enough large $n\in\mathbb{N}$,there are $g_n(x)\in\mathcal{L}^2(\sigma(\|T^n\|))$, $f_n(x)\in\mathcal{L}^{2}(\mathbb{R}_{+})$ and $y_0\in\mathbb{H}$ such that $x_0=g_n(\|T^n\|)\xi_n$,$y_0=g_n(\|T^n\|^{-1})\xi_n$ and $d\mu_{\|T^n\|}=f_n(x)\,dx$. $\\|T^nx_0\\|\\ =<T^{n}T^nx_0,x_0>\\ =<\|T^n\|^2g_n(\|T^n\|)\xi_n,g_n(\|T^n\|)\xi_n>\\ =<g_n(\|T^n\|)^{}\|T^n\|^2g_n(\|T^n\|)\xi_n,\xi_n>\\ =\int\limits_{\sigma(\|T^n\|)}{x^2g_n(x)\bar{g}(x)\,d\mu_{\|T^n\|}(x)}\\ =\int_{0}^{+\infty}{x^2\|g_n(x)\|^2f_n(x)\,dx}\\ =\int_{0}^{1}{x^2\|g_n(x)\|^2f_n(x)\,dx}+\int_{1}^{+\infty}{x^2\|g_n(x)\|^2f_n(x)\,dx}\\ =\int_{0}^{1}{x^2\|g_n(x)\|^2f_n(x)\,dx}+\int_{0}^{1}{x^{-4}\|g_n(x^{-1})\|^2f_n(x^{-1})\,dx}\\ \triangleq \int_{0}^{1}{\|g_n(x)\|^2f_n(x^{-1})\,dx}+\int_{0}^{1}{x^{-2}\|g_n(x^{-1})\|^2f_n(x)\,dx}\\ =\int_{1}^{+\infty}{x^{-2}\|g_n(x^{-1})\|^2f_n(x)\,dx}+\int_{0}^{1}{x^{-2}\|g_n(x^{-1})\|^2f_n(x)\,dx}\\ =\int_{0}^{+\infty}{x^{-2}\|g_n(x^{-1})\|^2f_n(x)\,dx}\\ =\int\limits_{\sigma(\|T^n\|)}{x^{-2}g_n(x^{-1})\bar{g}_n(x^{-1})\,d\mu_{\|T^n\|}(x)}\\ =<g_n(\|T^n\|)^{}\|T^n\|^{-2}g_n(\|T^n\|^{-1})\xi_n,\xi_n>\\ =<\|T^n\|^{-2}g_n(\|T^n\|^{-1})\xi_n,g_n(\|T^n\|^{-1})\xi_n>\\ =<\|T^n\|^{-2}y_0,y_0>\\ =<T^{-n}T^{-n}y_0,y_0>\\ =\\|T^{-n}y_0\\|.$ Where $\triangleq$ following the define of $f_n(x)$. $(2)$ If $\mathbb{H}$ is not $\mathcal{A}(\|T\|)$-cyclic, by Lemma $\ref{ncignsfenjiedingliyingyongyinli4}$ we get that for $\forall n\in\mathbb{N}$, there is a decomposition $\mathbb{H}=\bigoplus_i\mathbb{H}_i^{\|T^k\|}$, $\xi_k^{i}\in\mathbb{H},i,k\in\mathbb{N}$, where $\mathbb{H}_i^{\|T^k\|}=\overline{\mathcal{A}(\|T^k\|)\xi_k^{i}}$ is a sequence of $\mathcal{A}(\|T^k\|)$-invariant subspace, and do $(1)$ for $\mathbb{H}_i^{\|T^k\|}$. By $(1)(2)$ we get that $T$ is Li-Yorke chaotic if and only if $T^{-1}$ is Li-Yorke chaotic. \[lebesguesuanzifenbuhundundengjia8\] Let $T$ be a Lebesgue operator on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, then $T$ is $I$-distributionally chaotic (or $II$-distributionally chaotic or $III$-distributionally chaotic) if and only if $T^{-1}$ is $I$-distributionally chaotic (or $II$-distributionally chaotic or $III$-distributionally chaotic). \[lebesguesuanzicunzaixing9\] There is an invertible bounded linear operator $T$ on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, $T$ is Lebesgue operator but not is a normal operator. Let $0<a<b<+\infty$, then $\mathcal{L}^2([a,b])$ is a separable Hilbert space over $\mathbb{R}$, because any separable Hilbert space over $\mathbb{R}$ can be expanded to a separable Hilbert space over $\mathbb{C}$, it is enough to prove the conclusion on $\mathcal{L}^2([a,b])$. We prove the conclusion by six parts: $(1)$ Let $0<a<1<b<+\infty$,$M=\{[a,\frac{b-a}{2}],[\frac{b-a}{2},b]\}$. Construct measure preserving transformation on $[a,b]$. There is a Borel algebra $\xi(M)$ generated by $M$, define $\Phi:[a,b]\to[a,b]$, $\Phi([a,\frac{b-a}{2}])=[\frac{b-a}{2},b]$, $\Phi([\frac{b-a}{2},b])=[a,\frac{b-a}{2}]$. Then $\Phi$ is an invertible measure preserving transformation on the Borel algebra $\xi(M)$. By [@PeterWalters1982]P63,$U_{\Phi}\neq1$ is a unitary operator induced by $\Phi$, where $U_{\Phi}$ is the composition $U_{\Phi}h=h\circ\Phi,\forall h\in\mathcal{L}^2([a,b])$ on $\mathcal{L}^2([a,b])$. $(2)$ Define $M_xh=xh$ on $\mathcal{L}^2([a,b])$,then $M_x$ is an invertible positive operator. $(3)$ For $f(x)=\frac{\|\ln\,x\|}{x},x>0$, define $d\mu=f(x)d\,x$. then $f(x)$ is continuous and $f(x)>0,a.e.x\in[a,b]$, hence $d\,\mu$ that is absolutely continuous with respect to $d\,x$ is finite positive complete Borel measure, therefor $\mathcal{L}^2([a,b],d\,\mu)$ a separable Hilbert space over $\mathbb{R}$. Moreover $\mathcal{L}^2([a,b])$ and $\mathcal{L}^2([a,b],d\,\mu)$ are unitary equivalent. $(4)$ Let $T=U_{\Phi}M_x$,we get $T^{}T=U_{\Phi}TT^{}U_{\Phi}^{}$ and $U_{\Phi}\neq1$. Because of $U_{\Phi}M_{x}\neq M_{x}U_{\Phi}$ and $U_{\Phi}M_{x^2}\neq U_{\Phi}M_{x^2}$, we get that $T$ is not a normal operator and $\sigma(\|T\|)=[a,b]$. $(5)$ The operator $T=U_{\Phi}M_x$ on $\mathcal{L}^2([a,b])$ is corresponding to the operator $T^{\prime}$ on $\mathcal{L}^2([a,b],d\,\mu)$,$T^{\prime}$ is invertible bounded linear operator and is not a normal operator and $\sigma(\|T^{\prime}\|)=[a,b]$. $(6)$ From $\int_{a}^{b}x^{n}f(x)d\,x=\int_{a^n}^{b^n}tf(t^{\tfrac{1}{n}})\frac{1}{nt^{\tfrac{n-1}{n}}}d\,t$, let $f_n(t)=\frac{1}{n}I_{[a^n,b^n]}f(t^{\tfrac{1}{n}})\frac{1}{t^{\tfrac{n-1}{n}}}$, we get that $f_n(t)$ is continuous and almost everywhere positive, hence $f_n(t)d\,t$ is a finite positive complete Borel measure. For any $E\subseteq\mathbb{R}_{+}$ define $I_{E}=1$ when $x\in E$ else $I_{E}=0$,so $I_{E}$ is the identity function on $E$. $(i)$ $f_n(t^{-1})=\frac{1}{n}I_{[a^n,b^n]}f(t^{-\tfrac{1}{n}})\frac{1}{t^{-\tfrac{n-1}{n}}}$ $=\frac{1}{n}I_{[a^n,b^n]}\frac{\|\ln\,t^{-\tfrac{1}{n}}\|}{t^{-\tfrac{1}{n}}}\frac{1}{t^{-\tfrac{n-1}{n}}}$ $=\frac{1}{n}I_{[a^n,b^n]}t\|\ln\,t^{\tfrac{1}{n}}\|$ $(i)$ $t^2f_n(t)=\frac{1}{n}I_{[a^n,b^n]}f(t^{\tfrac{1}{n}})\frac{t^2}{t^{\tfrac{n-1}{n}}}$ $=\frac{1}{n}I_{[a^n,b^n]}\frac{\|\ln\,t^{\tfrac{1}{n}}\|}{t^{\tfrac{1}{n}}}\frac{t^2}{t^{\tfrac{n-1}{n}}}$ $=\frac{1}{n}I_{[a^n,b^n]}t{\|\ln\,t^{\tfrac{1}{n}}\|}$ By $(i)(ii)$ we get $x^2f_{n}(x)=f_{n}(x^{-1})$. From $\sigma(\|T^{\prime n}\|)=[a^n,b^n]$ and $\int_{a^n}^{b^n}t^2f(t^{\tfrac{1}{n}})\frac{1}{nt^{\tfrac{n-1}{n}}}d\,t=\int_{0}^{+\infty}t^2f_{n}(t)dt$, let $d\,\mu_{\|T^{\prime n}\|}=f_n(t)d\,t$,then $d\,\mu_{\|T^{\prime n}\|}$ is the finite positive complete Borel measure. For any given $0\neq h(x)\in\mathcal{L}^2([a,b])$ we get $0\neq h(x^{-1})\in\mathcal{L}^2([a,b])$. $I_{[a,b]}$ is a $\mathcal{A}(\|M_{x}^{n}\|)$-cyclic vector of the multiplication $M_{x}^{n}=M_{x^n}$, and $I_{[a^n,b^n]}$ is a $\mathcal{A}(\|T^{\prime n}\|)$-cyclic vector of $\|T^{\prime n}\|$, By Definition $\ref{lebesguesuanzileidingyi6}$ we get that $T^{\prime}$ is Lebesgus operator but not is a normal operator. \[lebesguesuanzicunzaizhengguisuanzi10\] There is an invertible bounded linear operator $T$ on the separable Hilbert space $\mathbb{H}$ over $\mathbb{C}$, $T$ is a Lebesgue operator and also is a positive operator. By the cyclic representation of $C^{}$ algebra and the GNS construction, also by the functional calculus of invertible bounded linear operator, we could study the operator by the integral on $\mathbb{R}$. This way neither change Li-Yorke chaotic nor the computing, but by the singular integral in mathematical analysis on the theoretical level we should find that there is a invertible bounded linear operator $T$ that is Li-Yorke chaotic but $T^{-1}$ is not Li-Yorke chaotic. [4. Cowen-Douglas function on Hardy space]{} For $\mathbb{D}=\{z\in\mathbb{C},\|z\|<1\}$, if $g$ is an complex analytic function on $\mathbb{D}$ and there is $\sup\limits_{r<1}\int_{-\pi}^{\pi}\|g(re^{i\theta})\|^2\,d\theta<+\infty$, then we denote $g\in\mathcal{H}^2(\mathbb{D})$, hence $\mathcal{H}^2(\mathbb{D})$ is a Hilbert space with the norm $\\|g\\|_{\mathbb{H}^2}^2=\sup\limits_{r<1}\int_{-\pi}^{\pi}\|g(re^{i\theta})\|^2\,\frac{d\theta}{2\pi}$, especially $\mathcal{H}^2(\mathbb{D})$ is denoted as a Hardy space. By the completion theory of complex analytic functions, the Hardy space $\mathcal{H}^2(\mathbb{D})$ is a special Hilbert space that is relatively easy not only for theoretical but also for computing, and we should give some properties about adjoint multiplier operators on $\mathcal{H}^2(\mathbb{D})$. Any given complex analytic function $g$ has a Taylor expansion $g(z)=\sum\limits_{n=0}^{+\infty}a_nz^n$, so $g\in\mathcal{H}^2(\mathbb{D})$ and $\sum\limits_{n=0}^{+\infty}a_n^2<+\infty$ are naturally isomorphic. For $\mathbb{T}=\partial\mathbb{D}$, if $\mathcal{L}^2(\mathbb{T})$ denoted the closed span of all Taylor expansions of functions in $\mathcal{L}^2(\mathbb{T})$, then $\mathcal{H}^{2}(\mathbb{T})$ is a closed subspace of $\mathcal{L}^2(\mathbb{T})$. From the naturally isomorphic of $\mathcal{H}^{2}(\mathbb{D})$ and $\mathcal{H}^{2}(\mathbb{T})$ by the properties of analytic function, we denote $\mathcal{H}^{2}(\mathbb{T})$ also as a Hardy space. By [@JohnBConway1990]P6 we get that any Cauchy sequence with the norm $\\|\bullet\\|_{\mathbb{H}^2}$ on $\mathcal{H}^2(\mathbb{D})$ is a uniformly Cauchy sequence on any closed disk in $\mathbb{D}$, in particular,we get that the point evaluations $f\to f(z)$ are continuous linear functional on $\mathcal{H}^2(\mathbb{D})$, by the Riesz Representation Theorem[@JohnBConway1990]P13,for any $g(s)\in\mathcal{H}^2(\mathbb{D})$, there is a unique $f_z(s)\in\mathcal{H}^2(\mathbb{D})$ such that $g(z)=<g(s),f_z(s)>$, so define that $f_z$ is a reproducing kernel at $z$. Let $\mathcal{H}^{\infty}(\mathbb{D})$ denote the set of all bounded complex analytic function on $\mathbb{D}$, for any given $\phi\in\mathcal{H}^{\infty}(\mathbb{D})$, it is easy to get that $\\|\phi\\|_{\infty}=\sup\{\|\phi(z)\|;\|z\|<1\}$ is a norm on $\phi\in\mathcal{H}^{\infty}(\mathbb{D})$. for any given $g\in\mathcal{H}^{2}(\mathbb{D})$, the multiplication operator $M_{\phi}(g)=\phi g$ associated with $\phi$ on $\mathcal{H}^{2}(\mathbb{D})$ is a bounded linear operator, and by the norm on $\mathcal{H}^{2}(\mathbb{D})$ we get $\\|M_{\phi}(g)\\|\leq\\|\phi\\|_{\infty}\\|g\\|_{\mathbb{H}^2}$. If we denote $\mathcal{H}^{\infty}(\mathbb{T})$ as the closed span of all Taylor expansions of functions in $\mathcal{L}^{\infty}(\mathbb{T})$, then $\mathcal{H}^{2}(\mathbb{T})$ is a closed subspace of $\mathcal{L}^2(\mathbb{T})$ and $\mathcal{H}^{\infty}(\mathbb{D})$ and $\mathcal{H}^{\infty}(\mathbb{T})$ are naturally isomorphic by the properties of complex analytic functions [@HenriCartanyujiarong2008]P55P97 and the Dirichlet Problem [@HenriCartanyujiarong2008]P103. There are more properties about Hardy space in [@FBayartEMatheron2009]P7, [@JohnBGarnett2007]P48, [@KennethHoffman1962]P39, [@RonaldGDouglas1998]P133 and [@WilliamArveson2002]P106. \[cowendouglassuanzidingyi2\] For a connected open subset $\Omega$ of $\mathbb{C}$,$n\in\mathbb{N}$, let $\mathcal{B}_n(\Omega)$ denotes the set of all bounded linear operator $T$ on $\mathbb{H}$ that satisfies: $(a)$ $\Omega\in\sigma(T)=\{\omega\in\mathbb{C}:T-\omega \text{not invertible}\}$; $(b)$ $ran(T-\omega)=\mathbb{H}$ for $\omega\in\Omega$; $(c)$ $\bigvee\ker\limits_{\omega\in\Omega}(T-\omega)=\mathbb{H}$; $(d)$ $\dim\ker(T-\omega)=n$ for $\omega\in\Omega$. If $T\in\mathcal{B}_n(\Omega)$, then say that $T$ is a Cowen-Douglas operator. \[cowendouglassuanzidingyidevaneyliyrokefenbuqianghunhedingli2\] For a connected open subset $\Omega$ of $\mathbb{C}$,$T\in\mathcal{B}_n(\Omega)$,we get $(1)$ If $\Omega\bigcap\mathcal{T}\neq\emptyset$,then $T$ is Devaney chaotic. $(2)$ If $\Omega\bigcap\mathcal{T}\neq\emptyset$,then $T$ is distributionally chaotic. $(3)$ If $\Omega\bigcap\mathcal{T}\neq\emptyset$,then $T$ strong mixing. \[waihanshudingyi46\] Let $\mathcal{P}(z)$ be the set of all polynomials about $z$,where $z\in\mathbb{T}$. Define a function $h(z)\in\mathcal{H}^2(\mathbb{T})$ is an outer function if $cl[h(z)\mathcal{P}(z)]=\mathcal{H}^2(\mathbb{T})$. \[waihanshukeyidexingzhi49\] A function $h(z)\in\mathcal{H}^{\infty}(\mathbb{T})$ is invertible on the Banach algebra $\mathcal{H}^{\infty}(\mathbb{T})$, if and only if $h(z)\in\mathcal{L}^{\infty}(\mathbb{T})$ and $h(z)$ is an outer function. \[chengfasuanzishimanshedangqiejindangshiwaihanshu45\] Let $\mathcal{P}(z)$ be the set of all polynomials about $z$,where $z\in\mathbb{D}$. then $h(z)\in\mathcal{H}^2(\mathbb{D})$ is an outer function if and only if $\mathcal{P}(z)h(z)=\{p(z)h(z);p\in\mathcal{P}(z)\}$ is dense in $\mathcal{H}^2(\mathbb{D})$. Let $\phi$ is a non-constant complex analytic function on $\mathbb{D}$, for any given $z_0\in\mathbb{D}$, by [@HenriCartanyujiarong2008]P29 we get that there exists $\delta_{z_0}>0$, exists $k_{z_0}\in\mathbb{N}$, when $\|z-z_0\|<\delta_{z_0}$, there is $$\begin{aligned} \phi(z)-\phi(z_0)=(z-z_0)^{k_{z_0}}h_{z_0}(z)\end{aligned}$$ where $h_{z_0}(z)$ is complex analytic on a neighbourhood of $z_0$ and $h_{z_0}(z_0)\neq0$. \[nyejiexihanshudedingyijitoubujiaobu51\] Let $\phi$ is a non-constant complex analytic function on $\mathbb{D}$, for any given $z_0\in\mathbb{D}$, there exists $\delta_{z_0}>0$ such that $$\begin{aligned} \phi(z)-\phi(z_0)\|_{\|z-z_0\|<\delta_{z_0}}=p_{n_{z_0}}(z)h_{z_0}(z)\|_{\|z-z_0\|<\delta_{z_0}},\end{aligned}$$ $h_{z_0}(z)$ is complex analytic on a neighbourhood of $z_0$ and $h_{z_0}(z_0)\neq0$, $p_{n_{z_0}}(z)$ is a $n_{z_0}$-th polynomial and the $n_{z_0}$-th coefficient is equivalent $1$. By the Analytic Continuation Theorem [@HenriCartanyujiarong2008]P28, we get that there is a unique complex analytic function $h_{z_0}(z)$ on $\mathbb{D}$ such that $\phi(z)-\phi(z_0)=p_{n_{z_0}}(z)h_{z_0}(z)$, then define $h_{z_0}(z)$ is a rooter function of $\phi$ at the point $z_0$. If for any given $z_0\in\mathbb{D}$, the rooter function $h_{z_0}(z)$ has non-zero point but the roots of $p_{n_{z_0}}(z)$ are all in $\mathbb{D}$ and $n_{z_0}\in\mathbb{N}$ is a constant on $\mathbb{D}$ that is equivalent $m$, then define $\phi$ is a $m$-folder complex analytic function on $\mathbb{D}$. \[youjiejiexihanshuliCowenDouglashanshudingyi47\] Let $\phi(z)\in\mathcal{H}^{\infty}(\mathbb{D}),n\in\mathbb{N}$, $M_{\phi}$ is the multiplication by $\phi$ on $\mathcal{H}^{2}(\mathbb{D})$. if the adjoint multiplier $M_{\phi}^{}\in\mathcal{B}_n(\bar{\phi}(\mathbb{D}))$, then define $\phi$ is a Cowen-Douglas function. By Definition $\ref{youjiejiexihanshuliCowenDouglashanshudingyi47}$ we get that any constant complex analytic function is not a Cowen-Douglas function \[cowendouglussuanziyujiexichengfasuanzijigezhonghundundeguanxi24\] Let $\phi(z)\in\mathcal{H}^{\infty}(\mathbb{D})$ be a $m$-folder complex analytic function, $M_{\phi}$ is the multiplication by $\phi$ on $\mathcal{H}^{2}(\mathbb{D})$. If for any given $z_0\in\mathbb{D}$, the rooter functions of $\phi$ at $z_0$ is a outer function, then $\phi$ is a Cowen-Douglas function, that is,the adjoint multiplier $M_{\phi}^{}\in\mathcal{B}_m(\bar{\phi}(\mathbb{D}))$. By the definition of $m$-folder complex analytic function Definition $\ref{nyejiexihanshudedingyijitoubujiaobu51}$ we get that $\phi$ is not a constant complex analytic function. For any given $z\in\mathbb{D}$, let $f_z\in\mathcal{H}^2(\mathbb{D})$ be the reproducing kernel at $z$. We confirm that $M_{\phi}^{}$ is valid the conditions of Definition $\ref{cowendouglassuanzidingyi2}$ one by one. $(1)$ For any given $z\in\mathbb{D}$, $f_z$ is an eigenvector of $M_{\phi}^{}$ with associated eigenvalue $\lambda=\bar{\phi}(z)$.Because for any $g\in\mathcal{H}^2(\mathbb{D})$ we get $$\begin{aligned} <g,M_{\phi}^{}(f_z)>_{\mathcal{H}^2}=<\phi g,(f_z)>_{\mathcal{H}^2}=\phi(z)f(z)=<g,\bar{\phi}(z)f_z>_{\mathcal{H}^2}\end{aligned}$$ By the Riesz Representation Theorem[@JohnBConway1990]P13 of bounded linear functional in the form of inner product on Hilbert space, we get $M_{\phi}^{}(f_z)=\bar{\phi}(z)f_z=\lambda f_z$, that is,$f_z$ is an eigenvector of $M_{\phi}^{}$ with associated eigenvalue$\lambda=\bar{\phi}(z)$. $(2)$ For any given $\bar{\lambda}\in\mathbb{\phi(D)}$, because of $0\neq\phi\in\mathcal{H}^{\infty}(\mathbb{D})$, we get that the multiplication operator $M_{\phi}-\lambda$ is injection by the properties of complex analysis, hence $\ker({M_{\phi}-\lambda})=0$. Because of $\mathcal{H}^2(\mathbb{D})=\ker(M_{\phi}-\lambda)^{\bot}=cl[ran(M_{\phi}^{}-\bar{\lambda})]$, we get that $ran(M_{\phi}^{}-\bar{\lambda})$ is a second category space. By [@JohnBConway1990]P305 we get that $M_{\phi}^{}-\bar{\lambda}$ is a closed operator, also by [@JohnBConway1990]P93 or [@zhanggongqinglinyuanqu2006]P97 we get $ran(M_{\phi}^{}-\bar{\lambda})=\mathcal{H}^2(\mathbb{D})$. $(3)$ Suppose that $span\{f_z;z\in\phi(\mathbb{D})\}=span\{\frac{1}{1-\bar{z}s};z\in\phi(\mathbb{D})\}$ is not dense in $\mathcal{H}^2(\mathbb{D})$, By the definition of reproducing kernel $f_z$ and because of $0\neq\phi\in\mathcal{H}^{\infty}(\mathbb{D})$, we get that there exists $0\neq g\in\mathcal{H}^2(\mathbb{D})$,for any given $z\in\mathbb{D}$, we have $$\begin{aligned} 0=<g,\bar{\phi}(z)f_z>_{\mathcal{H}^2}=\phi(z)g(z)=<\phi(z)g(z),f_z>_{\mathcal{H}^2}.\end{aligned}$$ So we get $g=0$ by the Analytic Continuation Theorem [@HenriCartanyujiarong2008]P28,that is a contradiction for $g\neq0$. Therefor we get that $span\{f_z;z\in\phi(\mathbb{D})\}$ is dense in $\mathcal{H}^2(\mathbb{D})$, that is,$\bigvee\ker\limits_{\bar{\lambda}\in\phi(\mathbb{D})}(M_{\phi}^{}-\bar{\lambda})=\mathcal{H}^2(\mathbb{D})$. $(4)$ By Definition $\ref{nyejiexihanshudedingyijitoubujiaobu51}$ and the conditions of this theorem, for any given $\lambda\in\phi(\mathbb{D})$, there exists $z_0\in\mathbb{D}$, exists $m$-th polynomial $p_m(z)$ and outer function $h(z)$ such that $$\begin{aligned} \left. \begin{array}{l} \phi(z)-\lambda=\phi(z)-\phi(z_0)=p_m(z)h(z), \end{array} \right.\end{aligned}$$ We give $\dim\ker(M_{\phi(z)}^{}-\bar{\lambda})=m$ by the following $(i)(ii)(iii)$ assertions. $(i)$ Let the roots of $p_m(z)$ are $z_0,z_1,\cdots,z_{m-1}$, then there exists decomposition $p_m(z)=(z-z_0)(z-z_1)\cdots(z-z_{m-1})$, and denote $p_{m,z_0z_1\cdots z_{m-1}}(z)$ is the decomposition of $p_m(z)$ by the permutation of $z_0,z_1,\cdots,z_{m-1}$, the following to get $\dim\ker M_{p_{m,z_0z_1\cdots z_{m-1}}}^{}=m$. By the Taylor expansions of functions in $\mathcal{H}^2(\mathbb{T})$, we get there is a naturally isomorphic $$\begin{aligned} \left. \begin{array}{l} F_{s}:\mathcal{H}^2(\mathbb{D})\to\mathcal{H}^2(\mathbb{D}-s),F_{s}(g(z))\to g(z+s),\text{ÆäÖÐ}s\in\mathbb{C}. \end{array} \right.\end{aligned}$$ It is easy to get that $G=\{F_s;s\in\mathbb{C}\}$ is a abelian group by the composite operation $\circ$, hence for $0\leq n\leq m-1$,there is $$\begin{array}{rcl} \mathcal{H}^2(\mathbb(D)) & \underrightarrow{\qquad\quad~M_{z-z_n}~~\quad\qquad} & \mathcal{H}^2(\mathbb(D))\\ F_{z_n}\downarrow & & \downarrow F_{z_n}\\ \mathcal{H}^2(\mathbb{D}-z_n)&\overrightarrow{\qquad\qquad~M_{z}^{'}\qquad\qquad}& \mathcal{H}^2(\mathbb{D}-z_n) \end{array}$$ Let $T$ is the backward shift operator on the Hilbert space $\mathcal{L}^2(\mathbb{N})$, that is, $T(x_1,x_2,\cdots)=(x_2,x_3,\cdots)$. With the naturally isomorphic between $\mathcal{H}^2(\mathbb{D}-z_n)$ and $\mathcal{H}^2(\partial(\mathbb{D}-z_n))$, $M_{z}^{'}$ is equivalent the backward shift operator $T$ on $\mathcal{H}^2(\partial(\mathbb{D}-z_n))$, that is,$M_{z}^{'}$ is a surjection and $\dim\ker M_{z}^{'}=1$, hence $M_{z-z_n}^{}$ is a surjection and $\dim\ker M_{z-z_n}^{}=1$, where $0\leq n\leq m-1$. By the composition of $F_{z_{m-1}}\circ F_{z_{m-2}}\circ\cdots\circ F_{z_{0}}$, $M_{p_{m,z_0z_1\cdots z_{m-1}}}^{'}$ is equivalent $T^m$. that is,$M_{p_{m,z_0z_1\cdots z_{m-1}}}^{'}$ is a surjection and $\dim\ker M_{p_{m,z_0z_1\cdots z_{m-1}}}^{'}=m$, hence $M_{p_{m,z_0z_1\cdots z_{m-1}}}^{}$ is a surjection and $\dim\ker M_{p_{m,z_0z_1\cdots z_{m-1}}}^{}=m$. $(ii)$ Because $\mathcal{H}^{\infty}$ is a abelian Banach algebra, $M_{p_{m}}$ is independent to the permutation of $1$-th factors of $p_m(z)$, that is,$M_{p_{m}}^{}$ is independent to the $1$-th factors multiplication of $p_m(z)=(z-z_0)(z-z_1)\cdots(z-z_{m-1})$. Because $G=\{F_s;s\in\mathbb{C}\}$ is a abelian group by composition operation $\circ$, for $0\leq n\leq m-1$, $F_{z_{m-1}}\circ F_{z_{m-2}}\circ\cdots\circ F_{z_{0}}$ is independent to the permutation of composition. Hence $M_{p_{m}}^{}$ is a surjection and $$\begin{aligned} \left. \begin{array}{l} \dim\ker M_{p_{m}}^{}=\dim\ker M_{p_{m,z_0z_1\cdots z_{m-1}}}^{}=m. \end{array} \right.\end{aligned}$$ $(iii)$ By Definition $\ref{waihanshudingyi46}$ and Theorem $\ref{chengfasuanzishimanshedangqiejindangshiwaihanshu45}$, also by [@JohnBConway1990]P93 or [@zhanggongqinglinyuanqu2006]P97 and by [@JohnBConway1990]P305 we get that the multiplication operator $M_{h}$ is surjection that associated with the outer function $h$. Hence we get $$\begin{aligned} \left. \begin{array}{l} \ker M_{h(z)}^{}=(ranM_{h(z)})^{\bot}=(\mathcal{H}^2(\mathbb{D}))^{\bot}=0. \end{array} \right.\end{aligned}$$ Because there exists decomposition $M_{p_m(z)h(z)}^{}=M_{h(z)}^{}M_{p_m(z)}^{}$ on $\mathcal{H}^2(\mathbb{D})$, we get $$\begin{aligned} \left. \begin{array}{l} \dim\ker(M_{\phi}^{}-\bar{\lambda})=\dim\ker M_{p_m(z)h(z)}^{}=\dim\ker M_{p_m(z)}^{}=m. \end{array} \right.\end{aligned}$$ By $(1)(2)(3)(4)$ we get the adjoint multiplier operator $M_{\phi}^{}\in\mathcal{B}_m(\bar{\phi}(\mathbb{D}))$. By Theorem $\ref{cowendouglussuanziyujiexichengfasuanzijigezhonghundundeguanxi24}$ and Lemma $\ref{waihanshukeyidexingzhi49}$, we get \[nyejiexihanshushicowendouglashanshudetiaojianqigenhanshukeni52\] Let $\phi\in\mathcal{H}^{\infty}(\mathbb{D})$ is a $m$-folder complex analytic function, for any given $z_0\in\mathbb{D}$, if the rooter function of $\phi$ at $z_0$ is invertible in the Banach algebra $\mathcal{H}^{\infty}(\mathbb{D})$, then $\phi$ is a Cowen-Douglas function. Especially,for any given $n\in\mathbb{D}$,if $a$ and $b$ are both non-zero complex, then $a+bz^n\in\mathcal{H}^{\infty}(\mathbb{D})$ is a Cowen-Douglas function. The following gives some properties about the adjoint multiplier of Cowen-Douglas functions. \[hardykongjianshangchengfasuanziyuyuanzhoujiaofeikongdedingli17\] If $\phi\in\mathcal{H}^{\infty}(\mathbb{D})$ is a Cowen-Douglas function, $M_{\phi}$ is the multiplication by $\phi$ on $\mathcal{H}^{2}(\mathbb{D})$, Then the following assertions are equivalent $(1)$ $M_{\phi}^{}$ is Devaney chaotic; $(2)$ $M_{\phi}^{}$ is distributionally chaotic; $(3)$ $M_{\phi}^{}$ is strong mixing; $(4)$ $M_{\phi}^{}$ is Li-Yorke chaotic; $(5)$ $M_{\phi}^{}$ is hypercyclic; $(6)$ $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$. By Example $\ref{chaoxunhuanyudanweikaiyuanpanjiaofeikongdeyinyongyinli34}$ we get that $M_{\phi}^{}$ is hypercyclic if and only if $\phi$ is non-constant and $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$, hence $(6)$ is equivalent to $(5)$. First to get that $(6)$ imply $(1)(2)(3)(4)$. Because $\phi\in\mathcal{H}^{2}(\mathbb{D})$ is a Cowen-Douglas function, by Definition $\ref{youjiejiexihanshuliCowenDouglashanshudingyi47}$, $M_{\phi}^{}\in\mathcal{B}_n(\bar{\phi}(\mathbb{D}))$. By Theorem $\ref{cowendouglassuanzidingyidevaneyliyrokefenbuqianghunhedingli2}$ we get that if $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$, then $(1)(2)(3)$ is valid. On Banach spaces Devaney chaotic, distributionally chaotic and strong mixing imply Li-Yorke chaotic, respectively.Hence $(4)$ is valid.Because $\bar{\phi}(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$ and $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$ are mutually equivalent, $(6)$ imply $(1)(2)(3)(4)$. Then to get that $(1)(2)(3)(4)$ imply $(6)$.By $(1)(2)(3)$ imply $(4)$,respectively, it is enough to get that $(4)$ imply $(6)$. If $M_{\phi}^{}$ is Li-Yorke chaotic, then we get that $\phi$ is non-constant and by [@HouBLiaoGCaoY2012]Theorem3.5 we get $\sup\limits_{n\to+\infty}\\|M_{\phi}^{n}\\|\to\infty$, hence $\\|M_{\phi}\\|=\\|M_{\phi}^{}\\|>1$, that is,$\sup\limits_{z\in\mathbb{D}}\|\phi(z)\|>1$. Moreover,we also have $\inf\limits_{z\in\mathbb{D}}\|\phi(z)\|<1$, Indeed,if we assume that $\inf\limits_{z\in\mathbb{D}}\|\phi(z)\|\geq1$ then $\frac{1}{\phi}\in\mathcal{H}^{\infty}$ and $\\|M_{\frac{1}{\phi}}^{}\\|=\\|M_{\frac{1}{\phi}}\\|\leq1$. Hence for any $0\neq x\in\mathcal{H}^2(\mathbb{D})$ we get $\\|M_{\phi}^{n}x\\|\geq\frac{1}{\\|M_{\phi}^{-n}\\|}\\|x\\|\geq\frac{1}{\\|M_{\frac{1}{\phi}}^{}\\|^n}\\|x\\|\geq\\|x\\|$. It is a contradiction to $M_{\phi}^{}$ is Li-Yorke chaotic. Therefor that $M_{\phi}^{}$ is Li-Yorke chaotic imply $\inf\limits_{z\in\mathbb{D}}\|\phi(z)\|<1<\sup\limits_{z\in\mathbb{D}}\|\phi(z)\|$, By the properties of a simple connectedness argument of complex analytic functions we get $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$. Hence we get $(1)(2)(3)(4)$ both imply $(6)$. \[cowendouglussuanzideniyuhardykongjiandechengfasuanzi23\] If $\phi\in\mathcal{H}^{\infty}(\mathbb{D})$ is a invertible Cowen-Douglas function in the Banach algebra $\mathcal{H}^{\infty}(\mathbb{D})$, and let $M_{\phi}$ be the multiplication by $\phi$ on $\mathcal{H}^{2}(\mathbb{D})$. Then $M_{\phi}^{}$ is Devaney chaotic or distributionally chaotic or strong mixing or Li-Yorke chaotic if and only if $M_{\phi}^{-1}$ is. Because of $T=(T^{-1})^{-1}$, it is enough to prove that $M_{\phi}^{}$ is Devaney chaotic or distributionally chaotic or strong mixing or Li-Yorke chaotic imply $M_{\phi}^{-1}$ is. By Definition $\ref{youjiejiexihanshuliCowenDouglashanshudingyi47}$ we get $M_{\phi}^{}\in\mathcal{B}_n(\bar{\phi}(\mathbb{D}))$, with a simple computing we get $M_{\phi}^{-1}\in\mathcal{B}_n(\frac{1}{\bar{\phi}}(\mathbb{D}))$. If $M_{\phi}^{}$ is Devaney chaotic or distributionally chaotic or strong mixing or Li-Yorke chaotic, by Theoem $\ref{hardykongjianshangchengfasuanziyuyuanzhoujiaofeikongdedingli17}$ we get $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$, and by the properties of complex analytic functions we get $\frac{1}{\phi}(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$. Because of $M_{\phi}^{-1}\in\mathcal{B}_1(\frac{1}{\bar{\phi}}(\mathbb{D}))$ and by Theorem $\ref{hardykongjianshangchengfasuanziyuyuanzhoujiaofeikongdedingli17}$ we get $M_{\phi}^{-1}$ is Devaney chaotic or distributionally chaotic or strong mixing or Li-Yorke chaotic. [5. The chaos of scalars perturbation of an operator]{} We now study some properties about scalars perturbation of an operator inspired by [@HoubingzheTiangengShiluoyi2009] and [@BermudezBonillaMartinezGimenezPeiris2011] that research some properties about the compact perturbation of scalar operator. \[shuzhijiafasuanzidedingyi1\] Let $\lambda\in\mathbb{C}$,$T\in\mathcal{L}(\mathbb{H})$.Define $(i)$ Let $S_{LY}(T)$ denote the set such that $\lambda I+T$ is Li-Yorke chaotic for every $\lambda\in S_{LY}(T)$. $(ii)$ Let $S_{DC}(T)$ denote the set such that $\lambda I+T$ is distributionally chaotic for every $\lambda\in S_{DC}(T)$. $(iii)$ Let $S_{DV}(T)$ denote the set such that $\lambda I+T$ is Devaney chaotic for every $\lambda\in S_{DV}(T)$. $(iv)$ Let $S_{H}(T)$ denote the set such that $\lambda I+T$ is hypercyclic for every $\lambda\in S_{H}(T)$. By Definition $\ref{shuzhijiafasuanzidedingyi1}$ we get $S_{LY}(\lambda I+T)=\lambda+S_{LY}(T)$, $S_{DC}(\lambda I+T)=\lambda+S_{DC}(T)$, $S_{DV}(\lambda I+T)=\lambda+S_{DV}(T)$ and $S_{H}(\lambda I+T)=\lambda+S_{H}(T)$. \[zhengguisuanzishuzhipuweikong3\] Let $T\in\mathcal{L}(\mathbb{H})$ be a normal operator,then $S_{LY}(T)=\emptyset$. Because $T$ is a normal operator,$\lambda I+T$ is a normal operator,too. by [@HoubingzheTiangengShiluoyi2009] we get $S_{LY}(T)=\emptyset$. \[jinmilingsuanzishuzhipu4\] There is a quasinilpotent compact operator $T\in\mathcal{L}(\mathbb{H})$ such that $S_{LY}(T)=\mathbb{T}$ is closed and $S_{LY}(T^{})=\emptyset$,where $\mathbb{T}=\partial\mathbb{D},\mathbb{D}=\{z\in\mathbb{C},\|z\|<1\}$. Let $\{e_n\}_{n\in\mathbb{N}}$ be a orthonormal basis of $\mathcal{L}^2(\mathbb{N})$ and let $T$ be a weighted backward shift operator with weight sequence $\{\omega_n=\frac{1}{n}\}_{n=1}^{+\infty}$ such that $S_{\omega}(e_0)=0$, $S_{\omega}(e_n)=\omega_{n}e_{n-1}$, where $0<\|\omega_n\|<M<+\infty$,$\forall n>0$. By the Spectral Radius formula[@JohnBConway1990]P197 $r_{\sigma}(T)=\lim\limits_{n\to+\infty}\\|T^n\\|^{\frac{1}{n}}$ we get $\sigma(T)=\{0\}$, hence $\sigma(\lambda I+T)=\lambda$. $(1)$ If $\|\lambda\|<1$, we can select $\varepsilon>0$ such that $\|\lambda\|+\varepsilon<q<1$. Then $\forall 0\neq x\in\mathbb{H}$ we get $$\begin{aligned} \lim\limits_{n\to\infty}\\|(\lambda I+T)^nx\\| \leq\lim\limits_{n\to\infty}\\|(\lambda I+T)^n\\|\\|x\\| \leq\lim\limits_{n\to\infty}(\|\lambda\|+\varepsilon)^n\\|x\\|=0.\end{aligned}$$ $(2)$ If $\|\lambda\|>1$,because of $\sigma((\lambda I+T)^{-1})=\frac{1}{\lambda}$,then we get $$\begin{aligned} \lim\limits_{n\to\infty}\\|(\lambda I+T)^nx\\| \geq\lim\limits_{n\to\infty}\frac{1}{\\|(\lambda I+T)^{-n}\\|}\\|x\\| \geq\lim\limits_{n\to\infty}\frac{1}{\\|(\lambda I+T)^{-1}\\|^n}\\|x\\| \geq\\|x\\|\neq0.\end{aligned}$$ $(3)$ By Theorem $\ref{duijiaoxianyujiaquanyiweizhuanzhihunhedingliyinyong11}$ we get that if $\|\lambda\|=1$, then $\lambda I+T$ is mixing. Mixing imply Li-Yorke chaotic. $(4)$ Because of $\sigma(T)=\sigma(T^{})$, by $(1)(2)$ we get that if $\|\lambda\|\neq1$,then $\lambda+T^{}$ is not Li-Yorke chaotic. $(5)$ If $\|\lambda\|=1$,from the view of infinite matrix $\lambda I+T^{}$ is lower triangular matrix, then with a simple computing,for any $0\neq x\in\mathcal{L}^2(\mathbb{N})$, we get $\varliminf\limits_{n\to\infty}\\|\lambda I+S_{\omega}^{}x\\|>0$. Hence $\lambda I+T^{}$ is not Li-Yorke chaotic. By $(1)(2)(3)(4)(5)$ we get that $S_{LY}(T)=\mathbb{T}$ is closed and $S_{LY}(T^{})=\emptyset$. \[shuzhipukeyishikaiji5\] Let $T$ be the backward shift operator on $\mathcal{L}^2(\mathbb{N})$, $T(x_1,x_2,\cdots)$ $=(x_2,x_3,\cdots)$. Then $S_{LY}(T)=S_{DC}(T)=S_{DV}(T)=S_{H}(T)=2\mathbb{D}\setminus\{0\}$, $S_{LY}(2T)=S_{DC}(2T)=S_{DV}(2T)=S_{H}(2T)=3\mathbb{D}$, Hence $S_{LY}(T)$ and $S_{LY}(2T)$ are open sets. By [@JohnBConway1990]P209 we get $\sigma(T)=cl\mathbb{D}$ and $\sigma(2T)=cl2\mathbb{D}$, by Definition $\sigma(T)$ we get $\sigma(\lambda I+T)=\lambda+cl\mathbb{D}$. Because of the method to prove the conclusion is similarly for $T$ and $2T$, we only to prove the conclusion for $T$. By the naturally isomorphic between $\mathcal{H}^{2}(\mathbb{T})$ and $\mathcal{H}^{2}(\mathbb{D})$. Let $\mathcal{L}^2(\mathbb{N})=\mathcal{H}^{2}(\mathbb{T})$, by the definition of $T$ we get $(\lambda I+T)^{}$ is the multiplication operator $M_{f}$ by $f(z)=\bar{\lambda}+z$ on the Hardy space $\mathcal{H}^{2}(\mathbb{T})$. By the Dirichlet Problem [@HenriCartanyujiarong2008]P103 we get that $f(z)$ is associated with the complex analytic function $\phi(z)=\bar{\lambda}+z\in\mathcal{H}^{\infty}(\mathbb{D})$ determined by the boundary condition $\phi(z)\|_{\mathbb{T}}=f(z)$. By Corollary $\ref{nyejiexihanshushicowendouglashanshudetiaojianqigenhanshukeni52}$ we get that $\phi$ is a Cowen-Douglas function. Therefor by the natural isomorphic between $\mathcal{H}^{2}(\mathbb{T})$ and $\mathcal{H}^{2}(\mathbb{D})$, $\lambda I+T$ is naturally equivalent to the operator $M_{\phi}^{}$ on $\mathcal{H}^{2}(\mathbb{D})$. By Theorem $\ref{hardykongjianshangchengfasuanziyuyuanzhoujiaofeikongdedingli17}$ we get that $M_{\phi}^{}$ is hypercyclic or Devaney chaotic or distributionally chaotic or Li-Yorke chaotic if and only if $\phi(\mathbb{D})\bigcap\mathbb{T}\neq\emptyset$. Because of $\sigma(\lambda I+T)=\sigma(\bar{\lambda}I+T^{})$, we get $\sigma(\lambda I+T)=\sigma(M_{\phi}^{})=\sigma(M_{\phi})\supseteq\phi(\mathbb{D})$, hence $S_{LY}(T)=S_{DC}(T)=S_{DV}(T)=S_{H}(T)=2\mathbb{D}\setminus\{0\}$ is an open set. Therefor we can get Let $T$ be the backward shift operator on $\mathcal{L}^2(\mathbb{N})$, $T(x_1,x_2,\cdots)$ $=(x_2,x_3,\cdots)$. For $\lambda\neq0,a\neq0,n\in\mathbb{N}$, if $\lambda+aT^n$ is a invertible bounded linear operator, then $\lambda+aT^n$ is strong mixing or Devaney chaotic or distributionally chaotic or Li-Yorke chaotic if and only if $(\lambda+aT^n)^{-1}$ is. \[jibushikaijiyebushibiji6\] There is $T\in\mathcal{L}(\mathbb{H})$, $S_{LY}(T)$ is neither open nor closed. Let $T_1,T_2\in\mathcal{L}(\mathbb{H})$, because $T_1$ or $T_2$ is Li-Yorke chaotic if and only if $T_1\bigoplus T_2$ is, we get $S_{LY}(T_1\bigoplus T_2)=S_{LY}(T_1)\bigcup S_{LY}(T_2)$. By Lemma $\ref{jinmilingsuanzishuzhipu4}$,Lemma $\ref{shuzhipukeyishikaiji5}$ and Definition $\ref{shuzhijiafasuanzidedingyi1}$ we get the conclusion. [6. Examples that $T$ is chaotic but $T^{-1}$ is not]{} In the last we give some examples to confirm the theory giving by functional calculus on the begin that $T$ is chaotic but $T^{-1}$ is not. \[duijiaoxianyujiaquanyiweizhuanzhibuhundun12\] Let $\{e_n\}_{n\in\mathbb{N}}$ be a orthonormal basis of $\mathcal{L}^2(\mathbb{N})$ and let $S_{\omega}$ be a backward shift operator on $\mathcal{L}^2(\mathbb{N})$ with weight sequence $\omega=\{\omega_n\}_{n\geq1}$ such that $S_{\omega}(e_0)=0$, $S_{\omega}(e_n)=\omega_{n}e_{n-1}$,where $0<\|\omega_n\|<M<+\infty$, $\forall n>0$. $(1)$ If $\|\lambda\|=1$,then $\lambda I+S_{\omega}$ is Li-Yorke chaotic, but $\lambda I+S_{\omega}^{}$ and $(\lambda I+S_{\omega}^{})^{-1}$ are not Li-Yorke chaotic. $(2)$ Let ${(\lambda I+S_{\omega})}^{n}=U_{n}\|{(\lambda I+S_{\omega})}^{n}\|$ is the polar decomposition of ${(\lambda I+S_{\omega})}^{n}$, $\{U_{n}\}_{n=1}^{\infty}$ is not a constant sequence. $(1)$ By Theorem $\ref{duijiaoxianyujiaquanyiweizhuanzhihunhedingliyinyong11}$, we get that for $\|\lambda\|=1$,$\lambda I+S_{\omega}$ is mixing and mixing imply Li-Yorke chaotic, hence $\lambda I+S_{\omega}$ is Li-Yorke chaotic. From the view of infinite matrix,$\lambda I+S_{\omega}^{}$ and $(\lambda I+S_{\omega}^{})^{-1}$ are lower triangular matrix, with a simple computing,for any $0\neq x\in\mathcal{L}^2(\mathbb{N})$, we get $\varliminf\limits_{n\to\infty}\\|\lambda I+S_{\omega}^{}x\\|>0$,and $\varliminf\limits_{n\to\infty}\\|(\lambda I+S_{\omega}^{})^{-1}x\\|>0$. Hence $\lambda I+S_{\omega}^{}$ and $(\lambda I+S_{\omega}^{})^{-1}$ are not Li-Yorke chaotic. $(2)$ Let ${(\lambda I+S_{\omega})}^{n}=U_{n}\|{(\lambda I+S_{\omega})}^{n}\|$ is the polar decomposition of ${(\lambda I+S_{\omega})}^{n}$. If $\{U_{n}\}_{n=1}^{\infty}$ is a constant sequence, then by Theorem $\ref{hanshuyansuan5}$ we get that $(\lambda I+S_{\omega}^{})^{-1}$ is Li-Yorke chaotic. A contradiction. Hence $\{U_{n}\}_{n=1}^{\infty}$ is not a constant sequence. \[duijiaoxianyujxiaojinsuanzidingliyinyong13\] For any $\varepsilon>0$, there is a small compact operator $K_\varepsilon\in\mathcal{L}(\mathbb{H})$ and $\\|K_\varepsilon\\|<\varepsilon$ such that $I+K_\varepsilon$ is distributionally chaotic. In [@HoubingzheTiangengShiluoyi2009], $I+K_\varepsilon=\bigoplus\limits_{j=1}^{+\infty}(I_j+K_j)$ is distributionally chaotic. where $\mathbb{H}=\bigoplus\limits_{j=1}^{+\infty}\mathbb{H}_j$, $n_j=2m_j$,$\mathbb{H}_j$ is the $n_j$-dimension subspace of $\mathbb{H}$. On $\mathbb{H}_j$ define: $S_j=\left[ \begin{array}{cccc} 0 & 2\varepsilon_j & & \\ & \ddots &\ddots & \\ & &\ddots &2\varepsilon_j\\ & & &0 \end{array} \right]_{n_j\times n_j}$, $K_j=\left[ \begin{array}{cccc} -\varepsilon_j& 2\varepsilon_j & & \\ & \ddots &\ddots & \\ & &\ddots &2\varepsilon_j\\ & & &-\varepsilon_j \end{array} \right]_{n_j\times n_j}$. $I_j+K_j=\left[ \begin{array}{cccc} 1-\varepsilon_j& 2\varepsilon_j & & \\ & \ddots &\ddots & \\ & &\ddots &2\varepsilon_j\\ & & &1-\varepsilon_j \end{array} \right]_{n_j\times n_j}=(1-\varepsilon_j)I_j+S_j$. We can construct a invertible bounded linear operator $I+K_\varepsilon$ in the same way that is Li-Yorke chaotic, but $(I+K_\varepsilon)^{-1}$ is not. \[duijiaoxianyujxiaojinsuanzidenibuhundun14\] There is a invertible bounded linear operator $I+K_\varepsilon$ on $\mathbb{H}=\mathcal{L}^2(\mathbb{N})$ such that $I+K_\varepsilon$ is Li-Yorke chaotic, but $(I+K_\varepsilon)^{-1}$,$(I+K_\varepsilon)^{-1}$ and $(I+K_\varepsilon)^{}$ are not Li-Yorke chaotic. Let $\{e_i\}_{i=1}^{\infty}$ is a orthonormal basis of $\mathbb{H}=\mathcal{L}^2(\mathbb{N})$ and Let $\mathbb{H}=\bigoplus\limits_{j=1}^{+\infty}\mathbb{H}_j$, $j\in\mathbb{N}$, where $\mathbb{H}_{j}=\overline{span\{e_i\}},1+\tfrac{j(j-1)}{2}\leq i\leq\tfrac{j(j+1)}{2}$, $\mathbb{H}_j$ is $j$-dimension subspace of $\mathbb{H}$. For any given positive sequence $\{\varepsilon_j\}_{1}^{\infty}$ such that $\varepsilon_j\to0$ and $\sup\limits_{j\to\infty}(1+\varepsilon_j)^{j}\to+\infty$, on $\mathbb{H}_j$ define: $$\begin{aligned} S_j=\left[ \begin{array}{cccc} 0 & 2\varepsilon_j & & \\ & \ddots &\ddots & \\ & &\ddots &2\varepsilon_j\\ & & &0 \end{array} \right]_{j\times j}, K_j=\left[ \begin{array}{cccc} -\varepsilon_j& 2\varepsilon_j & & \\ & \ddots &\ddots & \\ & &\ddots &2\varepsilon_j\\ & & &-\varepsilon_j \end{array} \right]_{j\times j} \end{aligned}$$ . $$\begin{aligned} I_j+K_j=\left[ \begin{array}{cccc} 1-\varepsilon_j& 2\varepsilon_j & & \\ & \ddots &\ddots & \\ & &\ddots &2\varepsilon_j\\ & & &1-\varepsilon_j \end{array} \right]_{j\times j}=(1-\varepsilon_j)I_j+S_j \end{aligned}$$ . First to prove that $(I+K_\varepsilon)$ is Li-Yorke chaotic. Let $I+K_\varepsilon=\bigoplus\limits_{j=1}^{+\infty}(I+K_j)$, $f_j=\frac{1}{\sqrt{j}}(1_1,\cdots,1_j)$ and $f_{j,n}=\frac{1}{\sqrt{j}}(1_1,\cdots,1_n,0,\cdots,0)\in\mathbb{H}_j$. for any $1\leq n\leq j$ we get $\left. \begin{array}{l} \\|(I+K_\varepsilon)^n(f_j)\\|\\ =\\|(I_j+K_j)^n(f_j)\\|\\ =\\|((1-\varepsilon_j)I_j+S_j)^n(f_j)\\|\\ =\\|\sum\limits_{k=0}^{n}C_{n}^{k}(1-\varepsilon_j)^{k}S_j^{n-k}f_j\\|\\ \geq\\|\sum\limits_{k=0}^{n}C_{n}^{k}(1-\varepsilon_j)^{k}(2\varepsilon_j)^{n-k}(1_1,\cdots,1_n,0,\cdots,0)\\|\\ =(1+\varepsilon_j)^n\\|f_{j,n}\\|. \end{array}\right.$ Hence we get $(a)$ $\varlimsup\limits_{j\to\infty}\\|(I+K_\varepsilon)^j(f_j)\\|\geq\lim\limits_{n\to\infty}\\|f_j\\|(1+\varepsilon_j)^j=+\infty$. Because of $r_{\sigma}(I+K_\varepsilon)<1$,we get $(b)$ $\lim\limits_{n\to\infty}\\|(I+K_\varepsilon)^n(f_j)\\|=0$. By $(a)(b)$ and by Definition $\ref{liyorkechaoscriteriondingyi0}$ we get that $\lambda I+K_\varepsilon$ satisfies the Li-Yorke Chaos Criterion, by Theorem $\ref{liyorkechaoscriteriondingli0}$ we get that $\lambda I+K_\varepsilon$ is Li-Yorke chaotic. Then to prove that $(I+K_\varepsilon)^{-1}$ is not Li-Yorke chaotic. For convenience we define $n?^m$ by induction on $m$ for any given $n\in\mathbb{N}$. For any given $j\in\mathbb{N}$,define: $(1)$ $j?=1+2+\cdots+j$, $(2)$ If defined $j?^{n}$ , then define $j?^{n+1}=1?^n+2?^n+\cdots+j?^n$. Let $A=\bigoplus\limits_{j=1}^{+\infty}A_j$,where $A_j=(I_j+K_j)^{-1}$.Because of $A_j(I_j+K_j)=A_j(I_j+K_j)=I_j$, we get $A(I+K_\varepsilon)=(I+K_\varepsilon)A=\bigoplus\limits_{j=1}^{+\infty}I_j=I$. By the Banach Inverse Mapping Theorem [@JohnBConway1990]P91 we get that $A=(I+K_\varepsilon)^{-1}$ is a bounded linear operator, that is,$A=(I+K_\varepsilon)^{-1}$.Hence we get $$\begin{aligned} A_j=\frac{1}{1-\varepsilon_j}\left[ \begin{array}{cccccc} 1 & -2\varepsilon_j & &(-2)^{j-1}\varepsilon_{j}^{j-1} \\ & \ddots &\ddots & \\ & &\ddots &-2\varepsilon_j\\ & & &1 \end{array} \right]_{j\times j}\end{aligned}$$ . $$\begin{aligned} A_j^{2} =\frac{1}{(1-\varepsilon_j)^2}\left[ \begin{array}{cccccc} 1 & 2\cdot(-2)\varepsilon_j & &j\cdot(-2)^{j-1}\varepsilon_{j}^{j-1} \\ & \ddots &\ddots & \\ & &\ddots &2\cdot(-2)\varepsilon_j\\ & & &1 \end{array} \right]_{j\times j}\end{aligned}$$ . $$\begin{aligned} A_j^{3} =\frac{1}{(1-\varepsilon_j)^3}\left[ \begin{array}{cccccc} 1 & (1+2)(-2)\varepsilon_j & &j?\cdot(-2)^{j-1}\varepsilon_{j}^{j-1} \\ & \ddots &\ddots & \\ & &\ddots &2?(-2)\varepsilon_j\\ & & &1 \end{array} \right]_{j\times j}\end{aligned}$$ . For $m\geq3,m\in\mathbb{N}$,If defined $$\begin{aligned} A_j^{m} =\frac{1}{(1-\varepsilon_j)^m}\left[ \begin{array}{cccc} 1 & 2?^{m-2}(-2)\varepsilon_j & &j?^{m-2}(-2)^{j-1}\varepsilon_j^{j-1} \\ & \ddots &\ddots & \\ & &\ddots &2?^{m-2}(-2)\varepsilon_j\\ & & &1 \end{array} \right]_{j\times j}\end{aligned}$$ . Then define $$\begin{aligned} A_j^{(m+1)}=AA^m =\frac{1}{(1-\varepsilon_j)^{(m+1)}} \left[ \begin{array}{cccccc} 1 & 2?^{m-1}(-2)\varepsilon_j & &j?^{m-1}(-2)^{j-1}\varepsilon_j^{j-1} \\ & \ddots &\ddots & \\ & &\ddots &2?^{m-1}(-2)\varepsilon_j\\ & & &1 \end{array} \right]_{j\times j}\end{aligned}$$ . For any given $0\neq x_0=(x_1,x_2,\cdots,)\in\mathbb{H}$, Let $$\begin{aligned} \left.\begin{array}{l} y_j=(x_{(1+\frac{j(j-1)}{2})},\cdots,x_{\frac{j(j+1)}{2}});\\ y_j^{\prime}=(x_{(2+\tfrac{j(j-1)}{2})},\cdots,x_{(\frac{j(j+1)}{2}-1)});\\ z_{j,m}=x_{(1+\frac{j(j-1)}{2})}+\sum\limits_{k=2}^{j}k?^{(m-2)}(-2)^{k-1}\varepsilon_j^{k}. \end{array}\right.\end{aligned}$$. Following a brilliant idea of Zermelo,we shall give the conclusion by induction. $(1)$ If $y_1\neq0$,then we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|=\varliminf\limits_{n\to\infty}\sum\limits_{j=1}^{+\infty}\\|A_j^ny_j\\|\geq$ $\varliminf\limits_{n\to\infty}\\|A_1^ny_1\\|=\varliminf\limits_{n\to\infty}\frac{\|x_1\|}{(1-\varepsilon_1)^n}=+\infty$. $(2)$ If $y_1=0$,but $y_2=(x_2,x_3)\neq0$. $(i)$ If $x_3\neq0$, by $(1)$ we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\geq\varliminf\limits_{n\to\infty}\frac{\|x_3\|}{(1-\varepsilon_1)^n}\to+\infty$. $(ii)$ If $x_3=0$ and $\varepsilon_2>\frac{1}{2?^{(n-2)}}$,because of $y_2=(x_2,x_3)\neq0$,we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|=\varliminf\limits_{n\to\infty}\sum\limits_{j=1}^{+\infty}\\|A_j^ny_j\\|$ $\geq\varliminf\limits_{n\to\infty}\\|A_2^ny_2\\|$ $=\varliminf\limits_{n\to\infty}\frac{1}{(1-\varepsilon_1)^n}\sqrt{(x_2^2+(2?^{n-2}(-2)\varepsilon_2x_3)^2)}$ $\geq\varliminf\limits_{n\to\infty}\frac{\|x_2\|}{(1-\varepsilon_1)^n}$ $=+\infty$. $(3)$ Assume for $k\leq m-1$, there is $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\to+\infty$ for $A^{k}$ and $y_k\neq0$. Then for $k=m$ and $y_m\neq0$ we get $(i)$ If $x_{m?}\neq0$, by $(1)$ we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\geq\varliminf\limits_{n\to\infty}\frac{\|x_{m?}\|}{(1-\varepsilon_1)^n}=+\infty$. $(ii)$ If $x_{m?}=0$ and $\varepsilon_m>\frac{1}{m?^{(n-2)}}$,because of $y_m\neq0$,we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|=\varliminf\limits_{n\to\infty}\sum\limits_{j=1}^{+\infty}\\|A_j^ny_j\\|$ $\geq\varliminf\limits_{n\to\infty}\\|A_m^ny_m\\|$ $=\varliminf\limits_{n\to\infty}\frac{1}{(1-\varepsilon_1)^n}$ $\sqrt{z_{m,n}^2+\\|A_{m-1}^ny_m^{\prime}\\|^2}$. If $y_m^{\prime}\neq0$,by the induction hypothesis we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\geq\varliminf\limits_{n\to\infty}\\|A_{m-1}^ny_m^{\prime}\\|=+\infty$; If $y_m^{\prime}=0$,because of $y_m\neq0$ we get $x_{(1+\frac{m(m-1)}{2})}\neq0$. by $(1)$ we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\geq$ $\varliminf\limits_{n\to\infty}\frac{\\|z_{m,n}\\|}{(1-\varepsilon_1)^n}$ $=\varliminf\limits_{n\to\infty}\frac{\|x_{(1+\frac{m(m-1)}{2})}\|}{(1-\varepsilon_1)^n}=+\infty$. Therefor for $k=m$ and $y_m\neq0$, we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\to+\infty$, by the induction we get that for any $m\in\mathbb{N}$ and $y_m\neq0$,there is $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|\to+\infty$. From $(1)(2)(3)$ and $\mathbb{H}=\bigoplus\limits_{j=0}^{+\infty}\mathbb{H}_j$, we get that for any given $0\neq x_0=(x_1,x_2,\cdots,)\in\mathbb{H}$ we can find $m\in\mathbb{N}$ such that $y_m\neq0$. Hence for any given $0\neq x_0=(x_1,x_2,\cdots,)\in\mathbb{H}$ we get $\varliminf\limits_{n\to\infty}\\|A^nx_0\\|=+\infty$. Therefor $(I+K_\varepsilon)^{-1}$ is not Li-Yorke chaotic. From the view of infinite matrix, $(I+K_\varepsilon)^{}$ and $(I+K_\varepsilon)^{-1}$ are lower triangular matrix, for any $0\neq x\in\mathbb{H}$,with a simple computing we get $\varliminf\limits_{n\to\infty}\\|(I+K_\varepsilon)^{n}x\\|>0$, $\varliminf\limits_{n\to\infty}\\|(I+K_\varepsilon)^{-n}x\\|>0$. Hence $(I+K_\varepsilon)^{}$ and $(I+K_\varepsilon)^{-1}$ are not Li-Yorke chaotic. \[fenbuhundundenibushiliyorkehundunde17\] There is a invertible bounded linear operator $I+K_\varepsilon$ on $\mathbb{H}=\mathcal{L}^2(\mathbb{N})$ such that $I+K_\varepsilon$ is distributionally chaotic, but $(I+K_\varepsilon)^{-1}$,$(I+K_\varepsilon)^{-1}$ and $(I+K_\varepsilon)^{}$ are not distributionally chaotic. By the construction of Theorem $\ref{duijiaoxianyujxiaojinsuanzidingliyinyong13}$, it is only to give the conclusion by induction on $\{k_i\}_{i=1}^{\infty}$ as Example $\ref{duijiaoxianyujxiaojinsuanzidenibuhundun14}$. \[cunzaisuanzishuzhishiyiduankaiyuanshu7\] There is $T\in\mathcal{L}(\mathbb{H})$, $S_{LY}(T)=S_{DC}(T)=\omega$ is an open arc of $\mathbb{T}=\{\|\lambda\|=1;\lambda\in\mathbb{C}\}$, and for $\forall \lambda\in\omega$,we get that $(\lambda+T)^{}$,$(\lambda+T)^{-1}$ and $(\lambda+T)^{-1}$ are not Li-Yorke chaotic. As Example $\ref{duijiaoxianyujxiaojinsuanzidenibuhundun14}$,give the same $\mathbb{H}=\bigoplus\limits_{j=1}^{+\infty}\mathbb{H}_j$,$S_j$ and $K_j$, give positive sequence $\{\varepsilon_j\}_{1}^{\infty}$ such that $\varepsilon_j\to0$ and $\sup\limits_{j\to\infty}\|i+\varepsilon_j\|^{j}\to+\infty$,where $i\in\mathbb{C}$. Let $\lambda I+K_\varepsilon=\bigoplus\limits_{j=1}^{+\infty}(\lambda I+K_j)$, so $\sigma(\lambda I+K_\varepsilon)=\{\lambda -\varepsilon_j;j\in\mathbb{N}\}$. $(i)$ If $\|\lambda\|<1$,because of $\varepsilon_j\to0$, we get that $\exists N>0$ when $n>N$, $\|\lambda -\varepsilon_j\|<1$. With the introduction of this paper we get that Li-Yorke chaos is valid only on infinite Hilbert space. Loss no generally,for any $j\in\mathbb{N}$,let $\|\lambda -\varepsilon_j\|<1$, so $r_{\sigma(\lambda I+K_\varepsilon)}<1$. Hence for any $0\neq x\in\mathbb{H}$ there is $\lim\limits_{n\to\infty}\\|(I+K_\varepsilon)^n(x)\\|=0$. $(ii)$ If $\|\lambda\|>1$ or $\lambda\in[\frac{\pi}{2},\frac{3\pi}{2}]$, because of $\varepsilon_j>0$, for $j\in\mathbb{N}$ we get $\|\lambda -\varepsilon_j\|>1$, $\frac{1}{\|\lambda -\varepsilon_j\|}<1$, and $\sigma(\lambda I+K_\varepsilon)^{-1}=\{\frac{1}{\lambda -\varepsilon_j};j\in\mathbb{N}\}$. By Example $\ref{duijiaoxianyujxiaojinsuanzidenibuhundun14}$, for any given $x\neq0$ there is $y_m\neq0,m\in\mathbb{N}$,hence we get $\left.\begin{array}{l} \lim\limits_{n\to\infty}\\|(\lambda I+K_\varepsilon)^nx_0\\|\\ \geq\lim\limits_{n\to\infty}\\|(\lambda I_m+K_m)^ny_m\\|\\ \geq\lim\limits_{n\to\infty}\frac{1}{\\|(\lambda I_m+K_m)^{-n}\\|}\\|y_m\\|\\ \geq\lim\limits_{n\to\infty}\frac{1}{\\|(\lambda I_m+K_m)^{-1}\\|^n}\\|y_m\\|\\ \geq\\|y_m\\|>0. \end{array}\right.$ $(iii)$ For $\forall \lambda\in(-\frac{\pi}{2},\frac{\pi}{2})$, because of $ \varepsilon_j\to0$, there exists $N>0$, when $j>N$, we get $\|\lambda-\varepsilon_j\|<1$. Let $\mathbb{H}^{'}=\bigoplus\limits_{j>N}\mathbb{H}_j$, $(\lambda I+K_\varepsilon)^{'}=(\lambda I+K_\varepsilon)\|_{\bigoplus\limits_{j>N}\mathbb{H}_j}$, then for $f_j=\frac{1}{\sqrt{j}}(1_1,\cdots,1_j)$ and $f_{j,n}=\frac{1}{\sqrt{j}}(1_1,\cdots,1_n,0,\cdots,0)\in\mathbb{H}^{'}$, we get that when $1\leq n\leq j$,there is $\left. \begin{array}{l} \\|(\lambda I+K_\varepsilon)^n(f_j)\\|\\ =\\|(\lambda I_j+K_j)^n(f_j)\\|\\ =\\|((\lambda-\varepsilon_j)I_j+S_j)^n(f_j)\\|\\ \geq\\|\sum\limits_{k=0}^{n}C_{n}^{k}(\lambda-\varepsilon_j)^{k}(2\varepsilon_j)^{n-k}(1_1,\cdots,1_n,0,\cdots,0)\\|\\ =\|\lambda+\varepsilon_j\|^n\\|f_{j,n}\\|. \end{array}\right.$ By $(i)(ii)$,if $\|\lambda\|\neq1$ or $\lambda\in[\frac{\pi}{2},\frac{3\pi}{2}]$, $\lambda I+K_\varepsilon$ is not Li-Yorke chaotic. By $(iii)$ and by the property of the triangle,if $\lambda\in(-\frac{\pi}{2},\frac{\pi}{2})$ and $j>N$, we get $\|\lambda+\varepsilon_j\|>\|i+\varepsilon_j\|$ and $r_{\sigma}((\lambda I+K_\varepsilon)^{'})<1$. Hence we get $\lim\limits_{n\to\infty}\\|(\lambda I+K_\varepsilon)^{'n}(f_j)\\|=0$, and $\varlimsup\limits_{j\to\infty}\\|(I+K_\varepsilon)^{'j}(f_j)\\|$ $\geq\lim\limits_{j\to\infty}\\|f_j\\|\|\lambda+\varepsilon_j\|^j\geq\lim\limits_{j\to\infty}\|i+\varepsilon_j\|^j=+\infty$. By Definition $\ref{liyorkechaoscriteriondingyi0}$ we get that $\lambda I+K_\varepsilon$ satisfies the Li-Yorke Chaos Criterion, by Theorem $\ref{liyorkechaoscriteriondingli0}$ we get that $\lambda I+K_\varepsilon$ is Li-Yorke chaotic. Using the same proof of Example $\ref{duijiaoxianyujxiaojinsuanzidenibuhundun14}$ we get that $(\lambda I+K_\varepsilon)^{}$,$(\lambda I+K_\varepsilon)^{-1}$ and $(\lambda I+K_\varepsilon)^{-1}$ are not Li-Yorke chaotic. Using Corollary $\ref{fenbuhundundenibushiliyorkehundunde17}$ and Theorem $\ref{duijiaoxianyujxiaojinsuanzidingliyinyong13}$ we get that $\lambda I+K_\varepsilon$ is distributionally chaotic,but $(\lambda I+K_\varepsilon)^{}$,$(\lambda I+K_\varepsilon)^{-1}$ and $(\lambda I+K_\varepsilon)^{-1}$ are not Li-Yorke chaotic. For any given $m\in\mathbb{N}$, there exists $m$-folder complex analytic function $\phi(z)\in\mathcal{H}^{\infty}(\mathbb{D})$ such that $\phi$ is not a Cowen-Douglas function. Gives the equivalent characterization of a $m$-folder complex analytic function; Gives the equivalent characterization of a rooter function; Gives the equivalent characterization of a Cowen-Douglas function. Gives the relations between them. Let $M_{\phi}$ is the multiplication operator of the Cowen-Douglas function $\phi(z)\in\mathcal{H}^{\infty}(\mathbb{D})$ on the Hardy space $\mathcal{H}^{2}(\mathbb{D})$, then is $M_{\phi}^{}$ a Lebesgue operator? 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--- abstract: 'We have observed 70 galaxies belonging to 45 Hickson compact groups in the and lines, in order to determine their molecular content. We detected 57 galaxies, corresponding to a detection rate of 81%. We compare the gas content relative to blue and luminosities of galaxies in compact groups with respect to other samples in the literature, including various environments and morphological types. We find that there is some hint of enhanced / and / ratios in the galaxies from compact group with respect to our control sample, especially for the most compact groups, suggesting that tidal interactions can drive the gas component inwards, by removing its angular momentum, and concentrating it in the dense central regions, where it is easily detected. The molecular gas content in compact group galaxies is similar to that in pairs and starburst samples. However, the total luminosity of HCGs is quite similar to that of the control sample, and therefore the star formation efficiency appears lower than in the control galaxies. However this assumes that the FIR spatial distributions are similar in both samples which is not the case at radio frequencies. Higher spatial resolution FIR data are needed to make a valid comparison. Given their short dynamical friction time-scale, it is possible that some of these systems are in the final stage before merging, leading to ultra-luminous starburst phases. We also find for all galaxy samples that the content (derived from CO luminosity and normalised to blue luminosity) is strongly correlated to the luminosity, while the total gas content (+) is not.' author: - 'S. Leon' - 'F. Combes' - 'T. K. Menon' title: Molecular gas in galaxies of Hickson compact groups --- Introduction ============ Galaxies are gregarious systems, most of them are gathered in groups or clusters, while only 30% are isolated and 10% are binaries in the field. Nevertheless compact groups (CG) are quite rare and according to Hickson’s classification (Hickson, 1982) only 0.1 % of galaxies belong to CGs. Criteria of population (initially four galaxies in the group), isolation (dynamically independent systems) and compactness (separation between galaxies comparable to the sizes of the galaxies) are chosen by Hickson to build his catalog. With these criteria around one hundred CGs were found on the Palomar Observatory Sky Survey red prints. Compact groups are ideal sites to study the influence of strong dynamical evolution due to environment on molecular cloud formation and star formation efficiency. They appear in projection as the densest galaxy systems known, even denser than the cores of rich clusters, and they show multiple signs of interactions. Due to their high density, and relatively small velocity dispersion, these systems are unstable with regard to merging instability. The dynamical friction time-scale is of the order of 2.10$^8$ yrs, and N-body simulations predict their rapid evolution towards a single elliptical massive galaxy (e.g. Barnes 1989). The existence of many such compact groups is therefore a puzzle, and the physical reality of HCG has been questioned (e.g. Mamon 1986, 1987); but evidence of galaxy-galaxy interactions in those groups, either morphologic (Hickson 1990; Mendes de Oliveira 1992), or kinematic (Rubin et al. 1991), speaks in favour of their reality. Latest spectroscopic observations showed that 92 of the original 100 groups have at least three galaxies with recession velocities within 1000 of each other (Hickson et al. 1992). The presence of hot intergalactic gas, detected by X-ray emission centered on some HCGs, is a further confirmation of the existence of these compact groups (Pildis et al. 1995, Ponman et al. 1996).\ Most of galaxies that belong to groups are in fact in loose groups of 10-30 galaxies and about 50% of all galaxies belong to loose groups. But loose groups are in their great majority un-bound and un-virialised (Gourgoulhon et al. 1992) while their true dynamical state is ambiguous (expanding, collapsing, transient). Clusters of galaxies are more near equilibrium, specially in their centers (about 10% of all galaxies belong to clusters). However, the depth of their potential well leads to high relative velocities between galaxies that reduce the efficiency of galaxy-galaxy encounters. The influence of environment is revealed by the high proportion of ellipticals and lenticulars, and by the HI gas deficiency of spirals (Dressler 1984, Cayatte et al. 1991). This gas deficiency can be explained by ram-pressure as well as tidal interactions (Combes et al. 1988). No molecular gas deficiency has been detected, either in Virgo (Kenney & Young 1988), or in Coma (Casoli et al. 1991), which suggests that the inner parts of the galaxies are not affected by their environment, since the CO emission essentially comes from the galaxy central regions. However, there could be two compensating effects at play here: the enhancement of CO emission in interacting galaxies (cf Braine et Combes 1993, Combes et al. 1994), and the outer gas stripping, stopping the gas fueling of galaxies.\ In compact groups, some HI deficiency has also been reported (Williams & Rood 1987), but no CO emission deficiency, according to a first study by Boselli et al (1996) with the SEST telescope. It is further interesting to investigate whether HCGs are actually sampling the highest densities of galaxies in the Universe. It has been claimed that, even if the CGs are real, we are not sure of their high density, since they could correspond to loose groups with only a high [projected]{} density through chance alignment of filaments along the line of sight (e.g. Mamon 1992). But no loose groups are observed in HCG neighborhood in the majority (67%) of cases (Rood & Williams 1989). Hickson (1982) found that the groups contain fewer spirals than a comparable sample of field galaxies. The spiral fraction decreases from 60% in the least compact groups to 20% in the most compact. There is also a deficiency of faint galaxies with respect to rich clusters and field. This apparent deficiency is more severe in groups with elliptical first-ranked galaxies. Radio properties of compact groups have been studied by Menon & Hickson (1985) and Menon (1991, 1995). Although the far-infrared and radio luminosities are still highly correlated as for field galaxies, the total radio emission from HCG spirals is relatively lower by a factor 2 in compact group galaxies while the nuclear radio emission is enhanced by a factor of about 10 compared to isolated galaxies. The results suggest a scenario in which interactions among group galaxies produce inflow of gas towards the centers, elevating the star formation there, and consequently the radio and far-infrared emissions. But at the same time the removal of gas and magnetic fields from the extended disks of the galaxies results in a decrease of total radio-emission. Williams & Rood (1987) have observed 51 of the 100 Hickson groups in the HI line, and detected 34 of them. They find that on average a Hickson compact group contains half as much neutral hydrogen as a loose group with a similar distribution of galaxy luminosities and morphological types. This result supports the reality of compact groups as independent dynamical systems and not transient or projected configurations in loose groups. The recent ROSAT survey of HCGs by Ponman et al (1996) also confirms that the groups are intrinsically compact, and not the projection of loose groups. They infer that more than 75% of the HCGs possess hot intragroup gas.\ We present here a large CO survey of Hickson group galaxies with the IRAM 30m telescope, and compare the relative gas content and star formation efficiency of CG galaxies with other samples belonging to widely different environments. After describing the observations in section 2, the sample and the data in section 3, we discuss the main conclusions and possible interpretations in section 4. Observations ============ Observations were carried out on 4-10 September 1995 with the 30 meter radiotelescope of the Instituto de Radio Astronomia Milimetrica (IRAM) in Pico Veleta, near Granada in Spain. Single-sideband SIS receivers were tuned for the and transitions at respectively 115 and 230 GHz. Weather conditions were excellent during the run with typical effective system temperatures of 300-400 K ([ ]{}scale ) at 115 GHz and 500-800 K ([ ]{}scale) at 230 GHz.\ For each line a 512 $\times $1 MHz channel filter bank is used with a velocity resolution of 2.6 smoothed for each spectrum to 10.4 or 20.8 according the quality of each spectrum. At 115 GHz and 230 GHz we assume a HPBW of 22$^{\prime\prime}$ and 11$^{\prime \prime}$ respectively. Pointing was done frequently on continuum sources with corrections of the offsets up to 8$^{\prime \prime}$, providing an accuracy of 3$^{ \prime \prime}$ (Greve et al, 1996).\ The temperature-scale calibration was checked on the sources W3OH, ORIA and IRC+10216 (Mauersberger et al. 1989); except for a transient problem for the 3mm calibration, it remained in a reasonable range providing at least a 20% calibration accuracy. We use the wobbler with a switch cycle of 4 seconds and a beam throw of 90-240$^{\prime \prime}$ avoiding off position on another galaxy of the same compact group. Each 12 minutes a chopper wheel calibration was performed on a load at ambient temperature and on a cold load (77 K). The line temperatures are expressed in the [ ]{}scale, antenna temperature corrected for atmospheric attenuation and rear sidelobes. Baselines were flat allowing us to subtract only linear polynomials out of the spectra.\ Results ======= The observed sample ------------------- Our observed sample is composed of 70 galaxies towards 45 compact groups, taken from the catalog of Hickson Compact Groups (HCG, Hickson 1982) . We discarded afterwards 4 galaxies (11a,19b,73a,78a) which appear not to belong to Compact Groups (Hickson, 1992). The galaxies, mostly spirals, are selected for their radio continuum (Menon 1995) and IRAS (Hickson et al. 1989) detections. All the targets are northern sources with $\delta > 0^\circ$. The redshift range from 1200 up to 18500 (27b). If we use H$_0$=75 for the Hubble constant, as adopted in this paper, the mean distance of the sample is 95 Mpc with a standard deviation of 45 Mpc. We present in Fig. \[stat\_sample\] the statistical distribution of our sample for distance, type, far infrared () luminosity and median projected separation.\ In Table \[tab\_sample\_1\] we display the main properties of the sample. The column headings are the following: name is taken from Hickson classification, type is taken from Hickson et al. (1989), D is the distance computed with a correction for the galactic rotation using a solar galactic velocity of rotation of 250 , R is the median projected galaxy-galaxy separation, $D_B$ indicates the diameter in arcsecs at $\mu_B$ = 24.5 mag.arcsec$^{-2}$, from Hickson et al. (1989), blue luminosity L$_B$ is computed as follows =12.208-0.4B$^o_T$+log(1+z)+2log(D/Mpc), luminosity is computed using reprocessed IRAS data from Allam et al. (1996) and following Hickson et al. (1989) derivation, is a dust temperature indicator using an emissivity dependence as $\lambda^{-1}$ in the 60-100 $\mu$m IRAS- range, S$_{20cm}$ is the total radio continuum flux density from Menon (1995), neutral hydrogen content has been found mainly from Williams & Rood (1987), and some from Huchtmeier & Richter (1989). Concerning luminosity or HI mass, we indicate in Table \[tab\_sample\_1\] the whole group emission preceded by $\leq$, meaning that poor spatial resolution does not allow a separation per galaxy.\ -------------- ------ ------- ------- ------------------ ------------ ---------------- ------- ------------------- ------------- ----------------- -- Name type D R D$_B$ Log(L$_B$) Log(L$_{FIR}$) T$_d$ S$_{20\mbox{cm}}$ Log(M(HI)) Log(M$_{dust}$) (Mpc) (kpc) ($\prime\prime$) (K) (mJy) () () 2b cI 59.3 52.5 37.6 9.95 10.16 35.8 $\leq$10.11 6.61 3c Sd 103.0 77.0 19.8 9.93 9.93 43.3 $<$10.15 5.94 4a Sc 112.3 57.0 67.5 10.86 10.95 34.6 7.49 7a Sb 57.8 45.6 80.2 10.50 10.27 35.2 12.85 $\leq$9.28 6.76 7c SBc 57.8 45.6 91.0 10.19 9.69 29.0 9.64 6.68 10a SBb 66.6 92.9 93.6 10.76 9.01 9.90 10c Sc 66.6 92.9 49.2 10.18 9.84 32.0 2.47 9.61 6.58 11a$^{\dag}$ SBbc 73.1 - 91.9 10.70 9.64 27.7 6.77 14b E5 73.5 26.9 51.9 10.23 $<$9.12 $<$9.77 14c Sbc 73.5 26.9 15.9 9.26 $<$9.12 $<$9.77 16a SBab 52.5 44.6 77.1 10.50 10.53 29.6 33.18 9.48 7.47 16b Sab 52.5 44.6 61.3 10.30 $<$8.84 2.62 $\leq$10.07 16c Im 52.5 44.6 58.9 10.36 10.69 33.8 78.29 $\leq$10.07 7.28 16d Im 52.5 44.6 63.5 10.24 10.74 33.9 30.91 $\leq$10.07 7.33 19b$^{\dag}$ Scd 56.1 - 28.5 9.49 9.37 28.7 2.08 $\leq$9.28 6.40 21a Sc 99.8 134.9 44.9 10.51 10.36 30.8 7.20 23b SBc 64.0 66.1 48.7 10.00 10.09 31.2 8.01 $\leq$10.04 6.87 23d Sd 64.0 66.1 26.7 9.38 9.29 3.18 $<$9.26 25a SBc 84.6 47.9 51.1 10.47 9.93 32.6 4.70 $\leq$10.09 6.62 25c Sb 145.0 47.9 23.9 10.35 10.44 34.0 7.03 27b SBc 245.7 107.2 21.7 10.66 10.54 31.4 7.32 31a Sdm 57.0 49.0 33.6 9.74 $\leq$10.32 9.92 31c Im 57.0 49.0 74.5 10.68 $\leq$10.32 9.85 33c Sd 103.7 24.5 18.7 9.64 9.86 4.58 $\leq$10.18 34b Sd 121.8 15.5 22.2 9.71 $\leq$10.36 6.03 $<$10.00 37b Sbc 88.5 28.8 40.8 10.26 9.85 28.9 1.55 $<$8.90 6.85 38b SBd 115.1 58.9 38.4 10.38 $\leq$10.55 $\leq$9.70 40a E3 86.7 15.1 65.3 10.66 $\leq$10.09 $<$9.70 40c Sbc 86.7 15.1 36.5 9.98 10.07 29.4 6.03 $<$9.70 7.03 40d SBa 86.7 15.1 41.0 10.23 9.56 6.39 $<$9.70 40e Sc 86.7 15.1 17.8 9.36 9.63 54.2 $<$9.70 5.21 43a Sb 129.8 58.9 27.0 10.34 10.18 25.4 0.95 $<$10.16 7.56 43b SBcd 129.8 58.9 25.8 10.32 10.12 26.7 1.06 $<$10.16 7.34 44a Sa 17.3 38.0 130.5 10.03 9.31 31.4 4.77 8.64 6.09 44c SBc 17.3 38.0 91.5 9.62 8.94 32.2 2.45 8.41 5.66 -------------- ------ ------- ------- ------------------ ------------ ---------------- ------- ------------------- ------------- ----------------- -- [$^{\dag}$not included in our final HCG sample]{} -------------- ------ ------- ------- ------------------ ------------ ---------------- ------- ------------------- ------------- ----------------- -- -- Name type D R D$_B$ Log(L$_B$) Log(L$_{FIR}$) T$_d$ S$_{20\mbox{cm}}$ Log(M(HI)) Log(M$_{dust}$) (Mpc) (kpc) ($\prime\prime$) (K) (mJy) () () 44d Sd 17.3 38.0 70.3 9.40 8.65 39.0 2.95 8.88 4.90 47a SBb 125.3 36.3 40.3 10.52 10.27 32.1 12.25 $<$9.78 7.00 49a Scd 134.2 12.3 21.6 10.07 $\leq$10.12 0.68 49b Sd 134.2 12.3 15.9 9.90 $\leq$10.12 1.70 55a E0 212.0 19.1 23.1 10.64 $\leq$10.58 57d SBc 120.8 72.4 44.3 10.52 $<$10.17 5.09 58a Sb 81.5 89.1 58.2 10.56 10.59 34.3 20.71 9.75 7.14 59a Sa 53.0 21.4 38.1 9.80 10.16 46.4 9.02 9.26 6.03 59d Im 53.0 21.4 28.0 9.29 $<$10.16 $\leq$8.97 61c Sbc 51.9 28.8 57.4 10.18 10.41 33.9 $\leq$9.95 6.99 61d S0 51.9 28.8 38.7 9.95 $<$8.92 $\leq$9.95 67b Sc 96.8 49.0 48.1 10.58 10.28 29.7 6.70 10.20 7.22 67c Scd 96.8 49.0 38.7 10.11 $<$9.87 13.49 9.54 68c SBbc 33.0 33.1 134.3 10.43 9.79 28.3 2.10 9.65 6.86 69a Sc 117.9 30.2 39.4 10.33 $<$9.48 3.13 $\leq$10.15 69b SBb 117.9 30.2 23.8 10.07 10.67 41.4 6.60 $\leq$10.15 6.79 71b Sb 120.9 50.1 22.6 10.37 10.60 35.0 $<$10.27 7.11 73a$^{\dag}$ Scd 76.9 100.0 76.3 10.62 9.32 10.10 75b Sb 167.3 37.2 14.8 10.65 $<$9.89 3.10 75e Sa 167.3 37.2 15.1 10.07 $\leq$10.22 78a$^{\dag}$ SBb 116.8 - 51.1 10.56 10.36 31.6 10.29 7.12 79a E0 59.3 6.8 44.7 9.97 $\leq$9.82 $<$9.15 79c S0 59.3 6.8 33.8 9.82 $<$9.19 $<$9.15 80a Sd 126.4 25.1 25.4 10.46 10.84 33.9 20.62 7.43 82c Im 146.7 70.8 32.9 10.58 10.59 33.9 8.33 $<$10.09 7.17 88a Sb 82.3 67.6 57.3 10.72 10.01 24.8 0.90 $<$10.02 7.46 89c Scd 120.8 58.9 24.4 10.12 $<$9.94 $<$10.46 92c SBa 89.0 28.2 83.2 10.73 10.15 26.0 24.74 9.90 7.46 93b SBd 69.8 70.8 64.1 10.58 10.25 31.4 10.92 9.59 7.03 95b Scd 160.9 30.2 25.1 10.44 $\leq$10.69 3.86 95c Sm 160.9 30.2 26.3 10.50 $\leq$10.69 6.06 95d Sc 160.9 30.2 16.9 10.12 96a Sc 119.0 30.2 61.6 10.90 11.10 39.4 200.1 $<$10.16 7.33 96c Sa 119.0 30.2 19.8 10.04 $<$9.88 4.80 $<$10.16 100a Sb 73.3 38.0 48.9 10.43 10.273 32.6 9.17 9.74 6.95 -------------- ------ ------- ------- ------------------ ------------ ---------------- ------- ------------------- ------------- ----------------- -- -- [$^{\dag}$not included in our final HCG sample]{} Comparison samples ------------------ We compare our results on Hickson compact groups with a ’control’ sample, gathering most of the CO data obtained in the literature until now: about 200 galaxies observed by Young et al (1989, 1996) with the FCRAO 14m antenna, by Solomon & Sage (1988) with the FCRAO and NRAO Kitt Peak 12m telescopes, Tinney et al (1990) and by Sage (1993) with the NRAO antenna. This big control sample consists essentially of nearby bright galaxies, and includes a wide range of environment conditions, from isolated and field, to interacting systems. The mergers are also included, and correspond to the highest FIR luminosities of the ensemble. From these observations, we have however separated the Virgo galaxies, and added the results on Coma galaxies (Casoli et al 1991), to build the sample ’Cluster’. We also compare HCG data to more specific samples, such as the isolated pairs from Combes et al (1994), dwarfs from Sage et al (1992), Israel et al (1995) and Leon et al. (1997), and ellipticals from Wiklind et al. (1995), and from Sanders et al. (1991) for the starbursts. For this latter ensemble we separate the pair galaxies to include them in the pair sample. These are noted respectively ’pair’, ’dwarf’, ’elliptic’ and ’starburst’ in the various figures of the present work. We have summarized the size of the different samples in Table \[table\_sample\].\ sample number of galaxies references ----------- -------------------- ------------ control 193 7,8,9,10 starburst 73 5 pair 48 2,5 cluster 40 1,10 dwarf 25 3,4,6 elliptic 18 11 : Galaxy numbers in the comparison samples.[]{data-label="table_sample"} [(1) Casoli et al. (1991), (2) Combes et al. (1994), (3) Israel et al. (1995), (4) Leon et al. (1997), (5) Sanders et al. (1991), (6) Sage et al.(1992), (7) Sage (1993), (8) Solomon & Sage (1988), (9) Tinney et al. (1990), (10) Young et al. (1989,1996), (11) Wiklind et al. (1995) ]{} CO data ------- We detected 57 (or 53 in our final HCG sample) galaxies, corresponding to a 80 % detection rate, with 2 detections in (75e,89c) not detected in , probably due to dilution factor, given the small size of these two galaxies. Some galaxies were detected only in the line. We reached typically an rms temperature level of 1-8 mK at a smoothed velocity resolution of 10.4-20.8 according to the spectra quality. In Table \[tab\_co\_1\] we present the results of our CO observations: I$_{CO}$ is the velocity-integrated temperature $\int T_a^* dv$ for the line, or integrated CO intensity, in the [ ]{}scale, $\delta$I$_ {CO}$ is the standard error on I$_{CO}$, $T_p$ is the peak antenna temperature, v$_{CO}$ is the intensity-weighted mean heliocentric velocity, FWHM is the full width half maximum of the spectra, I$_{CO(2-1)}$ is the integrated intensity and M(H$_2$) is the molecular gas mass. The spectra of galaxies detected through the CO(1-0) line are displayed in Fig. \[fig\_spectra\].\ Upper limits of the CO intensities are computed at 3$\sigma$ in antenna temperature scale, and with a line width $\delta_{CO}$ guessed from other lines when available from lines (Williams & Rood 1987) , and taken as $\delta_{CO}$ = 200 otherwise. In the case of 40d and 95d we deduced I$_{CO(2-1)}$ and I$_{CO}$ intensities from the fit, because the emission was shifted at the edge of the band. To derive H$_2$ molecular gas content the standard H$_2$/CO conversion factor is adopted (Strong et al. 1988), i.e: $$N(H_2)=2.3\times10^{20}\int_{line} T_R dv \mbox{(mol.cm$^{-2}$)}$$ where $T_R$ is the radiation temperature. Following Gordon et al. (1992) to include size source correction, H$_2$ mass is derived using the expression (see appendix A): $$M(H_2)=5.86\times10^4D^2KI_{CO} ({\mbox{M$_{\odot}$}})$$ where D is the distance in Mpc and K a correction factor for the weighting of the source distribution by the antenna beam. When the source was larger we used a factor 1.38 which leads to a main beam scale. An exponential law was used for modelling radial distribution of molecular gas with a scale length h=D$_B$/10, the molecular gas following approximately the optical light distribution (Young & Scoville, 1982) i.e. the assumed gas surface density $\mu(r)$ is $$\mu (r)\propto e^{-\frac{r}{h}}$$ We could expect a more radially concentrated molecular distribution in these tidally perturbed galaxies, but if we compare with a gaussian distribution, there is at most a difference of a factor 1.5 on the factor K. Thus M(H$_2$) will only be slightly overestimated through this effect. We note that this assumption works well for the group 16 where we have mapped in CO the whole galaxies. For 75e and 80c a mean ratio I$_{CO}$/I$_{CO(2-1)}$=1.35 is used to derive I$_{CO}$ intensity in the line. The mean mass versus galaxy morphological type is presented in Fig. \[gas\_type\]. -------------- ---------- ------------------ ------- ---------- ------ --------------- --------------- -------- ------------------- Name I$_{CO}$ $\delta$I$_{CO}$ $T_p$ v$_{CO}$ FWHM I$_{CO(2-1)}$ Log(M(H$_2$)) Log(/) Log(/($M_{gas}$)) () () (mK) () () () () (/) (/) 2b 3.90 0.25 37.8 4391 105 1.63 9.14 1.02 3c $<$1.80 $<$2.02 $<$9.17 4a 16.6 0.53 170.7 8047 83 10.58 0.38 7a 15.70 0.80 54.5 4221 317 10.02 0.25 7c 4.73 0.58 34.7 4401 124 1.51 9.59 0.10 -0.23 10a 2.72 0.49 9.3 5277 339 $<$8.10 9.65 -0.63 -1.08 10c 7.14 0.36 21.8 4612 359 4.81 9.70 0.14 -0.12 11a$^{\dag}$ 1.84 0.23 11.8 5433 171 $<$2.70 9.41 0.23 14b 6.35 0.60 22.9 5867 383 $<$2.50 9.60 14c $<$1.50 $<$3.60 $<$8.76 16a 42.86 1.29 4056 9.98 0.55 0.43 16b 3.44 0.67 3871 9.21 16c 62.95 1.53 3835 10.15 0.54 16d 32.96 1.10 3878 9.87 0.88 19b$^{\dag}$ 1.23 0.30 8.1 4255 146 8.51 0.86 21a 14.86 1.20 53.0 7587 362 10.31 0.05 23b 9.70 0.60 46.1 4914 318 9.85 0.23 23d 2.15 0.20 24.8 4455 83 8.92 0.37 25a 2.56 0.20 13.3 6272 193 1.68 9.39 0.55 25c 6.03 0.49 21.8 10894 394 $<$1.62 10.07 0.37 27b 2.08 0.40 8.8 18516 311 10.01 0.52 31a $<$2.64 $<$8.84 31c 1.53 0.30 15.7 3987 148 8.64 33c 4.58 0.24 15.2 7787 362 19.1 9.63 0.23 34b 5.99 0.70 19.7 9393 364 3.68 9.86 37b 8.65 0.40 19.6 6763 552 5.93 10.00 -0.15 38b 7.87 1.40 33.5 8677 251 10.08 40a $<$1.32 $<$2.88 $<$8.90 40c 9.7 0.65 34.9 6375 233 9.94 0.14 40d 5.09 0.43 17.4 6729 295 3.09 9.57 -0.01 40e 1.62 0.40 9.4 6480 66 1.23 8.98 0.65 43a 4.73 0.38 17.3 10026 256 1.49 9.86 0.33 43b 2.63 0.35 13.3 9920 278 1.82 9.66 0.45 44a 10.5 0.60 35.2 1239 269 3.65 9.13 -0.18 0.06 44c 7.63 0.50 50.2 1222 165 $<$6.24 8.71 0.23 0.06 -------------- ---------- ------------------ ------- ---------- ------ --------------- --------------- -------- ------------------- [$^{\dag}$not included in our final HCG sample]{} -------------- ---------- ------------------ ------- ---------- ------ --------------- --------------- -------- ------------------- Name I$_{CO}$ $\delta$I$_{CO}$ $T_p$ v$_{CO}$ FWHM I$_{CO(2-1)}$ Log(M(H$_2$)) Log(/) Log(/($M_{gas}$)) () () (mK) () () () () (/) (/) 44d 1.79 0.19 18.5 1584 103 1.17 8.09 0.55 -0.29 47a 4.82 0.32 18.7 9630 321 1.23 9.92 0.35 49a $<$2.0 $<$3.06 $<$9.46 49b $<$1.7 $<$2.22 $<$9.33 55a $<$1.3 $<$9.68 57d 1.45 0.24 10.0 9014 110 0.66 9.29 58a 26.56 1.80 78.1 6132 510 10.41 0.18 0.10 59a 6.05 0.25 44.2 4130 125 9.20 0.96 0.63 59d $<$1.40 $<$1.80 $<$8.50 61c 15.5 1.40 40.3 4034 484 7.50 9.82 0.59 61d $<$1.5 $<$2.00 $<$8.51 67b 6.69 0.46 19.9 7599 298 $<$2.94 10.07 0.21 -0.16 67c 0.85 0.20 11.9 7485 64 $<$1.05 8.87 68c 7.02 0.42 32.2 2302 214 9.49 0.30 -0.09 69a 4.3 0.55 12.5 8848 386 $<$5.22 9.92 69b 3.27 0.17 14.7 8728 279 4.47 9.58 1.09 71b 4.88 0.65 24.3 9391 323 9.79 0.82 73a$^{\dag}$ 2.85 0.45 28.7 5695 127 9.44 -0.12 -0.87 75b $<$2.40 $<$4.86 $<$9.66 75e $<$1.60 12368 0.53 9.14 78a$^{\dag}$ 5.25 0.40 15.7 8628 363 5.00 9.97 0.39 -0.10 79a $<$1.70 $<$2.52 $<$8.68 79c $<$2.10 $<$3.90 $<$8.77 80a 10.40 0.60 29.7 9046 428 10.34 10.20 0.64 82c 5.14 0.60 23.0 10090 388 2.96 10.03 0.56 88a 5.96 0.80 16.2 5996 464 1.12 9.79 0.22 89c $<$2.22 8992 0.56 8.96 92c 4.95 0.70 17.2 6760 168 9.97 0.18 -0.09 93b 9.95 0.52 46.5 4701 212 9.82 9.94 0.31 0.15 95b $<$1.70 $<$2.28 $<$9.55 95c 3.96 0.30 17.7 11579 302 $<$2.46 10.02 95d 1.07 0.20 8.9 12074 173 $<$1.80 9.34 96a 17.4 1.10 116.5 8666 192 13.74 10.57 0.54 96c 2.69 0.40 11.4 8808 320 3.34 9.47 100a 5.38 0.80 21.7 5298 402 9.55 0.72 0.31 -------------- ---------- ------------------ ------- ---------- ------ --------------- --------------- -------- ------------------- [$^{\dag}$not included in our final HCG sample]{} Average gas content and star formation efficiency ------------------------------------------------- From Table \[tab\_co\_1\], average quantities can be derived to characterize the HCGs as a class. To avoid size effects and artificial correlations induced by uncertain distances, we used quantities normalised to the blue luminosity . All the quantities are displayed in logarithm in the Table \[average\_Table\]. The average / ratio is -0.16 $\pm$ 0.45 for HCG, which indicates a moderate star forming enhancement over isolated galaxies, as already found by Sulentic & de Mello Rabaça (1993). It should be emphasized here that the comparison among samples using the blue luminosity normalised values of the total and by Sulentic & De Mello Rabaça (1993) assumes that the spatial distributions of these two quantities are similar to the blue luminosity distributions. However as stressed by Menon (1995) in his comparison of radio properties of HCG spirals and isolated spirals the differences in spatial distributions have to be taken into account for any meaningful comparison of sample properties. The average / ratio is found to be -0.61 $\pm$ 0.39, which has to be compared with the -1.19 value of Boselli et al. (1996). This difference might be due to the small size of the Boselli et al. sample and also to the fact that they do not take into account the correction of galaxy-to-beam size ratio in deriving the content. Yun et al (1997) also reported an apparent CO emission deficiency in two HCG groups that they mapped with the OVRO interferometer (31c and 92c). However, their observations are missing extended CO emission, and they find 2 and 10 times less than the present work, for 31c and 92c respectively. The / ratio is widely used as an indicator of the star formation efficiency (SFE) (Young et al. 1986). While our present control has an average SFE of 0.67 $\pm$ 0.38, the HCG sample has an SFE of 0.39 $\pm$ 0.33, which confirms the only moderate triggering effect of the compact environment on the global star formation.This value has to be compared with the high ratio of 1.24 for the starburst sample which is about 7 times higher. If we take as indicator of SFE the ratio /(+), as proposed by Sulentic & de Mello Rabaça (1993), no star formation enhancement is observed with a mean ratio of -0.02 $\pm$ 0.40; however, this value is then uncertain, due to the available sample of only 14 galaxies with HI content known, due to the poor spatial resolution of the observations (Williams & Rood, 1987) and the lack of interferometric observations up to now. As already observed, we find a close correlation between the absolute content and the . This correlation is usually interpreted as a relation between the fuel for star formation (molecular gas) and the tracer of that star formation (FIR luminosity) (Young et al., 1986). On Fig. \[mh2\_fir\_correl\] we present versus superposed with the linear fit (in log): =40$^{0.88\pm 0.31}$. One of the galaxies (10a) exhibits a particular high content, without any counterpart of high luminosity, that could be due to a peculiar (not exponential-like) gas distribution, which would imply that our estimation of the total gas fails. Indeed that galaxy has a very large diameter relative to our beam. Mendes de Oliveira & Hickson (1994) remarked that it exhibits a peculiar profile. ----------- ------------- ------------- ------------ ------------- ------------- ------------- ----------- HCG -0.61(0.39) -0.16(0.45) 0.39(0.33) -0.02(0.40) -0.42(0.22) -3.42(0.36) 33.1(5.7) Control -0.78(0.58) -0.11(0.61) 0.67(0.38) 0.06(0.41) -0.34(0.35) -3.54(0.41) 35.0(5.4) Pairs -0.57(0.45) 0.33(0.48) 0.91(0.43) 1.04(0.37) -0.47(0.31) -3.37(0.40) 34.9(6.0) Starburst -0.61(0.43) 0.63(0.43) 1.24(0.39) 0.91(0.39) -0.36(0.40) -3.27(0.34) 40.4(6.2) Cluster -1.08(0.36) -0.31(0.40) 0.77(0.37) 0.42(0.35) -0.73(0.34) -3.78(0.30) 33.2(4.7) Dwarf -1.65(0.88) -0.34(0.59) 1.38(0.68) 0.13(0.43) -0.32(0.65) -4.11(0.67) 38.1(7.2) Elliptic -1.65(0.77) -0.46(0.62) 1.19(0.48) 0.39(0.32) -0.64(0.24) -4.07(0.63) 33.1(5.2) ----------- ------------- ------------- ------------ ------------- ------------- ------------- ----------- a) All averages are logarithmic Dust masses ----------- From the ratio of the IRAS and fluxes we have derived dust temperatures (cf Table \[tab\_sample\_1\]), assuming $\kappa_{\nu} \propto \nu$. The average for the HCG galaxies with detected CO emission is 33$\pm$6K. By comparison, the average dust temperature for the starburst sample is 40$\pm$6K. Knowing the dust temperature and the flux, we can derive the dust mass as $$\begin{aligned} M_{\rm dust} & = & 4.8 \times 10^{-11} \, {S_{\rm \nu}\,d_{\rm Mpc}^{\,2} \over \kappa _{\rm \nu }\,B_{\rm \nu}(T_{\rm d})}\ {\rm M}_{\odot } \\ & = & 5 \,S_{100}\,d_{\rm Mpc}^{\,2}\, \left\{\exp (144/{\hbox{$T_{\rm d}$}}) - 1 \right\}\ {\rm M}_{\odot },\end{aligned}$$ where $S_{\rm \nu}$ is the FIR flux measured in Jy, $\kappa _{\rm \nu}$ is the mass opacity of the dust, and $B_{\rm \nu}$() the Planck function. We used a mass opacity coefficient of 25cm$^{2}$g$^{-1}$ at 100$\mu$ (Hildebrand, 1983). In Table \[tab\_sample\_1\] we list the estimated dust masses, and in Fig. \[dust\_h2\] we plot the dust mass derived from the FIR flux versus the mass derived from the CO observations. The full line is a fit of the dust mass to the mass, corresponding to a simple proportionality between the two masses, with a molecular gas–to–dust mass ratio of 725 which is similar to the molecular gas–to–dust ratio of the control sample (741). The total gas–to–dust ratio is not reliable (high dispersion) given our small sample with FIR and available. We computed for all our samples the cumulative distribution of the ratio /(log). It is clear from Fig. \[df\_ratio\] that this ratio in all samples, except the dwarf and elliptic samples , follows the same distribution, the maximum with the HCG sample being 4.1 (probability 0.46) to be compared with the Table \[table\_test\]. Although there is some observational evidence in favor of a constant gas-to-dust ratio in galactic giant molecular clouds (GMCs), as shown by Sanders et al. (1991), a departure from this value could indicate that either a fraction of the FIR luminosity comes from dust associated with the diffuse atomic ISM or that the /CO conversion factor is galaxy-dependent. Sanders et al. (1991) also suggest that the opacity coefficients could underestimate dust masses. The CO(2-1)/CO(1-0) ratio ------------------------- We detected the line in 26 galaxies only (55% detection rate). In the other CO detected galaxies, we have only an upper limit of the CO(2-1)/CO(1-0) ratio. In most cases, we observed only one position per galaxy: since the beam sizes are different for the two lines, we cannot determine the true line ratio, without a precise model of the source distribution. In any case, we measure an average raw line ratio of 0.74 $\pm$ 0.2, without beam correction. The true ratio will be obtained by dividing by a factor between 1 and 4, because of the factor 2 between the CO(2-1) and CO(1-0) linear beam size. It is thus certain that the CO emission is in general sub-thermally excited, as is frequently the case at large scale in galaxies (e.g. Braine & Combes 1992). There is only one exception, the galaxy 33c, where the true CO(2-1)/CO(1-0) ratio could be of the order of 1. It has been shown that the CO line ratio varies little with the interaction class of the galaxy (Casoli et al. 1988, Radford et al. 1991); it is not a good temperature indicator, but rather a density indicator (it is higher in the galaxy centers, as expected). As a matter of fact this global ratio cannot disentangle in any way the different excitation CO conditions in the galaxies (hot cores, diffuse component). Correlation with the radio continuum flux ----------------------------------------- We have plotted versus the radio luminosity at 1.4 GHz of the detected compact group galaxies in Fig. \[radio\_CO\]. A clear correlation can be found, indicating that both are related to star-forming activity. Menon (1995) has shown that the total radio emission from the discs of HCG galaxies is significantly less than that of a comparable sample of isolated galaxies, while the reverse is true for the nuclear emission. He suggested that the nuclear radio emission is mainly due to star formation bursts and not due to nuclear activity. This is supported here from the good correlation between normalised molecular gas content and radio power. AGN-powered radio emission should perturb this correlation, but this appears negligible here. CO maps of a few objects ------------------------ Some of the Hickson groups are near enough to be resolved by our beam, and we mapped a few objects, in particular several galaxies in HCG16. This compact group appears as a unique condensation of active galaxies, containing one Seyfert 2 galaxy, two LINERS and three starbursts (Ribeiro et al 1996). The galaxy density is 217 gal Mpc$^{-3}$. Ponman et al (1996) detected a diffuse X-ray component corresponding to intra-cluster hot gas. We present in Fig. \[HCG16\] the CO spectra towards the HCG16 galaxies. Some of them (16a and 16b) reveal several velocity components, which can be attributed to overlapping galaxies. 16c and 16d present a clear enhancement of their molecular content with an / ratio equal respectively to -0.21 and -0.37 (log). These two galaxies exhibit optical starburst activity which indicates recent interaction, younger than $10^8$ yr (Ribeiro et al. 1996); while the galaxies 16a and 16b show tidal tails indicating a later interaction phase. They should already have suffered intense star formation and have consumed part of the fueling gas available. To our spatial resolution of 22 we do not observe such a high central concentration of the molecular gas as for radio continuum emission (Menon, 1995). About the compactness --------------------- We plot in Fig. \[mh2\_separation\] the mean / ratio versus the mean projected separation in the group. There is an enhancement of the content up to a mean separation of 30 kpc. This is an indication for the interaction intensity threshold to trigger inner gas flows by tidal interactions. Galaxies with a mean separation less than 25 kpc have /=-0.26 ($\pm$0.33), whereas galaxies with separation more than that distance have /=-0.66 ($\pm$0.37). It is interesting to note that the correlation is weaker when the closest projected separation is used instead of the mean separation in the group. This suggests that the enhancement is a function of the dynamics of the [whole]{} group in the case of the most compact groups, apart from possible strong binary interactions, as shown in HCG16. The / ratio does not exhibit any dependence with the mean separation. Discussion ========== Gas content ----------- To compare our HCG sample with the comparison samples we used the cumulative distributions for the different quantities. Performing Kolmogorov-Smirnov (KS) or test we check the hypothesis of a common underlying population for the different samples. One of the result derived from the comparisons of Fig. \[df\_mass\] concerns the CO emission: there does seem to be an enhancement of in HCG galaxies with respect to a control sample which does not share the same underlying distribution (KS: 0.21 (0.05 significance), : 15.66 (0.64 probability)). However HCG population seems to share the same distribution with starburst (KS: 0.12 (0.61), : 9.65 (0.68)) and pair (KS: 0.22 (0.16), : 11.30 (0.58)) samples. We present in Table \[table\_test\] the KS and results for all coupled samples. The distribution functions of the HCG and control sample exhibit the main difference for the low H$_2$ content galaxies. Although the HCG sample does not exhibit a global FIR enhancement for all galaxies, as shown in Fig. \[df\_fir\], it appears that tidal interactions are efficient in Compact Groups, at least in the most compact ones as we have shown in the previous section. These tidal torques could drive the gas inwards, which might be related to the enhancement of radio continuum emission in the very center of these galaxies. The enhancement does not appear to be a bias from our FIR selected sample since it has a FIR distribution close to that of the control sample (KS: 0.14 (0.49)) but we will discuss afterwards about the Malmquist bias which could be an important limitation in this issue. We can point out that in case of a perturbed molecular gas distribution, our extrapolation for the total mass should fail. But in this case the conclusion would remain similar, leading that time to an enhancement of the molecular content in the [center]{}. This enhancement of molecular gas is also supported by the enhancement of dust mass. Its cumulative distribution in HCG also follows that of the starburst (KS: 0.18 (0.38),: 8.65 (0.69)) and pair (KS: 0.11 (0.91),: 9.08 (0.72)) samples. In spite of the poor data, we can point out that the similarity between these three populations (HCG, pairs and starbursts) for the total gas (+)/ ratio is even tighter (KS: 0.12 (0.99), : 5.51 (0.79), gathering pair and starburst samples in one sample). The case of dwarfs, exhibiting a very peculiar molecular gas-to-dust ratio, can be interpreted in terms of the low metallicity of these objects and will be discussed in a forthcoming paper (Leon et al., 1997).\ Thus in a galaxy group two mechanisms are at play concerning the gas evolution: on one hand tidal interactions enhance the molecular content by driving gas inwards and on the other hand the Intra Cluster Medium strips off the outer gas, reducing the eventual molecular content. It appears that the former dominates for the most compact groups, while for the least compact groups the picture is mitigated. Nevertheless it can be emphasized that the enhancement of dust mass gives us a hint about evidence of tidal interactions in the HCG sample. HCG control starburst pair cluster dwarf elliptic ----------- ------------ ------------- ------------- ------------- ------------- ------------- ------------- HCG 15.66(0.64) 9.65(0.68) 11.30(0.58) 34.36(0.02) 47.75(0.01) 40.61(0.02) control 0.21(0.05) 19.08(0.52) 22.65(0.33) 32.00(0.10) 57.55(0.00) 63.06(0.00) starburst 0.12(0.71) 0.16(0.14) 9.98(0.66) 32.49(0.04) 52.06(0.00) 49.10(0.00) pair 0.22(0.16) 0.18(0.15) 0.10(0.95) 25.11(0.08) 39.22(0.03) 34.73(0.04) cluster 0.59(0.00) 0.40(0.00) 0.51(0.00) 0.50(0.00) 24.88(0.19) 18.96(0.26) dwarf 0.66(0.00) 0.50(0.00) 0.65(0.00) 0.67(0.00) 0.41(0.01) 16.90(0.49) elliptic 0.68(0.00) 0.53(0.00) 0.66(0.00) 0.61(0.00) 0.33(0.11) 0.18(0.89) Completeness of the distribution -------------------------------- We check the completeness of the / distribution function (top left of Fig. \[df\_mass\]) by simulating the distribution function taking into account the threshold of detection for that quantity. The minimum temperature detection is 4 mK in our observations. Then we consider the distance distribution for the galaxies to be uniform up to 150 Mpc or gaussian with the parameters of our sample. The linewidth distribution has been fitted to the blue luminosity with a power law (Tully-Fisher-like relation with ${\hbox{$L_{\rm B}$}}\propto {\Delta V}^{4.3}$). Then inclination angle is distributed uniformly between 0 and 90 degrees. The blue luminosity is distributed with a gaussian distribution ($<\log ({\hbox{$L_{\rm B}$}}) >=10.22$, $\sigma_{{\hbox{$L_{\rm B}$}}}=0.16 $). For each $\log ( {\hbox{$M_{{\rm H}_2}$}}/ {\hbox{$L_{\rm B}$}}) $ bin, the fraction of realisations above the threshold detection is computed to estimate the completeness of our sample. Results are displayed in Fig. \[plot\_completude\] for $10^4$ realisations. Gaussian and uniform distance distributions are two extremes chosen to estimate the weight of the distance parameter: the 50 % level of completeness is $ \log ( {\hbox{$M_{{\rm H}_2}$}}/ {\hbox{$L_{\rm B}$}}) =-1.8 $ for the uniform case and -1.5 for the gaussian case. Assuming that our control sample is complete, we compute a cumulative distribution biased by the completeness function of Fig. \[plot\_completude\]. Fig. \[plot\_cf\_simul\] shows the result where it appears that the Malmquist bias in our sample, spread over a large distance range, can explain part of the apparent enhancement of molecular gas in the HGGs. However the control sample, assumed to be complete, is not likely to be so for the low values of $\log(\mbox{{\hbox{$M_{{\rm H}_2}$}}/ {\hbox{$L_{\rm B}$}}}) )$ where the samples are the most different. Similarly the $\log(\mbox{{\hbox{$M_{\rm dust}$}}/{\hbox{$L_{\rm B}$}}})$ distribution is affected by the Malmquist bias (Verter, 1993), but the point is that the cumulative distribution is lower than the control distribution on the whole range of variation up to higher values, leading to a suggestion of a real enhancement of the dust material in the HCGs. The close correlation between dust and molecular gas suggests that the content is really enhanced in the HCGs. [As it has been shown previously, that enhancement is only significant for the most compact groups in their merging phase, confusing somewhat the question of molecular gas enhancement in the whole HCG sample.]{} High SFE due to artificially low gas content -------------------------------------------- What can also be seen in Fig. \[fir-H2-SFE-HI\](a) is the large dispersion of the SFE as defined by the / ratio: a large number of galaxies are deficient in CO emission, leading to a depressed / ratio and large SFE. This large dispersion is mainly due to the dwarf and elliptical samples, but also to small galaxies in the control sample; we have checked that the objects with high SFE at moderate / have a lower than average. This phenomenon disappears when the total gas content is considered instead of the mere H$_2$ content, as shown in fig \[fir-H2-SFE-HI\](b) and (d). It is striking that the total normalised gas content is almost a constant, independent of /. There is one exception for the cluster population where the stripping of the neutral and molecular content is at play. We find a deficiency of the molecular content in these galaxies, which seems related to a lower luminosity (Horellou et al. 1995). All that suggests the importance of in star formation as a source of fueling, through the conversion $\leftrightarrow$ , and the higher reliability of the star formation indicator /(+). A least square fit yields the relation ${\hbox{$L_{\rm FIR}$}}/({\hbox{$M_{{\rm H}_2}$}}+{\hbox{$M_{\rm HI}$}}) \simeq 2.4(\frac{{\hbox{$L_{\rm FIR}$}}}{{\hbox{$L_{\rm B}$}}})^{0.69 \pm 0.10}$ for all the samples gathered. The good correlation between the normalised and could be in a large part due to the dependence of both quantities on the metallicity and temperature of the interstellar medium. It is now well established that the CO to H$_2$ conversion ratio is strongly dependent on metallicity (e.g. Rubio et al 1993), as well as the dust-to-gas ratio (and therefore the FIR luminosity). As shown on Fig. \[df\_ratio\], all the cumulative distributions for the molecular gas-to-dust ratio are highly correlated, (: all, except dwarf and elliptic samples, $<$ 6. (probability $>$ 0.54)). While the dependence of on temperature is direct, that of the CO emission is more complex. Low brightness temperatures of the CO lines are obtained either for cold gas, or diffuse gas; the rotational levels of the molecule are excited by collision, and diffuse molecular clouds are generally sub-thermal. The use of a standard CO to H$_2$ conversion ratio is then problematic. For thermalised dense gas however, the CO emission is directly proportional to the gas temperature. The metal abundance and the gas recycling are closely related to the IMF and evolution of the MF. High mass stars (M$> 5{\mbox{M$_{\odot}$}}$) are active on short time scales ($<10^8$ yrs) whereas low mass stars have an influence on much longer time scales (Vigroux et al. 1996). Thus the CO abundance is very dependent on the star formation rate, and the CO/ conversion ratio could be very variable, particularly for galaxies with recent star formation episodes (e.g. Casoli et al 1992, Henkel & Mauersberger 1993). Far-infrared and star-forming activity -------------------------------------- Given the enhancement in CO emission detected in HCG with respect to a control sample and the strong correlation between the radio continuum, CO and the FIR for HCG spirals it would appear that the lack of enhancement of the total FIR emission in HCG galaxies is due to lack of spatial resolution of IRAS measurements. This is particularly important if the FIR is mainly enhanced in the central regions of the galaxies. ISO observations might allow to check this assertion. The consequence of an enhanced without FIR enhancement is a lower star forming efficiency for HCG, as displayed in Fig. \[df\_fir\]. This property might appear surprising, but disappears when the total gas content is taken into account (this result should be taken with caution, since only 14 galaxies have a well-defined content in our HCG sample). \[Td\] We have tested the correlations of the star formation indicators, with and without account of the atomic phase, with the dust temperature, in Fig. \[survey\_td\]. Both quantities correlate well with : for all samples together we find a relationship flatter than Young et al. (1989), i.e. /$\propto$ ${\hbox{$T_{\rm d}$}}^{3.8\pm 0.7}$, but with differences and large dispersions among categories: for the HCG sample alone /$\propto$ ${\hbox{$T_{\rm d}$}}^{2.5 \pm 2.5}$ and for the starburst sample /$\propto$ ${\hbox{$T_{\rm d}$}}^{3.2 \pm 1.8}$. Sage (1993) pointed out that a single dust temperature is an “average” of the cold dust and warm dust associated respectively with the quiescent molecular clouds and the clouds with massive star forming ones. Devereux & Young (1990) emphasized that high mass ($>$ 6 ) O and B stars are responsible for high ($> 10^9$ ) and H$_\alpha$ luminosities, advocating a two-component model: one dust component heated by high mass stars ($\sim$ 50-60 K) and the other heated by the interstellar radiation field ($\sim$ 16-20 K). The low mean temperature of the HCG sample ($\approx$ 33 K) suggests that the luminosity is coming from an important quiescent molecular phase, together with the “cirrus” phase associated with diffuse atomic hydrogen. It could explain why the [global]{} luminosity does not fit with a high star formation population, without excluding star formation towards the center, as revealed by radio continuum. In a recent study, Lisenfeld et al. (1996) find the same /[$L_{2.4GHz}$]{} ratio for starburst interacting and normal galaxies. From this, we could infer that the enhanced radio continuum emission from the center of HCG galaxies should be accompanied with an enhanced central FIR luminosity. The constancy of the /[$L_{2.4GHz}$]{} ratio has been interpreted as being due to a strong and fast ($10^7$ yrs) increase of the magnetic field at the beginning of the starburst, together with a time-scale of variation of the star-formation rate longer than some $10^7$ yrs. Fate of the HCG --------------- The enhancement of the content in the most compact groups suggests that tidal interactions in HCG are efficient in driving the gas inwards. This is also a confirmation that at least some groups are actually compact and not only projections along the line of sight. These very compact groups should merge through dynamical friction on a short time scale (a few $10^8$ yrs, cf Barnes 1989). A conclusion from the present work is that the most compact of these groups have concentrated an important amount of molecular gas without initiating yet important star formation. However they must correspond to a short duration phase just before merging and enhanced star-formation. It is tempting to identify the next phase of this process to the Ultra Luminous Infra Red Galaxies (ULIRGs). The latter have their infrared luminosity powered by massive star formation (Lutz et al. 1996) with an important consumption of molecular gas. Sanders et al (1988) have shown that many ULIRGs are interacting/merging galaxies; recently Clements & Baker (1996) extended that to the vast majority of the ULIRGs sample. Some of these systems, with luminous masses ranging up to few $10^{11} {\mbox{M$_{\odot}$}}$, should represent the remnant of some compact group which has undertaken multiple mergers on a very short time-scale ($\sim 10^8$ yrs). The FIR luminosity of the final ULIRGs should be a function of the spiral fraction in the parent group, the most powerful ULIRGs being the result of the merging of typically four gas-rich spirals. As pointed out earlier the spatial distribution of interstellar matter within galaxies will play an important role in the rate of fuelling of starbursts during interactions. But does the expected rate of ULIRGs formation via CG merging match the presently observed frequency of these objects? The answer is very uncertain, since the actual fraction of very compact groups, on the point of merging, is not known. This fraction cannot be close to 1, since there would be too large a discrepancy between the expected and observed number of ultra luminous galaxies and their remnants (e.g. Williams & Rood 1987, Sulentic & Rabaça 1994). Simulations of galaxy formation and large-scale structures evolution have suggested that some of the CGs could be filaments of galaxies seen end-on (Hernquist et al. 1995). This idea has been studied further by Pildis et al. (1996): if it is true that galaxies that will form a compact group spend a large fraction of their time first in a filament, this filament will appear as a CG in projection only for less than 20% of cases. Then the galaxies will fall into a real CG, and this phase corresponds to at least 30% of their lifetime. Although these figures are model dependent, they suggest that the majority of HCG in the sky are physically compact groups. The time-scale of merging can then depend highly on initial conditions, and in particular on the elliptical fraction, which may alleviate the over-merging problem (e.g. Governato et al. 1991, Garcia-Gomez et al. 1996). The threshold in mean galaxy separation for the enhancement of suggests that it corresponds to the last stage of the life of the compact group, when each galaxy is undertaking frequent and strong tidal effects. In Fig. \[histo\_sepa\] we plot the histogram of the mean projected separation of the whole sample of HCGs from Hickson et al. (1992): there is a significant cut-off at short separation at approximatively 20-30 kpc, which must correspond to a short life time for very compact configurations. We suggest that this cut-off corresponds to the acceleration of the merging process, and at the same time to significant inward gas flows, that account for our observations. Dynamically, such an exponential acceleration of the collapse is predicted by simulations and analytic models of satellite decay through dynamical friction (see for example Leeuwin & Combes 1997). This rapid acceleration has been interpreted through the excitation of numerous high-order resonances for a satellite at about twice the primary radius (e.g. Tremaine & Weinberg 1984) . Conclusion ========== From our survey of CO emission in 70 galaxies belonging to 45 Hickson compact groups, we have detected 57 objects. We find in average that the gas and dust contents / and / show evidence of enhancement with respect to our control sample. This result however is somewhat weakened due to the Malmquist bias in our sample. For the most compact groups the enhancement is more clear. On the contrary, the global far-infrared flux does not appear to be enhanced with respect to the control sample. The FIR and distribution indicates that the FIR luminosity is coming essentially from a cold dust component heated by the interstellar radiation field. From the general correlation between FIR and radio contiuum power we suggest that only the very centers of some groups are experiencing star formation and are sites of enhanced FIR emission. IRAS spatial resolution is not sufficient to show this directly. Statistical tests show that the HCG gas and dust contents are closer to that of pair and starburst galaxies, revealing the efficiency of tidal interactions in driving the gas inwards in compact group galaxies. We find a stronger enhancement for the CGs having a short mean separation ($< 30$ kpc). We suggest that these most compact, high- content groups, may be in a final merging phase, just before the starburst phase, that will lead them in a very short time-scale to the ULIRGs category.\ The comparison of the various samples suggest that the [total]{} gas content (+) should be taken into account to estimate the star formation efficiency. The corresponding SFE indicator, /( +), should be more reliable, and allow us to avoid some systematic effects depending on metallicity and temperature. Appendix: H$_2$ mass determination ================================== \[fig\_K\] We follow Gordon et al. (1992) to derive H$_2$ masses from line observations. Our temperature unit is expressed in [ ]{}antenna temperature scale which is corrected for atmospheric attenuation and rear sidelobes. The radiation temperature T$_R$ of the extragalactic source is then: $$\label{equa_Tr} T_R=\frac{4}{\pi}(\frac{\lambda}{D})^2\frac{K}{\eta _A} \frac{T_A}{\Omega_S}$$ where $\lambda$ is the observed wavelength (2.6 mm), D is the IRAM radio telescope diameter (30 m), K is the correction factor for the coupling of the source with the beam, $\eta_A$ is the apperture efficiency (0.55) at 115 GHz, T$_A$ is an antenna temperature which is $F_{eff}$T$_A^$ in the IRAM convention, explicitely T$_A$=0.92T$_A^$, and $\Omega_S$ is the source size. Without taking into account cosmological correction, because of low redshift, column density of molecular hydrogen it written down as $$N(H_2)=2.3\times10^{20}\int_{line} T_R dv \mbox{ (mol.cm$^{-2}$)}$$ where $dv$ is the velocity interval . In equation \[equa\_Tr\] K$\equiv \Omega_s / \Omega_{\Sigma}$ is the factor which corrects the measured antenna temperature for the weighting of the source distribution by the large antenna beam in case of a smaller source. We have defined the source solid angle $$\Omega_S\equiv \int_{source} \phi (\theta , \psi ) d\Omega$$ where $\phi (\theta , \psi$) is the normalized source brightness distribution function. The beam-weighted source solid angle is $$\Omega_{\Sigma}\equiv \int_{source} \phi (\theta , \psi ) f( \theta, \psi ) d\Omega$$ where $f( \theta, \psi )$ denotes the normalized antenna power pattern (Baars, 1973). Experiments have shown that we can approximate $f$ by a gaussian beam. As mentionned in section 3.3 an exponential law of scale length h=D$_B$/10 is taken to model the source distribution function. As long as the source size is smaller than the beam size we have $$\label{equa_K} K=\frac{\int^{\theta_s /2}_0 sin(\theta ) e^{-\frac{ 10 \theta }{\theta_s}} d\theta}{\int^ {\theta_s /2}_0 sin(\theta ) e^{-\frac{ 10 \theta }{\theta_s}-\mbox{ln}(2)( \frac{2 \theta}{\theta_b})^2} d\theta}$$ where $\theta_b$ and $\theta_s$ are the beam and the source sizes. We present in Fig. \[fig\_K\] the plot of K in the case of exponential and uniform source distributions, for the IRAM-30m beam at 115 GHz (22 ). In the case of a source larger than the beam size it is more difficult to compute the coupling of the beam to the source region since contributions from error beam can be quite significant to the resulting spectra. A simple representation of the overall beam, including the error beam, is not available to correct that coupling. 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--- abstract: 'Reducing hateful and offensive content in online social media pose a dual problem for the moderators. On the one hand, rigid censorship on social media cannot be imposed. On the other, the free flow of such content cannot be allowed. Hence, we require efficient abusive language detection system to detect such harmful content in social media. In this paper, we present our machine learning model, HateMonitor, developed for Hate Speech and Offensive Content Identification in Indo-European Languages (HASOC) [@hasoc2019overview], a shared task at FIRE 2019. We have used Gradient Boosting model, along with BERT and LASER embeddings, to make the system language agnostic. Our model came at First position for the German sub-task A. We have also made our model public [^1].' author: - Punyajoy Saha - Binny Mathew - Pawan Goyal - Animesh Mukherjee bibliography: - 'main.bib' title: 'HateMonitors: Language Agnostic Abuse Detection in Social Media' --- Introduction ============ In social media, abusive language denotes a text which contains any form of unacceptable language in a post or a comment. Abusive language can be divided into hate speech, offensive language and profanity. Hate speech is a derogatory comment that hurts an entire group in terms of ethnicity, race or gender. Offensive language is similar to derogatory comment, but it is targeted towards an individual. Profanity refers to any use of unacceptable language without a specific target. While profanity is the least threatening, hate speech has the most detrimental effect on the society. Social media moderators are having a hard time in combating the rampant spread of hate speech[^2] as it is closely related to the other forms of abusive language. The evolution of new slangs and multilingualism, further adding to the complexity. Recently, there has been a sharp rise in hate speech related incidents in India, the lynchings being the clear indication [@arun2019whatsapp]. Arun et al. [@arun2019whatsapp] suggests that hate speech in India is very complicated as people are not directly spreading hate but are spreading misinformation against a particular community. Hence, it has become imperative to study hate speech in Indian language. For the first time, a shared task on abusive content detection has been released for Hindi language at HASOC 2019. This will fuel the hate speech and offensive language research for Indian languages. The inclusion of datasets for English and German language will give a performance comparison for detection of abusive content in high and low resource language. In this paper, we focus on the detection of multilingual hate speech detection that are written in Hindi, English, and German and describe our submission (HateMonitors) for HASOC at FIRE 2019 competition. Our system concatenates two types of sentence embeddings to represent each tweet and use machine learning models for classification. Related works ============= Analyzing abusive language in social media is a daunting task. Waseem et al. [@waseem2017understanding] categorizes abusive language into two sub-classes – hate speech and offensive language. In their analysis of abusive language, Classifying abusive language into these two subtypes is more challenging due to the correlation between offensive language and hate speech [@davidson2017automated]. Nobata et al. [@nobata2016abusive] uses predefined language element and embeddings to train a regression model. With the introduction of better classification models [@qian2018hierarchical; @stammbach2018offensive] and newer features [@alorainy2018enemy; @davidson2017automated; @unsvaag2018effects], the research in hate and offensive speech detection has gained momentum. Silva et al. [@silva2016analyzing] performed a large scale study to understand the target of such hate speech on two social media platforms: Twitter and Whisper. These target could be the Refugees and Immigrants [@ross2017measuring], Jews [@bilewicz2013harmful; @finkelstein2018quantitative] and Muslims [@awan2016islamophobia; @vidgen2018detecting]. People could become the target of hate speech based on Nationality [@erjavec2012you], sex [@bartlett2014misogyny; @saha2018hateminers], and gender [@reddy2002perverts; @gatehouse2018troubling] as well. Public expressions of hate speech affects the devaluation of minority members [@greenberg1985effect], the exclusion of minorities from the society [@mullen2003ethnophaulisms], and tend to diffuse through the network at a faster rate [@mathew2019spread]. One of the key issues with the current state of the hate and offensive language research is that the majority of the research is dedicated to the English language on [@fortuna2018survey]. Few researchers have tried to solve the problem of abusive language in other languages [@ross2017measuring; @sanguinetti2018italian], but the works are mostly monolingual. Any online social media platform contains people of different ethnicity, which results in the spread of information in multiple languages. Hence, a robust classifier is needed, which can deal with abusive language in the multilingual domain. Several shared tasks like HASOC [@hasoc2019overview], HaSpeeDe [@bosco2018overview], GermEval [@wiegand2018overview], AMI [@fersini2018overview], HatEval [@basile2019semeval] have focused on detection of abusive text in multiple languages recently. Dataset and Task description ============================ The dataset at HASOC 2019 [^3] were given in three languages: Hindi, English, and German. Dataset in Hindi and English had three subtasks each, while German had only two subtasks. We participated in all the tasks provided by the organisers and decided to develop a single model that would be language agnostic. We used the same model architecture for all the three languages. Datasets -------- We present the statistics for HASOC dataset in Table \[tab:dataset\_statistics\]. From the table, we can observe that the dataset for the German language is highly unbalanced, English and Hindi are more or less balanced for sub-task A. For sub-task B German dataset is balanced but others are unbalanced. For sub-task C both the datasets are highly unbalanced. Language ------------ ------- ------ ------- ------ ------- ------ Sub-Task A Train Test Train Test Train Test HOF 2261 288 407 136 2469 605 NOT 3591 865 3142 714 2196 713 Total 5852 1153 3819 850 4665 1318 Sub-Task B Train Test Train Test Train Test HATE 1141 124 111 41 556 190 OFFN 451 71 210 77 676 197 PRFN 667 93 86 18 1237 218 Total 2261 288 407 136 2469 605 Sub-Task C Train Test Train Test Train Test TIN 2041 245 - - - - 1545 542 UNT 220 43 - - - - 924 63 Total 2261 288 - - - - 2469 605 : This table shows the initial statistics about the training and test data[]{data-label="tab:dataset_statistics"} Tasks ----- Sub-task A consists of building a binary classification model which can predict if a given piece of text is hateful and offensive (HOF) or not (NOT). A data point is annotated as HOF if it contains any form of non-acceptable language such as hate speech, aggression, profanity. Each of the three languages had this subtask. Sub-task B consists of building a multi-class classification model which can predict the three different classes in the data points annotated as HOF: Hate speech (HATE), Offensive language (OFFN), and Profane (PRFN). Again all three languages have this sub-task. Sub-task C consists of building a binary classification model which can predict the type of offense: Targeted (TIN) and Untargeted (UNT). Sub-task C was not conducted for the German dataset. System Description ================== In this section, we will explain the details about our system, which comprises of two sub-parts- feature generation and model selection. Figure \[Train\] shows the architecture of our system. Feature Generation ------------------ ### Preprocessing: We preprocess the tweets before performing the feature extraction. The following steps were followed: - We remove all the URLs. - Convert text to lowercase. This step was not applied to the Hindi language since Devanagari script does not have lowercase and uppercase characters. - We did not normalize the mentions in the text as they could potentially reveal important information for the embeddings encoders. - Any numerical figure was normalized to a string ‘number’. We did not remove any punctuation and stop-words since the context of the sentence might get lost in such a process. Since we are using sentence embedding, it is essential to keep the context of the sentence intact. ### Feature vectors: The preprocessed posts are then used to generate features for the classifier. For our model, we decided to generate two types of feature vector: BERT Embeddings and LASER Embeddings. For each post, we generate the BERT and LASER Embedding, which are then concatenated and fed as input to the final classifier. Multilingual BERT embeddings: Bidirectional Encoder Representations from Transformers(BERT) [@DBLP:journals/corr/abs-1810-04805] has played a key role in the advancement of natural language processing domain (NLP). BERT is a language model which is trained to predict the masked words in a sentence. To generate the sentence embedding[^4] for a post, we take the mean of the last 11 layers (out of 12) to get a sentence vector with length of 768. LASER embeddings: Researchers at Facebook released a language agnostic sentence embeddings representations (LASER) [@DBLP:journals/corr/abs-1812-10464], where the model jointly learns on 93 languages. The model takes the sentence as input and produces a vector representation of length 1024. The model is able to handle code mixing as well [@VERMA1976153]. ![Architecture of our system[]{data-label="Train"}](figures/HASOC_training.pdf){width="60.00000%"} We pass the preprocessed sentences through each of these embedding models and got two separate sentence representation. Further, we concatenate the embeddings into one single feature vector of length 1792, which is then passed to the final classification model. Our Model --------- The amount of data in each category was insufficient to train a deep learning model. Building such deep models would lead to overfitting. So, we resorted to using simpler models such as SVM and Gradient boosted trees. Gradient boosted trees [@DBLP:journals/corr/ChenG16] are often the choice for systems where features are pre-extracted from the raw data[^5]. In the category of gradient boosted trees, Light Gradient Boosting Machine (LGBM) [@Ke2017LightGBMAH] is considered one of the most efficient in terms of memory footprint. Moreover, it has been part of winning solutions of many competition [^6]. Hence, we used LGBM as model for the downstream tasks in this competition. Results ======= The performance of our models across different languages for sub-task A are shown in table \[tab2\]. Our model got the first position in the German sub-task with a macro F1 score of 0.62. The results of sub-task B and sub-task C is shown in table \[tab3\] and \[tab4\] respectively. Language English German Hindi ---------- --------- -------- ------- HOF 0.59 0.36 0.76 NOT 0.79 0.87 0.79 Total 0.69 0.62 0.78 : This table gives the language wise result of sub-task A by comparing the macro F1 values[]{data-label="tab2"} Language English German Hindi ---------- --------- -------- ------- HATE 0.28 0.04 0.29 OFFN 0.00 0.0 0.29 PRFN 0.31 0.19 0.59 NONE 0.79 0.87 0.79 Total 0.34 0.28 0.49 : This table gives the language wise result of sub-task B by comparing the macro F1 values []{data-label="tab3"} Language English Hindi ---------- --------- ------- TIN 0.51 0.63 UNT 0.11 0.17 NONE 0.79 0.79 Total 0.47 0.53 : This table gives the language wise result of sub-task C by comparing the macro F1 values[]{data-label="tab4"} Discussion ========== In the results of subtask A, models are mainly affected by imbalance of the dataset. The training dataset of Hindi dataset was more balanced than English or German dataset. Hence, the results were around 0.78. As the dataset in German language was highly imbalanced, the results drops to 0.62. In subtask B, the highest F1 score reached was by the profane class for each language in table \[tab3\]. The model got confused between OFFN, HATE and PRFN labels which suggests that these models are not able to capture the context in the sentence. The subtask C was again a case of imbalanced dataset as targeted(TIN) label gets the highest F1 score in table \[tab4\]. Conclusion ========== In this shared task, we experimented with zero-shot transfer learning on abusive text detection with pre-trained BERT and LASER sentence embeddings. We use an LGBM model to train the embeddings to perform downstream task. Our model for German language got the first position. The results provided a strong baseline for further research in multilingual hate speech. We have also made the models public for use by other researchers[^7]. [^1]: <https://github.com/punyajoy/HateMonitors-HASOC> [^2]: <https://tinyurl.com/y6tgv865> [^3]: <https://hasoc2019.github.io/> [^4]: We use the BERT-base-multilingual-cased which has 104 languages, 12-layer, 768-hidden, 12-heads and 110M parameters [^5]: <https://tinyurl.com/yxmuwzla> [^6]: <https://tinyurl.com/y2g8nuuo> [^7]: <https://github.com/punyajoy/HateMonitors-HASOC>
--- abstract: 'Spin current injection from sputtered yttrium iron garnet (YIG) films into an adjacent platinum layer has been investigated by means of the spin pumping and the spin Seebeck effects. Films with a thickness of 83 and 96 nanometers were fabricated by on-axis magnetron rf sputtering at room temperature and subsequent post-annealing. From the frequency dependence of the ferromagnetic resonance linewidth, the damping constant has been estimated to be $(7.0\pm1.0)\times 10^{-4}$. Magnitudes of the spin current generated by the spin pumping and the spin Seebeck effect are of the same order as values for YIG films prepared by liquid phase epitaxy. The efficient spin current injection can be ascribed to a good YIG$\mid$Pt interface, which is confirmed by the large spin-mixing conductance $(2.0\pm0.2)\times 10^{18}$ m$^{-2}$.' author: - 'J. Lustikova' - 'Y. Shiomi' - 'Z. Qiu' - 'T. Kikkawa' - 'R. Iguchi' - 'K. Uchida' - 'E. Saitoh' title: 'Spin current generation from sputtered Y$_3$Fe$_5$O$_{12}$ films' --- Introduction ============ Spintronics is an aspiring field of electronics which incorporates the spin degree of freedom into charge-based devices. Among the main interests in spintronics are generation, manipulation and detection of spin current, the flow of spin angular momentum. Pure spin current unaccompanied by charge current has high potential to open a path to new information technology free from the Joule heating. For spin current generation in thin-film systems, two dynamical methods are the spin pumping [@silsbee; @tserkovnyak-PRL; @mizukami; @azevedo; @saitoh; @costache; @kajiwara; @ando] and the spin Seebeck effect.[@uchida2008; @uchidanmat2010; @jaworski; @uchidaAPL; @kirihara; @kikkawa] In spin pumping \[SP, Fig. \[fig1\](a)\], spin current is generated by magnetization dynamics in the ferromagnet. The magnetization vector of a ferromagnet irradiated by a microwave precesses when the ferromagnetic resonance condition is fulfilled. This precession motion relaxes not only by damping processes inside the ferromagnet (F), but also by emission of spin current into the adjacent non-magnetic conductor (N) by exchange interaction at the F$\mid$N interface.[@silsbee; @tserkovnyak-PRL; @mizukami] In the spin Seebeck effect (SSE), spin current is generated in the presence of a temperature gradient across the ferromagnet. The simplest setup for SSE is the so-called longitudinal configuration \[Fig. \[fig1\](b)\], where the temperature difference is applied parallel to the direction of spin injection. Given that the ferromagnet is attached to a non-magnetic conductor, spin current is emitted from the ferromagnet into the neighbouring non-magnetic metal by thermal spin pumping.[@adachi; @rezende] Notably, these two mechanisms of spin current generation do not require that the ferromagnet be a conductor. The use of an insulator enables generation of pure spin currents and limits transport mediated by conduction electrons to the adjacent non-magnetic metal. The ferrimagnet yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$, YIG) is a material of choice as a spin current injector due to its highly insulating properties and high Curie temperature (550 K).[@glass] In addition, its low magnetic loss properties at microwave frequencies make it ideal for efficient spin injection. The magnetization damping in YIG is two orders of magnitude lower than that in ferromagnetic metals.[@sparks] ![\[fig1\] Schematic illustrations of the experimental setup for spin pumping (SP) and the spin Seebeck effect (SSE). (a) SP and the inverse spin Hall effect (ISHE). $\mathbf{H}$, $\mathbf{h}_{\text{ac}}$, $\mathbf{M}(t)$, $\mathbf{j}_s$ and $\boldsymbol{\sigma}$ denote the static magnetic field, the microwave magnetic field, the magnetization vector, the direction of spin current generated by SP and the spin-polarization vector of the spin current, respectively. The bent arrows in the Pt layer denote the motion of the electrons under the influence of the spin-orbit coupling which leads to the appearance of a transverse electromotive force (ISHE). (b) Longitudinal SSE in a YIG$\mid$Pt bilayer film. $\boldsymbol{\nabla} T$ denotes the temperature gradient. Spin current is generated along $\boldsymbol{\nabla} T$ due to SSE and the electromotive force by ISHE in Pt appears in a direction perpendicular both to the sample magnetization and to the temperature gradient. ](fig1.jpg){width="8cm"} Among the various fabrication methods of YIG, liquid-phase epitaxy (LPE) is known for its ability to produce high-quality single-crystal films[@levinstein; @glass] which have been used extensively in spintronics experiments.[@kajiwara; @uchidanmat2010; @kurebayashi] However, it is difficult to produce films thinner than a few hundred nanometers by the LPE method.[@castel] Since it has been shown that the interface damping due to SP increases with decreasing thickness of the ferromagnetic film,[@tserkovnyak-PRB; @tserkovnyak-PRL] synthesis of YIG films with thickness below 100 nm is desirable for the study of interface effects. Conversely, the increase and saturation of the spin Seebeck signal with increasing YIG thickness has been interpreted as evidence that SSE originates in bulk magnonic spin currents. [@kehlberger] Therefore, thin YIG films are also useful for probing the physics of the spin Seebeck effect. In quest of controlling YIG thickness at nanometer scale, the growth of thin films by pulsed-laser deposition (PLD)[@krockenberger; @sun; @althammer; @kelly; @sun-wu] and sputtering[@okamura; @jang; @jang2; @kang0; @kang; @mwu; @boudiar; @yamamoto; @wang1; @wang2; @wang3; @wang4; @wang5; @wang6; @marmion] has attracted interest. In the sputtering method, the growth of crystals can be realized either by direct epitaxial growth via sputtering at high temperatures [@okamura; @jang2; @wang1; @mwu] or by sputtering at room temperature and subsequent post-annealing. [@jang; @yamamoto; @jang2; @boudiar; @kang0; @kang; @mwu] Direct epitaxial growth at high temperature can provide crystals of excellent quality.[@wang1] However, the sputtering rates are usually very low [@jang; @kang0] and the sample quality sensitive to the conditions during deposition. In contrast, sputtering at ambient temperature is technologically more accessible as it does not require a high process temperature and enables faster deposition.[@jang] Although there are various industrial advantages to the sputtering method, such as high compatibility with the semiconductors technology, suitability for coating of large areas, and dryness of the preparation process, there are only a limited number of reports on the use of sputtered YIG films in spintronics experiments. [@wang1; @wang2; @wang3; @wang4; @wang5; @wang6; @marmion] In this work, we grow thin YIG films by sputtering and subsequent post-annealing and confirm epitaxial growth by transmission electron microscopy (TEM). By measuring SP and SSE, we demonstrate that the obtained YIG films are an efficient spin current generator comparable to LPE films. Methods ======= YIG films were deposited by on-axis magnetron rf sputtering on gadolinium gallium garnet (111) (Gd$_3$Ga$_5$O$_{12}$, GGG) substrates with a thickness of 500 $\mu$m. The choice of substrate was due to the close match of the lattice constants and of the thermal expansion coefficients of GGG and YIG. [@boudiar] The sputtering target had a nominal composition of Y$_{3}$Fe$_{5}$O$_{12}$. The base pressure was $2.3\times 10^{-5}$ Pa. The substrate remained at ambient temperature during sputtering. The pressure of the pure argon atmosphere was $1.3$ Pa. The deposition rate was fairly high at $2.7$ nm/min with a sputtering power of 100 W. The as-deposited films were non-magnetic; according to Refs. such films are amorphous. Crystalization was realized by post-annealing in air at $850$ $^{\circ}$C for 24 hours. In this study, we focus on films with a thickness of 83 and 96 nanometers. The thickness was determined by X-ray reflection (XRR) and TEM. The structure of the samples was characterized by high-resolution TEM. X-ray photoelectron spectroscopy confirmed a Y:Fe stoichiometry 3:4.4. Microwave properties were analyzed using a 9.45-GHz TE$_{011}$ cylindrical microwave cavity and a coplanar transmission waveguide in the 3-10 GHz range. The waveguide had a 2-mm-wide signal line and was designed to a 50-$\Omega$ impedance. The width and length of the samples were $w=1$ mm and $l=3$ mm, respectively. For the spin injection experiments, the annealed YIG samples were coated by a platinum film by rf sputtering. Spin current injected into the platinum layer was detected electrically using the inverse spin Hall effect \[ISHE, Fig. \[fig1\](a)\]. ISHE originates in the spin-orbit interaction which bends the trajectories of electrons with opposite spins and opposite velocities in the same direction and produces an electric field transverse to the direction of the spin current. [@saitoh; @costache; @azevedo; @valenzuela] Platinum was chosen for its high conversion efficiency from spin current to charge current.[@ando] Spin pumping was performed at room temperature in a cylindrical 9.45-GHz TE$_{011}$ cavity at a microwave power $P_{\text{MW}}=1$ mW (corresponding to a microwave field $\mu_0h_{\text ac}=0.01$ mT) in a setup illustrated in Fig. \[fig1\](a). The sample was placed in the centre of the cavity where the electric field component of the microwave is minimized while the magnetic field component is maximized and lies in the plane of the sample surface. A static magnetic field was applied perpendicular to the direction of the microwave field and to the direction in which the voltage was measured.[@ando] Measurements were performed on a set of three samples. The thickness of the Pt layer was $d_{N}=14$ nm, the thickness of the YIG layer $d_{F}=96$ nm. The SSE experiment was performed in a longitudinal setup identical to that of Ref. on three YIG(83 nm)$\mid$Pt(10 nm) samples. The length, the width, and the thickness of the samples were $L_V=6$ mm, $w=1$ mm, and $L_T=0.5$ mm, respectively. The sample was sandwiched between two insulating AlN plates with high thermal conductivity. The upper AlN plate (on top of the Pt layer) was thermally connected to a Cu block held at room temperature. The bottom AlN plate (under the GGG substrate) was placed on a Peltier module. The width of the upper AlN plate (5 mm) was slightly shorter than the sample length (6 mm) in order to take electrical contacts with tungsten needles. The samples were placed in a $10^{-2}$ Pa vacuum in order to prevent heat exchange with the surrounding air. A static magnetic field was applied in the plane of the sample surface perpendicular to the direction in which the voltage was measured. Results and discussion ====================== Structural and microwave properties ----------------------------------- Figures \[fig2\](a)-(e) present the structural properties of the 96-nm-thick films observed by TEM. A magnified view of the GGG$\mid$YIG interface and the diffraction pattern at this interface are shown in Figs. \[fig2\](a) and \[fig2\](b), respectively. The YIG grows epitaxially on the GGG substrate. Neither defects nor misalignment in the lattice planes were observed in the TEM images \[Fig. \[fig2\](a)\]. As shown in Fig. \[fig2\](b), the diffraction pattern consists of a single reciprocal lattice confirming perfect alignment of the GGG and YIG structures. \[\] [![image](fig2.jpg){width="12.1cm"}]{} An image of the whole cross section of a GGG$\mid$YIG$\mid$Pt sample is shown in Fig. \[fig2\](c). The YIG film contains spherical defects with a diameter of roughly 10 nm. However, these are suppressed in the vicinity of the YIG$\mid$Pt interface. TEM imaging of as-deposited films revealed a uniform amorphous Y-Fe-O layer indicating that the spherical structures emerge during post-annealing. A magnified view of these objects is given in Fig. \[fig2\](d). They do not possess crystalline structure. This can be also inferred from the fact that only a single reciprocal lattice, corresponding to epitaxial growth, was observed in the diffraction pattern. We speculate that these defects are voids which appear due to the volume change in the transition from amorphous to crystalline phase. There is a possibility that these structures contain residual amorphous material left over in the crystallization. We expect that these structures, due to their location inside the film, do not affect spin injection efficiency because spin injection originates in the spin transport at the F$\mid$N interface.[@tserkovnyak-PRB; @tserkovnyak-PRL] The YIG$\mid$Pt interface is magnified in Fig. \[fig2\](e). One can see that the YIG maintains its crystal structure up to the top of the layer. The surface of the YIG film is flat with a roughness less than 1 nm. This clean interface is of advantage for efficient spin injection. [@qiu] XRR measurement on a bare YIG film yielded a YIG surface roughness of $(0.008\pm 0.002)$ nm. In contrast, the GGG$\mid$YIG interface roughness was $(0.6\pm0.1)$ nm. The fact that the roughness at the GGG$\mid$YIG interface was many times larger than that at the YIG surface can be ascribed to substrate damage caused by on-axis sputtering. Figure \[fig2\](f) shows the ferromagnetic resonance (FMR) derivative absorption spectrum $dI/dH$ of a 96-nm-thick YIG film measured in a microwave cavity at $P_{\text{MW}}=1$ mW. It consists of a single Lorentzian peak derivative with a peak-to-peak linewidth $W=0.38$ mT. This corresponds to a single FMR mode with damping proportional to the linewidth. The linewidth in a broad set of samples varied in the range of $0.4-0.6$ mT. These values are among the lowest reported on sputtered YIG films.[@kang; @wang1; @mwu] The effective saturation magnetization $M_{\text{eff}}$ was determined from the dependence of the FMR field on the direction of the static magnetic field with respect to the sample plane.[@ando] The obtained value $M_{\text{eff}}=(103\pm4)$ kA/m is lower than the saturation magnetization value for bulk YIG crystal (140 kA/m).[@spin-waves] The decrease in the saturation magnetization might be a result of a deficiency in Fe atoms indicated by the off-stoichiometry. Figure \[fig2\](g) shows the frequency $f$ dependence of the peak-to-peak linewidth $W$ measured on a 83-nm-thick YIG film using a coplanar waveguide. Using a linear fit [@spin-waves] $$W=W_0+\frac{4\pi}{\sqrt{3}}\frac{\alpha}{\gamma}f \label{linewidth}$$ with gyromagnetic ratio $\gamma=1.78\times10^{11}$ T$^{-1}$s$^{-1}$ determined from the frequency dependence of the FMR field,[@iguchi] we obtain a damping constant $\alpha=(7.0\pm1.0)\times 10^{-4}$. This value is more than ten times larger than the value for bulk single crystals ($3\times 10^{-5}$),[@sparks] but is slightly smaller than other values reported on films prepared by sputtering [@wang4; @mwu] and only three times higher than values reported on LPE films.[@castel; @pirro] The increase in the damping constant is probably due to two-magnon scattering on defects in the film.[@arias] Spin pumping and spin Seebeck effect ------------------------------------ The results of the SP experiment are shown in Fig. \[fig3\]. Figure \[fig3\](a) compares the integrated FMR spectra of the plain YIG film and the YIG$\mid$Pt bilayer measured on one YIG sample prior to and after Pt coating. The linewidth in the YIG$\mid$Pt bilayer increases on average by 30% as compared to the linewidth in the bare YIG layer. This corresponds to enhanced damping of the magnetization precession in the YIG$\mid$Pt sample. This enhancement is caused by the transfer of spin angular momentum to conduction electrons in Pt near the YIG$\mid$Pt interface, indicating successful spin injection. ![\[fig3\] Results of the spin pumping measurement on a YIG(96 nm)$\mid$Pt(14 nm) bilayer at $P_{\text{MW}}=1$ mW ($\mu_0 h_{\text{ ac}}=0.01$ mT). (a) The integrated FMR spectrum of a YIG sample before and after Pt coating (YIG and YIG$\mid$Pt, respectively). (b) ISHE voltage signal measured on the Pt layer at $\theta_H=-90^{\circ}$ overlaid with a Lorentzian fit. (c), (d) FMR derivative spectra \[(c)\] and spectral shapes of the voltage signals \[(d)\] at selected $\theta_H$ values. (e) Angle dependence of the peak value of the ISHE voltage (“data”) overlaid with the calculated dependence (“calc”). The inset shows the definition of $\theta_H$. ](fig3.jpg){width="8.5cm"} Simultaneously with the FMR peak of the ferromagnet, a voltage signal appears across the Pt layer, as shown in Fig. \[fig3\](b). The spectral shape of the voltage signal is a Lorentzian with the same centre and full-width-at-half-maximum as the FMR spectrum of the YIG \[Fig. \[fig3\](a)\]. This is an expected behaviour in ISHE, where the generated voltage at field $H$ is proportional to the microwave absorption intensity $I(H)$.[@ando] The fact that the detected voltage signal is due to ISHE is confirmed by the $\theta_H$ dependence, where $\theta_H$ is the angle between the surface normal and the magnetic field \[see inset of Fig. \[fig3\](e)\]. The FMR derivative spectra and the spectral shapes of the voltage signals at selected values of $\theta_H$ are shown in Figs. \[fig3\](c) and (d), respectively. The spectral shape of the voltage copies the shape of $I(H)$ even when the magnetic field is tilted out of the sample plane ($\theta_H=\pm 45^\circ$). The sign of the voltage reverses by reversing the direction of the magnetic field and the signal vanishes when the magnetic field is perpendicular to the sample surface ($\theta_H=0^{\circ}$). This is a signature of ISHE, where the electromotive force is generated along the vector product of the spin polarization and the spin current, $\mathbf{E}_{\text{ISHE}}\propto \mathbf{j}_{s} \times \boldsymbol{\sigma}$. [@saitoh; @ando] Figure \[fig3\](e) shows the $\theta_H$ dependence of the peak value of the voltage signal. The black curve is the expected $\theta_H$ dependence of the ISHE voltage calculated following the procedure in Ref. . The magnitude of the ISHE voltage is proportional to the injected spin current, $V_{\text{ISHE}} \propto j_s\sin \theta_M$, where the spin current magnitude $j_s$ is given by Eq. (12) in Ref. $$j_s=\frac{g_r^{\uparrow \downarrow} \gamma^2 (\mu_0 h_{\text{ac}})^2 \hbar \left[ {\mu_0 M_{\text{eff}}} \gamma \sin^2 \theta_M + \sqrt{({\mu_0 M_{\text{eff}}})^2 \gamma^2 \sin^4 \theta_M + 4 \omega^2} \right] }{8\pi\alpha^2 \left[ ({\mu_0 M_{\text{eff}}})^2 \gamma^2 \sin^4 \theta_M + 4\omega^2 \right] }. \label{sc}$$ Here $\theta_M$ is the angle between the magnetization vector and the surface normal, and $g_r^{\uparrow \downarrow}$ the real part of the spin mixing conductance. The relation between $\theta_H$ and $\theta_M$ is determined by the resonance condition $ \left( \omega/\gamma \right)^2= \left[ \mu_0 H_{\text{R}} \cos (\theta_H-\theta_M) -{\mu_0 M_{\text{eff}}} \cos 2\theta_M \right] \times \left[ \mu_0 H_{\text{R}} \cos (\theta_H-\theta_M) -{\mu_0 M_{\text{eff}}} \cos^2 \theta_M \right] $ \[Eq. (9) in Ref. \] and the static equilibrium condition $ 2\mu_0 H\sin(\theta_H-\theta_M)+{\mu_0 M_{\text{eff}}} \sin 2 \theta_M=0 $ \[Eq. (6) in Ref. \]. To numerically calculate Eq. , we used $\omega=5.94\times10^{10}$ s$^{-1}$, $\gamma=1.78\times10^{11}$ s$^{-1}$T$^{-1}$ and [$M_{\text{eff}}=103$ kA/m]{}. The result of the calculation is in very good agreement with the data, providing another piece of evidence that the observed voltage is due to ISHE. Figure \[fig4\] shows the results of the longitudinal SSE measurement. Figure \[fig4\](a) gives the magnetic field $\mu_0H$ dependence of the voltage signal $V$ measured on the Pt layer for selected values of temperature difference $\Delta T$ between the bottom and the top of the sample. We observed a voltage signal whose sign is reversed by reversing the direction of the magnetic field. Upon increasing the magnetic field, the magnitude of the signal increases monotonically until reaching a saturation value. This $\mu_0H$ dependence of $V$ reflects the magnetization curve of YIG.[@kikkawa] The saturation value of the voltage increases with increasing temperature gradient. No signal was observed for $\Delta T=0$ K. As shown in Fig. \[fig4\](b), the magnitude of the voltage at $\mu_0 H=30$ mT is linear in $\Delta T$. This behaviour is consistent with ISHE induced by SSE, where the spin current generated across the YIG$\mid$Pt interface is proportional to the temperature gradient $\boldsymbol{\nabla} T$.[@uchidaAPL] ![\[fig4\] Results of the SSE measurement on a YIG(83 nm)$\mid$Pt(10 nm) bilayer. (a) Magnetic field $\mu_0 H$ dependence of the voltage signal $V$ on the Pt layer measured for different values of temperature difference $\Delta T$ across the GGG$\mid$YIG$\mid$Pt sample in the longitudinal SSE configuration. (b) $\Delta T$ dependence of the voltage magnitude at $\mu_0H=30$ mT. ](fig4.jpg){width="8.5cm"} Spin injection efficiency ------------------------- The normalized value of the ISHE electromotive force observed in the SP experiment is $E_{\text {ISHE}}/(\mu_0 h_{\text {ac}})=(150\pm30)$ $\mu$V/(mm$\cdot$mT). This is a few times higher than the value reported on a 4.5 $\mu$m-thick LPE film, $E_{\text {ISHE}}/(\mu_0 h_{\text {ac}})=39$ $\mu$V/(mm$\cdot$mT), measured at the same equipment, [@qiu] and comparable with values reported for LPE films of 1.2-$\mu$m thickness \[160 $\mu$V/(mm$\cdot$mT) in Ref. \]. As for the ISHE voltage in the SSE measurement, using the experimental value $V=(5.6 \pm 1.2)$ $\mu$V at $\Delta T=10$ K, we obtain a normalized voltage $V\times L_T/L_V =(0.47\pm0.10)$ $\mu$V. This is also of the same order as the value for YIG prepared by LPE (1 $\mu$V in Ref. for a 4.5-$\mu$m-thick film). Finally, we estimate the spin mixing conductance at the YIG$\mid$Pt interface. The efficiency of the transfer of spin angular momentum at the F$\mid$N interface is described by the real part of the spin-mixing conductance $g_r^{\uparrow \downarrow}$,[@tserkovnyak; @kajiwara; @xia; @jia] which is given by [@tserkovnyak; @mosendz; @burrowes] $$g_r^{\uparrow \downarrow}= \frac{4\pi {M_{\text{s}}} d_F}{g \mu_{\rm B}} \frac{\sqrt{3}\gamma}{2\omega} \left( W_{F/N}-W_{F} \right). \label{gr}$$ Here, $g$ is the $g$-factor, $\mu_{\rm B}= e \hbar/(2m_e)=9.27\times10^{-24}$ J$\cdot$T$^{-1}$ the Bohr magneton, $M_\text{s}$ the saturation magnetization; and $W_F$ and $W_{F/N}$ are the peak-to-peak linewidth of the FMR spectrum in the bare ferromagnetic film and in the F$\mid$N bilayer, respectively. Using $g=2.12$, $M_\text{s}\approx M_{\text{eff}}=103$ kA/m, $\omega/\gamma=0.334$ T, $d_F=96$ nm, $W_F=(0.40\pm0.03)$ mT and $W_{F/N}=(0.52\pm0.02)$ mT, we obtain a spin-mixing conductance $g_r^{\uparrow \downarrow}=(2.0\pm 0.2)\times10^{18}$ m$^{-2}$. This value is of the same order as those in other reports on the YIG$\mid$Pt interface, e.g. $g_r^{\uparrow \downarrow}=1.3\times10^{18}$ m$^{-2}$ in Ref. . Thus, both in SP and in SSE, we have obtained spin injection efficiencies comparable to those reported at LPE-made-YIG$\mid$Pt bilayer samples. This result suggests that defects inside the YIG film do not significantly affect the transfer of spin angular momentum at the interface with Pt. The high spin injection efficiency is promoted by the presence of a regular garnet structure in the vicinity of the interface with Pt as well as the clean interface, as observed by TEM imaging. It is worth noting that based on Eqs. and , a decrease in magnetization from 140 kA/m to 103 kA/m should lead to a 28 % decrease in the injected spin current. A corresponding suppression of the inverse spin Hall voltage should be observed. However, the errors in the spin pumping and the SSE voltage measurements were 20 % and 21 %, respectively. This degree of error does not allow to discuss the effect of the decreased magnetization on spin pumping. To conclude, we estimate the spin Hall angle of Pt from the obtained data. The injected spin current determined from Eq. is $j_s=5.3\times10^{-10}$ J/m$^{2}$, where we have used $g_{r}^{\uparrow \downarrow}=2.0\times 10^{18}$ m$^{-2}$, $\mu_0 h_{\text{ac}}=0.01$ mT, $\alpha=7.0\times 10^{-4}$, $\gamma=1.78\times 10^{11}$ T$^{-1}$s$^{-1}$, $\omega=5.94\times 10^{10}$ s$^{-1}$, $M_{\text{eff}}=103$ kA/m, and $\theta_H=\theta_M=-90$ $^\circ$. The peak value of the inverse spin Hall voltage is given by[@ando] $$V_{\text{ISHE}}=\frac{d_E\theta_{\text{SHE}}\lambda_N \tanh(d_N/2\lambda_N)}{d_N\sigma_N} \left( \frac{2e}{\hbar} \right)j_s,$$ where $d_E$ is the distance of the electrodes, $\theta_{\text{SHE}}$ the spin Hall angle, $\lambda_N$ the spin diffusion length in Pt, $d_N$ the thickness of the Pt layer, and $\sigma_N$ the conductivity of Pt. Using $V_{\text{ISHE}}=3.9$ $\mu$V, $d_E=2.6$ mm, $d_N=14$ nm, $\sigma_N=3.1\times10^6$ $\Omega^{-1}$m$^{-1}$, and $j_s=5.3\times10^{-10}$ J/m$^{2}$, we obtain $\theta_{\text{SHE}}=0.029$ or $0.007$ for Pt spin diffusion length $\lambda_N=1.4$ nm,[@lliu] or 10 nm,[@vila] respectively. Both values are within the range of spin Hall angles reported for Pt. Conclusions =========== In summary, we have prepared YIG films by the sputtering method and investigated their structural and microwave properties as well as spin current generation from these films. The results show that the presented fabrication method, consisting of sputtering at room temperature and post-annealing in air, provides epitaxial YIG films with thickness below 100 nm which have excellent microwave properties in spite of defects in the structure. The spin injection efficiency observed in spin pumping and in the spin Seebeck effect is comparable with that for high-quality films prepared by liquid phase epitaxy. The above preparation of garnet films is relatively straightforward and the technological requirements modest. Moreover, this method offers a possibility to control the YIG thickness at the nanometer scale. These results are of potential use in spintronics research. We thank S. Ito from Analytical Research Core for Advanced Materials, Institute for Materials Research, Tohoku University, for performing transmission electron microscopy on our samples. 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--- author: - \| A. Caldwell\ Max Planck Institute for Physics, Munich\ E-mail: - \| F. Keeble\ University College London\ E-mail: - \| E. Simpson Dore\ University College London\ E-mail: - \| [^1]\ University College London\ E-mail: title: 'Physics case of the very high energy electron–proton collider, VHEeP' --- Introduction ============ Simulations of proton-driven plasma wakefield acceleration [@pdpwa] have shown that acceleration of electrons to TeV scales over km scales is possible. The AWAKE experiment [@awake] at CERN is investigating and measuring the effect for the first time. Assuming the current and future investigations at AWAKE are successful, possible applications of the AWAKE acceleration scheme are being considered in which an electron beam is accelerated to high energies over short distances. Making strong use of CERN infrastructure, a very high energy electron–proton (VHEeP) collider [@vheep] has been proposed with an electron energy of $E_e = 3$TeV and $E_p =7$TeV, leading to a centre-of-mass energy of about 9TeV. The proposed scheme for VHEeP leads to a baseline centre-of-mass energy a factor of 30 times higher than HERA and a corresponding extension to low Bjorken $x$ of a factor of 1000, with a similar extension to high photon virtuality, $Q^2$, depending on the luminosity. The layout for VHEeP leads to modest luminosities, with an integrated luminosity during the collider’s lifetime of $10 - 100$pb$^{-1}$. Studies showed that the nature of the strong force and its explanation within QCD is not well understood at values of Bjorken $x$ down to $10^{-8}$, corresponding to $Q^2$ of about 1 GeV$^2$. It is expected that effects of saturation of the structure of the proton and hadronic cross sections will be observed given the unphysical nature of various models when extrapolated to VHEeP energies. This can be studied in inclusive deep inelastic scattering events, by measuring the total photon–proton cross section, as well as tagging vector mesons. Given the rise of the cross sections to low Bjorken $x$, even with modest luminosities, numerous events are expected in all of these reactions. Exotic, high-$Q^2$ physics can also be studied at VHEeP, with particular sensitivity to leptoquark production up to the kinematic limit of 9TeV, as well as quark substructure. The physics case of VHEeP developed in the original publication [@vheep] was briefly described in the previous paragraph, with the rest of the proceedings concentrating on developments since. In Section \[sec:recent\], recent developments are discussed, in particular the use of modern Monte Carlo tools and comparison with the previous studies. In Section \[sec:workshop\], a workshop dedicated to the particle physics case for VHEeP is briefly summarised. The proceedings are concluded in Section \[sec:summary\]. Recent studies and Monte Carlo development {#sec:recent} ========================================== Studies published previously [@vheep] used the [Ariadne]{} [@ariadne] Monte Carlo programme to investigate $ep$ collisions at 9TeV. The [Ariadne]{} Monte Carlo programme is written in Fortran and was used standalone and so a more modern framework and programmes were investigated. The [Ariadne]{} predictions were also made using the CTEQ2L [@cteq2l] proton parton density functions (PDFs); more modern Monte Carlo programmes are interfaced to LHAPDF [@lhapdf] and so allow the use of more recent PDFs. The [Rivet]{} toolkit (Robust Independent Validation of Experiment and Theory) [@rivet] is a system for validation of Monte Carlo generators and so once a routine has been developed to investigate VHEeP collisions, then several Monte Carlo programmes can be used; the Monte Carlo programmes can also be compared to existing data and so their model(s) validated. Within the Rivet framework, simulations of high energy $ep$ collisions were performed with the [Rapgap]{} [@rapgap] and [Herwig]{} [@herwig] Monte Carlo programmes. These were run with more modern proton PDFs than used for [Ariadne]{}, namely CT10 [@ct10], MRST2004 FF4 LO [@mrst2004] and NNPDF 3.0 LO [@nnpdf]. The programmes could also simulate events at very low Bjorken $x$, whereas the results from previous simulations with [Ariadne]{} were restricted to a minimum Bjorken $x$ of $10^{-7}$. The results of the simulations and comparison to previous results are shown in Fig. \[fig:q2-y\]. The variables photon virtuality, $Q^2$, and inelasticity, $y$, are shown for [Ariadne]{} (Fig. \[fig:q2-y\](a,c)) and [Rapgap 3.3]{} and [Herwig 7.0]{} (Fig. \[fig:q2-y\](b,d)). The shapes of the three simulations are similar versus $Q^2$, most clearly shown for [Rapgap 3.3]{} and [Herwig 7.0]{}. The shapes are also similar versus $y$, although a difference between [Rapgap 3.3]{} and [Herwig 7.0]{} can be seen. ![Kinematic distributions for (a,b) photon virtuality, $Q^2$, and (c,d) inelasticity, $y$, for a small sample of events at VHEeP. (a,c) show 100k events generated [@vheep] using the [Ariadne]{} Monte Carlo programme with the CTEQ2L proton PDFs for $Q^2>1$GeV$^2$, $x>10^{-7}$ and $W^2> 5$GeV$^2$. (b,d) show 500k events generated using either [Rapgap 3.3]{} (orange line) or [Herwig 7.0]{} (blue line) Monte Carlo programmes with the NNPDF 3.0 LO proton PDFs for $Q^2>1$GeV$^2$.[]{data-label="fig:q2-y"}](VHEeP-MW-Plots_1-Q2 "fig:"){width="46.00000%"} (-30,150)[(0,0)\[tl\][(a)]{}]{} (195,150)[(0,0)\[tl\][(b)]{}]{} ![Kinematic distributions for (a,b) photon virtuality, $Q^2$, and (c,d) inelasticity, $y$, for a small sample of events at VHEeP. (a,c) show 100k events generated [@vheep] using the [Ariadne]{} Monte Carlo programme with the CTEQ2L proton PDFs for $Q^2>1$GeV$^2$, $x>10^{-7}$ and $W^2> 5$GeV$^2$. (b,d) show 500k events generated using either [Rapgap 3.3]{} (orange line) or [Herwig 7.0]{} (blue line) Monte Carlo programmes with the NNPDF 3.0 LO proton PDFs for $Q^2>1$GeV$^2$.[]{data-label="fig:q2-y"}](Q2_comparison "fig:"){width="52.00000%"} ![Kinematic distributions for (a,b) photon virtuality, $Q^2$, and (c,d) inelasticity, $y$, for a small sample of events at VHEeP. (a,c) show 100k events generated [@vheep] using the [Ariadne]{} Monte Carlo programme with the CTEQ2L proton PDFs for $Q^2>1$GeV$^2$, $x>10^{-7}$ and $W^2> 5$GeV$^2$. (b,d) show 500k events generated using either [Rapgap 3.3]{} (orange line) or [Herwig 7.0]{} (blue line) Monte Carlo programmes with the NNPDF 3.0 LO proton PDFs for $Q^2>1$GeV$^2$.[]{data-label="fig:q2-y"}](VHEeP-MW-Plots_1-y "fig:"){width="46.00000%"} (-30,150)[(0,0)\[tl\][(c)]{}]{} (195,150)[(0,0)\[tl\][(d)]{}]{} ![Kinematic distributions for (a,b) photon virtuality, $Q^2$, and (c,d) inelasticity, $y$, for a small sample of events at VHEeP. (a,c) show 100k events generated [@vheep] using the [Ariadne]{} Monte Carlo programme with the CTEQ2L proton PDFs for $Q^2>1$GeV$^2$, $x>10^{-7}$ and $W^2> 5$GeV$^2$. (b,d) show 500k events generated using either [Rapgap 3.3]{} (orange line) or [Herwig 7.0]{} (blue line) Monte Carlo programmes with the NNPDF 3.0 LO proton PDFs for $Q^2>1$GeV$^2$.[]{data-label="fig:q2-y"}](y_comparison "fig:"){width="52.00000%"} A further comparison between [Rapgap 3.3]{} and [Herwig 7.0]{} and between the central values of different proton PDFs, CT10 LO, MRST2004 FF4 LO and NNPDF3.0 LO, using [Rapgap 3.3]{} is shown for the cross section versus Bjorken $x$ in Fig. \[fig:x\]. The predictions from [Rapgap 3.3]{} with different proton PDFs are very similar for Bjorken $x > 10^{-5}$, where they are well constrained by current data. The PDFs deviate from each other to lower Bjorken $x$, where they are extrapolated under different assumptions. Given the lack of data at very low Bjorken $x$, the true uncertainty is larger than the spread in these predictions, however, these predictions can be used as representative event simulations. Differences between [Herwig 7.0]{} and [Rapgap 3.3]{}, using the same proton PDF, are observed over the full Bjorken $x$ range, even in the region well constrained by data. This indicates differences in the simulations which should be investigated further. It should be noted that the Monte Carlo simulations can be validated with HERA data within [Rivet]{}, although the number of analyses from H1 and ZEUS number just a few at the time of writing. It is hoped this will improve in the future and such differences between simulations can be understood. ![Comparison of the Bjorken $x$ distribution when using either [Rapgap 3.3]{} (orange line) or [Herwig 7.0]{} (dark blue line) Monte Carlo programmes with the NNPDF 3.0 LO proton PDFs and also the [Rapgap 3.3]{} Monte Carlo programme with either MRST2004 FF4 LO or CT10 LO proton PDFs.[]{data-label="fig:x"}](x_comparison_PDFS){width="70.00000%"} VHEeP workshop {#sec:workshop} ============== A workshop [@vheep-workshop], “Prospects for a very high energy $ep$ and $eA$ collider”, was held soon after the DIS 2017 workshop. The workshop covered two full days, held in the Max Planck Institute for Physics, Munich, with experts in the field invited to further explore the particle physics case for VHEeP. The VHEeP baseline parameters with ranges which physics studies should consider are the following: - collisions are nominally electron–proton, but electron–ion (e.g. electron–lead) collisions are foreseen; - there should be the possibility of accelerating positrons and having positron–proton collisions; - polarisation of the lepton beam should be possible and polarised protons should be considered; - the nominal energies are $E_e = 3$TeV and $E_p = 7$TeV, giving a centre-of-mass energy of $\sqrt{s} = 9.2$TeV; - the electron beam energy can be varied in the range 0.1 to 5TeV, giving $\sqrt{s} = 1.7 - 11.8$TeV; - the integrated luminosity is in the range 10 – 1000pb$^{-1}$. If different parameters are required, this will need to be considered with the constraints imposed by the acceleration scheme. A brief summary of the workshop is given here and the interested reader is referred to the slides on the workshop web-page [@vheep-workshop]. The workshop started with an introduction to plasma wakefield acceleration and its application to higher energy physics (Caldwell), a report on the status of AWAKE (Muggli) and an introduction to VHEeP (Wing). Of particular note were the presentations on QCD and hadronic cross sections (Bartels, Mueller, Schildknecht and Stodolsky) in which theoretical expectations show that saturation will be observed at VHEeP and will also be at a scale where QCD calculations are perturbative. Low-$x$ physics was related to other areas, including cosmic rays (Stasto) and black holes and gravity (Erdmenger), as well as being used as a testing-bed for new physics descriptions (Dvali and Kowalski). The needs of polarisation and electron–ion physics were discussed (Aschenauer and Mäntysaari) as well as what has been learnt from the HERA data at low $x$ (Myronenko). Finally, the status of Monte Carlo simulations for $ep$ and $eA$ physics was presented (Plätzer) as well as the general kinematics and challenges faced by the detector as well as the use of simulations (Keeble). Summary {#sec:summary} ======= These proceedings have highlighted recent progress on studies for a very high energy electron–proton (VHEeP) collider since the proposal was published. Simulations have been extended and can be used to investigate the physics potential and aid design of the detector and accelerator. A workshop to further investigate the physics potential of electron–proton and electron–ion collisions at very high energies was held in June and was here briefly summarised. It should also be investigated how VHEeP could fit into the future world particle physics programme and complement other similar projects investigating deep inelastic scattering. [99]{} A. Caldwell et al., Nature Phys. [5]{} (2009) 363;\ A. Caldwell and K. Lotov, Phys. Plasmas [18]{} (2011) 103101. AWAKE Coll., R. Assmann et al., Plasma Phys. Control. Fusion [56]{} (2014) 084013;\ AWAKE Coll., A. Caldwell et al., Nucl. Instrum. Meth. [A 829]{} (2016) 3;\ AWAKE Coll., E. Gschwendtner et al., Nucl. Instrum. Meth. [A 829]{} (2016) 76; A. Caldwell and M. Wing, Eur. Phys. J. [C 76]{} (2016) 463. L. Lönnblad, Comp. Phys. Comm. [71]{} (1992) 15; L. Lönnblad, Z. Phys. [C 65]{} (1995) 285. CTEQ Coll., J. Botts et al., Phys. Lett. [B 304]{} (1993) 159. A. Buckley et al., Eur. Phys. J. [C 75]{} (2015) 132;\ D. Bourilkov, R.C. Group and M.R. Whalley, arXiv:hep-ph/0605240;\ M.R. Whalley, D. Bourilkov and R.C. Group, arXiv:hep-ph/0508110. A. Buckley et al., Comp. Phys. Comm. [184]{} (2013) 2803. hepforge site, [https://rapgap.hepforge.org]{};\ H. Jung, Comp. Phys. Comm. [86]{} (1995) 147. J. Bellm et al., Eur. Phys. J. [C 76 ]{} (2016) 196. NNPDF Coll., R.D. Ball et al., JHEP [1504]{} (2015) 040. H.-L. Lai et al., Phys. Rev. [D 82]{} (2010) 074024. A.D. Martin et al., Phys. Lett. [B 636]{} (2006) 259. Workshop on “Prospects for a very high energy $ep$ and $eA$ collider”, MPI Munich, June 2017, [https://indico.mpp.mpg.de/event/5222/overview]{} [^1]: Also at DESY.
--- abstract: 'If a compact closed category has finite products or finite coproducts then it in fact has finite biproducts, and so is semi-additive.' author: - \| Robin Houston[^1]\ School of Computer Science, University of Manchester bibliography: - 'cs.bib' title: Finite Products are Biproducts in a Compact Closed Category --- Introduction ============ Compact closed categories with biproducts have recently attracted renewed attention from theoretical computer scientists, because of their role in the abstract approach to quantum information initiated by Abramsky and Coecke [@AbramskyCoecke]. Perhaps surprisingly, it seems to have gone unnoticed that finite products or coproducts in a compact closed category necessarily carry a biproduct structure. Here we prove that this is so. In fact we prove a more general result, viz: Let ${\mathbb{C}}$ be a monoidal category with finite products and coproducts, and suppose that for every object $A \in {\mathbb{C}}$, the functor $A{\otimes}\mathord-$ preserves products and the functor $\mathord-{\otimes}A$ preserves coproducts. Then ${\mathbb{C}}$ has finite biproducts. A category with finite biproducts is necessarily semi-additive, i.e. enriched over commutative monoids. In other words, each homset has the structure of a commutative monoid, and composition preserves the commutative monoid structure. The converse is also true: a semi-additive category with finite products or coproducts in fact has finite biproducts. Therefore an equivalent statement of our conclusion would be that ${\mathbb{C}}$ is semi-additive. The gist of the argument is as follows. Let ${\mathbb{C}}$ be as in the statement of the proposition. The object $0{\otimes}1$ is initial because $\mathord-{\otimes}1$ preserves initiality, and terminal because $0{\otimes}\mathord-$ preserves terminality. So it is a zero object. The binary case is similar, though more intricate. Let $A$, $B$, $C$, $D\in{\mathbb{C}}$ and consider the object $(A+B){\otimes}(C\times D)$. Since the ${\otimes}$ distributes over both the $+$ and the $\times$ in this expression, it may be multiplied out as either $$\label{mo1} (A{\otimes}C \times A{\otimes}D)+(B{\otimes}C \times B{\otimes}D)$$ or $$\label{mo2} (A{\otimes}C + B{\otimes}C)\times(A{\otimes}D + B{\otimes}D),$$ hence (\[mo1\]) is isomorphic to (\[mo2\]). Letting $C=D=I$ shows that $$\label{isom} A^2+B^2\cong(A+B)^2,$$ and it may be verified (Lemmas \[l1.1\]–\[l1.2\]) that the canonical natural map $$A^2+B^2\to(A+B)^2,$$ denoted $t_{A,B}$ below, is equal to the left-to-right direction of (\[isom\]). It follows that $t_{A,B}$ is invertible. From this we derive, via Lemma \[l1.3\], that the natural map $A+B\to A\times B$ is also invertible, which implies the desired conclusion. The remainder of this paper contains the detailed proof. The next section recalls the basic facts about finite products and coproducts, and some simple properties of compact closed categories: it will not tax the experienced reader, who may prefer to skip directly to §\[s-main\]. Background ========== This short paper uses only elementary ideas of category theory, which we briefly recall so as to fix our notation. In a category with finite products, we denote the given terminal object $1$, and suppose that for every pair $A$, $B$ of objects there is a given product cone $(\pi_1: A\times B\to A, \pi_2: A\times B\to B)$. For any pair of maps $f: X\to A$, $g: Y\to B$, we denote their pairing as $\langle f,g\rangle: X\to A\times B$, i.e. $\langle f,g\rangle$ is the unique map for which $\pi_1\after\langle f,g\rangle=f$ and $\pi_2\after\langle f,g\rangle=g$. Given $f: A\to B$ and $g: C\to D$, we write $f\times g$ for the map $$\langle f\after\pi_1, g\after\pi_2\rangle: A\times C\to B\times D.$$ Note that this definition makes $\times$ into a functor, in such a way that $\pi_1$ and $\pi_2$ constitute natural transformations. For example $\pi_1\after(f\times g) = \pi_1\after\langle f\after\pi_1, g\after\pi_2\rangle = f\after\pi_1$. A functor $F$ is said to preserve products if the image under $F$ of a product cone is always a product cone (not necessarily the chosen one). We take it to include the nullary case also, i.e. the image of a terminal object must be terminal. If the categories ${\mathbb{C}}$ and ${\mathbb{D}}$ have finite products and $F:{\mathbb{C}}\to{\mathbb{D}}$ preserves products then the morphism $$F(A\times B) \rTo^{\langle F\pi_1, F\pi_2\rangle} FA\times FB$$ is invertible. The case of coproducts is dual to the above. In a category that has finite coproducts, we assume that there is an initial object $0$ and that for every pair of objects $A$, $B$, there is a given coproduct cocone $(i_1: A\to A+B, i_2: B\to A+B)$. Given maps $f: A\to Y$ and $g: B\to Y$, we write their co-pairing as $$[f,g]: A+B\to Y;$$ if ${\mathbb{C}}$ and ${\mathbb{D}}$ have finite coproducts and $F:{\mathbb{C}}\to{\mathbb{D}}$ preserves coproducts then the map $$FA + FB \rTo^{[Fi_1, Fi_2]} F(A+B)$$ is invertible. Now suppose we are in a category that has both finite products and finite coproducts. A morphism $$f: A+B \to C\times D$$ is determined by the four maps $$\begin{array}{l@{\;}l@{\qquad}l@{\;}l} f_{11} := \pi_1\after f\after i_1:& A\to C, & f_{12} := \pi_1\after f\after i_2:& B\to C\\ f_{21}:= \pi_2\after f\after i_1:& A\to D, & f_{22}:= \pi_2\after f\after i_2:&B\to D, \end{array}$$ since $f = [\langle f_{11}, f_{21} \rangle,\langle f_{12}, f_{22}\rangle] = \langle[f_{11}, f_{12}], [f_{21}, f_{22}]\rangle$. We refer to this as the matrix representation of $f$, and write it as $$f = \Bigl[\begin{array}[c]{cc} f_{11} & f_{12} \\ f_{21} & f_{22} \end{array}\Bigr].$$ A technique that is used several times below is to check that two maps are equal by calculating and comparing their matrix representations. There are several equivalent ways of defining what it means for a category to have finite biproducts. The one most convenient for our purposes is as follows (see Exercise VIII.2.4 of Mac Lane [@MacLane]). A category ${\mathbb{C}}$ has finite biproducts if it has finite products and finite coproducts, such that: - the unique morphism $0\to 1$ is invertible, thus there is a (unique) zero map $0_{A,B}: A \to 1 \cong 0 \to B$ between any objects $A$ and $B$, and - the morphism $$\Bigl[\begin{array}{cc}1_A&0_{B,A}\\0_{A,B}&1_B\end{array}\Big]: A+B \to A\times B$$ is invertible for all $A$ and $B$ in ${\mathbb{C}}$. Compact closed categories were first defined (almost in passing) by Kelly [@MVFC], and later studied in depth by Kelly and Laplaza [@KL]. The reader may consult either of those references for the precise definition. For the purposes of this paper, it suffices to know that a compact closed category is a monoidal category $({\mathbb{C}},{\otimes},I)$ that has – among other things – the following two properties: - ${\mathbb{C}}$ is self-dual, i.e. ${\mathbb{C}}$ is equivalent to ${\mathbb{C}}^{\mathrm op}$, - for every object $A\in{\mathbb{C}}$, the functors $A{\otimes}\mathord-$ and $\mathord-{\otimes}A$ have both a left and a right adjoint. Examples include the category $\mathrm{Rel}$ of sets and relations, with the tensor as cartesian product, and the category $\mathrm{FinVect}$ of finite-dimensional vector spaces, with the usual tensor product of vector spaces. Main Result {#s-main} =========== Our main result is as follows. \[theorem\] Let ${\mathbb{C}}$ be a compact closed category. If ${\mathbb{C}}$ has finite products (or coproducts) then it has finite biproducts. We shall deduce the theorem from a somewhat more general proposition: \[prop\] Let ${\mathbb{C}}$ be a monoidal category with finite products and coproducts, and suppose that for every object $A \in {\mathbb{C}}$, the functor $A{\otimes}\mathord-$ preserves products and the functor $\mathord-{\otimes}A$ preserves coproducts. Then ${\mathbb{C}}$ has finite biproducts. The nullary case may be dispensed with immediately: The functor $0{\otimes}\mathord{-}$ preserves products, thus $0{\otimes}1$ is terminal. But also the functor $\mathord-{\otimes}1$ preserves coproducts, so $0{\otimes}1$ is also initial. Therefore $0$ is isomorphic to $1$, and the claim follows. From now on, we assume that we have a category that satisfies the conditions of Proposition \[prop\], and which therefore has a zero object. We shall omit the subscripts when referring to a zero map, since the type is always obvious from the context. We have no further occasion to refer explicitly to an initial object, so the symbol ‘$0$’ below always denotes a zero map. Also we shall follow the common practice of abbreviating the identity morphism $1_A$ to $A$. Since $A{\otimes}-$ preserves products, we know that for all objects $A$,$B$,$C$, the distribution map $$\langle A{\otimes}\pi_1, A{\otimes}\pi_2\rangle: A{\otimes}(B \times C) \to (A{\otimes}B) \times (A{\otimes}C)$$ is invertible, and since $-{\otimes}C$ preserves coproducts, we know that for all objects $A$,$B$,$C$, the distribution map $$[i_1{\otimes}C, i_2{\otimes}C]: (A{\otimes}C) + (B{\otimes}C) \to (A + B){\otimes}C$$ is invertible. \[l1.1\] For all objects $A_1$, $A_2$, $B_1$, $B_2$, the canonical map ($= [i_1 \times i_1, i_2 \times i_2] = \langle \pi_1 + \pi_1, \pi_2 + \pi_2\rangle$) of type $$\begin{array}{lcr} \begin{array}{l} ((A_1{\otimes}B_1) \times (A_1{\otimes}B_2))\\ \hskip 3em+ ((A_2{\otimes}B_1) \times (A_2{\otimes}B_2)) \end{array} &\!\!\to\!\!& \begin{array}{l} ((A_1{\otimes}B_1) + (A_2{\otimes}B_1)) \\ \hskip 3em\times ((A_1{\otimes}B_2) + (A_2{\otimes}B_2)) \end{array} \end{array}$$ is invertible. We’ll show that $()$ is equal to the map $y$ defined as the composite $$\begin{array}{l} ((A_1{\otimes}B_1) \times (A_1{\otimes}B_2)) + ((A_2{\otimes}B_1) \times (A_2{\otimes}B_2))\\ \hskip 3em \to (A_1{\otimes}(B_1\times B_2)) + (A_2{\otimes}(B_1\times B_2))\\ \hskip 5em \to (A_1 + A_2){\otimes}(B_1 \times B_2)\\ \hskip 7em \to ((A_1 + A_2){\otimes}B_1) \times ((A_1 + A_2){\otimes}B_2)\\ \hskip 9em \to ((A_1{\otimes}B_1) + (A_2{\otimes}B_1)) \times ((A_1{\otimes}B_2) + (A_2{\otimes}B_2)) \end{array}$$ of distribution maps and their inverses. Clearly $y$ is invertible, since it is composed of isomorphisms. Take $j$, $k \in \{1,2\}$ and consider the diagram in Fig. \[fig-1\]. [r]{}((A\_1B\_1) (A\_1B\_2))\ + ((A\_2B\_1) (A\_2B\_2)) &.& (A\_1 + A\_2)(B\_1 B\_2) &.& [r]{}( (A\_1B\_1) + (A\_2B\_1))\ ((A\_1B\_2) + (A\_2B\_2)) \ \^\~ & & (1,2)\_ [r]{} (A\_1+A\_2)\_1,\ (A\_1+A\_2)\_2 \^\~ (1,2)\_ [r]{}\ \^\~ & >[\_k]{}\ [r]{} (A\_jB\_1)\ (A\_jB\_2) & [r]{}(A\_1(B\_1 B\_2))\ +(A\_2(B\_1B\_2)) && [r]{}((A\_1+A\_2)B\_1)\ ((A\_1+A\_2)B\_2) & [r]{}(A\_1B\_k)\ + (A\_2B\_k) \ &(1,2)\_[A\_j\_1, A\_j\_2]{}\^\~ (0,4)<[\_k]{} >[i\_j]{} && &&&& (1,2)<[i\_jB\_k]{}\ A\_jB\_k && \_[A\_jB\_k]{} && A\_jB\_k (.3)[i\_j(B\_1B\_2)]{} (.7)[(A\_1+A\_2)\_k]{} All the regions commute for obvious reasons, so the outside commutes and $\pi_k\after y\after i_j = i_j\after \pi_k$. Since this is true for all $j$ and $k$, it follows that $y = ()$, as required. Given objects $A$ and $B$, let $t_{A,B}$ denote the map $$\Bigl[\begin{array}{cc} i_1\after\pi_1 & i_2\after\pi_1 \\ i_1\after\pi_2 & i_2\after\pi_2 \end{array}\Bigr]: (A\times A) + (B\times B) \to (A+B) \times (A+B)$$ \[l1.2\] For all objects $A$, $B$, the map $t_{A,B}$ is invertible. Use Lemma \[l1.1\] with $A_1=A$, $A_2=B$ and $B_1=B_2=I$, and apply the right-unit isomorphism. Given objects $A$ and $B$, let $e_{A,B}$ denote the composite $$(A\times A)+(B\times B) \rTo^{\pi_1 + \pi_2} A+B \rTo^{\langle A,0\rangle+\langle 0,B\rangle} (A\times A)+(B\times B)$$ which is clearly an idempotent that splits on $A + B$, and let $e'_{A,B}$ denote the composite $$(A+B)\times(A+B) \rTo^{[A,0]\times[0,B]} A\times B \rTo^{i_1\times i_2} (A+B) \times (A+B)$$ which is an idempotent that splits on $A\times B$. \[l1.3\] $t_{A,B}$ is a map of idempotents from $e_{A,B}$ to $e'_{A,B}$, i.e. the diagram (AA)+(BB) & \^[t\_[A,B]{}]{} & (A+B)(A+B)\ \ (AA)+(BB) & \_[t\_[A,B]{}]{} & (A+B)(A+B) commutes. We claim that both paths have the matrix representation $$\Bigl[\begin{array}{cc} i_1\after\pi_1 & 0 \\ 0 & i_2\after\pi_2 \end{array}\Bigr].$$ Consider the diagram A\^2+B\^2 & \^[\_1 + \_2]{} & A+B & \^[A,0+0,B]{} & A\^2+B\^2 & \^[t\_[A,B]{}]{} & (A+B)\^2\ &&&&>[i\_1]{}\ \ &&&\^[A,0]{} &>[\_1]{}\ A\^2 & \_[\_1]{} &A & \^A & A & \_[i\_1]{} & A+B\ &&>[i\_1]{} & \_[\[A,0\]]{}\ && >[\_1]{}\ &&&(0,5)<[\_1]{}\ &&\ &&B &\_0&A\ B\^2&(2,1)\^[\_2]{} &&&&(2,1)\^[i\_2]{}& A+B\ &(0,5)<[i\_2]{}(2,1)\_[\_1]{}&B&\^0&A&(2,1)\_[i\_1]{} (0,5)<[\_1]{}\ &&>[i\_2]{}&\_[\[A,0\]]{}\ &&A+B&&>[\_1]{}\ &&>[\_1]{}\ A\^2+B\^2&\_[t\_[A,B]{}]{}&(A+B)\^2&\_[\[A,0\]]{}&AB &\_[i\_1i\_2]{} & (A+B)\^2. We can now complete the proof of Proposition \[prop\], and hence of Theorem \[theorem\]. By Lemma \[l1.3\], we know that the map $c_{A,B} :=$ $$A+B \rTo^{\langle A,0\rangle + \langle 0,B\rangle} (A\times A) + (B\times B) \rTo^{t_{A,B}} (A+B) \times (A+B) \rTo^{[A,0] \times [0,B]} A\times B$$ is invertible with inverse $$A\times B \rTo_{i_1\times i_2} (A+B) \times (A+B) \rTo_{t_{A,B}^{-1}} (A\times A) + (B\times B) \rTo_{\pi_1 + \pi_2} A+B,$$ so it suffices to check that $c_{A,B} = [\langle A,0\rangle, \langle0,B\rangle]$. But that’s easy to check: for example, the diagram &&&\ & \_[A,0]{} & A\^2(1,1)\_[\_1]{} && A+B(1,1)\_[i\_1]{} & \_[\[A,0\]]{} &\ &&\ A+B & \_[A,0+ 0,B]{} & A\^2 + B\^2 & \_[t\_[A,B]{}]{} & (A+B)\^2 & \_[\[A,0\]]{} & AB 12 23 shows that $\pi_1\after c_{A,B}\after i_1$ is the identity on $A$, and the diagram &&&\ & \_[A,0]{} & A\^2(1,1)\_[\_2]{} && A+B(1,1)\_[i\_1]{} & \_[\[0,B\]]{} &\ &&\ A+B & \_[A,0+ 0,B]{} & A\^2 + B\^2 & \_[t\_[A,B]{}]{} & (A+B)\^2 & \_[\[A,0\]]{} & AB 12 23 shows that $\pi_2\after c_{A,B}\after i_1 = 0$. The other two cases are similar. A compact closed category is equivalent to its opposite, therefore has finite coproducts iff it has finite products. For every object $A$, the functors $A{\otimes}-$ and $-{\otimes}A$ have both a left and a right adjoint, hence preserve limits and colimits. So Proposition \[prop\] applies, in particular, to a compact closed category that has finite products (or coproducts). Final Remarks ============= It is significant that the zero object plays a crucial role in our argument. A compact closed category may very well have finite non-empty products and coproducts that are not biproducts. A simple example, due to Masahito Hasegawa, is the ordered group of integers under addition. Indeed any linearly ordered abelian group constitutes an example, for the following reason. A partially ordered abelian group may be regarded as a compact closed category: the underlying partial order is regarded as a category in the usual way, the group operation provides a symmetric tensor product, and the adjoint of an object is its group inverse. If in fact the group is linearly ordered then every non-empty finite set of elements has a minimum (which is their product) and a maximum (coproduct). This degenerate example may also be used to construct non-degenerate examples, by taking its product with $\mathrm{Rel}$, say. One last observation: Proposition \[prop\]’s requirement that ${\mathbb{C}}$ be a monoidal category is stronger than necessary. We didn’t actually need the associativity of tensor, nor the left unit isomorphism. So instead of the full monoidal structure it suffices merely to have a functor ${\otimes}: {\mathbb{C}}\times {\mathbb{C}}\to {\mathbb{C}}$ with a right unit. Acknowledgements {#acknowledgements .unnumbered} ================ I am indebted to Peter Selinger for bringing this question to my attention, and to Robin Cockett for pointing out how to simplify my original proof. I have used Paul Taylor’s diagrams package. [^1]: This work is supported by an EPSRC PhD studentship.
--- abstract: \| Human mobility is one of the key factors at the basis of the spreading of diseases in a population. Containment strategies are usually devised on movement scenarios based on coarse-grained assumptions. Mobility phone data provide a unique opportunity for building models and defining strategies based on very precise information about the movement of people in a region or in a country. Another very important aspect is the underlying social structure of a population, which might play a fundamental role in devising information campaigns to promote vaccination and preventive measures, especially in countries with a strong family (or tribal) structure. In this paper we analyze a large-scale dataset describing the mobility and the call patterns of a large number of individuals in Ivory Coast. We present a model that describes how diseases spread across the country by exploiting mobility patterns of people extracted from the available data. Then, we simulate several epidemics scenarios and we evaluate mechanisms to contain the epidemic spreading of diseases, based on the information about people mobility and social ties, also gathered from the phone call data. More specifically, we find that restricting mobility does not delay the occurrence of an endemic state and that an information campaign based on one-to-one phone conversations among members of social groups might be an effective countermeasure. author: - 'A. Lima' - 'M. De Domenico' - 'V. Pejovic' - 'M. Musolesi' bibliography: - 'd4d.bib' title: \| Exploiting Cellular Data for Disease Containment and\ Information Campaigns Strategies in Country-Wide Epidemics[^1] --- The authors thank Charlotte Sophie Mayer for useful and fruitful discussions. This work was supported through the EPSRC Grant “The Uncertainty of Identity: Linking Spatiotemporal Information Between Virtual and Real Worlds” (EP/J005266/1). [^1]: Appeared in Proceedings of NetMob 2013. Boston, MA, USA. May 2013.\
--- abstract: 'The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale $M$. A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strong convergent. A type of generalized martingales associated with $M$ are introduced and their convergence theorems are established. Some applications are also shown.' address: - 'School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China ' - 'School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China' author: - Caishi Wang - Jinshu Chen title: 'Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales' --- Introduction {#sec-1} ============ Hida’s white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion [@hida; @huang; @kuo; @obata]. In 1988, Ito [@ito] introduced his analysis of generalized Poisson functionals, which can be viewed as a theory of infinite dimensional calculus on generalized functionals of Poisson martingale. It is known that both Brownian motion and Poisson martingale are continuous-time normal martingales. There are theories of white noise analysis for some other continuous-time processes (see, e.g., [@albe; @barhoumi; @di; @hu; @lee]). Discrete-time normal martingales [@privault] also play an important role in many theoretical and applied fields. For example, the classical random walk (a special discrete-time normal martingale) is used to establish functional central limit theorems in probability theory [@dud; @rud]. It would then be interesting to develop a theory of infinite dimensional calculus on generalized functionals of discrete-time normal martingales. Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale satisfying some mild conditions. In a recent paper [@wang-chen], we constructed generalized functionals of $M$, and introduced a transform, called the Fock transform, to characterize those functionals. In this paper, we apply the Fock transform [@wang-chen] to investigate generalized functional sequences of $M$. First, by using the Fock transform, we obtain a necessary and sufficient condition for a generalized functional sequence of $M$ to be strong convergent. Then we introduce a type of generalized martingales associated with $M$, called $M$-generalized martingales, which are a special class of generalized functional sequences of $M$ and include as a special case the classical martingales with respect to the filtration generated by $M$. We establish a strong-convergent criterion in terms of the Fock transform for $M$-generalized martingales. Some other convergence criteria are also obtained. Finally we show some applications of our main results. Our one interesting finding is that for an $M$-generalized martingale, its strong convergence is just equivalent to its strong boundedness. Throughout this paper, $\mathbb{N}$ designates the set of all nonnegative integers and $\Gamma$ the finite power set of $\mathbb{N}$, namely $$\label{eq-1-1} \Gamma = \{\,\sigma \mid \text{$\sigma \subset \mathbb{N}$ and $\#(\sigma) < \infty$} \,\},$$ where $\#(\sigma)$ means the cardinality of $\sigma$ as a set. In addition, we always assume that $(\Omega, \mathcal{F}, P)$ is a given probability space with $\mathbb{E}$ denoting the expectation with respect to $P$. We denote by $\mathcal{L}^{2}(\Omega, \mathcal{F}, P)$ the usual Hilbert space of square integrable complex-valued functions on $(\Omega, \mathcal{F}, P)$ and use $\langle\cdot,\cdot\rangle$ and $\\|\cdot\\|$ to mean its inner product and norm, respectively. By convention, $\langle\cdot,\cdot\rangle$ is conjugate-linear in its first argument and linear in its second argument. Generalized functionals {#sec-2} ======================= Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale on $(\Omega, \mathcal{F}, P)$ that has the chaotic representation property and $Z=(Z_n)_{n\in \mathbb{N}}$ the discrete-time normal noise associated with $M$ (see Appendix). We define $$\label{eq-2-1} Z_{\emptyset}=1;\quad Z_{\sigma} = \prod_{i\in \sigma}Z_i,\quad \text{$\sigma \in \Gamma$, $\sigma \neq \emptyset$}.$$ And, for brevity, we use $\mathcal{L}^2(M)$ to mean the space of square integrable functionals of $M$, namely $$\label{eq-2-2} \mathcal{L}^2(M) = \mathcal{L}^2(\Omega, \mathcal{F}_{\infty}, P),$$ which shares the same inner product and norm with $\mathcal{L}^2(\Omega, \mathcal{F}, P)$, namely $\langle\cdot,\cdot\rangle$ and $\\|\cdot\\|$. We note that $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ forms a countable orthonormal basis for $\mathcal{L}^2(M)$ (see Appendix). \[lem-2-1\][@wang-z] Let $\sigma\mapsto\lambda_{\sigma}$ be the $\mathbb{N}$-valued function on $\Gamma$ given by $$\label{eq-2-3} \lambda_{\sigma}= \left\{ \begin{array}{ll} \prod_{k\in\sigma}(k+1), & \hbox{$\sigma\neq \emptyset$, $\sigma\in\Gamma$;}\\ 1, & \hbox{$\sigma=\emptyset$, $\sigma\in\Gamma$.} \end{array} \right.$$ Then, for $p>1$, the positive term series $\sum_{\sigma\in\Gamma}\lambda^{-p}_{\sigma}$ converges and moreover $$\label{eq-2-4} \sum_{\sigma\in\Gamma}\lambda^{-p}_{\sigma}\leq \exp\bigg[\sum_{k=1}^{\infty}k^{-p}\bigg]<\infty.$$ Using the $\mathbb{N}$-valued function defined by (\[eq-2-3\]), we can construct a chain of Hilbert spaces consisting of functionals of $M$ as follows. For $p\geq 0$, we define a norm $\\|\cdot\\|_p$ on $\mathcal{L}^2(M)$ through $$\label{eq-2-5} \\|\xi\\|_{p}^2=\sum_{\sigma\in \Gamma}\lambda_{\sigma}^{2p}\|\langle Z_{\sigma}, \xi\rangle\|^{2},\quad \xi \in \mathcal{L}^2(M)$$ and put $$\label{eq-2-6} \mathcal{S}_p(M) = \big\{\, \xi \in \mathcal{L}^2(M) \mid \\|\xi\\|_{p}< \infty\,\big\}.$$ It is not hard to check that $\\|\cdot\\|_{p}$ is a Hilbert norm and $\mathcal{S}_p(M)$ becomes a Hilbert space with $\\|\cdot\\|_{p}$. Moreover, the inner product corresponding to $\\|\cdot\\|_{p}$ is given by $$\label{eq-2-7} \langle \xi,\eta\rangle_p = \sum_{\sigma\in \Gamma}\lambda_{\sigma}^{2p}\overline{\langle Z_{\sigma},\xi\rangle} \langle Z_{\sigma}, \eta\rangle,\quad \xi,\, \eta \in \mathcal{S}_p(M).$$ Here $\overline{\langle Z_{\sigma},\xi\rangle}$ means the complex conjugate of $\langle Z_{\sigma},\xi\rangle$. \[lem-2-2\][@wang-chen] For each $p\geq 0$, one has $\{Z_{\sigma}\mid \sigma\in\Gamma\} \subset \mathcal{S}_p(M)$ and moreover the system $\{\lambda^{-p}_{\sigma}Z_{\sigma}\mid \sigma\in\Gamma\}$ forms an orthonormal basis for $\mathcal{S}_p(M)$. It is easy to see that $\lambda_{\sigma}\geq 1$ for all $\sigma\in \Gamma$. This implies that $\\|\cdot\\|_p \leq \\|\cdot\\|_q$ and $\mathcal{S}_q(M)\subset \mathcal{S}_p(M)$ whenever $0\leq p \leq q$. Thus we actually get a chain of Hilbert spaces of functionals of $M$: $$\label{eq-2-8} \cdots \subset \mathcal{S}_{p+1}(M) \subset \mathcal{S}_p(M)\subset \cdots \subset \mathcal{S}_1(M) \subset \mathcal{S}_0(M)=\mathcal{L}^2(M).$$ We now put $$\label{eq-2-9} \mathcal{S}(M)=\bigcap ^{\infty}_{p=0}\mathcal{S}_{p}(M)$$ and endow it with the topology generated by the norm sequence $\{\\|\cdot \\|_{p}\}_{p\geq 0}$. Note that, for each $ p\geq 0$, $\mathcal{S}_p(M)$ is just the completion of $\mathcal{S}(M)$ with respect to $\\|\cdot\\|_{p}$. Thus $\mathcal{S}(M)$ is a countably-Hilbert space [@becnel; @gelfand-vi]. The next lemma, however, shows that $\mathcal{S}(M)$ even has a much better property. \[lem-2-3\][@wang-chen] The space $\mathcal{S}(M)$ is a nuclear space, namely for any $p\geq 0$, there exists $q> p$ such that the inclusion mapping $i_{pq}\colon \mathcal{S}_{q}(M) \rightarrow \mathcal{S}_p(M)$ defined by $i_{pq}(\xi)=\xi$ is a Hilbert-Schmidt operator. For $p\geq 0$, we denote by $\mathcal{S}_p^(M)$ the dual of $\mathcal{S}_p(M)$ and $\\|\cdot\\|_{-p}$ the norm of $\mathcal{S}_p^(M)$. Then $\mathcal{S}_p^(M)\subset \mathcal{S}_q^(M)$ and $\\|\cdot\\|_{-p} \geq \\|\cdot\\|_{-q}$ whenever $0\leq p \leq q$. The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see, e.g., [@becnel] or [@gelfand-vi]). \[lem-2-4\][@wang-chen] Let $\mathcal{S}^(M)$ the dual of $\mathcal{S}(M)$ and endow it with the strong topology. Then $$\label{eq-2-10} \mathcal{S}^(M)=\bigcup_{p=0}^{\infty}\mathcal{S}_p^(M)$$ and moreover the inductive limit topology on $\mathcal{S}^(M)$ given by space sequence $\{\mathcal{S}_p^(M)\}_{p\geq 0}$ coincides with the strong topology. We mention that, by identifying $\mathcal{L}^2(M)$ with its dual, one comes to a Gel’fand triple $$\label{eq-2-11} \mathcal{S}(M)\subset \mathcal{L}^2(M)\subset \mathcal{S}^(M),$$ which we refer to as the Gel’fand triple associated with $M$. \[lem-2-5\][@wang-chen] The system $\{Z_{\sigma} \mid \sigma \in \Gamma\}$ is contained in $\mathcal{S}(M)$ and moreover it serves as a basis in $\mathcal{S}(M)$ in the sense that $$\label{eq-2-12} \xi = \sum_{\sigma \in \Gamma} \langle Z_{\sigma}, \xi\rangle Z_{\sigma}, \quad \xi \in \mathcal{S}(M),$$ where $\langle\cdot,\cdot\rangle$ is the inner product of $\mathcal{L}^2(M)$ and the series converges in the topology of $\mathcal{S}(M)$. \[def-2-1\][@wang-chen] Elements of $\mathcal{S}^(M)$ are called generalized functionals of $M$, while elements of $\mathcal{S}(M)$ are called testing functionals of $M$. Denote by $\langle\!\langle \cdot,\cdot\rangle\!\rangle$ the canonical bilinear form on $\mathcal{S}^(M)\times \mathcal{S}(M)$, namely $$\label{eq-2-13} \langle\!\langle \Phi,\xi\rangle\!\rangle = \Phi(\xi),\quad \Phi\in \mathcal{S}^(M),\, \xi\in \mathcal{S}(M),$$ where $\Phi(\xi)$ means $\Phi$ acting on $\xi$ as usual. Note that $\langle\cdot,\cdot\rangle$ denotes the inner product of $\mathcal{L}^2(M)$, which is different from $\langle\!\langle \cdot,\cdot\rangle\!\rangle$. \[def-2-2\][@wang-chen] For $\Phi \in \mathcal{S}^(M)$, its Fock transform is the function $\widehat{\Phi}$ on $\Gamma$ given by $$\label{eq-2-14} \widehat{\Phi}(\sigma) = \langle\!\langle \Phi, Z_{\sigma}\rangle\!\rangle,\quad \sigma \in \Gamma,$$ where $\langle\!\langle \cdot,\cdot\rangle\!\rangle$ is the canonical bilinear form. It is easy to to verify that, for $\Phi$, $\Psi \in \mathcal{S}^(M)$, $\Phi=\Psi$ if and only if $\widehat{\Phi}=\widehat{\Psi}$. Thus a generalized functional of $M$ is completely determined by its Fock transform. The following theorem characterizes generalized functionals of $M$ through their Fock transforms. \[lem-2-6\][@wang-chen] Let $F$ be a function on $\Gamma$. Then $F$ is the Fock transform of an element $\Phi$ of $\mathcal{S}^(M)$ if and only if it satisfies $$\label{eq-2-15} \|F(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma$$ for some constants $C\geq 0$ and $p\geq 0$. In that case, for $q> p+\frac{1}{2}$, one has $$\label{eq-2-16} \\|\Phi\\|_{-q} \leq C\bigg[\sum_{\sigma \in \Gamma}\lambda_{\sigma}^{-2(q-p)}\bigg]^{\frac{1}{2}}$$ and in particular $\Phi \in \mathcal{S}_q^(M)$. Convergence theorems for generalized functional sequences {#sec-3} ========================================================= Let $M=(M_n)_{n\in \mathbb{N}}$ be the same discrete-time normal martingale as described in Section \[sec-2\]. In the present section, we apply the Fock transform (see Definition \[def-2-2\]) to establish convergence theorems for generalized functionals of $M$. In order to prove our main results in a convenient way, we first give a preliminary proposition, which is an immediate consequence of the general theory of countably normed spaces, especially nuclear spaces [@becnel; @gelfand-shi; @gelfand-vi], since $\mathcal{S}(M)$ is a nuclear space (see Lemma \[lem-2-3\]). \[prop-3-1\] Let $\Phi$, $\Phi_n \in \mathcal{S}^(M)$, $n\geq 1$, be generalized functionals of $M$. Then the following conditions are equivalent: (i) The sequence $(\Phi_n)$ converges weakly to $\Phi$ in $\mathcal{S}^(M)$; (ii) The sequence $(\Phi_n)$ converges strongly to $\Phi$ in $\mathcal{S}^(M)$; (iii) There exists a constant $p\geq 0$ such that $\Phi$, $\Phi_n \in \mathcal{S}_p^(M)$, $n\geq 1$, and the sequence $(\Phi_n)$ converges to $\Phi$ in the norm of $\mathcal{S}_p^(M)$. Here we mention that “$(\Phi_n)$ converges strongly (resp. weakly) to $\Phi$” means that $(\Phi_n)$ converges to $\Phi$ in the strong (resp. weak) topology of $\mathcal{S}^(M)$. For details about various topologies on the dual of a countably normed space, we refer to [@becnel; @gelfand-shi]. The next theorem is one of our main results, which offers a criterion in terms of the Fock transform for checking whether or not a sequence in $\mathcal{S}^(M)$ is strongly convergent. \[thr-3-1\] Let $\Phi$, $\Phi_n \in \mathcal{S}^(M)$, $n\geq 1$, be generalized functionals of $M$. Then the sequence $(\Phi_n)$ converges strongly to $\Phi$ in $\mathcal{S}^(M)$ if and only if it satisfies: 1. $\widehat{\Phi_n}(\sigma)\rightarrow \widehat{\Phi}(\sigma)$ for all $\sigma\in \Gamma$; 2. There are constants $C\geq 0$ and $p\geq 0$ such that $$\label{} \sup_{n\geq 1}\|\widehat{\Phi_n}(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ The “only if ” part. Let $(\Phi_n)$ converge strongly to $\Phi$ in $\mathcal{S}^(M)$. Then, we obviously have $$\widehat{\Phi_n}(\sigma)=\langle\!\langle \Phi_n,Z_{\sigma}\rangle\!\rangle \rightarrow \langle\!\langle \Phi,Z_{\sigma}\rangle\!\rangle=\widehat{\Phi}(\sigma),\quad \sigma\in \Gamma,$$ because $\{Z_{\sigma}\mid \sigma \in \Gamma\}\subset \mathcal{S}(M)$ and $(\Phi_n)$ also converges weakly to $\Phi$. On the other hand, by Proposition \[prop-3-1\], we know that there exists $p\geq 0$ such that $\Phi$, $\Phi_n \in \mathcal{S}_p^(M)$, $n\geq 1$, and $(\Phi_n)$ converges to $\Phi$ in the norm of $\mathcal{S}_p^(M)$, which implies that $C\equiv \sup_{n\geq 1}\\|\Phi_n\\|_{-p} < \infty$. Therefore $$\sup_{n\geq 1}\|\widehat{\Phi_n}(\sigma)\| = \sup_{n\geq 1}\|\langle\!\langle \Phi_n,Z_{\sigma}\rangle\!\rangle\| \leq \sup_{n\geq 1}\\|\Phi_n\\|_{-p}\\|Z_{\sigma}\\|_p = C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ The “if” part. Let $(\Phi_n)$ satisfy conditions (1) and (2). Then, by taking $q > p+ \frac{1}{2}$ and using Lemma \[lem-2-6\], we get $$\label{eq-3-2} \sup_{n\geq 1}\\|\Phi_n\\|_{-q} \leq C\bigg[\sum_{\sigma \in \Gamma}\lambda_{\sigma}^{-2(q-p)}\bigg]^{\frac{1}{2}},$$ in particular $\Phi_n \in \mathcal{S}_q^(M)$, $n\geq 1$. On the other hand, $\{Z_{\sigma} \mid \sigma \in \Gamma\}$ is total in $\mathcal{S}_q(M)$, which, together with (\[eq-3-2\]) as well as the property $$\langle\!\langle \Phi_n,Z_{\sigma}\rangle\!\rangle =\widehat{\Phi_n}(\sigma)\rightarrow \widehat{\Phi}(\sigma) = \langle\!\langle \Phi,Z_{\sigma}\rangle\!\rangle,\quad \sigma\in \Gamma,$$ implies that $\Phi \in \mathcal{S}_q^(M)$ and $$\langle\!\langle \Phi_n,\xi\rangle\!\rangle \rightarrow \langle\!\langle \Phi,\xi\rangle\!\rangle,\quad \forall\, \xi\in \mathcal{S}_q(M).$$ Thus $(\Phi_n)$ converges weakly to $\Phi$ in $\mathcal{S}^(M)$, which together with Proposition \[prop-3-1\] implies that $(\Phi_n)$ converges strongly to $\Phi$ in $\mathcal{S}^(M)$. In a similar way we can prove the following theorem, which is slightly different form Theorem \[thr-3-1\], but more convenient to use. \[thr-3-2\] Let $(\Phi_n) \subset \mathcal{S}^(M)$ be a sequence of generalized functionals of $M$. Suppose $\big(\widehat{\Phi_n}(\sigma)\big)$ converges for all $\sigma\in \Gamma$, and moreover there are constants $C\geq 0$ and $p\geq 0$ such that $$\label{} \sup_{n\geq 1}\|\widehat{\Phi_n}(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ Then there exists a generalized functional $\Phi\in \mathcal{S}^(M)$ such that $(\Phi_n)$ converges strongly to $\Phi$. $M$-generalized martingales and their convergence theorems ========================================================== In this section, we first introduce a type of generalized martingales associated with $M$, which we call $M$-generalized martingales, and then we use the Fock transform to a give necessary and sufficient condition for such a generalized martingale to be strongly convergent. Some other convergence results are also obtained. For a nonnegative integer $n\geq 0$, we denote by $\Gamma\!_{n]}$ the power set of $\{0,1,\cdots, n\}$, namely $$\label{eq-4-1} \Gamma\!_{n]}=\{\,\sigma \mid \sigma\subset \{0,1, \cdots, n\}\,\}.$$ Clearly $\Gamma\!_{n]} \subset \Gamma$. We use $\mathbf{I}_{n]}$ to mean the indicator of $\Gamma\!_{n]}$, which is a function on $\Gamma$ given by $$\label{eq-4-2} \mathbf{I}_{n]}(\sigma)= \left\{ \begin{array}{ll} 1, & \hbox{$\sigma \in \Gamma\!_{n]}$;} \\ 0, & \hbox{$\sigma \notin \Gamma\!_{n]}$.} \end{array} \right.$$ \[def-4-1\] A sequence $(\Phi_n)_{n\geq 0} \subset \mathcal{S}^(M)$ is called an $M$-generalized martingale if it satisfies that $$\label{eq-4-3} \widehat{\Phi_n}(\sigma) = \mathbf{I}_{n]}(\sigma)\widehat{\Phi_{n+1}}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0,$$ where $\mathbf{I}_{n]}$ mean the indicator of $\Gamma\!_{n]}$ as defined by (\[eq-4-2\]). Let $\mathfrak{F}=(\mathcal{F}_n)_{n\geq 0}$ be the filtration on $(\Omega, \mathcal{F}, P)$ generated by $Z=(Z_n)_{n\geq 0}$, namely $$\label{} \mathcal{F}_n = \sigma\{Z_k \mid 0\leq k \leq n\},\quad n\geq 0.$$ The following theorem justifies Definition \[def-4-1\]. \[thr-4-1\] Suppose $(\xi_n)_{n\geq 1}\subset \mathcal{L}^2(M)$ is a martingale with respect to filtration $\mathfrak{F}=(\mathcal{F}_n)_{n\geq 0}$. Then $(\xi_n)_{n\geq 1}$ is an $M$-generalized martingale. By the assumptions, $(\xi_n)_{n\geq 1}$ satisfies that the following conditions $$\label{eq-4-5} \xi_n = \mathbb{E}[\xi_{n+1}\,\|\, \mathcal{F}_n],\quad n\geq 0,$$ where $\mathbb{E}[\,\cdot \mid\! \mathcal{F}_n]$ means the conditional expectation given $\sigma$-algebra $\mathcal{F}_n$. Note that $$\mathbb{E}[Z_{\tau}\,\|\, \mathcal{F}_n] = \mathbf{I}_{n]}(\tau)Z_{\tau},\quad \tau\in \Gamma,$$ which, together with (\[eq-4-5\]) and the expansion $\xi_{n+1} = \sum_{\tau \in \Gamma}\langle Z_{\tau}, \xi_{n+1}\rangle Z_{\tau}$, gives $$\xi_n = \mathbb{E}[\xi_{n+1}\,\|\, \mathcal{F}_n] = \sum_{\tau \in \Gamma}\langle Z_{\tau}, \xi_{n+1}\rangle \mathbb{E}[Z_{\tau}\,\|\, \mathcal{F}_n] = \sum_{\tau \in \Gamma}\langle Z_{\tau}, \xi_{n+1}\rangle \mathbf{I}_{n]}(\tau)Z_{\tau}.$$ Taking Fock transforms yields $$\widehat{\xi_n}(\sigma) = \sum_{\tau \in \Gamma}\langle \xi_{n+1}, Z_{\tau}\rangle \mathbf{I}_{n]}(\tau)\widehat{Z_{\tau}}(\sigma) = \langle \xi_{n+1}, Z_{\sigma}\rangle \mathbf{I}_{n]}(\sigma) = \mathbf{I}_{n]}(\sigma)\widehat{\xi_{n+1}}(\sigma),$$ where $\sigma \in \Gamma$. Thus $(\xi_n)_{n\geq 1}$ is an $M$-generalized martingale. The next theorem gives a necessary and sufficient condition in terms of the Fock transform for an $M$-generalized martingale to be strongly convergent. \[thr-4-2\] Let $(\Phi_n)_{n\geq 1} \subset \mathcal{S}^(M)$ be an $M$-generalized martingale. Then the following two conditions are equivalent: 1. $(\Phi_n)_{n\geq 1}$ is strongly convergent in $\mathcal{S}^(M)$; 2. There are constants $C\geq 0$ and $p\geq 0$ such that $$\label{} \sup_{n\geq 1}\|\widehat{\Phi_n}(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ By Theorem \[thr-3-1\], we need only to prove “$(2)\Rightarrow (1)$”. Let $\sigma \in \Gamma$ be taken. Then by the definition of $M$-generalized martingales (see Definition \[def-4-1\]) we have $$\widehat{\Phi_m}(\sigma) = \mathbf{I}_{m]}(\sigma)\widehat{\Phi_{m+k}}(\sigma),\quad m,\, k\geq 0.$$ Now take $n_0\geq 0$ such that $\sigma\in \Gamma_{n_0]}$. Then $\mathbf{I}_{{n_0}]}(\sigma)=1$ and moreover $$\widehat{\Phi_{n_0}}(\sigma) = \mathbf{I}_{{n_0}]}(\sigma)\widehat{\Phi_{n}}(\sigma) = \widehat{\Phi_{n}}(\sigma),\quad n>n_0,$$ which implies $\big(\widehat{\Phi_n}(\sigma)\big)$ converges. Thus, by Theorem \[thr-3-2\], $(\Phi_n)_{n\geq 1}$ is strongly convergent in $\mathcal{S}^(M)$. \[thr-4-3\] Let $D$ be a subset of $\mathcal{S}^(M)$. Then the following two conditions are equivalent: 1. There is a constant $p\geq 0$ such that $D$ is contained and bounded in $\mathcal{S}_p^(M)$; 2. There are constants $C\geq 0$ and $p\geq 0$ such that $$\label{} \sup_{\Phi \in D}\|\widehat{\Phi}(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ Obviously, condition (1) implies condition (2). We now verify the inverse implication relation. In fact, under condition (2), by using Lemma \[lem-2-6\] we have $$\sup_{\Phi \in D} \\|\Phi\\|_{-q} \leq C\bigg[\sum_{\sigma \in \Gamma}\lambda_{\sigma}^{-2(q-p)}\bigg]^{\frac{1}{2}},$$ where $q>p+\frac{1}{2}$, which clearly implies condition (1). The next theorem shows that for an $M$-generalized martingale, its strong (weak) convergence is just equivalent to its strong (weak) boundedness. \[thr-4-4\] Let $(\Phi_n)_{n\geq 1} \subset \mathcal{S}^(M)$ be an $M$-generalized martingale. Then the following conditions are equivalent: 1. $(\Phi_n)_{n\geq 1}$ is strongly convergent in $\mathcal{S}^(M)$; 2. $(\Phi_n)_{n\geq 1}$ is weakly bounded in $\mathcal{S}^(M)$; 3. $(\Phi_n)_{n\geq 1}$ is strongly bounded in $\mathcal{S}^(M)$; 4. $(\Phi_n)_{n\geq 1}$ is bounded in $\mathcal{S}_p^(M)$ for some $p\geq 0$. Clearly, conditions (2), (3) and (4) are equivalent each other because $\mathcal{S}(M)$ is a nuclear space (see Lemma \[lem-2-3\]). Using Theorems \[thr-4-2\] and \[thr-4-3\], we immediately know that conditions (1) and (4) are also equivalent. Applications ============ In the last section we show some applications of our main results. Recall that the system $\{Z_{\sigma}\mid \sigma\in \Gamma\}$ is an orthonormal basis of $\mathcal{L}^2(M)$. Now if we write $$\label{eq-5-1} \Psi_n^{(0)} = \sum_{\tau \in \Gamma\!_{n]}}Z_{\tau},\quad n\geq 0,$$ then $\big(\Psi_n^{(0)}\big)_{n\geq 0} \subset \mathcal{L}^2(M)$, and moreover $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is a martingale with respect to filtration $\mathfrak{F}=(\mathcal{F}_n)_{n\geq 0}$. However, $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is not convergent in $\mathcal{L}^2(M)$ since $$\label{eq-5-2} \\|\Psi_n^{(0)}\\|= \sqrt{\#(\Gamma\!_{n]})} = 2^{\frac{n+1}{2}} \rightarrow \infty\quad (\text{as $n\rightarrow \infty$}),$$ where $\#(\Gamma\!_{n]})$ means the cardinality of $\Gamma\!_{n]}$ as a set and $\\|\cdot\\|$ the norm in $\mathcal{L}^2(M)$. \[prop-5-1\] The sequence $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ defined above is an $M$-generalized martingale, and moreover it is strongly convergent in $\mathcal{S}^(M)$. According to Theorem \[thr-4-1\], $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is certainly an $M$-generalized martingale. On the other hand, in viewing the relation between the canonical bilinear form on $\mathcal{S}^(M)\times \mathcal{S}(M)$ and the inner product in $\mathcal{L}^2(M)$, we have $$\label{eq-5-3} \widehat{\Psi_n^{(0)}}(\sigma) = \langle\!\langle \Psi_n^{(0)}, Z_{\sigma} \rangle\!\rangle = \langle \Psi_n^{(0)}, Z_{\sigma} \rangle = \mathbf{I}_{n]}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0,$$ which implies that $$\sup_{n\geq 0}\big\|\widehat{\Psi_n^{(0)}}(\sigma)\big\| \leq C \lambda_{\sigma}^p,\quad \sigma \in \Gamma$$ with $C=1$ and $p=0$. It then follows from Theorem \[thr-4-2\] that $\big(\Psi_n^{(0)}\big)_{n\geq 0}$ is strongly convergent in $\mathcal{S}^(M)$. Recall that [@wang-chen], for two generalized functionals $\Phi_1$, $\Phi_2 \in \mathcal{S}^(M)$, their convolution $\Phi_1\ast\Phi_2$ is defined by $$\label{} \widehat{ \Phi_1\ast\Phi_2}(\sigma) = \widehat{ \Phi_1}(\sigma)\widehat{\Phi_2}(\sigma),\quad \sigma \in \Gamma.$$ The next theorem provides a method to construct an $M$-generalized martingale through the $M$-generalized martingale$\big(\Psi_n^{(0)}\big)_{n\geq 0}$ defined in (\[eq-5-1\]). \[thr-5-1\] Let $\Phi \in \mathcal{S}^(M)$ be a generalized functional and define $$\label{} \Phi_n = \Psi_n^{(0)} \ast\Phi,\quad n\geq 0.$$ Then $(\Phi_n)_{n\geq 0}$ is an $M$-generalized martingale, and moreover it converges strongly to $\Phi$ in $\mathcal{S}^(M)$. By Lemma \[lem-2-6\], there exist some constants $C\geq 0$ and $p\geq 0$ such that $$\label{eq-5-6} \|\widehat{\Phi}(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ On the other hand, by using (\[eq-5-3\]), we get $$\label{eq-5-7} \widehat{\Phi_n}(\sigma) =\widehat{\Psi_n^{(0)}}(\sigma)\widehat{\Phi}(\sigma) =\mathbf{I}_{n]}(\sigma)\widehat{\Phi}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0,$$ which, together with the fact $\mathbf{I}_{n]}(\sigma)\mathbf{I}_{n+1]}(\sigma)=\mathbf{I}_{n]}(\sigma)$, gives $$\widehat{\Phi_n}(\sigma)= \mathbf{I}_{n]}(\sigma)\widehat{\Phi_{n+1}}(\sigma),\quad \sigma \in \Gamma,\, n\geq 0.$$ Thus $(\Phi_n)_{n\geq 0}$ is an $M$-generalized martingale. Additionally, it easily follows from (\[eq-5-6\]) and (\[eq-5-7\]) that $\widehat{\Phi_n}(\sigma)\rightarrow \widehat{\Phi}(\sigma)$ for each $\sigma \in \Gamma$ and $$\sup_{n\geq 0}\|\widehat{\Phi_n}(\sigma)\| =\sup_{n\geq 0}\big[\mathbf{I}_{n]}(\sigma)\big]\|\widehat{\Phi}(\sigma)\| \leq C\lambda_{\sigma}^p,\quad \sigma \in \Gamma.$$ Therefore, by Theorem \[thr-3-2\], we finally find $(\Phi_n)_{n\geq 0}$ converges strongly to $\Phi$. Appendix {#appendix .unnumbered} ======== In this appendix, we provide some basic notions and facts about discrete-time normal martingales. For details we refer to [@privault; @wang-lc]. Let $(\Omega, \mathcal{F}, P)$ be a given probability space with $\mathbb{E}$ denoting the expectation with respect to $P$. We denote by $\mathcal{L}^{2}(\Omega, \mathcal{F}, P)$ the usual Hilbert space of square integrable complex-valued functions on $(\Omega, \mathcal{F}, P)$ and use $\langle\cdot,\cdot\rangle$ and $\\|\cdot\\|$ to mean its inner product and norm, respectively. [Definition A.1.]{}\[def-A-1\] A stochastic process $M=(M_n)_{n\in \mathbb{N}}$ on $(\Omega, \mathcal{F}, P)$ is called a discrete-time normal martingale if it is square integrable and satisfies: (i) $\mathbb{E}[M_0 \| \mathcal{F}_{-1}] = 0$ and $\mathbb{E}[M_n \| \mathcal{F}_{n-1}] = M_{n-1}$ for $n\geq 1$; (ii) $\mathbb{E}[M_0^2 \| \mathcal{F}_{-1}] = 1$ and $\mathbb{E}[M_n^2 \| \mathcal{F}_{n-1}] = M_{n-1}^2 +1$ for $n\geq 1$, where $\mathcal{F}_{-1}=\{\emptyset, \Omega\}$, $\mathcal{F}_n = \sigma(M_k; 0\leq k \leq n)$ for $n \in \mathbb{N}$ and $\mathbb{E}[\cdot \| \mathcal{F}_{k}]$ means the conditional expectation. Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale on $(\Omega, \mathcal{F}, P)$. Then one can construct from $M$ a process $Z=(Z_n)_{n\in \mathbb{N}}$ as $$Z_0=M_0,\quad Z_n = M_n-M_{n-1},\quad n\geq 1. \eqno(\mathrm{A}.1)$$ It can be verified that $Z$ admits the following properties: $$\mathbb{E}[Z_n \| \mathcal{F}_{n-1}] =0\quad \text{and}\quad \mathbb{E}[Z_n^2 \| \mathcal{F}_{n-1}] =1,\quad n\in \mathbb{N}. \eqno(\mathrm{A}.2)$$ Thus, it can be viewed as a discrete-time noise. [Definition A.2.]{} \[def-A-2\] Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale. Then the process $Z$ defined by (A.2) is called the discrete-time normal noise associated with $M$. The next lemma shows that, from the discrete-time normal noise $Z$, one can get an orthonormal system in $\mathcal{L}^2(\Omega, \mathcal{F}, P)$, which is indexed by $\sigma \in \Gamma$. [Lemma A.1.]{} Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale and $Z=(Z_n)_{n\in \mathbb{N}}$ the discrete-time normal noise associated with $M$. Define $Z_{\emptyset}=1$, where $\emptyset$ denotes the empty set, and $$Z_{\sigma} = \prod_{i\in \sigma}Z_i,\quad \text{$\sigma \in \Gamma$, $\sigma \neq \emptyset$}. \eqno(\mathrm{A}.3)$$ Then $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ forms a countable orthonormal system in $\mathcal{L}^2(\Omega, \mathcal{F}, P)$. Let $\mathcal{F}_{\infty}=\sigma(M_n; n\in \mathbb{N})$, the $\sigma$-field over $\Omega$ generated by $M$. In the literature, $\mathcal{F}_{\infty}$-measurable functions on $\Omega$ are also known as functionals of $M$. Thus elements of $\mathcal{L}^2(\Omega, \mathcal{F}_{\infty}, P)$ can be called square integrable functionals of $M$. For brevity, we usually denote by $\mathcal{L}^2(M)$ the space of square integrable functionals of $M$, namely $$\mathcal{L}^2(M)=\mathcal{L}^2(\Omega, \mathcal{F}_{\infty}, P). \eqno(\mathrm{A}.4)$$ [Definition A.3.]{} \[def-A-3\] The discrete-time normal martingale $M$ is said to have the chaotic representation property if the system $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ defined by (A.3) is total in $\mathcal{L}^2(M)$. So, if the discrete-time normal martingale $M$ has the chaotic representation property, then the system $\{Z_{\sigma}\mid \sigma \in \Gamma\}$ is actually an orthonormal basis for $\mathcal{L}^2(M)$, which is a closed subspace of $\mathcal{L}^2(\Omega, \mathcal{F}, P)$ as is known. [Remark A.1.]{} Émery [@emery] called a $\mathbb{Z}$-indexed process $X=(X_n)_{n \in \mathbb{Z}}$ satisfying (A.2) a novation and introduced the notion of the chaotic representation property for such a process. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported by National Natural Science Foundation of China (Grant No. 11461061). [99]{} S. Albeverio, Yu.L. Daletsky, Yu.G. Kondratiev and L. Streit, Non-Gaussian infinite dimensional analysis, J. Funct. Anal. 138 (1996), 311–350. A. Barhoumi, H. Ouerdiane and A, Riahi, Pascal white noise calculus, Stochastics 81 (2009), 323–343. Jeremy J. Becnel, Equivalence of topologies and Borel fields for countably-Hilbert spaces, Proc. Amer. Math. Soc. 134 (2006), 581–590. G. Di Nunno, B. Oksendal and F. Proske, White noise analysis for Lévy processes, J. Funct. Anal. 206 (2004), 109–148. R.X. Dudley, Uniform Central Limit Theorems, Cambridge University Press, Cambridge 1999. M. Émery, A discrete approach to the chaotic representation property, in: Séminaire de Probabilités, XXXV, Lecture Notes in Mathematics 1755, 123–138, Springer, Berlin 2001. I.M. Gel’fand and G.E. Shilov, Generalized Functions, vol. 2, Spaces of Fundamental and Generalized Functions, Academic Press, New York 1968. I.M. Gel’fand and N.Ya. Vilenkin, Generalized Functions, vol. 4, Applications of Harmonic Analysis, Academic Press, New York 1964. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, White Noise: An Infinite Dimensional Calculus, Kluwer Academic, Dordrecht 1993. H. Holden, B. Oksendal, J. Uboe and T. Zhang, Stochastic Partial Differential Equations, A modeling, White Noise Functional Approach, Birkhauser, Boston 1996. Y. Hu and B. Oksendal, Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 1–32. Z.Y. Huang and J.A. Yan, Introduction to Infinite Dimensional Stochastic Analysis, Kluwer Academic, Dordrecht 1999. Y. Ito, Generalized Poisson functionals, Probab. Theory and Related Fields 77 (1988), 1–28. H.H. Kuo, White Noise Distribution Theory, CRC, Boca Raton 1996. Y.-J. Lee and H.-H. Shih, The Segal-Bargmann transform for Lévy functionals, J. Funct. Anal. 168 (1999), 46–83. N. Obata, An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan 45 (1993), 421–445. J. Potthoff and L. Streit, A characterization of Hida distributions, J. Funct. Anal. 101 (1991), 212–229. N. Privault, Stochastic analysis of Bernoulli processes, Probab. Surv. 5 (2008), 435–483. J. Rudnick and G. Gaspari, Elements of the Random Walk, Cambridge University Press, Cambridge 2004. C.S. Wang, J.S. Chen, Characterization theorems for generalized functionals of discrete-time normal martingale, Journal of Function Spaces, Volume 2015, Article ID 714745. C.S. Wang, Y.C. Lu and H.F. Chai, An alternative approach to Privault’s discrete-time chaotic calculus, J. Math. Anal. Appl. 373 (2011), 643–654. C.S. Wang and J.H. Zhang, Wick analysis for Bernoulli noise functionals, Journal of Function Spaces, Volume 2014, Article ID 727341, 2014.
--- author: - Rohit Tripathy - Ilias Bilionis - Marcial Gonzalez bibliography: - 'bibliography.bib' title: 'Gaussian processes with built-in dimensionality reduction: Applications in high-dimensional uncertainty propagation' --- Ilias Bilionis and Marcial Gonzalez acknowledge the startup support provided by the School of Mechanical Engineering at Purdue University.
--- author: - 'J. Knoll$^{1}$, Yu. B. Ivanov$^{1,2}$ and D. N. Voskresensky$^{1,3}$' title: 'Exact Conservation Laws of the Gradient Expanded Kadanoff–Baym Equations' --- [exact-cons-fig]{} Introduction ============ Non-equilibrium Green function techniques, developed by Schwinger, Kadanoff, Baym and Keldysh [@Schw; @Kad62; @Keld64; @LP], provide the appropriate concepts to study the space–time evolution of many-particle quantum systems. This formalism finds now applications in various fields, such as quantum chromo-dynamics [@Land; @BI00], nuclear physics, in particular heavy ion collisions [@Dan84; @Dan90; @Toh; @Bot90; @MSTV; @Vos93; @Hen; @Knoll95; @IKV; @Bozek98; @Knoll98; @IKV99; @Cass99; @Leupold; @Mosel; @IKHV00; @HK00], astrophysics [@MSTV; @VS87; @Keil], cosmology [@CalHu], spin systems [@Manson], lasers [@Korenman], physics of plasma [@Bez; @Kraft], physics of liquid $^{3}$He [@SerRai], critical phenomena, quenched random systems and disordered systems [@Chou], normal metals and super-conductors [@VS87; @Rammer; @Fauser], semiconductors [@LipS], tunneling and secondary emission [@Noziers], etc. For actual calculations certain approximation steps are necessary. In many cases perturbative approaches are insufficient, like for systems with strong couplings as treated in nuclear physics. In such cases, one has to resum certain sub-series of diagrams in order to obtain a reasonable approximation scheme. In contrast to perturbation theory, for such resummations one frequently encounters the fact that the scheme may no longer be conserving, although for each diagram considered the conservation laws are implemented at each vertex. Thus, the resulting equations of motion may no longer comply with the conservation laws, e.g., of currents, energy and momentum. This is a problem of particular importance for recent studies of particles with broad damping width such as resonances [@IKV99]. The problem of conservation laws in such resummation schemes has first been considered in two pioneering papers by Baym and Kadanoff [@KadB; @Baym] discussing the response to an external perturbation of quantum systems in thermodynamic equilibrium. Baym, in particular, showed [@Baym] that any approximation, in order to be conserving, must be based on a generating functional $\Phi$. This functional was first considered by Luttinger and Ward [@Luttinger] in the context of the thermodynamic potential, cf. [@Abrikos], and later reformulated in terms of path integrals [@CJT]. For truncated self-consistent Dyson resummations this functional method provides conserved Noether currents and the conservation of total energy and momentum at the expectation value level. In our previous paper [@IKV] we extended the concept to the real-time Green function technique and relativistic systems, constructing conserved 4-currents and local energy–momentum tensor for any chosen approximation to the $\Phi$ functional. While $\Phi$-derivable Dyson resummations formulated in terms of the integro-differential Kadanoff-Baym (KB) equations indeed provide [exact]{} conservation laws, these equations are usually not directly solvable. Therefore many applications involve further approximations to the KB equations: the [gradient]{} approximation and often the [quasi-particle]{} approximation leading to differential equations of mean field and transport type. Any improvement of the quasi-particle approximation beyond the mean-field level, e.g., through inclusion of energy-momentum-dependent self-energies, again leads to difficulties with conservation laws. Various attempts to remedy this problem were undertaken, see refs. [@LipS; @LSV; @Bonitz; @Bornath; @SCFNW; @Jeon; @VBRS] and references therein. An essential progress within the quasi-particle approximation was achieved by the ansätze of refs. [@LipS; @Bornath]. Interested in the dynamics of particles with broad mass width like resonances we like to discuss the question of conservation laws for transport problems at a much more general level, than usually considered. We call this the [quantum transport]{} level. It completely avoids the quasi-particle approximation, and solely rests on the first-order gradient approximation of the KB equations. This concept was first addressed by Kadanoff and Baym [@Kad62] in the chapter “Slowly varying disturbances”, Eqs. (9-25). Motivated by applications for the description of heavy-ion collisions further attempts were recently suggested [@IKV99; @Cass99; @Leupold; @Mosel; @IKHV00; @HK00] based on the so called Botermans–Malfliet (BM) substitution [@Bot90]. Within the BM choice of the quantum kinetic equations a number of desired properties including an H-theorem for a local entropy current related to these equations were derived [@IKV99], however no strict realization of the conservation laws for the Noether currents were obtained[@IKV99; @Leupold]. Due to the approximation steps involved one may expect the quantum transport equations both, in the KB and BM forms, to possess only approximate conservation laws though in line with the level of approximation. Such approximate nature of conservation laws may be well acceptable theoretically. Nevertheless, both from a principle perspective and also from a practical point of view this situation is less satisfactory. Quantum kinetic equations which possess exact conservation laws related to the symmetries of the problem could serve as a natural extension of the quasiparticle transport phenomenology to broad resonances, e.g. applicable to high energy heavy ion collisions. In this paper we give a proof that the quantum kinetic equations in the form originally derived by Kadanoff and Baym in fact possess the generic feature of exact conservation laws at the expectation value level. This holds provided all self-consistent self-energies are generated from a $\Phi$-functional and all possible memory effects due to internal vertices within the self-energy diagrams are also consistently expanded to first-order gradients. In sect. \[QKE\] we review the derivation of the quantum kinetic equations in a notation suitable for our later derivation of the conservation laws in sect. \[CL\]. Sect. \[Gadient-Apprx\] deals with the general gradient approximation and its representation in terms of diagrams. Finally, in sect. \[Phi-Conserv\] within the $\Phi$-derivable method we give a diagrammatic proof of the exact conservation laws of the quantum kinetic equations for the KB choice. We restrict the presentation to physical systems described by complex quantum fields of different constituents interacting via local couplings. The kinematics can be either relativistic or non-relativistic. Extension to real boson fields, as well as to relativistic fermions is straight forward though tedious in the latter case. We also exclude theories with derivative couplings. Kadanoff–Baym equations and complete gradient approximation {#QKE} =========================================================== We assume the reader to be familiar with the real-time formulation of non-equilibrium field theory on the so called closed time contour, Fig. \[Fcontour\]. Since we will deal with general multi-point functions we use the more convenient $\{-+\}$ contour-vertex notation of refs. [@LP; @IKV99]. Some rules are summarized in Appendix \[Contour\]. The set of coupled KB equations on the time contour in $-+$ notation[^1] reads $$\begin{aligned} \label{KB-Eq} &&\left(\Gr^{-1}_{0}(-\ii\partial_1)-\Gr^{-1}_{0}(-\ii\partial_2)\right) \Gr^{-+}(1,2) \cr && =\oint\di 3 \left(\Se(1^-,3 )\Gr(3 ,2^+) -\Gr(1^-,3 )\Se(3,2^+)\right)%\cr %&&=\left[\Se\cdot\Gr-\Gr\cdot\Se\right]^{-+}_{1,2}\\ %&& \equiv \MP C(1^-,2^+) %%\\\end{aligned}$$ with Fourier transform of the inverse free Green function $$\begin{aligned} \Gr^{-1}_{0}(p)=\left\{ \begin{array}{ll} p^2-m^2\quad&\mbox{for relativistic bosons}\\ p_0-{\vec p}^2/(2m)\quad&\mbox{for non-rel. fermions or bosons.} \end{array}\right.\end{aligned}$$ Here and below the upper/lower signs refer to fermions or bosons respectively, $\Gr_0$ and $\Gr$ correspondingly denote the free and full Green functions, labels for the different species and internal quantum numbers are suppressed. The driving term on the r.h.s. of Eq. (\[KB-Eq\]), summarized by $C$, is a functional of the Green functions through the self-energies $\Se$ contour folded with $\Gr$. The real-time integration contour, cf. Fig. \[Fcontour\], is denoted by ${\cal C}$. The step towards transport equations is provided by introducing the four-dimensional Wigner transforms for all two-point functions through $$\begin{aligned} \label{Wigner} F(x,y)=\intp e^{-\ii p(x-y)} F({\textstyle\frac{x+y}2},p).\end{aligned}$$ The KB Eq. (\[KB-Eq\]) then transforms to $$\begin{aligned} \label{KB-Eq-Wigner} v^{\mu}\partial_{\mu}(\MP\ii)\Gr^{-+}(X,p)&=& C^{-+}(X,p;\{\Gr\}) \quad\mbox{with} \quad v^{\mu}=\frac{\partial}{\partial p_{\mu}}G_0^{-1}(p),\end{aligned}$$ where the r.h.s. is also expressed in terms of the Wigner transforms of all Green functions through (\[Wigner\]). The final step is to expand the complicated r.h.s. of Eq. (\[KB-Eq-Wigner\]) to the first-order gradients. Then the [local]{} part of this r.h.s., $C^{-+}_{\mathrm{(loc)}}$, consists of [non-gradient terms]{}, where one replaces the different mean positions $(x_i +x_j )/2$ occurring in the various Green functions by the externally given mean position $X$ of the l.h.s., i.e. $X=(x_1+x_2)/2$ and evaluates the diagrams as in momentum representation. The corrections for the displacement to the true coordinates of each Green function are then accounted for to the first order in the gradients. Here we simply abbreviate the gradient terms by a $\Diamond$ operator acting on the [local]{} diagram expression $$\begin{aligned} \label{Grad-KB-Eq} v^{\mu}\partial_{\mu}(\MP\ii)\Gr^{-+}(X,p) &=& (1+{\textstyle\frac{\ii}{2}}\Diamond) \left\{C^{-+}_{\mathrm{(loc)}}(X,p)\right\},\end{aligned}$$ where $$\begin{aligned} \label{C-loc} C^{-+}_{\mathrm{(loc)}}(X,p)&=&C^{-+}(X,p;\{\Gr_{\mathrm{(loc)}}\}) ,\end{aligned}$$ is a functional of the local Green functions $\Gr_{\mathrm{(loc)}} \equiv \Gr(X,p)$. Here and for all further considerations below, both, Green functions $\Gr(X,p)$ and self-energies $\Se(X,p)$, whenever quoted in their Wigner function form, are taken in [local]{} approximation, i.e. with $X$ given by the external coordinate and $\Se(X,p)$ void of any gradient correction terms. The explicit definition of the $\Diamond$-operator is deferred to sect. \[Gadient-Apprx\]. All what we need to know at this level is that it consists of terms where in the diagrams defining $C_{\mathrm{(loc)}}$ pairs of Green functions are replaced by their space-time and momentum derivatives, respectively, just leading to equations linear in space-time gradients. Naturally the result of the $\Diamond$-operation depends on the explicit form, i.e. diagrammatic structure, of the functional on which it operates. Therefore for the non-gradient term in Eq. (\[Grad-KB-Eq\]), which defines the [local]{} collision term $$\begin{aligned} \label{C-loc-dia} C^{-+}_{\mathrm{(loc)}} &{\begin{array}[t]{c}=\\[-1mm] {\!\!\!\!\!\scr{diagram}\!\!\!\!\!}\end{array}} & \MP \Se^{-k}(X,p)\sigma_{kl}\Gr^{l+}(X,p) -(\MP) \Gr^{-k}(X,p)\sigma_{kl}\Se^{l+}(X,p)\\ \label{C-loc-value} &{\begin{array}[t]{c}=\\[-1mm] {\!\!\!\scr{value}\!\!\!}\end{array}} & \,\underbrace{\MP \ii\Se^{-+}(X,p)\ii\Gr^{+-}(X,p)}_{\mbox{gain}} \, -\, \underbrace{(\MP)\ii\Gr^{-+}(X,p)\ii\Se^{+-}(X,p)}_{\mbox{loss}} ,\end{aligned}$$ we give both, the diagram expression (\[C-loc-dia\]) and the normally quoted [value]{} expression (\[C-loc-value\]). The latter simplifies due to a cancellation of terms which however survive for the order sensitive gradient operation. Here $\sigma_{ik}=\sigma^{ik}=\mathrm{diag}(1,-1)$ defines the “contour metric”, which accounts for the integration sense, and summation over the contour labels $k,l\in \{-,+\}$ is implied, cf. (\[sig\]) ff. The above quantum kinetic equation (\[Grad-KB-Eq\]) has to be supplemented by a [local]{} Dyson equation for the retarded Green function [@Kad62] $$\begin{aligned} \label{retarded-Eq} \left(\Gr^R(X,p)\right)^{-1}=\left(\Gr^R_0(p)\right)^{-1}-\Sigma^R(X,p),\end{aligned}$$ which together with Eq. (\[Grad-KB-Eq\]) provides the simultaneous solution to $\Gr^{+-}$. The full retarded Green function $\Gr^R$ depends on the retarded self-energy $\Se^R=\Se^{--}-\Se^{-+}=\Se^{+-}-\Se^{++}$ again in [local]{} approximation. $\Gr^R_0$ is the free retarded Green function. Please note that equation (\[retarded-Eq\]) is just algebraic although it is obtained in the framework of the first-order gradient approximation. In most presentations of the gradient approximation to the KB equations, Eq. (\[Grad-KB-Eq\]) is rewritten such that the gradient terms are subdivided into two parts, consisting of Poisson bracket terms describing drag- and back-flow effects, on the one side, and a memory collision term $C^{\scr{mem}}$, on the other side, in cases when the self-energy contains internal vertices $$\begin{aligned} \label{Grad-Sep-KB-Eq} v^{\mu}\partial_{\mu}(\MP)\ii\Gr^{-+}(X,p) &=& \Pbr{\Re\Se^R,\MP\ii\Gr^{-+}}+\Pbr{\MP\ii\Se^{-+},\Re\Gr^{R}}\cr &&+ C^{-+}_{\mathrm{(mem)}}(X,p)+C^{-+}_{\mathrm{(loc)}}(X,p),\end{aligned}$$ where $$\begin{aligned} \label{C-mem}\nonumber &&\hspace{-15mm}\MP C^{-+}_{\mathrm{(mem)}}(X,p)\\ &=&\Se^{-k}_{(\scr{mem})}(X,p)\sigma_{kl}\Gr^{l+}(X,p) -\Gr^{-k}(X,p)\sigma_{kl} \Se_{(\scr{mem})}^{l+}(X,p) \\ \label{C-mem-value} &\!\!\begin{array}[t]{c}=\\[-1mm] {\scr{value}}\end{array}\!\!& {-\Se_{(\scr{mem})}^{-+}(X,p)\Gr^{+-}(X,p)} + {\Gr^{-+}(X,p)\Se_{(\scr{mem})}^{+-}(X,p)}\end{aligned}$$ and $$\begin{aligned} \label{S-mem} &&\hspace{-15mm}\Se_{(\scr{mem})}(X,p)= {\textstyle\frac{\ii}{2}}\Diamond\left\{\Se(X,p)\right\}. \end{aligned}$$ One may further introduce the spectral function $A(X,p)=-2\Im \Gr^R(X,p)$ determined by the retarded equation (\[retarded-Eq\]), as well as the four-phase-space distribution function $f(X,p)$, by means of $\MP\ii G^{-+}(X,p)=f(X,p) A(X,p)$, whose evolution is governed by transport Eq. (\[Grad-Sep-KB-Eq\]) or equivalently by (\[Grad-KB-Eq\]). This defines a generalized quantum transport scheme which is void of the usual quasi-particle assumption. The time evolution is completely determined by initial instantaneous values of the Green functions and their gradients at each space–time point, which means it is [Markovian]{}, since the memory part of the collision term is kept only up to first-order gradient terms. Within its validity range this transport scheme is capable to describe slow space-time evolutions of particles with broad damping width, such as resonances, within a transport dynamics, now necessarily formulated in the four-dimensional phase-space. Depending on the questions raised, the above separation (\[Grad-Sep-KB-Eq\]) may not be always useful. For the derivation of the conservation laws only a unified treatment of both, the Poisson brackets and memory collision terms, reveals the symmetry among these terms, which then displays the necessary cancellation of certain contributions such that the conservation laws emerge. Therefore, in the forthcoming considerations we will mostly refer to the quantum kinetic equation in the form (\[Grad-KB-Eq\]). Conservation laws {#CL} ================= Conservations of charge and energy–momentum result from taking the charge or four–momentum weighted traces of the transport equations (\[Grad-KB-Eq\]). These traces include the integration over four-momentum, as well as sums over internal quantum numbers and all types of species $a$ with charges $e_{a}$ $$\begin{aligned} \label{Conserv-Eq} \hspace{-5mm} &&\hspace{-10mm}\partial_{\mu}\sum_a\intp\displaystyle{e_a\choose p^{\nu}} v^{\mu}(\MP\ii)\Gr_a^{-+}(X,p) \cr &&=\sum_a\intp {e_a\choose p^{\nu}} \left(1+{\textstyle\frac{\ii}{2}}\Diamond\right) \left\{C^{--}_{a\;\mathrm{(loc)}}(X,p)\right\} \equiv{{Q}(X)\choose {T}^{\nu}(X)}.\end{aligned}$$ The the charge and four-momentum leaks on the r.h.s, abbreviated as $Q$ and $T^{\nu}$, can be represented by closed diagrams, where the two end points ($x_1^-$ and $x_2^+$ in Eq. (\[KB-Eq\])) coalesce, i.e. $x_1=x_2=X$. In coordinate representation the r.h.s. corresponds to contour integrals of the type (\[diffrules0\]) - (\[diffrules1\]) for which entirely retarded terms drop out. Therefore, one even can place the two end points on the same contour side (respecting the fixed order for Tad-pole terms). For definiteness we have chosen the time ordered ($-$) branch. The external point $x_1^{-}=x_2^{-}=X$ is then the reference point with respect to which the gradients are to be evaluated. In line with causality requirements this reference point $X=(t,{\vec x})$ is also the [retarded point]{}. This implies that in a real-time contour representation any contribution to (\[Conserv-Eq\]) from internal integrations with physical times larger than $t$ drops out, cf. [@Chou; @Dan90]. Using the explicit form of the contour metric $\sigma$ (cf. Eqs. (\[Fij\]) and (\[H=FG\])) one obtains $$\begin{aligned} \label{SeGr-GrSe} \hspace{-3mm} C^{--}_{a\;\mathrm{(loc)}}(X,p) &=& \mp\left(\Se^a_{-k}(X,p)\Gr_a^{k-}(X,p) -\Gr_a^{-k}(X,p)\Se^a_{k-}(X,p)\right).\end{aligned}$$ In the proof given in sect. \[Phi-Conserv\] we show that the local parts of the r.h.s. of (\[Conserv-Eq\]) as given by $$\begin{aligned} \label{dotQ} {{Q}_{\mathrm{loc}}(X)\choose {T}^{\nu}_{\mathrm{loc}}(X)} = \sum_a\intp {e_a\choose p^{\nu}} C^{--}_{a\mathrm{(loc)}}(X,p)\end{aligned}$$ entirely drop out. Also the gradient terms of $Q$ cancel, while the gradients of the $T^{\nu}$ term compile to a complete divergence $$\begin{aligned} \label{Conserv-Result} {Q}(X)\equiv 0,\quad\quad {T}^{\nu}(X)=g^{\mu\nu} \partial_{\mu} \left(\E^{\mathrm{pot}}(X)-\E^{\mathrm{int}}(X)\right),\end{aligned}$$ provided the self-energies are derived from a so called $\Phi$-functional [@Baym]. This implies exact conservation laws for the Noether currents and the energy–momentum tensor given by $$\begin{aligned} \label{Q-E-M} \partial_{\mu}J^{\mu}(X)&=&0,\quad\partial_{\mu} \Theta_{\scr{loc}}^{\mu\nu}(X)=0 \hspace{1cm}\mbox{with} \\ J^{\mu}(X)&=&\sum_a\intp{e_a}v^{\mu}(\MP\ii)\Gr_a^{-+}(X,p),\\ \Theta_{\scr{loc}}^{\mu\nu}(X) &=&\sum_a\intp v^{\mu}p^{\nu}(\MP\ii)\Gr_a^{-+}(X,p) \cr &&\hspace{5mm}+g^{\mu\nu}\left(\E^{\mathrm{int}}_{\scr{loc}}(X)- \E^{\mathrm{pot}}_{\scr{loc}}(X)\right).\end{aligned}$$ Here $\Gr$ is the self-consistent propagator solving the coupled set of quantum transport equations (\[Grad-KB-Eq\]) and (\[retarded-Eq\]). Furthermore, $\Theta^{\mu\nu}_{\scr{loc}}(X)$ is the [local]{} version of energy–momentum tensor which for the $\Phi$-derivable approximation to the KB equations has been constructed in our previous papers [@IKV; @IKV99]. Thereby, $\E^{\mathrm{int}}_{\scr{loc}}(X)$ and $\E^{\mathrm{pot}}_{\scr{loc}}(X)$ define the interaction and single-particle potential energy densities, respectively, also taken in the local approximation. In the subsequent sections we formally define the complete first-order gradient expansion for any two point function, which contains internal vertices, and specify the corresponding diagrammatic rules. Finally, in sect. \[Phi-Conserv\] we prove the conservation laws (\[Q-E-M\]), using the $\Phi$-derivable properties and the gradient rules. Complete Gradient Approximation {#Gadient-Apprx} =============================== Let $M(1,2)$ be any two-point function with complicated internal structure. We are looking for its Wigner function $M(X,p)$ with $X=\frac{1}{2}(x_1+x_2)$ to first-order gradient approximation. The zero-order term is just given by evaluating $M(1,2)$ with the Wigner functions of [all]{} Green functions taken at the same space-time point $X=(x_1+x_2)/2$ and the momentum integrations being done as in the momentum representation of a homogeneous system. To access the gradient terms related to any Green function $G(i,j)$ involved in $M(1,2)$, its Wigner function $G(\frac{1}{2}(x_i+x_j),p)$ is to be Taylor expanded with respect to the reference point $X=(x_1+x_2)/2$, i.e. $$\begin{aligned} G(\frac{x_i+x_j}{2},p)\approx G(X,p) +\frac{1}{2}\left[(x^{\mu}_i-x_1^{\mu})+(x_j^{\mu}-x_2^{\mu})\right] \frac{\partial}{\partial X^{\mu}}G(X,p). \end{aligned}$$ Both the space derivatives of Green functions and the factors $(x_i-x_1)$ and $(x_j-x_2)$ accompanying them can be taken as special two-point functions, and we therefore assign them special diagrams 0.75mm $$\begin{aligned} \label{x-dashed} \Gdashed \;&=&\;{\textstyle\frac{1}{2}}\left(\partial _i +\partial_j\right)\Gr(i,j) \longrightarrow \partial_X \Gr(X,p), \\ \label{p-dashed} \Pdashed{i}{j} \;&=&\;-\ii\left(x_i-x_j\right) \longrightarrow -(2\pi)^4\frac{\partial}{\partial p}\delta(p) \end{aligned}$$ with the corresponding Wigner functions at the right hand side. Then the gradient terms of a complicated two-point function (given in different notation) can graphically be represented by the following two diagrams on the r.h.s. $$\begin{aligned} \label{Gradient-Diag} \Diamond \left\{M(1,2)\right\}&=& \Diamond \FBox{$1$}{$2$}{$M$}\! \equiv\!\FBox{$1$}{$2$}{$\Diamond M$} \!=\! \FBoxL+\FBoxR\end{aligned}$$ Here the diamond operator $\Diamond$, as above, formally defines the gradient approximation of the two-point function $M$ to its right with respect to the two external points $(1,2)$ displayed by full dots. The diagrammatic rules are then the following. For any $\Gr(3,4)$ in $M$, take the spatial derivative $\partial_X\Gr(X,p)$ (double line) and construct the two diagrams, where external point 1 is linked to 3 by an oriented dashed line, and where point 2 is linked to 4, respectively. Interchange of these links provides the same result. Here $M'$ is a four-point function generated by opening $M(1,2)$ with respect to any propagator $G(3,4)$, i.e. $$\begin{aligned} M'(1,2;3,4)=\MP\frac{\delta M(1,2)}{\delta \;\ii G(4,3)}.\end{aligned}$$ The diagrams in Eq. (\[Gradient-Diag\]) are then to be evaluated in the local approximation, i.e. with all Wigner Green functions taken at the same space-time point $X$. The dashed line (\[p-dashed\]) adds a new loop integration to the diagram, which, if integrated, leads to momentum derivatives of the Green functions involved in that loop. Both, double and dashed lines have four-vector properties, and the rule implies a four-scalar product between them. The explicit properties of the dashed line (\[p-dashed\]) permit to decompose it into two or several dashed lines through the algebra 0.75mm $$\begin{aligned} \label{p-dashed-addition} \Pdashed{1}{3}\;&=&\;\Pdashed{1}{2}\;+\;\;\Pdashed{2}{3} \quad\quad{\rm and}\quad\quad \Ploop =0.\end{aligned}$$ In momentum-space representation these rules correspond to the partial integration. They imply that $\Diamond \{M(1,2)\}=0$, if $M$ contains no internal vertices. Applying rule (\[p-dashed-addition\]) to the convolution of two two-point functions $$\begin{aligned} \label{Convolution} C(1,2)=\oint \di 3 A(1,3)B(3,2)\end{aligned}$$ leads to the following convolution theorem for the gradient approximation $$\begin{aligned} \label{Poisson} \hspace{-5mm}\Diamond \{C(X,p)\}=&\Diamond& \left\{\vphantom{\int}\right.\!\DRhomb{A}{B}\!\left.\vphantom{\int}\right\}\cr =&& \DPoisson \cr &+&\DRhomb{$A$}{$\Diamond B$} + \DRhomb{$\Diamond A$}{$B$} \\[3mm] =&& \Pbr{A(X,p),B(X,p)}\cr &&+A(X,p)\Diamond\{B(X,p)\} +\Diamond\{A(X,p)\}B(X,P).\end{aligned}$$ Besides the standard Poisson bracket expression $\Pbr{A,B}$ it leads to further gradients within each of the two functions (note that the $\Diamond$ operator acts only on the two-point function in the immediate braces to its right). Applied to the r.h.s. of Eq. (\[Grad-KB-Eq\]), this rule indeed provides the decomposition into Poisson bracket and memory terms for the more conventional formulation of the quantum kinetic equation (\[Grad-Sep-KB-Eq\]). $\Phi$-derivable scheme and exact conservation laws {#Phi-Conserv} =================================================== In this section we first review the $\Phi$-derivable properties at the level of self-consistent Dyson or KB equations, then proceed towards the implications for the quantum kinetic equations (\[Grad-KB-Eq\]) and to the proof of the corresponding exact conservation laws (\[Q-E-M\]). In a $\Phi$-derivable scheme the self-energies are generated from a functional $\Phi\{\Gr,\lambda\}$ through the following functional variation, cf. [@IKV] $$\begin{aligned} \label{varphdl1} -\ii \Se(x,y) &=&\mp\frac{\delta\ii \Phi\{\Gr,\lambda\}}{\delta \ii\Gr(y,x)}\times \left\{ \begin{array}{ll} 2\quad&\mbox{for real fields}\\ 1\quad&\mbox{for complex fields} \end{array}\right. .\end{aligned}$$ The $\Phi$ functional itself is given by two-particle irreducible ($2PI$) closed diagrams in terms of [full]{} Green functions of the underlying field theory, i.e. $$\begin{aligned} \label{Phi-def} \ii\Phi\{\Gr,\lambda\}= \left<\exp\left(\ii\oint\di^4 x \lambda(x) {\cal L}^{\rm int}\right)\right>_{2PI}. \end{aligned}$$ The interaction strength $\lambda(x)$, the physical value of which is $\lambda=1$, allows the definition of the interaction energy density $$\begin{aligned} \label{eps-int} {\cal E}^{\scr{int}}(x)&=&\left<-\Lint(x)\right> =-\left.\frac{\delta\ii\Phi}{\delta\ii\lambda(x)}\right\|_{\lambda=1}.\end{aligned}$$ The single-particle potential energy density is defined as $$\begin{aligned} \label{eps-pot} %\!\!\!\!\!\! {\cal E}^{\scr{pot}}(x) &=& \frac{1}{2} \oint\di^4 y \left[ \Se(x,y) (\MP\ii)\Gr(y,x)+(\MP\ii)\Gr(x,y)\Se(y,x)\right] \end{aligned}$$ for complex fields. The local approximants to both ${\cal E}^{\scr{int}}$ and ${\cal E}^{\scr{pot}}$ enter the energy–momentum tensor (\[Q-E-M\]). The diagrammatic series of $\Phi$ given by Eq. (\[Phi-def\]) can be truncated at any level. Keeping the variational property (\[varphdl1\]), this defines a truncated self-consistent scheme for the KB equations (\[KB-Eq\]). The so constructed self-energies lead to a coupling between the different species $a$, which obey detailed balance[^2]. It has been shown [@Baym], see also [@IKV], that such a self-consistent scheme is exactly conserving at the expectation value level and thermodynamically consistent at the same time. A prominent example is the particle-hole ring resummation in Fermi liquid theory in the limit of zero range four-fermion coupling with the following diagrams $$\begin{aligned} \label{F-liquid} \Phi&=&{1\over 2}\PhiHartree+{1\over 4}\PhiSandwich+ \;\sum_{n>2}\frac{1}{2n}\PhiRing{4},\\ {\cal E}^{\scr int}(X)&=& {1\over 2}\EintHartree+{1\over 2}\EintSandwich+\;\sum_{n>2} \;{1\over 2}\;\EintRing{4},\\ \Sigma(x,y)&=&\phantom{{1\over 2}}\SigmaHartree+\phantom{{1\over 2}} \SigmaSandwich+\;\sum_{n>2}\;\,\SigmaRing{3}\end{aligned}$$ Here $n$ counts the number of vertices in the diagram, the full dots denote the external points $X$ and $(x,y)$. Note that compared to ${\cal E}^{\scr{int}}(X)$ all coordinates are integrated in $\Phi$, and therefore its diagrams attain an extra combinatorial factor $1/n$ [@Luttinger]. These diagrams illustrate various levels of approximation. Restricting $\Phi$ just to the one-point term provides the Hartree approximation. From the two-point level on the collision term becomes finite, which also leads to finite damping widths of the particles. Terms, where $\Phi$ has more than two internal vertices, give rise to memory contributions to the self-energies, cf. (\[C-mem\]), due to the intermediate times of the internal vertices in $\Sigma(x,y)$. For this example the memory terms give rise to the famous $T^3\ln T$ term in the specific heat of of liquid $^3$He [@Riedel; @Carneiro; @Baym91] at low temperatures $T$. It is possible to transcribe the variational rules to the local approximation defining a local $\Phi$-functional replacing everywhere $\Gr$ by its local approximant, cf. (\[C-loc\]), $$\begin{aligned} \label{Phi-loc} \Phi_{\scr{loc}}(X)=\left.\Phi\{\Gr,\lambda^{\mp}\} \right\|_{\Gr=\Gr_{\scr{loc}}=\Gr(X,p)}.\end{aligned}$$ Here $X$ is an externally given parameter defining the reference point for the local approximation, and $\lambda^{\mp}$ denotes the scaling factors of the vertices on the time or anti-time ordered branches of the contour. The variational rules (\[varphdl1\]) and (\[eps-int\]) then transcribe to $$\begin{aligned} \label{var-Phi-loc} -\ii \Se_{ik}(X,p) &=&\mp\frac{\delta\ii \Phi_{\scr{loc}}(X)}{\delta \ii\Gr^{ki}(X,p)}\times \left\{ \begin{array}{ll} 2\quad&\mbox{for real fields}\\ 1\quad&\mbox{for complex fields} \end{array}\right. ,\\ \label{eps-int-loc} {\cal E}^{\scr{int}}_{\scr{loc}}(X)&=& -\left.\frac{\delta\ii\Phi_{\scr{loc}}(X)}{\delta\ii\lambda^-} \right\|_{\lambda^-=1} =\left.\frac{\delta\ii\Phi_{\scr{loc}}(X)}{\delta\ii\lambda^+} \right\|_{\lambda^+=1}.\end{aligned}$$ In order to prove that the exact conserving properties indeed survive in the first-order gradient expansion (\[Grad-KB-Eq\]), we shall use the conserving properties of the $\Phi$ diagrams at each internal vertex together with the variational property (\[varphdl1\]). Properties of diagrams ---------------------- We return to the r.h.s. terms of the conservation laws (\[dotQ\]). Due to the variational property (\[varphdl1\]), it is evident that they are given by closed diagrams of the same topology as those of $\Phi$, however, with one point not contour integrated but placed on the time-ordered ($-$) branch with coordinate $X$. To further reveal this relation between the “two-point” representation, as given by the r.h.s. of Eq. (\[dotQ\]), and the [retarded]{} one-point representation of $\Phi$ with respect to a chosen reference point $X^-$, we have to resolve the internal structure of the different diagrams contributing to $\Phi$. Therefore, we first decompose $\Phi$ into the terms of different diagrammatic topology $$\begin{aligned} \label{Phi_D} \ii\Phi_{\scr{loc}}(X)=\sum_{D}\ii\Phi_{\scr{loc}}^{D}(X)\end{aligned}$$ and discuss the features of any such term $\Phi_{\scr{loc}}^D$. We then enumerate the different vertices ($r=1,\dots,n_\lambda$) in each $\Phi_{\scr{loc}}^D$, $n_\lambda$ denoting the number of vertices in $\Phi_{\scr{loc}}^D$. For each such retarded vertex $r^-$ we define a one-point function $\Phi_{\scr{loc}}^D(X;r^-)$, which is obtained by integration and $\{-+\}$ summation over all other vertices except $r$ and by putting $r$ on the time ordered ($-$) branch at coordinate $X$. Subsequently we enumerate the few Green functions attached to $r^-$ and label them as $\Gr^{-k}_\gamma$ or $\Gr^{k-}_{\bar{\gamma}}$, depending on the line sense pointing towards or away from $r^-$, respectively. Thus, the retarded function $\Phi^D(X;r^-)$ can be represented by a diagram, where the retarded point is explicitly pulled out. We draw this as a kind of “parachute” diagram which in [local]{} approximation becomes $$\begin{aligned} \label{parachute} \hspace{-10mm} \ii\Phi_{\scr{loc}}^D(X;r^-)&=& \unitlength1.0mm \Parachute \cr &=& \int\dpi{p_1}\cdots\dpi{\bar{p}_1}\cdots {\cal C}^{Dr}_{k_1\cdots \bar{k}_1\cdots} (X;p_1,\dots,\bar{p}_1,\dots)\cr &&\hspace{10mm}\times\;(\mp\ii) G^{-k_1}_1(X,p_1)\cdots \ii G^{\bar{k}_1-}_{\bar{1}}(X,\bar{p}_1)\cdots \end{aligned}$$ The Green functions attached to the external retarded point $r^-$, form the “suspension cords”. The “canopy” part ${\cal C}^{Dr}$ specifies the rest of the diagram which is not explicitly drawn[^3]. Here all Green functions are taken in the local approximation, i.e. with coordinate $X$ given by the reference point. From the variational principle (\[varphdl1\]) it is clear that $\Phi^D(X;r^-)$ can be interpreted in different equivalent ways depending on which line attached to $r^-$ being opened $$\begin{aligned} \label{parachute1} \hspace{-8mm} \ii\Phi_{\scr{loc}}^D(X;r^-)&=&\MP \int\frac{\di^4 p}{(2\pi)^4}\Gr^{-k}_\gamma(X,p) \Se^{Dr\gamma}_{k-}(X,p) \\ \label{parachute2} &=&\MP\int\frac{\di^4 p}{(2\pi)^4} \Se^{Dr\bar{\gamma}}_{-k}(X,p) \Gr^{k-}_{\bar{\gamma}}(X,p) ,\end{aligned}$$ Here $\Gr_\gamma$ (or $\Gr_{\bar{\gamma}}$) is one of the suspension cords with the arrow pointing towards (or away) from $r$. [No]{} summation over $\gamma$ is implied by these relations! The self-energy terms $\Se^{Dr\gamma}$ and $\Se^{Dr\bar{\gamma}}$ are given by those subdiagrams of $\Phi_{\scr{loc}}^{D}(r^-)$ complementary to $\Gr_\gamma$ and $\Gr_{\bar{\gamma}}$, respectively. According to the variational rule (\[varphdl1\]) they contribute to the self-energy of a given species $a$ as $$\begin{aligned} \label{Dl-sum} -\ii\Se^a_{k-}(X,p) &=& \MP\frac{\delta\ii\Phi_{\scr{loc}}}{\delta \ii\Gr_a^{-k}(X,p)} = -\sum_{D}\sum_{r\in D}\sum_{\gamma}\ii\Se^{Dr\gamma}_{k-}(X,p)\delta_{a\gamma}\end{aligned}$$ and similarly for $\Se^{Dr\bar{\gamma}}_{-k}$ leading to $\Se^a_{-k}$. The Kronecker symbol $\delta_{a\gamma}$ projects on species $a$. The summation in $r$ runs over all vertices in the $\Phi_{\scr{loc}}^{D}$ diagram. It is clear that each Green function in $\Phi_{\scr{loc}}^D$ appears once in the count of $\Se_{k-}$ and once in the count of $\Se_{-k}$. This precisely matches with the two terms required in Eq. (\[SeGr-GrSe\]). We further note that the variation (\[eps-int\]) of $\Phi$ with respect to the coupling strength $\lambda(X)$ can be represented as $$\begin{aligned} \label{var-lambda} {\cal E}^{\scr{int}}(X) =-\frac{\delta\ii\Phi_{\scr{loc}}(X)}{\delta \ii\lambda^-} &=& -\sum_{D}\sum_{r\in D}\Phi_{\scr{loc}}^D(X;r^-).\end{aligned}$$ Charge-current conservation --------------------------- The discussion above shows that any term of the local part of the charge leak $Q_{\mathrm{loc}}(X)$ in Eq. (\[dotQ\]) is given by a closed diagram of the same topology as the $\Phi_{\scr{loc}}^{D}(r^-)$. Indeed, weighting $\Phi_{\scr{loc}}^D(X;r^-)$ with the charges of the in- and out-going Green functions reproduces piece by piece the local gain and loss terms in Eq. (\[SeGr-GrSe\]). The sum over all possible retarded vertices $r$ in diagram $D$ and the sum over all diagrams indeed exactly construct $Q_{\scr{loc}}$ as $$\begin{aligned} \label{Q.loc-1} \hspace{-4mm}Q_{\scr{loc}}(X)&=& \intp\sum_a e_a C^{--}_{a,\;\mathrm{loc}}(X,p) \cr &=& \sum_{D}\sum_{r\in D} \left(\vphantom{\sum_i}\right. \underbrace{ \sum_{\bar{\gamma}} e_{\bar{\gamma}}-\sum_\gamma e_{\gamma} }_{\equiv 0} \left.\vphantom{\sum_i}\right) \ii\Phi_{\scr{loc}}^D(X;r^-).\end{aligned}$$ Since the charge sum vanishes at each vertex, $Q_{\scr{loc}}$ identically vanishes. The fact that the local collision term part vanishes is indeed trivial. The point here is that the cancellation occurs diagram by diagram in terms of $\Phi_{\scr{loc}}^D(X;r^-)$. This has the important consequence that the gradient terms exactly cancel out, too, i.e. $$\begin{aligned} \label{gradQ=0} \sum_a\intp e_a \Diamond C^{--}_{a,\;\mathrm{loc}}(X,p) = \Diamond Q_{\scr{loc}}\equiv 0, \end{aligned}$$ since they are generated by applying linear differential operations to the integrand of $\Phi_{\scr{loc}}^D(X;r^-)$, while the charge factors are constants. Therefore, we have verified that $Q(X)\equiv 0$. This proves current conservation, where the current has the original Noether form. This proof applies to any conserved current of the underlying field theory, which relates to a global symmetry. Energy-Momentum Tensor ---------------------- In a similar way as above the four-momentum weighted terms become $$\begin{aligned} \label{T.2} \hspace{-0.5cm} &&T^{\nu}_{\mathrm{loc}}(X)=\sum_{D}\sum_{r\in D} \int\dpi{p_1}\cdots\dpi{\bar{p}_1}\cdots \left(\vphantom{\sum_i}\right. \underbrace{ \sum_{\bar{\gamma}} p^{\nu}_{\bar{\gamma}}-\sum_\gamma p^{\nu}_{\gamma} }_{\equiv 0} \left.\vphantom{\sum_i}\right) \\\nonumber &&\times {\cal C}^{Dr}_{k_1\cdots \bar{k}_1\cdots} (X;p_1,\dots,\bar{p}_1,\dots) (\mp\ii) G^{-k_1}_1(X,p_1)\cdots \ii G^{\bar{k}_1-}_{\bar{1}}(X,\bar{p}_1)\cdots . \end{aligned}$$ Here we have used the integrand form (\[parachute\]) of the parachute diagram $\Phi(X;r^-)$, since now the weights are momentum dependent. This local collision-term part again drops out in the same way as above. However, the gradient correction to ${T}^{\nu}$ now involves momentum derivatives which can act on the $p^{\nu}$-factors in Eq. (\[Conserv-Eq\]) through partial integrations. Indeed, [only]{} those momentum derivatives survive that act on any of the $p^{\nu}$-factors, since momentum derivatives acting on any of the Green functions leave the vanishing pre-factor in Eq. (\[T.2\]) untouched. The actual evaluation of the gradients is subtle and depends on the detailed topological structure of the diagram. For this purpose we use the diagrammatic rules (\[Gradient-Diag\]) for the gradient terms with the double line as $\partial^{\mu}G(X,p)$, dashed lines as $\propto\partial\delta(p)/\partial p^{\mu}$. For the subsequent analysis we also introduce a new diagrammatic element, a genuine two-point function which gives the four-momentum $p^\nu$-factor “flowing” through the line $$\begin{aligned} \label{p-graf} p^\nu=\Pnu \quad\quad\mbox{with}\quad \PnuLr = -\PnuLl = \frac{\partial p^\nu}{\partial p_\mu}=g^{\mu\nu}. \end{aligned}$$ The last relation results from partial integration and depends only on the sense of the dashed line. Note that the $p^\nu$ factors in the r.h.s. of Eq. (\[Conserv-Eq\]) occur outside the gradient expression. Using the convolution theorem (\[Poisson\]), we first pull the $p^\nu$ factors into the gradient terms in the following manner (using $g^{\mu\nu}={\partial p^\nu}/{\partial p_\mu}$) $$\begin{aligned} \label{convol-pnu} &&\Diamond\left\{\Se\cdot\Gr\right\}p^\nu -p^\nu\Diamond\left\{\Gr\cdot\Se\right\} \cr &&\hspace{5mm} = \underbrace{g^{\mu\nu} \partial_{\mu} \left(\Se\cdot\Gr+\Gr\cdot\Se \right)}_{\MP E_1^{\nu}(X,p)} + \underbrace{ \Diamond \left\{\Se\cdot\Gr\cdot p^\nu-p^\nu\cdot\Gr\cdot\Se \right\}}_{\MP E_2^{\nu} (X,p)}.\end{aligned}$$ Please, note the order of the terms under the $\Diamond$ operator as they do [not commute]{}! The first term, $E_1^\nu$, arose from the Poisson bracket term in Eq. (\[Poisson\]). In accord with Eq. (\[eps-pot\]), the four-momentum integration obviously provides the potential energy density term $$\begin{aligned} \label{E-1} {\textstyle\frac{\ii}{2}}\intp E_1^{\nu}(X,p)=g^{\mu\nu} \partial_\mu {\cal E}^{\scr{pot}}_{\scr{loc}}(X)\end{aligned}$$ of the energy–momentum tensor (\[Q-E-M\]). Thus, we expect the second $E_2^\nu$ term to generate the remaining interaction energy density part $g^{\mu\nu}{\cal E}^{\scr{int}}_{\scr{loc}}(X)$ of $\Theta^{\mu\nu}$. The four-momentum integration of the $E_2^\nu$ term again leads to a coalescence of the two external points. Thus the corresponding diagrams are of the parachute type 0.75mm $$\begin{aligned} \label{Grad-p} \intp E_2^\nu(X,p)= \sum_{D,r\in D} \left( \Diamond \ParachL +\dots + \Diamond \ParachR \right) ,\end{aligned}$$ where both reference points for the gradient expansion coalesce to $r^-$, cf. Eqs. (\[parachute1\]) and (\[parachute2\]). Here the $p^\nu$ factor occurs in sequence at each of the suspension cords reflecting the sum over $\gamma$ and $\bar{\gamma}$. In order to exploit the fact that thereby $\sum_\gamma p^\nu_\gamma-\sum_{\bar{\gamma}} p^\nu_{\bar{\gamma}}$ vanishes, all the different diagrams in the bracket in (\[Grad-p\]) have to be evaluated in the same way. Omitting all labels, one obtains for the first diagram in Eq. (\[Grad-p\]) $$\begin{aligned} \label{grad-p-para} \hspace{-7mm}\Diamond \ParachAll&=&\Paracha+\Parachb_{\vphantom{\int}}\cr &&+\Parachc+\Parachd+\dots+\Parachf_{\vphantom{\int}}\\ \label{Par-survive} &\Longrightarrow&2\;\Parachalpha\;+\; 2\;\Parachbeta\end{aligned}$$ where ${{\cal C}'}=\MP\delta {\cal C}/\delta\ii\Gr$ and the four-scalar product between the double and dashed lines is implied in each diagram. The sequence (a) to (f) defines all diagrams resulting from the gradient expansion. The two diagrams ($\alpha$) and ($\beta$) shown in (\[Par-survive\]) are the only ones that finally survive the $p^{\nu}$ sum, i.e. the sum over “suspension cords” in Eq. (\[Grad-p\]). In detail: diagrams (a) and (b) specify the gradient terms arising from the space-time derivatives acting on Green functions within the canopy part. Using addition theorem (\[p-dashed-addition\]) the dashed lines can be first linked from the bottom point to one definitely chosen upper suspension points [f]{}, as shown in diagram ($\alpha$), and from there then further linked to the end points of the double line. However, the latter terms lead to $p$-derivatives entirely within the canopy, which finally drop out due to the vanishing $p^\nu$ sum. Diagrams (c) to (f) sequentially take the space-time gradients of the Green functions in the suspension cords converting them to a double line in each case. Also here only terms survive, where the momentum derivative acts on the $p^\nu$ factor, leading to digram ($\beta$). One can now compile the terms in Eq. (\[Grad-p\]) provided point [f]{} is kept fixed. Then from ($\alpha$) only a single term survives, namely that where the $p^\nu$ factor is on the line linking to [f]{}, while from ($\beta$) each term survives. The net result leads to a common $g^{\mu\nu}$-factor, cf. (\[p-graf\]), times a total derivative of the entire parachute diagram, i.e. $$\begin{aligned} \label{Grad-p-new} {\textstyle\frac{\ii}{2}}\intp E_2^\nu(X,p)&=& -\ii\sum_{D,r\in D} g^{\mu\nu}\partial_{\mu} \ParachEint \cr &=&-\ii g^{\mu\nu}\partial_{\mu} \underbrace{\sum_{D,r\in D}\ii\Phi_{\scr{loc}}^D(X,r^-)} _{\displaystyle \ii\frac{\delta\ii\Phi_{\scr{loc}}^D}{\delta\ii\lambda^-}}\\ &=&-g^{\mu\nu}\partial_{\mu}{\cal E}^{\scr{int}}(X)\end{aligned}$$ from Eq. (\[var-lambda\]). Together with relation (\[E-1\]), this gives $$\begin{aligned} \label{R(p)-Diag2} T^{\nu}(X)= -\partial^{\nu} \left({\cal E}^{\rm int}_{\rm loc}(X)-{\cal E}^{\rm pot}_{\rm loc}(X)\right)\end{aligned}$$ for the r.h.s of conservation law (\[Conserv-Eq\]). It is seen to be determined by the full divergence of the difference between the interaction energy and single-particle potential energy densities defined by Eqs. (\[eps-int\]) and (\[eps-pot\]), now however evaluated in the [local]{} approximation, i.e. with no gradient terms in the Wigner representation: $$\begin{aligned} \label{eps-pot-local} {\cal E}^{\scr{pot}}_{\rm loc}(X) &=& \int\dpi{p} \left[ \Re\Sa^R(X,p)(\MP\ii)\Gr^{-+}(X,p)\right.\cr &&\hspace{17mm}+\left.\Re\Gr^R(X,p)(\MP\ii)\Sa^{-+}(X,p) \right]\\ &=&- \int\dpi{p} \left[ \Re\left(\frac{\delta\Phi_{\scr{loc}}}{\delta\ii\Gr(X,p)}\right)^R \ii\Gr^{-+}(X,p)\right.\cr &&\hspace{19mm}+\left.\Re\Gr^{R}(X,p)\left(\frac{\delta\Phi_{\scr{loc}}} {\delta\ii\Gr(X,p)}\right)^{-+} \right].\end{aligned}$$ Here the superscript $R$ denotes the corresponding retarded function. For each diagram $\Phi_{\scr{loc}}^D$ the contribution to ${\cal E}^{\scr{int}}_{\rm loc}(X)$ results from the corresponding terms of ${\cal E}^{\scr{pot}}_{\rm loc}(X)$ just by scaling each term by the number of vertices $n_\lambda$ over the number of Green functions $n_G$. Concluding remarks ================== The quantum transport equations in the form originally proposed by Kadanoff and Baym, (\[Grad-KB-Eq\]) or equivalently (\[Grad-Sep-KB-Eq\]), have very pleasant generic features. As possible memory effects in the collision term are to be included only up to first-order space–time gradients, they are local in time to the extent that only the knowledge of the Green functions and their space-time and four-momentum derivatives at this time is required to determine the future evolution. They further preserve the retarded relations among the various real-time components of the Green function. In this paper we have shown that they also possess [exact]{} rather than approximate conservation laws, related to global symmetries of the system, if the scheme is $\Phi$-derivable. The same Noether currents and the same energy-momentum-tensor [@IKV] as those for the original KB equations, however now in their local approximation forms, are exactly conserved for the complete gradient-expanded KB equation, i.e. the quantum transport equations (\[Grad-KB-Eq\]). Thus, $$\begin{aligned} \label{C-j-mu} \partial_{\mu}J^{\mu}(X)&=&0,\quad\quad \label{C-Th-mu-nu} \partial_{\mu}\Theta_{\scr{loc}}^{\mu\nu}(X)=0, \quad\quad\mbox{with}\\ \label{j-mu} J^{\mu}(X)&=&\sum_a e_a\int\dpi{p}p^{\mu}f_a(X,p)A_a(X,p), \\ \label{Th-mu-nu} \Theta_{\scr{loc}}^{\mu\nu}(X)&=&\sum_a \underbrace{\int\dpi{p}v^{\mu}p^{\nu}f_a(X,p)A_a(X,p)} _{\mbox{\footnotesize sum of single particle terms}}\cr &&\hspace{30mm} +g^{\mu\nu} \left({\cal E}^{\rm int}_{\rm loc}(X)-{\cal E}^{\rm pot}_{\rm loc}(X)\right),\end{aligned}$$ here written in terms of the product of phase-space occupation and spectral functions $f_a(X,p)A_a(X,p)=(\MP\ii)\Gr^{-+}_a(X,p)$, are exact consequences of the equations of motion. In order to preserve this exact conserving property, two conditions have to be met. First, the original KB equations should be based on a $\Phi$-derivable approximation scheme that guarantees that the KB equations themselves are conserving [@Baym; @IKV99]. The second condition is that the gradient expansion has to be done systematically, whereby it is important that no further approximations are applied that violate the balance between the different first-order gradient terms. Indeed, all gradient terms residing in the Poisson brackets and in the memory collision term cancel each other for the conserved currents such that the original Noether expression (\[j-mu\]) remains conserved. This implies the compensation of drag-flow terms by all the other gradient terms (back-flow and memory flow), cf. the discussion given in ref. [@IKV99]. For the energy–momentum tensor the gradient terms result into the divergence of the difference between interaction energy density and single-particle potential energy density, cf. (\[Th-mu-nu\]). Thereby ${\cal E}^{\rm int}_{\rm loc}(X)$ and ${\cal E}^{\rm pot}_{\rm loc}(X)$ are obtained from the same $\Phi$-functional in the local approximation as the self-energies driving the equations of motion (\[Grad-KB-Eq\]) and (\[retarded-Eq\]). The so obtained energy–momentum tensor is general and applies to any local coupling scheme. The energy component $\Theta_{\scr{loc}}^{00}$ has a simple interpretation. The first term determines the single-particle energy, which consists of the kinetic and single-particle potential energy parts. Evidently this part by itself is not conserved[^4]. Rather its potential energy part is compensated by the last term, i.e. ${\cal E}^{\rm pot}_{\rm loc}(X)$, such that finally the total kinetic plus interaction energy survive. A typical example for the imbalance of gradient terms is the case, where one neglects the second Poisson bracket term in (\[Grad-Sep-KB-Eq\]). This implies that drag-flow effects contained in the first Poisson bracket remain uncompensated. Also possible memory effects $C_{\scr{mem}}$ in the collision term should not be omitted. Otherwise the Poisson brackets $\Pbr{\Re\Se_a^R,\ii\Gr_a^{-+}} + \Pbr{\ii\Se_a^{-+},\Re\Gr_a^{R}}$ remain uncompensated and, as a consequence, the conservation laws are again violated already in zero-order gradients. A less evident example is the modification of the gradient terms after the formal gradient expansion, as it has been suggested by Botermans and Malfliet [@Bot90], see also [@IKV99]. There one simplifies those self-energy terms that are involved in the Poisson brackets $\Pbr{\Re\Se_a^R,\ii\Gr_a^{-+}}+\Pbr{\ii\Se_a^{-+},\Re\Gr_a^{R}}$, employing quasi-equilibrium relations. This modification implies deviations at second-order gradients only[^5], which is quite acceptable from the formal point of view, when one considers a slow space-time dynamics. However, such kind of modifications violate the strict balance between the gradient terms and thus lead to approximate conservation laws though within first-order gradients. Although even in the BM case an exact conservation law can be formulated for an effective charge [@Leupold], which however only approximately coincides with the true (Noether) one, exact conservation laws of energy and momentum could not be derived yet. The presence of exact conservations puts the Kadanoff–Baym formulation of quantum transport to the level of a generic phenomenological concept. It permits to define phenomenological models for the dynamical description of particles with broad damping widths, such as resonances, with built-in consistency and exact conservation laws, which for practical simulations of complex dynamical systems may even be applied in cases, where the smallness of the gradients can not always be guaranteed. This opens applications to the strong non-equilibrium dynamics of high-energy nuclear collisions. We considered here systems of relativistic bosons and/or non-relativistic particles with a local interaction. The generalization to relativistic fermions with local interactions is straightforward but involves extra complications resulting from the spinor structure of the kinetic equations. Non-relativistic systems with instantaneous interaction at finite spatial distance (two-body potentials) also possess exact conservation laws, if given by a $\Phi$-derivable approximation. Such systems can equivalently be described by a local field theory with mesons mediating the interactions. The appropriate limit towards a potential picture is obtained by treating these interactions instantaneously (non-relativistic limit), i.e. without retardation. This amounts to reduce the corresponding meson Dyson equation to a Poisson equation over the sources of the meson fields and finally eliminating those meson fields. As the local field theory is conserving, the corresponding non-relativistic picture with potentials is conserving too. Note, however, that in this latter case the spatially local energy–momentum tensor does not exist, since energies and momenta are transfered at finite distances through the potentials and it is only possible to formulate the conservation of the [total]{} space-integrated energy and momentum. An important example of such a conserving approximation is given by the ring diagrams $$\begin{aligned} \label{ring} \ii\Phi=&\frac{1}{2}\PhiHartreeT&+\sum_{n=2}^{\infty}\frac{1}{2n} \PhiRingT{8},\\[3mm]\label{Sigma-ring} -\ii\Sigma(x,y)=&\;\;\SigmaHartreeT &+ \;\;\sum_{n=2}^{\infty}\;\SigmaRingT{6}, \end{aligned}$$ which is just the finite range analog of the Fermi-liquid example (\[F-liquid\]), now for the particle-particle resummation channel. Here $n$ counts the number of interaction (dashed) lines representing the two-body potential $V(x_i-x_k)$. The first terms in both expressions are the usual Hartree terms. The remaining sum leads to a conserving $T$-matrix type of approximation for the self-energies (\[Sigma-ring\]), which, e.g., provides thermodynamically consistent description of the nuclear matter [@Bozek; @Dickhoff; @Dewulf]. In the dilute limit it expresses the self-energies through the vacuum scattering $T$-matrix[@Lenz; @Dover71; @IKV]. In this limit it provides a collision term given by vacuum scattering cross sections and at the same time the gradient terms account for the appropriate virial corrections. The latter indirectly depend on the energy variations of the corresponding phase shifts, which give rise to delay time effects [@DP; @IKV] and the corresponding changes of the underlying equation of state (energy–momentum tensor) [@BethU; @DMB; @Mekjian; @VPrakash]. Furthermore, the $s$-channel bosonization of the interactions in the particle-hole channel, cf. ref. [@IKV99], leads to the RPA-approximation. Further applications and considerations of the quantum transport equations will be discussed in a forthcoming paper. A case that still requires a separate treatment is that of derivative coupling of relativistic fields, as e.g., in the case of the pion–nucleon interaction. The reason is that derivative couplings produce extra terms in the currents and the energy–momentum tensor, which require special treatment. Besides all this success at the one-particle expectation value level, one has to keep in mind that partial Dyson resummations still may violate the symmetries at the two-body correlator level and beyond. In particular, it means that the corresponding Ward-Takahashi identities are not necessarily fulfilled within the $\Phi$-derivable Dyson resummation scheme. On the other side the $\Phi$-derivable scheme provides the tools to construct the driving terms and kernels of the corresponding higher order vertex equations (Bethe–Salpeter equations, etc.) which precisely recover the conservation laws at the correlator level [@KadB; @Baym; @Hees-Thesis; @HK01]. So far such equations, however, could mostly be solved in drastically simplified cases (e.g., by RPA-type resummation). A particular challenge represents the inclusion of vector or gauge bosons into a self-consistent Dyson scheme beyond the mean-field level, i.e. at the propagator level, since partial Dyson resummations violate the four-dimensional transversality of the propagators. A practical way out of this difficulty has recently been suggested in refs. [@HK00; @Hees-Thesis]. A further virtue of the $\Phi$-derivable scheme is that it apparently permits a renormalization of the non-perturbative self-consistent self-energies with temperature- and density-independent counter terms [@Hees-Thesis]. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to G. Baym, P. Danielewicz, H. Feldmeier, B. Friman, H. van Hees, C. Greiner, E.E. Kolomeitsev and S. Leupold for fruitful discussions on various aspects of this research. Two of us (Y.B.I. and D.N.V.) highly appreciate the hospitality and support rendered to us at Gesellschaft für Schwerionenforschung. This work has been supported in part by DFG (project 436 Rus 113/558/0). Y.B.I and D.N.V. were partially supported by RFBR grant NNIO-00-02-04012. Y.B.I. was also partially supported by RFBR grant 00-15-96590. Contour Matrix Notation {#Contour} ======================= In calculations that apply the Wigner transformations, it is necessary to decompose the full contour into its two branches—the [time-ordered]{} and [anti-time-ordered]{} branches. One then has to distinguish between the physical space-time coordinates $x,\dots$ and the corresponding contour coordinates $x^{\cal C}$ which for a given $x$ take two values $x^-=(x^-_{\mu})$ and $x^+=(x^+_{\mu})$ ($\mu\in\{0,1,2,3\}$) on the two branches of the contour (see figure 1). Closed real-time contour integrations can then be decomposed as $$\begin{aligned} \label{C-int} \oint\di x^{\cal C} \dots &=&\int_{t_0}^{\infty}\di x^-\dots +\int^{t_0}_{\infty}\di x^+\dots \cr &=&\int_{t_0}^{\infty}\di x^-\dots -\int_{t_0}^{\infty}\di x^+\dots, \end{aligned}$$ where only the time limits are explicitly given. The extra minus sign of the anti-time-ordered branch can conveniently be formulated by a $\{-+\}$ “metric” with the metric tensor in $\{-+\}$ indices $$\begin{aligned} \label{sig} \left(\sigma^{ij}\right)&=& \left(\sigma_{ij}\vphantom{\sigma^{ij}}\right)= {\footnotesize\left(\begin{array}{cc}1&0\\ 0& -1\end{array}\right)}\end{aligned}$$ which provides a proper matrix algebra for multi-point functions on the contour with “co”- and “contra”-contour values. Thus, for any two-point function $F$, the contour values are defined as $$\begin{aligned} \label{Fij} F^{ij}(x,y)&:=&F(x^i,y^j), \quad i,j\in\{-,+\},\quad\mbox{with}\cr F_i^{~j}(x,y)&:=&\sigma_{ik}F^{kj}(x,y),\quad F^i_{~j}(x,y):=F^{ik}(x,y)\sigma_{ki}\cr F_{ij}(x,y)&:=&\sigma_{ik}\sigma_{jl}F^{kl}(x,y), \quad\sigma_i^k=\delta_{ik}\end{aligned}$$ on the different branches of the contour. Here summation over repeated indices is implied. Then contour folding of contour two-point functions, e.g. in Dyson equations, simply becomes $$\begin{aligned} \label{H=FG} H(x^i,y^k)=H^{ik}(x,y)&=&\oint\di z^{\cal C} F(x^i,z^{\cal C})G(z^{\cal C},y^k) \cr &=&\int\di z F^i_{~j}(x,z)G^{jk}(z,y)\end{aligned}$$ in the matrix notation. For any multi-point function the external point $x_{max}$, which has the largest physical time, can be placed on either branch of the contour without changing the value, since the contour-time evolution from $x_{max}^-$ to $x_{max}^+$ provides unity. Therefore, one-point functions have the same value on both sides of the contour. Due to the change of operator ordering, genuine multi-point functions are, in general, discontinuous, when two contour coordinates become identical. In particular, two-point functions like $\ii F(x,y)=\left<\Tc {\widehat A(x)}\medhat{B}(y)\right>$ become $$\begin{aligned} \label{Fxy} \hspace{-0.5cm}\ii F(x,y) &=& \left(\begin{array}{ccc} \ii F^{--}(x,y)&&\ii F^{-+}(x,y)\\[3mm] \ii F^{+-}(x,y)&&\ii F^{++}(x,y) \end{array}\right) \cr &=& \left(\begin{array}{ccc} \left<{\cal T}\medhat{A}(x)\medhat{B}(y)\right>&\hspace{5mm}& \mp \left<\medhat{B}(y)\medhat{A}(x)\right>\\[5mm] \left<\medhat{A}(x)\medhat{B}(y)\right> &&\left<{\cal T}^{-1}\medhat{A}(x)\medhat{B}(y)\right> \end{array}\right), \end{aligned}$$ where ${\cal T}$ and ${\cal T}^{-1}$ are the usual time and anti-time ordering operators. Since there are altogether only two possible orderings of the two operators, in fact given by the Wightman functions $F^{-+}$ and $F^{+-}$, which are both continuous, not all four components of $F$ are independent. Eq. (\[Fxy\]) implies the following relations between non-equilibrium and usual retarded and advanced functions $$\begin{aligned} \label{Fretarded} F^R(x,y)&:=&\Theta(x_0-y_0)\left(F^{+-}(x,y)-F^{-+}(x,y)\right),\nonumber\\ &=&F^{--}(x,y)-F^{-+}(x,y)=F^{+-}(x,y)-F^{++}(x,y)\nonumber\\ F^A (x,y)&:=&-\Theta(y_0-x_0)\left(F^{+-}(x,y)-F^{-+}(x,y)\right)\nonumber\\ &=&F^{--}(x,y)-F^{+-}(x,y)=F^{-+}(x,y)-F^{++}(x,y),\end{aligned}$$ where $\Theta(x_0-y_0)$ is the step function of the time difference. The rules for the co-contour functions $F_{--}$ etc. follow from Eq. (\[Fij\]). Discontinuities of a two-point function may cause problems for differentiations, in particular, since they often occur simultaneously in products of two or more two-point functions. The proper procedure is, first, with the help of Eq. (\[Fretarded\]) to represent the discontinuous parts in $F^{--}$ and $F^{++}$ by the continuous $F^{-+}$ and $F^{+-}$ times $\Theta$-functions, then to combine all discontinuities, e.g. with respect to $x_0-y_0$, into a single term proportional to $\Theta(x_0-y_0)$, and finally to apply the differentiations. One can easily check that in the following particularly relevant cases $$\begin{aligned} \label{diffrules0} && \oint\di z\left( F(x^i,z)G(z,x^j) - G(x^i,z)F(z,x^j)\right), \\ \label{diffrules} && \frac{\partial}{\partial x_{\mu}} \oint\di z\left( F(x^i,z)G(z,x^j)+G(x^i,z)F(z,x^j)\right), %\quad\mbox{\rm and} \\ \label{diffrules1} &&\left[\left(\frac{\partial}{\partial x_{\mu}} -\frac{\partial}{\partial y_{\mu}}\right) \oint\di z \vphantom{\frac{\partial}{\partial x_{\mu}} -\frac{\partial}{\partial y_{\mu}}} \left(F(x^i,z)G(z,y^j)-G(x^i,z)F(z,y^j)\right)\right]_{x=y}\end{aligned}$$ [all discontinuities exactly cancel]{}. Thereby, these values are independent of the placement of $x^i$ and $x^j$ on the contour, i.e. the values are only functions of the physical coordinate $x$. For such two point functions complex conjugation implies $$\begin{aligned} \label{ComplexConjugate} \left(\ii F^{-+}(x,y)\right)^&=&\ii F^{-+}(y,x) \quad\Rightarrow\quad \ii F^{-+}(X,p)=\mbox{real},\nonumber\\ \left(\ii F^{+-}(x,y)\right)^&=&\ii F^{+-}(y,x) \quad\Rightarrow\quad \ii F^{+-}(X,p)=\mbox{real},\nonumber\\ \left(\ii F^{--}(x,y)\right)^&=&\ii F^{++}(y,x) \quad\Rightarrow\quad \left(\ii F^{--}(X,p)\right)^=\ii F^{++}(X,p),\nonumber\\ \left(F^R(x,y)\right)^&=&F^A(y,x) \quad\hspace{3.5mm}\Rightarrow\quad \left(F^R(X,p)\right)^=F^A(X,p),\end{aligned}$$ where the right parts specify the corresponding properties in the Wigner representation. 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[^2]: generalizing the special recipes for broad resonances given in ref.[@DB91] [^3]: For definiteness we have drawn a diagram with 4 propagators linking to the retarded point $r^-$. All considerations, however, are independent of the coupling scheme which can even vary from vertex to vertex. For simple $\Phi$ diagrams, such as those with only two vertices at all, the canopy part reduces simply to a single point, cf. the second diagram in Eq. (\[F-liquid\]). [^4]: Contrary to the constructions given in ref. [@Cass00]. [^5]: This freedom of choice is due to the fact that various redundant combinations of the KB equations lead to non-redundant equations after gradient approximation due to the asymmetric treatment of sums and differences of the KB equations and their adjoint ones in the gradient approximation. The so called mass-shell equation indeed agrees in the first-order gradient terms with the here considered transport Eq. (\[Grad-KB-Eq\]), cf. [@IKV99], however they differ in higher orders of gradients.
--- abstract: 'We consider two kinds of periodicities of mutations in cluster algebras. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the sequence is ‘regular’, they are particularly natural generalizations of the known ‘classic’ T- and Y-systems. Furthermore, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. We prove the identities when exchange matrices are skew symmetric.' author: - Tomoki Nakanishi title: Periodicities in cluster algebras and dilogarithm identities --- Primary 13F60; Secondary 17B37 cluster algebras, T-systems, Y-systems, dilogarithm Introduction ============ Cluster algebras were introduced by Fomin and Zelevinsky [@Fomin02]. They naturally appear in several different areas of mathematics, for example, in geometry of surfaces, in coordinate rings of algebraic varieties related to Lie groups, in the representation theory of algebras, and also in the representation theory of quantum groups, etc. See [@Fock05; @Gekhtman05; @Geiss07; @Caldero06; @Buan06; @DiFrancesco09a], to name a few. The simplest and the most tractable class of cluster algebras are the cluster algebras with finitely many seeds. They are called the [cluster algebras of finite type]{} and play a fundamental role in many applications. Fomin and Zelevinsky classified the cluster algebras of finite type by the Dynkin diagrams in their pioneering works [@Fomin03a; @Fomin03b]. They also clarified the intimate relation between cluster algebras of finite type and root systems of finite type. In particular, a remarkable periodicity property of mutations of seeds was discovered and proved; it is related to the Coxeter elements of the Weyl groups [@Fomin03a; @Fomin03b; @Fomin07]. In this paper we focus on two kinds of periodicities of mutations in general cluster algebras. The first one is the periodicity of [exchange matrices (or quivers)]{} under a sequence of mutations. In other words, the [exchange relations]{} of clusters and coefficient tuples are periodic under such a sequence of mutations. The second one is the periodicity of [seeds]{} under a sequence of mutations. The latter periodicity implies the former one, but the converse is not true. Let us briefly explain the background for this study. Many examples of such periodicities appeared in connection with systems of algebraic relations called [T-systems and Y-systems]{} [@Zamolodchikov91; @Klumper92; @Kuniba92; @Ravanini93; @Kuniba94a; @Hernandez07a] and the [dilogarithm identities]{} [@Kirillov86; @Kirillov89; @Bazhanov90; @Kirillov90; @Kuniba93a; @Gliozzi95] which originated in the study of integrable models in two dimensions. In retrospect, they are part of the cluster algebraic structure which appeared prior to the notion of cluster algebra itself. Naturally and inevitably, their cluster algebraic nature has been gradually revealed recently [@Fomin03b; @Chapoton05; @Fomin07; @Keller08; @DiFrancesco09a; @Hernandez09; @Kuniba09; @Inoue10c; @Nakajima09; @Keller10; @Nakanishi09; @Inoue10a; @Inoue10b; @Nakanishi10a; @Nakanishi10b], and it turned out that the cluster algebraic formulation and machinery are very powerful and essential to understanding their properties. Since examples of periodicities of exchange relations and seeds are accumulating, it may be a good time to reverse the viewpoint, namely, to formulate T-systems, Y-systems, and dilogarithm identities in a more general and unified setting, starting from general cluster algebras with such periodicity properties. This is the subject of the paper. Let us summarize the main result of the paper. For any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. When the sequence is ‘regular’, they are particularly natural generalizations of the known ‘classic’ T- and Y-systems. The definition of the term ‘regular’ is found in Definition \[def:regular\]. Furthermore, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. We prove the identities when exchange matrices are skew symmetric. We expect that there will be several applications of the result in various areas related to cluster algebras. We mention that many examples of periodicities of exchange matrices were constructed by Fordy and Marsh [@Fordy09] and the associated T-systems were also introduced. The relation between cluster algebras and the dilogarithm was also studied earlier by Fock and Goncharov [@Fock09; @Fock07]. In fact, Proposition \[prop:local\] is motivated by their formulas. However, there is a subtle but important difference; that is, we use [$F$-polynomials]{} of [@Fomin07] in . See Section \[subsec:local\] for more details. The organization of the paper is as follows. In Section 2 we introduce the basic notions for periodicities of exchange matrices and seeds. In Section 3 we present some of known examples of periodicities of exchange matrices (or quivers) and seeds, most of which are connected to ‘classic’ T- and Y-systems. In Section 4 we give Restriction/Extension Theorem of periodicities of seeds. After these preparations, in Section 5, for any sequence of mutations under which exchange matrices are periodic, we define the associated T- and Y-systems. Here we do not require the periodicity of seeds. Special attention is paid to the case when the sequence is regular. In Section 6, for any sequence of mutations under which seeds are periodic, we formulate the associated dilogarithm identity. Then, we prove the identities when exchange matrices are skew symmetric in Theorem \[thm:DI\], which is the main theorem of the paper. Acknowledgments {#acknowledgments .unnumbered} =============== I thank Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, Roberto Tateo, and Junji Suzuki for sharing their insights in the preceding joint works. I am grateful to Frédéric Chapoton, Vladimir Fock, Bernhard Keller, Robert Marsh, Pierre-Guy Plamondon, and Andrei Zelevinsky for kindly communicating their works and also for useful comments. Finally, I thank Andrzej Skowronski for his kind invitation to this proceedings volume. Periodicities of exchange matrices and seeds ============================================ Cluster algebras with coefficients {#subsec:cluster} ---------------------------------- In this subsection we recall the definition of the cluster algebras with coefficients and some of their basic properties, following the convention in [@Fomin07] with slight change of notations and terminology. See [@Fomin07] for more details and information. Fix an arbitrary semifield $\mathbb{P}$, i.e., an abelian multiplicative group endowed with a binary operation of addition $\oplus$ which is commutative, associative, and distributive with respect to the multiplication [@Hutchins90]. Let $\mathbb{Q}\mathbb{P}$ denote the quotient field of the group ring $\mathbb{Z}\mathbb{P}$ of $\mathbb{P}$. Let $I$ be a finite set, and let $B=(b_{ij})_{i,j\in I}$ be a skew symmetrizable (integer) matrix; namely, there is a diagonal positive integer matrix $D$ such that ${}^{t}(DB)=-DB$. Let $x=(x_i)_{i\in I}$ be an $I$-tuple of formal variables, and let $y=(y_i)_{i\in I}$ be an $I$-tuple of elements in $\mathbb{P}$. For the triplet $(B,x,y)$, called the [initial seed]{}, the [cluster algebra $\mathcal{A}(B,x,y)$ with coefficients in $\mathbb{P}$]{} is defined as follows. Let $(B',x',y')$ be a triplet consisting of skew symmetrizable matrix $B'=(b'_{ij})_{i,j\in I}$, an $I$-tuple $x'=(x'_i)_{i\in I}$ with $x'_i\in \mathbb{Q}\mathbb{P}(x)$, and an $I$-tuple $y'=(y'_i)_{i\in I}$ with $y'_i\in \mathbb{P}$. For each $k\in I$, we define another triplet $(B'',x'',y'')=\mu_k(B',x',y')$, called the [mutation of $(B',x',y')$ at $k$]{}, as follows. [(i) Mutation of matrix.]{} $$\begin{aligned} \label{eq:Bmut} b''_{ij}= \begin{cases} -b'_{ij}& \mbox{$i=k$ or $j=k$},\\ b'_{ij}+\frac{1}{2} (\|b'_{ik}\|b'_{kj} + b'_{ik}\|b'_{kj}\|) &\mbox{otherwise}. \end{cases}\end{aligned}$$ [(ii) Exchange relation of coefficient tuple.]{} $$\begin{aligned} \label{eq:coef} y''_i = \begin{cases} \displaystyle {y'_k}{}^{-1}&i=k,\\ \displaystyle y'_i \frac{1}{(1\oplus {y'_k}^{-1})^{b'_{ki}}}& i\neq k,\ b'_{ki}\geq 0,\\ y'_i (1\oplus y'_k)^{-b'_{ki}}& i\neq k,\ b'_{ki}\leq 0.\\ \end{cases}\end{aligned}$$ [(iii) Exchange relation of cluster.]{} $$\begin{aligned} \label{eq:clust} x''_i = \begin{cases} \displaystyle \frac{y'_k \prod_{j: b'_{jk}>0} {x'_j}^{b'_{jk}} + \prod_{j: b'_{jk}<0} {x'_j}^{-b'_{jk}} }{(1\oplus y'_k)x'_k} & i= k,\\ {x'_i}&i\neq k.\\ \end{cases}\end{aligned}$$ It is easy to see that $\mu_k$ is an involution, namely, $\mu_k(B'',x'',y'')=(B',x',y')$. Now, starting from the initial seed $(B,x,y)$, iterate mutations and collect all the resulting triplets $(B',x',y')$. We call $(B',x',y')$ a [seed]{}, $y'$ and $y'_i$ a [coefficient tuple]{} and a [coefficient]{}, $x'$ and $x'_i$, a [cluster]{} and a [cluster variable]{}, respectively. The [cluster algebra $\mathcal{A}(B,x,y)$ with coefficients in $\mathbb{P}$]{} is the $\mathbb{Z}\mathbb{P}$-subalgebra of the rational function field $\mathbb{Q}\mathbb{P}(x)$ generated by all the cluster variables. Similarly, the [coefficient group $\mathcal{G}(B,y)$ with coefficients in $\mathbb{P}$]{} is the multiplicative subgroup of the semifield $\mathbb{P}$ generated by all the coefficients $y'_i$ together with $1\oplus y'_i$. It is standard to identify a [skew symmetric]{} (integer) matrix $B=(b_{ij})_{i,j\in I}$ with a [quiver $Q$ without loops or 2-cycles]{}. The set of the vertices of $Q$ is given by $I$, and we put $b_{ij}$ arrows from $i$ to $j$ if $b_{ij}>0$. The mutation $Q''=\mu_k(Q')$ of a quiver $Q'$ is given by the following rule: For each pair of an incoming arrow $i\rightarrow k$ and an outgoing arrow $k\rightarrow j$ in $Q'$, add a new arrow $i\rightarrow j$. Then, remove a maximal set of pairwise disjoint 2-cycles. Finally, reverse all arrows incident with $k$. Let $\mathbb{P}_{\mathrm{univ}}(y)$ be the [universal semifield]{} of the $I$-tuple of generators $y=(y_i)_{i\in I}$, namely, the semifield consisting of the [subtraction-free]{} rational functions of formal variables $y$ with usual multiplication and addition in the rational function field $\mathbb{Q}(y)$. We write $\oplus$ in $\mathbb{P}_{\mathrm{univ}}(y)$ as $+$ for simplicity when it is not confusing. [From now on, unless otherwise mentioned, we set the semifield $\mathbb{P}$ for $\mathcal{A}(B,x,y)$ to be $\mathbb{P}_{\mathrm{univ}}(y)$, where $y$ is the coefficient tuple in the initial seed $(B,x,y)$.]{} Let $\mathbb{P}_{\mathrm{trop}}(y)$ be the [tropical semifield]{} of $y=(y_i)_{i\in I}$, which is the abelian multiplicative group freely generated by $y$ endowed with the addition $\oplus$ $$\begin{aligned} \label{eq:trop} \prod_i y_i^{a_i}\oplus \prod_i y_i^{b_i} = \prod_i y_i^{\min(a_i,b_i)}.\end{aligned}$$ There is a canonical surjective semifield homomorphism $\pi_{\mathbf{T}}$ (the [tropical evaluation]{}) from $\mathbb{P}_{\mathrm{univ}}(y)$ to $\mathbb{P}_{\mathrm{trop}}(y)$ defined by $\pi_{\mathbf{T}}(y_i)= y_i$ and $\pi_{\mathbf{T}}(\alpha)=1$ ($\alpha \in \mathbb{Q}_+$). For any coefficient $y'_i$ of $\mathcal{A}(B,x,y)$, let us write $[y'_i]_{\mathbf{T}}:= \pi_{\mathbf{T}}(y'_i)$ for simplicity. We call $[y'_i]_{\mathbf{T}}$’s the [tropical coefficients]{} (the [principal coefficients]{} in [@Fomin07]). They satisfy the exchange relation by replacing $y'_i$ with $[y'_i]_{\mathbf{T}}$ with $\oplus$ being the addition in . We also extend this homomorphism to the homomorphism of fields $\pi_{\mathbf{T}}:(\mathbb{Q}\mathbb{P}_{\mathrm{univ}}(y))(x) \rightarrow (\mathbb{Q}\mathbb{P}_{\mathrm{trop}}(y))(x)$. To each seed $(B',x',y')$ of $\mathcal{A}(B,x,y)$ we attach the [$F$-polynomials]{} $F'_i(y)\in \mathbb{Q}(y)$ ($i\in I$) by the specialization of $[x'_i]_{\mathbf{T}}$ at $x_j=1$ ($j\in I$). It is, in fact, a polynomial in $y$ with integer coefficients due to the Laurent phenomenon [@Fomin07 Proposition 3.6]. For definiteness, let us take $I=\{1,\dots,n\}$. Then, $x'$ and $y'$ have the following factorized expressions [@Fomin07 Proposition 3.13, Corollary 6.3] by the $F$-polynomials. $$\begin{aligned} \label{eq:gF} x'_i &= \left( \prod_{j=1}^n x_j^{g'_{ji}} \right) \frac{ F'_i(\hat{y}_1, \dots,\hat{y}_n) } { F'_i(y_1, \dots,y_n) }, \quad \hat{y}_i=y_i\prod_{j=1}^n x_j^{b_{ji}}, \\ \label{eq:Yfact} y'_i&= [y'_i]_{\mathbf{T}} \prod_{j=1}^n F'_j(y_1,\dots,y_n)^{b'_{ji}}.\end{aligned}$$ The integer vector $\mathbf{g}'_i=(g'_{1i},\dots,g'_{ni})$ ($i=1,\dots,n$) uniquely determined by for each $x'_i$ is called the [$g$-vector]{} for $x'_i$. \[conj:pos\] For any cluster algebra $\mathcal{A}(B,x,y)$ with skew symmetrizable matrix $B$, the following properties hold. \(a) Each tropical coefficient $[y'_i]_{\mathbf{T}}$ is not $1$, and, either positive or negative Laurent monomial in $y$. \(b) (the ‘sign coherence’) For each $i$, the $i$th components of $g$-vectors, $g'_{ij}$ ($j=1,\dots,n+1$) in , are simultaneously nonpositive or nonnegative. \(c) Each $F$-polynomial $F'_i(y)$ has a constant term 1. This conjecture was proved in the skew symmetric case by [@Derksen10; @Plamondon10b; @Nagao10] with the result of [@Fomin07 Proposition 5.6]. \[thm:pos\] Conjecture \[conj:pos\] is true for any skew symmetric matrix $B$. Let $\mathbf{i}=(i_1,\dots,i_r)$ be an $I$-sequence, namely, $i_1,\dots,i_r\in I$. We define the [composite mutation]{} $\mu_{\mathbf{i}}$ by $\mu_{\mathbf{i}}=\mu_{i_r} \cdots \mu_{i_2} \mu_{i_1}$, where the product means the composition. For $I$-sequences $\mathbf{i}$ and $\mathbf{i}'$, we write $\mathbf{i}{\sim}_B \mathbf{i}'$ if $\mu_{\mathbf{i}}(B,x,y)= \mu_{\mathbf{i}'}(B,x,y)$. The following fact will be used implicitly and frequently. \[lem:order\] Let $B=(b_{ij})_{i,j\in I}$ be a skew symmetrizable matrix and let $\mathbf{i}=(i_1,\dots,i_r)$ be an $I$-sequence. Suppose that $b_{{i_a}{i_b}}=0$ for any $1\leq a,b \leq r$. Then, the following facts hold. \(a) For any permutation $\sigma$ of $\{1,\dots, r\}$, we have $$\begin{aligned} {\mathbf{i}}\sim_B {(i_{\sigma(1)},\dots,i_{\sigma(r)})}.\end{aligned}$$ \(b) Let $B'=\mu_{\mathbf{i}}(B)$. Then, $b'_{{i_a}{i_b}}=0$ holds for any $1\leq a,b \leq r$. \(c) Let $(B',x',y')=\mu_{\mathbf{i}}(B,x,y)$. Then, $(B,x,y)=\mu_{\mathbf{i}}(B',x',y')$. The facts (a) and (b) are easily verified from –. The fact (c) follows from (a), (b), and the involution property of each mutation $\mu_i$. Periodicities of exchange matrices and seeds -------------------------------------------- \[def:period\] Let $\mathcal{A}(B,x,y)$ be a cluster algebra, $(B',x',y')$ be a seed of $\mathcal{A}(B,x,y)$, $\mathbf{i}=(i_1,\dots,i_r)$ be an $I$-sequence, $(B'',x'',y'')=\mu_{\mathbf{i}}(B',x',y')$, and $\nu:I \rightarrow I$ be a bijection. \(a) We call the sequence $\mathbf{i}$ a [$\nu$-period]{} of $B'$ if $b''_{\nu(i)\nu(j)}=b'_{ij}$ ($i,j\in I$) holds; furthermore, if $\nu=\mathrm{id}$, we simply call it a [period]{} of $B'$. \(b) We call the sequence $\mathbf{i}$ a [$\nu$-period]{} of $(B',x',y')$ if $$\begin{aligned} \label{eq:periodBxy} b''_{\nu(i)\nu(j)}=b'_{ij}, \quad x''_{\nu(i)}=x'_i, \quad y''_{\nu(i)}=y'_i \quad (i,j\in I)\end{aligned}$$ holds; furthermore, if $\nu=\mathrm{id}$, we simply call it a [period]{} of $(B',x',y')$. For any seed $(B',x',y')$ of $\mathcal{A}(B,x,y)$, $\mathcal{A}(B',x',y')$ is isomorphic to $\mathcal{A}(B,x,y)$ as a cluster algebra. Therefore, by resetting the initial seed if necessary, we may concentrate on the situation where $(B',x',y')=(B,x,y)$ in the above without losing generality. If $\mathbf{i}$ is a $\nu$-period of $(B,x,y)$, then, of course, it is a $\nu$-period of $B$. However, the converse does not hold, in general. If $\mathbf{i}$ is a $\nu$-period of $B$ and there is a nontrivial automorphism $\omega:I\rightarrow I$ of $B$, i.e., $b_{\omega(i)\omega(j)}=b_{ij}$, then $\mathbf{i}$ is also an $\nu\omega$-period. On the other hand, there is no such ambiguity for a $\nu$-period of $(B,x,y)$, since cluster variables $x_i$ $(i\in I)$ are algebraically independent. If $\mathbf{i}$ and $\mathbf{i}'$ are a $\nu$-period and a $\nu'$-period of $B$, or $(B,x,y)$, respectively, then the concatenation of sequences $$\begin{aligned} \mathbf{i}\,\|\, \nu(\mathbf{i}') :=(i_1,\dots,i_r, \nu(i'_1),\dots,\nu(i'_{r'}))\end{aligned}$$ is a $\nu\nu'$-period of $B$, or $(B,x,y)$. In particular, $\mathbf{i}\,\|\, \nu(\mathbf{i})\,\|\, \cdots \,\|\, \nu^{p-1}(\mathbf{i})$ is a $\nu^p$-period of $B$, or $(B,x,y)$, for any positive integer $p$. Since $\nu$ acts on a finite set $I$, it has a finite order, say, $g$. Define $$\begin{aligned} \mathbf{j}(\mathbf{i},\nu):= \mathbf{i}\,\|\, \nu(\mathbf{i})\,\|\, \cdots \,\|\, \nu^{g-1}(\mathbf{i}).\end{aligned}$$ Then, $\mathbf{j}(\mathbf{i},\nu)$ is a period of $B$, or $(B,x,y)$. Examples of periodicities of exchange matrices (or quivers) and seeds will be given in Section \[sec:examples\]. \[prop:opposite\] If $\mathbf{i}$ is a period of $(B,x,y)$ in $\mathcal{A}(B,x,y)$, then $\mathbf{i}$ is also a period of $(-B,x,y)$ in $\mathcal{A}(-B,x,y)$. This is due to the duality between the exchange $B\leftrightarrow -B$, $y\leftrightarrow y^{-1}$. Namely, the correspondence of seeds $(B',x',y')$ in $\mathcal{A}(B,x,y)$ and $(-B',x',y'^{-1})$ in $\mathcal{A}(-B,x,y^{-1})$ commutes with mutations and yields the isomorphism of cluster algebras. When $B$ is skew symmetric, the transformation $B\leftrightarrow -B$ corresponds to the transformation $Q\leftrightarrow Q^{\mathrm{op}}$, where $Q^{\mathrm{op}}$ is the opposite quiver of $Q$. Criterion of periodicity of seeds for skew symmetric case --------------------------------------------------------- In general, checking the condition directly is a [very]{} difficult task. However, at least when $B$ is [skew symmetric]{}, one can reduce the condition drastically, thanks to the existence of the categorification with 2-Calabi-Yau property by Plamondon [@Plamondon10a; @Plamondon10b]. \[thm:tropperiod\] Assume that the matrix $B$ in Definition \[def:period\] is skew symmetric. Then, the condition holds if and only if the following condition holds: $$\begin{aligned} \label{eq:periody} [y''_{\nu(i)}]_{\mathbf{T}}=[y'_i]_{\mathbf{T}} \quad (i\in I).\end{aligned}$$ We expect that Theorem \[thm:tropperiod\] holds for any skew symmetrizable matrix $B$. Let us also mention that the application software by Bernhard Keller [@Keller08c] is a practical and versatile tool to check and explore periodicities of quivers and seeds. Examples {#sec:examples} ======== We present some of known examples of periodicities of exchange matrices (or quivers) and seeds. Examples of periodicities of exchange matrices ---------------------------------------------- There are plenty of examples of periodicities of exchange matrices. We identify a skew symmetric matrix $B$ and the corresponding quiver $Q$ as in Section \[subsec:cluster\]. \[ex:preperiod\] [Cluster algebras for bipartite matrices with alternating property [@Fomin07].]{} Assume that a skew symmetrizable matrix $B$ is [bipartite]{}; namely, the index set $I$ of $B$ admits the decomposition $I=I_+\sqcup I_-$ such that, for any pair $(i,j)$ with $b_{ij}\neq 0$, either $i\in I_+$, $j\in I_-$ or $i\in I_-$, $j\in I_+$ holds. Assume further that $B$ has the following ‘alternating’ property: $b_{ij}>0$ only if $i\in I_+$, $j\in I_-$. (In the quiver picture, $i\in I_+$ is a source and $i\in I_-$ is a sink so that the quiver is alternating.) Let $\mathbf{i}_+$ and $\mathbf{i}_-$ be the sequences of all the distinct elements of $I_+$ and $I_-$, respectively, where the order of the sequence is chosen arbitrarily thanks to Lemma \[lem:order\]. Then, $\mu_{\mathbf{i}_+}(B) =-B$ and $\mu_{\mathbf{i}_-}(-B) = B$. Thus, $\mathbf{i}=\mathbf{i}_+ \, \| \, \mathbf{i}_-$ is a period of $B$. The associated Y-system and ‘T-system’ (in our terminology) were studied in detail in [@Fomin07]. (A matrix $B$ here is called a ‘bipartite matrix’ in [@Fomin07].) (300,288)(0,-15) (0,0) [ (0,60) (0,135) (0,210) (30,0) (30,15) (30,30) (30,45) (30,60) (30,75) (30,90) (30,105) (30,120) (30,135) (30,150) (30,165) (30,180) (30,195) (30,210) (30,225) (30,240) (30,255) (30,270) (30,3)[(0,1)[9]{}]{} (30,27)[(0,-1)[9]{}]{} (30,33)[(0,1)[9]{}]{} (30,57)[(0,-1)[9]{}]{} (30,63)[(0,1)[9]{}]{} (30,87)[(0,-1)[9]{}]{} (30,93)[(0,1)[9]{}]{} (30,117)[(0,-1)[9]{}]{} (30,123)[(0,1)[9]{}]{} (30,147)[(0,-1)[9]{}]{} (30,153)[(0,1)[9]{}]{} (30,177)[(0,-1)[9]{}]{} (30,183)[(0,1)[9]{}]{} (30,207)[(0,-1)[9]{}]{} (30,213)[(0,1)[9]{}]{} (30,237)[(0,-1)[9]{}]{} (30,243)[(0,1)[9]{}]{} (30,267)[(0,-1)[9]{}]{} (0,132)[(0,-1)[69]{}]{} (0,138)[(0,1)[69]{}]{} (3,51)[(1,-2)[24]{}]{} (27,19)[(-2,3)[23]{}]{} (3,58)[(1,-1)[24]{}]{} (27,47)[(-2,1)[23]{}]{} (3,60)[(1,0)[24]{}]{} (27,73)[(-2,-1)[23]{}]{} (3,62)[(1,1)[24]{}]{} (27,101)[(-2,-3)[23]{}]{} (3,69)[(1,2)[24]{}]{} (27,135)[(-1,0)[24]{}]{} (3,201)[(1,-2)[24]{}]{} (27,169)[(-2,3)[23]{}]{} (3,208)[(1,-1)[24]{}]{} (27,197)[(-2,1)[23]{}]{} (3,210)[(1,0)[24]{}]{} (27,223)[(-2,-1)[23]{}]{} (3,212)[(1,1)[24]{}]{} (27,251)[(-2,-3)[23]{}]{} (3,219)[(1,2)[24]{}]{} (3,-2) [ (-17,60)[$-$]{} (-17,135)[$+$]{} (-17,210)[$-$]{} ]{} (4,-2) [ (30,0)[$+$]{} (30,15)[$-$]{} (30,30)[$+$]{} (30,45)[$-$]{} (30,60)[$+$]{} (30,75)[$-$]{} (30,90)[$+$]{} (30,105)[$-$]{} (30,120)[$+$]{} (30,135)[$-$]{} (30,150)[$+$]{} (30,165)[$-$]{} (30,180)[$+$]{} (30,195)[$-$]{} (30,210)[$+$]{} (30,225)[$-$]{} (30,240)[$+$]{} (30,255)[$-$]{} (30,270)[$+$]{} ]{} ]{} (70,0) [ (0,60) (0,135) (0,210) (30,0) (30,15) (30,30) (30,45) (30,60) (30,75) (30,90) (30,105) (30,120) (30,135) (30,150) (30,165) (30,180) (30,195) (30,210) (30,225) (30,240) (30,255) (30,270) (30,3)[(0,1)[9]{}]{} (30,27)[(0,-1)[9]{}]{} (30,33)[(0,1)[9]{}]{} (30,57)[(0,-1)[9]{}]{} (30,63)[(0,1)[9]{}]{} (30,87)[(0,-1)[9]{}]{} (30,93)[(0,1)[9]{}]{} (30,117)[(0,-1)[9]{}]{} (30,123)[(0,1)[9]{}]{} (30,147)[(0,-1)[9]{}]{} (30,153)[(0,1)[9]{}]{} (30,177)[(0,-1)[9]{}]{} (30,183)[(0,1)[9]{}]{} (30,207)[(0,-1)[9]{}]{} (30,213)[(0,1)[9]{}]{} (30,237)[(0,-1)[9]{}]{} (30,243)[(0,1)[9]{}]{} (30,267)[(0,-1)[9]{}]{} (0,63)[(0,1)[69]{}]{} (0,207)[(0,-1)[69]{}]{} (27,19)[(-2,3)[23]{}]{} (3,58)[(1,-1)[24]{}]{} (27,47)[(-2,1)[23]{}]{} (3,60)[(1,0)[24]{}]{} (27,73)[(-2,-1)[23]{}]{} (3,62)[(1,1)[24]{}]{} (27,101)[(-2,-3)[23]{}]{} (3,133)[(2,-1)[24]{}]{} (27,135)[(-1,0)[23]{}]{} (3,137)[(2,1)[24]{}]{} (27,169)[(-2,3)[23]{}]{} (3,208)[(1,-1)[24]{}]{} (27,197)[(-2,1)[23]{}]{} (3,210)[(1,0)[24]{}]{} (27,223)[(-2,-1)[23]{}]{} (3,212)[(1,1)[24]{}]{} (27,251)[(-2,-3)[23]{}]{} (3,-2) [ (-17,60)[$+$]{} (-17,135)[$-$]{} (-17,210)[$+$]{} ]{} (4,-2) [ (30,0)[$+$]{} (30,15)[$-$]{} (30,30)[$+$]{} (30,45)[$-$]{} (30,60)[$+$]{} (30,75)[$-$]{} (30,90)[$+$]{} (30,105)[$-$]{} (30,120)[$+$]{} (30,135)[$-$]{} (30,150)[$+$]{} (30,165)[$-$]{} (30,180)[$+$]{} (30,195)[$-$]{} (30,210)[$+$]{} (30,225)[$-$]{} (30,240)[$+$]{} (30,255)[$-$]{} (30,270)[$+$]{} ]{} ]{} (140,0) [ (0,60) (0,135) (0,210) (30,0) (30,15) (30,30) (30,45) (30,60) (30,75) (30,90) (30,105) (30,120) (30,135) (30,150) (30,165) (30,180) (30,195) (30,210) (30,225) (30,240) (30,255) (30,270) (30,3)[(0,1)[9]{}]{} (30,27)[(0,-1)[9]{}]{} (30,33)[(0,1)[9]{}]{} (30,57)[(0,-1)[9]{}]{} (30,63)[(0,1)[9]{}]{} (30,87)[(0,-1)[9]{}]{} (30,93)[(0,1)[9]{}]{} (30,117)[(0,-1)[9]{}]{} (30,123)[(0,1)[9]{}]{} (30,147)[(0,-1)[9]{}]{} (30,153)[(0,1)[9]{}]{} (30,177)[(0,-1)[9]{}]{} (30,183)[(0,1)[9]{}]{} (30,207)[(0,-1)[9]{}]{} (30,213)[(0,1)[9]{}]{} (30,237)[(0,-1)[9]{}]{} (30,243)[(0,1)[9]{}]{} (30,267)[(0,-1)[9]{}]{} (0,132)[(0,-1)[69]{}]{} (0,138)[(0,1)[69]{}]{} (3,58)[(1,-1)[24]{}]{} (27,47)[(-2,1)[23]{}]{} (3,60)[(1,0)[24]{}]{} (27,73)[(-2,-1)[23]{}]{} (3,62)[(1,1)[24]{}]{} (27,107)[(-1,1)[23]{}]{} (3,133)[(2,-1)[24]{}]{} (27,135)[(-1,0)[23]{}]{} (3,137)[(2,1)[24]{}]{} (27,163)[(-1,-1)[23]{}]{} (3,208)[(1,-1)[24]{}]{} (27,197)[(-2,1)[23]{}]{} (3,210)[(1,0)[24]{}]{} (27,223)[(-2,-1)[23]{}]{} (3,212)[(1,1)[24]{}]{} (3,-2) [ (-17,60)[$-$]{} (-17,135)[$+$]{} (-17,210)[$-$]{} ]{} (4,-2) [ (30,0)[$+$]{} (30,15)[$-$]{} (30,30)[$+$]{} (30,45)[$-$]{} (30,60)[$+$]{} (30,75)[$-$]{} (30,90)[$+$]{} (30,105)[$-$]{} (30,120)[$+$]{} (30,135)[$-$]{} (30,150)[$+$]{} (30,165)[$-$]{} (30,180)[$+$]{} (30,195)[$-$]{} (30,210)[$+$]{} (30,225)[$-$]{} (30,240)[$+$]{} (30,255)[$-$]{} (30,270)[$+$]{} ]{} ]{} (210,0) [ (0,60) (0,135) (0,210) (30,0) (30,15) (30,30) (30,45) (30,60) (30,75) (30,90) (30,105) (30,120) (30,135) (30,150) (30,165) (30,180) (30,195) (30,210) (30,225) (30,240) (30,255) (30,270) (30,3)[(0,1)[9]{}]{} (30,27)[(0,-1)[9]{}]{} (30,33)[(0,1)[9]{}]{} (30,57)[(0,-1)[9]{}]{} (30,63)[(0,1)[9]{}]{} (30,87)[(0,-1)[9]{}]{} (30,93)[(0,1)[9]{}]{} (30,117)[(0,-1)[9]{}]{} (30,123)[(0,1)[9]{}]{} (30,147)[(0,-1)[9]{}]{} (30,153)[(0,1)[9]{}]{} (30,177)[(0,-1)[9]{}]{} (30,183)[(0,1)[9]{}]{} (30,207)[(0,-1)[9]{}]{} (30,213)[(0,1)[9]{}]{} (30,237)[(0,-1)[9]{}]{} (30,243)[(0,1)[9]{}]{} (30,267)[(0,-1)[9]{}]{} (0,63)[(0,1)[69]{}]{} (0,207)[(0,-1)[69]{}]{} (27,47)[(-2,1)[23]{}]{} (3,60)[(1,0)[24]{}]{} (27,73)[(-2,-1)[23]{}]{} (3,128)[(2,-3)[24]{}]{} (27,107)[(-1,1)[23]{}]{} (3,133)[(2,-1)[24]{}]{} (27,135)[(-1,0)[23]{}]{} (3,137)[(2,1)[24]{}]{} (3,142)[(2,3)[23]{}]{} (27,163)[(-1,-1)[24]{}]{} (27,197)[(-2,1)[23]{}]{} (3,210)[(1,0)[24]{}]{} (27,223)[(-2,-1)[23]{}]{} (3,-2) [ (-17,60)[$+$]{} (-17,135)[$-$]{} (-17,210)[$+$]{} ]{} (4,-2) [ (30,0)[$+$]{} (30,15)[$-$]{} (30,30)[$+$]{} (30,45)[$-$]{} (30,60)[$+$]{} (30,75)[$-$]{} (30,90)[$+$]{} (30,105)[$-$]{} (30,120)[$+$]{} (30,135)[$-$]{} (30,150)[$+$]{} (30,165)[$-$]{} (30,180)[$+$]{} (30,195)[$-$]{} (30,210)[$+$]{} (30,225)[$-$]{} (30,240)[$+$]{} (30,255)[$-$]{} (30,270)[$+$]{} ]{} ]{} (280,0) [ (0,60) (0,135) (0,210) (30,0) (30,15) (30,30) (30,45) (30,60) (30,75) (30,90) (30,105) (30,120) (30,135) (30,150) (30,165) (30,180) (30,195) (30,210) (30,225) (30,240) (30,255) (30,270) (30,3)[(0,1)[9]{}]{} (30,27)[(0,-1)[9]{}]{} (30,33)[(0,1)[9]{}]{} (30,57)[(0,-1)[9]{}]{} (30,63)[(0,1)[9]{}]{} (30,87)[(0,-1)[9]{}]{} (30,93)[(0,1)[9]{}]{} (30,117)[(0,-1)[9]{}]{} (30,123)[(0,1)[9]{}]{} (30,147)[(0,-1)[9]{}]{} (30,153)[(0,1)[9]{}]{} (30,177)[(0,-1)[9]{}]{} (30,183)[(0,1)[9]{}]{} (30,207)[(0,-1)[9]{}]{} (30,213)[(0,1)[9]{}]{} (30,237)[(0,-1)[9]{}]{} (30,243)[(0,1)[9]{}]{} (30,267)[(0,-1)[9]{}]{} (0,132)[(0,-1)[69]{}]{} (0,138)[(0,1)[69]{}]{} (3,60)[(1,0)[24]{}]{} (27,77)[(-1,2)[23]{}]{} (3,128)[(2,-3)[24]{}]{} (27,107)[(-1,1)[23]{}]{} (3,133)[(2,-1)[24]{}]{} (27,135)[(-1,0)[23]{}]{} (3,137)[(2,1)[24]{}]{} (27,163)[(-1,-1)[23]{}]{} (3,142)[(2,3)[24]{}]{} (27,193)[(-1,-2)[23]{}]{} (3,210)[(1,0)[24]{}]{} (3,-2) [ (-17,60)[$-$]{} (-17,135)[$+$]{} (-17,210)[$-$]{} ]{} (4,-2) [ (30,0)[$+$]{} (30,15)[$-$]{} (30,30)[$+$]{} (30,45)[$-$]{} (30,60)[$+$]{} (30,75)[$-$]{} (30,90)[$+$]{} (30,105)[$-$]{} (30,120)[$+$]{} (30,135)[$-$]{} (30,150)[$+$]{} (30,165)[$-$]{} (30,180)[$+$]{} (30,195)[$-$]{} (30,210)[$+$]{} (30,225)[$-$]{} (30,240)[$+$]{} (30,255)[$-$]{} (30,270)[$+$]{} ]{} ]{} (10,-20)[$Q_1$]{} (80,-20)[$Q_2$]{} (150,-20)[$Q_3$]{} (220,-20)[$Q_4$]{} (290,-20)[$Q_5$]{} \[ex:G5\] [Cluster algebras for T- and Y-systems for quantum affinizations of tamely laced quantum Kac-Moody algebras [@Hernandez07a; @Kuniba09; @Nakanishi10a].]{} The cluster algebras for T- and Y-systems for quantum affinizations of tamely laced quantum Kac-Moody algebras provide a rich family of more complicated periodicities of exchange matrices. As a typical example, let $Q$ be the quiver in Figure \[fig:quiver\], where the right columns in the five quivers $Q_1$, …, $Q_5$ are identified. We remark that $Q$ is not bipartite in the sense of Example \[ex:preperiod\]. Let $\mathbf{i}^{\bullet}_+$ (resp. $\mathbf{i}^{\bullet}_-$) be the sequence of all the distinct elements of the vertices in $Q$ with property $(\bullet,+)$ (resp. $(\bullet,-)$), and let $\mathbf{i}^{\circ}_{+,k}$ (resp. $\mathbf{i}^{\circ}_{-,k}$) be a sequence of all the distinct elements of the vertices in $Q_k$ with property $(\circ,+)$ (resp. $(\circ,-)$), where the order of the sequence is chosen arbitrarily thanks to Lemma \[lem:order\]. Let $$\begin{aligned} \begin{split} \mathbf{i}&= \mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,1} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,4} \,\|\, \mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,3} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,2} \,\|\, \mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,5} \\ & \quad \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{-,1} \,\|\, \mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{-,4} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{-,3} \,\|\, \mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{-,2} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{-,5}. \end{split}\end{aligned}$$ Let $\sigma$ be the permutation of $\{1,\dots,5\}$, $$\begin{aligned} \sigma= \left( \begin{matrix} 1&2& 3& 4 & 5\\ 3&1& 5& 2 & 4\\ \end{matrix} \right).\end{aligned}$$ Let $\nu:I \rightarrow I$ be the bijection of order $5$ such that each vertex in $Q_i$ maps to the vertex in $Q_{\sigma(i)}$ in the same position. In particular, every vertex with $\bullet$ is a fixed point of $\nu$, and every vertex with $\circ$ has the $\nu$-orbit of length $5$. Then $\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,1} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,4} $ is a $\nu$-period of $Q$, and $\mathbf{i}$ is a period of $Q$. Note that $\mathbf{i}\sim_Q \mathbf{j}(\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,1} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,4},\nu)$. The quiver $Q$ corresponds to the level $\ell=4$ T- and Y-systems for the quantum affinization of the quantum Kac-Moody algebra whose Cartan matrix is represented by the following Dynkin diagram. $$\begin{aligned} \begin{picture}(20,25)(0,-15) % % A_r % % B_r \put(0,0){ \put(0,0){\circle{6}} \put(20,0){\circle{6}} \drawline(2,-4)(18,-4) \drawline(2,-2)(18,-2) \drawline(2,2)(18,2) \drawline(2,4)(18,4) \drawline(3,0)(17,0) \drawline(7,6)(13,0) \drawline(7,-6)(13,0) \put(-2,-15){\small $1$} \put(18,-15){\small $2$} } % \end{picture}\end{aligned}$$ See [@Nakanishi10a] for more details. \[ex:fordy\] [Cluster algebras with $\rho^m$-period of $Q$ for cyclic permutation $\rho$ [@Fordy09].]{} For a cyclic permutation $\rho$ of $I$, many examples of quivers with $\rho^m$-period were constructed and partially classified in [@Fordy09]. For instance, let $Q$ be the following quiver: $$\begin{aligned} \label{eq:fordy} \raisebox{-45pt}{ \begin{picture}(70,80)(0,-15) \put(20,0){\circle{6}} \put(50,0){\circle{6}} \put(0,30){\circle{6}} \put(70,30){\circle{6}} \put(20,60){\circle{6}} \put(50,60){\circle{6}} % horizontal \put(3,27){\vector(2,-3){15}} \put(17,57){\vector(-2,-3){15}} \put(53,3){\vector(2,3){15}} \put(67,33){\vector(-2,3){15}} \put(5,31.5){\vector(1,0){60}} \put(5,28.5){\vector(1,0){60}} \put(46,59){\vector(-3,-2){40}} \put(46,1){\vector(-3,2){40}} \put(64,27){\vector(-3,-2){40}} \put(64,33){\vector(-3,2){40}} \put(22,4){\vector(1,2){26}} \put(22,56){\vector(1,-2){26}} \put(20,4){\vector(0,1){52}} \put(50,56){\vector(0,-1){52}} % % % \put(48,67){\small $1$} \put(75,27){\small $2$} \put(48,-13){\small $3$} \put(18,-13){\small $4$} \put(-10,27){\small $5$} \put(18,67){\small $6$} \end{picture} }\end{aligned}$$ Let $$\begin{aligned} \mathbf{i}=(1,2,3,4,5,6).\end{aligned}$$ Let $\rho$ be the cyclic permutation of $\{1,\dots,6\}$, $$\begin{aligned} \rho= \left( \begin{matrix} 1&2& 3& 4 & 5& 6\\ 2&3& 4& 5 & 6& 1\\ \end{matrix} \right).\end{aligned}$$ Then, $(1,2)$ is a $\rho^2$-period of $Q$, and $\mathbf{i}$ is a period of $Q$. Note that $\mathbf{i}=\mathbf{j}((1,2),\rho^2)$. The quiver $Q$ is a special case of a family of ‘period 2 solutions’ of [@Fordy09 Section 7.4] with $m_1=m_3=-m_2=1$ and $m_{\bar{1}}=0$ therein. It is also the quiver for the quiver gauge theory on the del Pezzo 3 surface [@Feng01]. See [@Fordy09] for more details. Examples of periodicities of seeds ---------------------------------- All the examples of periodicities of seeds below can be proved by verifying the condition in Theorem \[thm:tropperiod\], case by case, with the help of relevant Coxeter elements. \[ex:finite\] [Cluster algebras of finite type]{} [@Fomin03a; @Fomin03b]. For a skew symmetrizable matrix $B$, define a matrix $C=C(B)$ by $$\begin{aligned} C_{ij}= \begin{cases} 2 & i=j\\ -\|b_{ij}\| & i \neq j.\\ \end{cases}\end{aligned}$$ Then, it is known that the cluster algebra $\mathcal{A}(B,x,y)$ is of finite type if and only if $B$ is mutation equivalent to a skew-symmetric matrix $B'$ such that $C(B')$ is a direct sum of Cartan matrices of finite type. Suppose that $C(B)$ is a Cartan matrix of finite type. Since there are only a finite number of seeds of $\mathcal{A}(B,x,y)$, for any $I$-sequence $\mathbf{i}$, there is some $p$ such that the $p$-fold concatenation $\mathbf{i}^p$ of $\mathbf{i}$ is a period of $\mathcal{A}(B,x,y)$. Among them there is some distinguished period of $(B,x,y)$, which is closely related to the Coxeter element of the Weyl group for $C$. For type $A_n$, for example, they are given as follows. \(a) Type $A_n$ ($n$: odd). For odd $n$, let $Q$ be the following alternating quiver with index set $I=\{1,\dots,n\}$: (80,25)(0,-15) (0,0) (20,0) (40,0) (60,0) (80,0) (3,0)[(1,0)[14]{}]{} (37,0)[(-1,0)[14]{}]{} (43,0)[(1,0)[14]{}]{} (77,0)[(-1,0)[14]{}]{} (-3,-15)[$1$]{} (17,-15)[$2$]{} (37,-15)[$3$]{} (77,-15)[$n$]{} Let $\mathbf{i}=\mathbf{i}_+\,\|\, \mathbf{i}_-$, where $\mathbf{i}_+=(1,3,\dots,n)$, $\mathbf{i}_-=(2,4,\dots,n-1)$. Let $\omega:I \rightarrow I$ be the left-right reflection, which is a quiver automorphism of $Q$. Then $\mathbf{i}$ is a period of $Q$. Furthermore, $\mathbf{i}^{(n+3)/2}$ is an $\omega$-period of $(Q,x,y)$, and $\mathbf{i}^{n+3}$ is a period of $(Q,x,y)$. We note that $n+3=h(A_{n})+2$, where $h(X)$ is the [Coxeter number]{} of type $X$. For simply laced $X$, $h(X)$ coincides with the [dual Coxeter number]{} $h^{\vee}(X)$ of type $X$. \(b) Type $A_n$ ($n$: even). For even $n$, let $Q$ be the following quiver with index set $I=\{1,\dots,n\}$: (100,25)(0,-15) (0,0) (20,0) (40,0) (60,0) (80,0) (100,0) (3,0)[(1,0)[14]{}]{} (37,0)[(-1,0)[14]{}]{} (43,0)[(1,0)[14]{}]{} (77,0)[(-1,0)[14]{}]{} (83,0)[(1,0)[14]{}]{} (-3,-15)[$1$]{} (17,-15)[$2$]{} (37,-15)[$3$]{} (97,-15)[$n$]{} Let $\mathbf{i}=\mathbf{i}_+\,\|\, \mathbf{i}_-$, where $\mathbf{i}_+=(1,3,\dots,n-1)$, $\mathbf{i}_-=(2,4,\dots,n)$. Let $\nu:I \rightarrow I$ be the left-right reflection, which is [not]{} a quiver automorphism of $Q$. Then $\mathbf{i}_+$ is an $\nu$-period of $Q$ and $\mathbf{i}$ is a period of $Q$. Furthermore, $\mathbf{i}^{n/2+1}\,\|\, \mathbf{i}_+$ is a $\nu$-period of $(Q,x,y)$, and $\mathbf{i}^{n+3}$ is a period of $(Q,x,y)$. Note that $\mathbf{i}\sim_Q \mathbf{j}(\mathbf{i}_+,\nu)$ and $\mathbf{i}^{n+3}\sim_Q\mathbf{j}(\mathbf{i}^{n/2+1}\mathbf{i}_+,\nu)$. \[ex:affine\] [Cluster algebras for T- and Y-systems of quantum affine algebras]{} [@Keller08; @Keller10; @DiFrancesco09a; @Inoue10a; @Inoue10b; @Inoue10c]. With each pair $(X,\ell)$ of a Dynkin diagram $X$ of finite type and an integer $\ell \geq 2$, one can associate a quiver $Q=Q(X,\ell)$. They are related to the T- and Y-systems of a quantum affine algebra of type $X$, and provide a family of periodicities of seeds. Let us give typical examples for simply laced and nonsimply laced ones. \(a) [Simply laced case:]{} $(X,\ell)=(A_4,4)$. Let $Q$ be the following quiver with index set $I$: $$\begin{aligned} \label{eq:qA} \raisebox{-40pt}{ \begin{picture}(90,80)(0,-5) \put(0,0){\circle{6}} \put(30,0){\circle{6}} \put(60,0){\circle{6}} \put(90,0){\circle{6}} \put(0,30){\circle{6}} \put(30,30){\circle{6}} \put(60,30){\circle{6}} \put(90,30){\circle{6}} \put(0,60){\circle{6}} \put(30,60){\circle{6}} \put(60,60){\circle{6}} \put(90,60){\circle{6}} % vertical \put(0,3){\vector(0,1){24}} \put(0,57){\vector(0,-1){24}} \put(30,27){\vector(0,-1){24}} \put(30,33){\vector(0,1){24}} \put(60,3){\vector(0,1){24}} \put(60,57){\vector(0,-1){24}} \put(90,27){\vector(0,-1){24}} \put(90,33){\vector(0,1){24}} % horizontal \put(27,0){\vector(-1,0){24}} \put(33,0){\vector(1,0){24}} \put(87,0){\vector(-1,0){24}} \put(3,30){\vector(1,0){24}} \put(57,30){\vector(-1,0){24}} \put(63,30){\vector(1,0){24}} \put(27,60){\vector(-1,0){24}} \put(33,60){\vector(1,0){24}} \put(87,60){\vector(-1,0){24}} % % % \put(-12,3){$+$} \put(18,3){$-$} \put(48,3){$+$} \put(78,3){$-$} \put(-12,33){$-$} \put(18,33){$+$} \put(48,33){$-$} \put(78,33){$+$} \put(-12,63){$+$} \put(18,63){$-$} \put(48,63){$+$} \put(78,63){$-$} \end{picture} }\end{aligned}$$ Let $\mathbf{i}_+$ and $\mathbf{i}_-$ be as before. Let $\mathbf{i}=\mathbf{i}_+\,\|\, \mathbf{i}_-$. Let $\nu:I \rightarrow I$ be the left-right reflection, and let $\omega :I \rightarrow I$ be the top-bottom reflection, so that $\omega\nu=\nu\omega$. Then $\mathbf{i}_+$ is a $\nu$-period of $Q$, and $\mathbf{i}$ is a period of $Q$. Furthermore, $\mathbf{i}^{4}\,\|\, \mathbf{i}_+$ is a $\nu\omega$-period of $(Q,x,y)$, and $\mathbf{i}^{9}$ is a period of $(Q,x,y)$, where $9=5+4=h(A_4)+\ell$. Note that $\mathbf{i}\sim_Q \mathbf{j}(\mathbf{i}_+,\nu)$ and $\mathbf{i}^{9}\sim_Q\mathbf{j}(\mathbf{i}^{4}\mathbf{i}_+,\nu\omega)$. \(b) [Nonsimply laced case:]{} $(X,\ell)=(B_4,4)$. Let $Q$ be the following quiver with index set $I$: $$\begin{aligned} \label{eq:qB} \setlength{\unitlength}{1pt} \raisebox{-45pt}{ \begin{picture}(180,105)(0,-20) \put(0,0){\circle{6}} \put(30,0){\circle{6}} \put(60,0){\circle{6}} \put(90,0){\circle{6}} \put(120,0){\circle{6}} \put(150,0){\circle{6}} \put(180,0){\circle{6}} \put(90,15){\circle{6}} % \put(3,0){\vector(1,0){24}} \put(57,0){\vector(-1,0){24}} \put(87,0){\vector(-1,0){24}} \put(93,0){\vector(1,0){24}} \put(147,0){\vector(-1,0){24}} \put(153,0){\vector(1,0){24}} % % \put(0,30) { \put(0,0){\circle{6}} \put(30,0){\circle{6}} \put(60,0){\circle{6}} \put(90,0){\circle{6}} \put(120,0){\circle{6}} \put(150,0){\circle{6}} \put(180,0){\circle{6}} \put(90,15){\circle{6}} % \put(27,0){\vector(-1,0){24}} \put(33,0){\vector(1,0){24}} \put(87,0){\vector(-1,0){24}} \put(93,0){\vector(1,0){24}} \put(123,0){\vector(1,0){24}} \put(177,0){\vector(-1,0){24}} } % \put(0,60) { \put(0,0){\circle{6}} \put(30,0){\circle{6}} \put(60,0){\circle{6}} \put(90,0){\circle{6}} \put(120,0){\circle{6}} \put(150,0){\circle{6}} \put(180,0){\circle{6}} \put(90,15){\circle{6}} % \put(3,0){\vector(1,0){24}} \put(57,0){\vector(-1,0){24}} \put(87,0){\vector(-1,0){24}} \put(93,0){\vector(1,0){24}} \put(147,0){\vector(-1,0){24}} \put(153,0){\vector(1,0){24}} } \put(90,-15){\circle{6}} % % vertical arrows \put(90,-12){\vector(0,1){9}} \put(90,12){\vector(0,-1){9}} \put(90,18){\vector(0,1){9}} \put(90,42){\vector(0,-1){9}} \put(90,48){\vector(0,1){9}} \put(90,72){\vector(0,-1){9}} % \put(63,-2){\vector(2,-1){24}} \put(63,2){\vector(2,1){24}} \put(63,58){\vector(2,-1){24}} \put(63,62){\vector(2,1){24}} \put(117,28){\vector(-2,-1){24}} \put(117,32){\vector(-2,1){24}} % %\put(-30,3){\vector(0,1){24}} \put(0,27){\vector(0,-1){24}} \put(30,3){\vector(0,1){24}} \put(60,27){\vector(0,-1){24}} \put(120,3){\vector(0,1){24}} \put(150,27){\vector(0,-1){24}} \put(180,3){\vector(0,1){24}} %\put(210,27){\vector(0,-1){24}} % %\put(-30,57){\vector(0,-1){24}} \put(0,33){\vector(0,1){24}} \put(30,57){\vector(0,-1){24}} \put(60,33){\vector(0,1){24}} \put(120,57){\vector(0,-1){24}} \put(150,33){\vector(0,1){24}} \put(180,57){\vector(0,-1){24}} %\put(210,33){\vector(0,1){24}} % \put(-14,1) { \put(2,2){\small $-$} \put(2,32){\small $+$} \put(2,62){\small $-$} \put(32,2){\small $+$} \put(32,32){\small $-$} \put(32,62){\small $+$} \put(62,2){\small $-$} \put(62,32){\small $+$} \put(62,62){\small $-$} \put(92,-11){\small $+$} \put(92,2){\small $-$} \put(92,17){\small $+$} \put(92,32){\small $-$} \put(92,49){\small $+$} \put(92,62){\small $-$} \put(92,75){\small $+$} \put(122,2){\small $+$} \put(122,32){\small $-$} \put(122,62){\small $+$} \put(152,2){\small $-$} \put(152,32){\small $+$} \put(152,62){\small $-$} \put(182,2){\small $+$} \put(182,32){\small $-$} \put(182,62){\small $+$} } \end{picture} }\end{aligned}$$ Let $\mathbf{i}^{\bullet}_+$ (resp. $\mathbf{i}^{\bullet}_-$, $\mathbf{i}^{\circ}_+$, $\mathbf{i}^{\circ}_-$) be a sequence of all the distinct elements of $I$ with property $(\bullet,+)$ (resp. $(\bullet,-)$, $(\circ,+)$, $(\circ,-)$), where the order of the sequence is chosen arbitrarily. Let $$\begin{aligned} \label{eq:slice22} \mathbf{i}= (\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_+ \,\|\, \mathbf{i}^{\bullet}_-)\,\|\, (\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_- \,\|\, \mathbf{i}^{\bullet}_-).\end{aligned}$$ Let $\nu:I \rightarrow I$ be the left-right reflection, and let $\omega :I \rightarrow I$ be the top-bottom reflection, so that $\omega\nu=\nu\omega$. Then $\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_+ \,\|\, \mathbf{i}^{\bullet}_- $ is a $\nu$-period of $Q$, and $\mathbf{i}$ is a period of $Q$. Furthermore, $\mathbf{i}^{5}\,\|\, (\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_+ \,\|\, \mathbf{i}^{\bullet}_-)$ is a $\nu\omega$-period of $(Q,x,y)$, and $\mathbf{i}^{11}$ is a period of $(Q,x,y)$, where $11=7+4=h^{\vee}(B_4)+\ell$. Note that $\mathbf{i}\sim_Q \mathbf{j}(\mathbf{i}^{\bullet}_+ \, \| \, \mathbf{i}^{\circ}_+\, \| \, \mathbf{i}^{\bullet}_-,\nu)$ and $\mathbf{i}^{11}\sim_Q\mathbf{j}( \mathbf{i}^{5}\,\|\, (\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_+ \,\|\, \mathbf{i}^{\bullet}_-),\nu\omega)$. See [@Inoue10a] for more details. [Cluster algebras for sine-Gordon T- and Y-systems]{} [@Nakanishi10b]. \[ex:sine\] The T- and Y-systems which originated from the sine-Gordon model provide another family of periodicities of seeds. Let us give an example. Let $Q$ be the following quiver with index set $I=\{1, \dots, 13\}$. Here all the vertices with $\bullet$ in the same position in the quivers $Q_1$,…,$Q_{6}$ are identified. The quiver $Q$ is mutation equivalent to the quiver of type $D_{13}$. (330,110)(0,-25) (0,0) [ (0,45) (30,0) (30,15) (30,30) (30,45) (30,60) (15,75) (30,75) (30,12)[(0,-1)[9]{}]{} (30,18)[(0,1)[9]{}]{} (30,42)[(0,-1)[9]{}]{} (30,48)[(0,1)[9]{}]{} (30,72)[(0,-1)[9]{}]{} (18,72)[(1,-1)[9]{}]{} (27,4)[(-2,3)[24]{}]{} (3,-2) [ ]{} (4,-2) [ (30,0)[$-$]{} (30,15)[$+$]{} (30,30)[$-$]{} (30,45)[$+$]{} (30,60)[$-$]{} (30,75)[$+$]{} (-3,75)[$+$]{} ]{} ]{} (60,0) [ (0,45) (30,0) (30,15) (30,30) (30,45) (30,60) (15,75) (30,75) (30,12)[(0,-1)[9]{}]{} (30,18)[(0,1)[9]{}]{} (30,42)[(0,-1)[9]{}]{} (30,48)[(0,1)[9]{}]{} (30,72)[(0,-1)[9]{}]{} (18,72)[(1,-1)[9]{}]{} (3,43)[(1,-1)[24]{}]{} (27,32)[(-2,1)[23]{}]{} (3,-2) [ ]{} (4,-2) [ (30,0)[$-$]{} (30,15)[$+$]{} (30,30)[$-$]{} (30,45)[$+$]{} (30,60)[$-$]{} (30,75)[$+$]{} (-3,75)[$+$]{} ]{} ]{} (120,0) [ (0,45) (30,0) (30,15) (30,30) (30,45) (30,60) (15,75) (30,75) (30,12)[(0,-1)[9]{}]{} (30,18)[(0,1)[9]{}]{} (30,42)[(0,-1)[9]{}]{} (30,48)[(0,1)[9]{}]{} (30,72)[(0,-1)[9]{}]{} (18,72)[(1,-1)[9]{}]{} (3,45)[(1,0)[24]{}]{} (27,58)[(-2,-1)[23]{}]{} (3,-2) [ ]{} (4,-2) [ (30,0)[$-$]{} (30,15)[$+$]{} (30,30)[$-$]{} (30,45)[$+$]{} (30,60)[$-$]{} (30,75)[$+$]{} (-3,75)[$+$]{} ]{} ]{} (180,0) [ (0,45) (30,0) (30,15) (30,30) (30,45) (30,60) (15,75) (30,75) (30,12)[(0,-1)[9]{}]{} (30,18)[(0,1)[9]{}]{} (30,42)[(0,-1)[9]{}]{} (30,48)[(0,1)[9]{}]{} (30,72)[(0,-1)[9]{}]{} (18,72)[(1,-1)[9]{}]{} (3,48)[(1,1)[25]{}]{} (27,58)[(-2,-1)[23]{}]{} (2,48)[(1,2)[11]{}]{} (3,-2) [ ]{} (4,-2) [ (30,0)[$-$]{} (30,15)[$+$]{} (30,30)[$-$]{} (30,45)[$+$]{} (30,60)[$-$]{} (30,75)[$+$]{} (-3,75)[$+$]{} ]{} ]{} (240,0) [ (0,45) (30,0) (30,15) (30,30) (30,45) (30,60) (15,75) (30,75) (30,12)[(0,-1)[9]{}]{} (30,18)[(0,1)[9]{}]{} (30,42)[(0,-1)[9]{}]{} (30,48)[(0,1)[9]{}]{} (30,72)[(0,-1)[9]{}]{} (18,72)[(1,-1)[9]{}]{} (3,45)[(1,0)[24]{}]{} (27,32)[(-2,1)[23]{}]{} (3,-2) [ ]{} (4,-2) [ (30,0)[$-$]{} (30,15)[$+$]{} (30,30)[$-$]{} (30,45)[$+$]{} (30,60)[$-$]{} (30,75)[$+$]{} (-3,75)[$+$]{} ]{} ]{} (300,0) [ (0,45) (30,0) (30,15) (30,30) (30,45) (30,60) (15,75) (30,75) (30,12)[(0,-1)[9]{}]{} (30,18)[(0,1)[9]{}]{} (30,42)[(0,-1)[9]{}]{} (30,48)[(0,1)[9]{}]{} (30,72)[(0,-1)[9]{}]{} (18,72)[(1,-1)[9]{}]{} (27,4)[(-2,3)[23]{}]{} (3,43)[(1,-1)[24]{}]{} (3,-2) [ ]{} (4,-2) [ (30,0)[$-$]{} (30,15)[$+$]{} (30,30)[$-$]{} (30,45)[$+$]{} (30,60)[$-$]{} (30,75)[$+$]{} (-3,75)[$+$]{} ]{} ]{} (10,-20)[$Q_1$]{} (70,-20)[$Q_2$]{} (130,-20)[$Q_3$]{} (190,-20)[$Q_4$]{} (250,-20)[$Q_5$]{} (310,-20)[$Q_6$]{} (-3,25)[$1$]{} (57,25)[$2$]{} (117,25)[$3$]{} (177,25)[$4$]{} (237,25)[$5$]{} (297,25)[$6$]{} (43,-3)[$7$]{} (43,12)[$8$]{} (43,27)[$9$]{} (43,42)[$10$]{} (43,57)[$11$]{} (43,72)[$12$]{} (-10,72)[$13$]{} Let $\mathbf{i}^{\bullet}_+$ and $\mathbf{i}^{\bullet}_-$ be the $I$-sequences as before. Let $$\begin{aligned} \mathbf{i}= (\mathbf{i}^{\bullet}_+\,\|\, (1) \,\|\, \mathbf{i}^{\bullet}_- )\,\|\, (\mathbf{i}^{\bullet}_+\,\|\, (2) \,\|\, \mathbf{i}^{\bullet}_- )\,\|\, \cdots \,\|\, (\mathbf{i}^{\bullet}_+\,\|\, (6) \,\|\, \mathbf{i}^{\bullet}_- ).\end{aligned}$$ Let $\nu:I \rightarrow I$ be the bijection of order $6$ cyclically mapping the vertices 1, 2, …, 6 to 2, 3, …, 1 and fixing the rest. Let $\omega:I \rightarrow I$ be the involution exchanging the top two vertices with $\bullet$ and fixing the rest, which is a quiver automorphism of $Q$. Note that $\omega\nu=\nu\omega$. Then $\mathbf{i}^{\bullet}_+\,\|\, (1) \,\|\, \mathbf{i}^{\bullet}_-$ is a $\nu$-period of $Q$, and $\mathbf{i}$ is a period of $Q$. Furthermore, $\mathbf{i}^{2}\,\|\, (\mathbf{i}^{\bullet}_+\,\|\, (1) \,\|\, \mathbf{i}^{\bullet}_-)$ is a $\nu\omega$-period of $(Q,x,y)$, and $\mathbf{i}^{13}$ is a period of $(Q,x,y)$, where $13=(12+2+10+2)/2 = (h(D_7)+2+h(D_6)+2)/2$. Note that $\mathbf{i}\sim_Q \mathbf{j}(\mathbf{i}^{\bullet}_+ \, \| \, ( 1) \, \| \, \mathbf{i}^{\bullet}_-,\nu)$ and $\mathbf{i}^{13}\sim_Q\mathbf{j}( \mathbf{i}^{2}\,\|\, (\mathbf{i}^{\bullet}_+\,\|\, ( 1) \,\|\, \mathbf{i}^{\bullet}_-),\nu\omega)$. See [@Nakanishi10b] for more details. In summary, we make two plain observations in these examples. - There are a variety of patterns of periodicities of seeds. - It is hard to tell whether a repetition of a given period of exchange matrices yields a period of seeds just by looking at the shape of its quiver. We expect that these examples are just a tip of iceberg of the whole class of periodicities of seeds and their classification would be very challenging but interesting. Restriction/Extension Theorem ============================= In this section we present Restriction/Extension Theorem on periodicities of seeds. We first state a very general theorem on the relation between $g$-vectors and tropical coefficients. Let $I=\{1,\dots,n\}$. For a given seed $(B',x',y')$ of $\mathcal{A}(B,x,y)$, let $\mathbf{g}'_i=(g'_{1i},\dots,g'_{ni})$ ($i\in I$) be the $g$-vectors defined by . We also introduce integers $c'_{ij}$ ($i,j=1,\dots,n$) by $$\begin{aligned} [y'_i]_{\mathbf{T}}= \prod_{j=1}^n y_j^{c'_{ji}}.\end{aligned}$$ Consider the matrices $C'=(c'_{ij})_{i,j=1}^n$ and $G'=(g'_{ij})_{i,j=1}^n$; that is, each column of $C'$ is the exponents of a tropical coefficient, and each column of $G'$ is a $g$-vector. We remark that $C'$ is the bottom part of the extended exchange matrix $\tilde{B}'$ with the principal coefficients in [@Fomin07 Eq. (2.14)]. \[thm:CG\] Assume that $B$ is skew symmetric. Then, the matrices ${C'}^{T}$ and $G'$ are inverse to each other, where ${C'}^{T}$ is the transpose of $C'$. We prove it by the induction on mutations. When $(B',x',y')=(B,x,y)$, the claim holds because $C'=G'=I$. Suppose that $(B'',x'',y'')=\mu_k(B',x',y')$. Let $C''$ and $G''$ be the corresponding matrices. Then, the following recursion relations hold [@Fomin07 Eq.(5.9) & Proposition 6.6] $$\begin{aligned} \label{eq:cmut} c''_{ij}&= \begin{cases} -c'_{ik}& j=k\\ c'_{ij} + \frac{1}{2}(\|c'_{ik}\|b'_{kj}+ c'_{ik}\|b'_{kj}\|) & j \neq k, \end{cases} \\ \label{eq:gmut} g''_{ij} &= \begin{cases} \displaystyle -g'_{ik} + \sum_{p=1}^n g'_{ip} [b'_{pk}]_+ - \sum_{p=1}^n b_{ip} [c'_{pk}]_+ & j=k,\\ g'_{ij}& j\neq k, \end{cases}\end{aligned}$$ where $[a]_+=a$ if $a> 0$ and 0 otherwise. Also the following relation holds [@Fomin07 Eq.(6.14)] $$\begin{aligned} \sum_{p=1}^n g'_{ip} b'_{pj} =\sum_{p=1}^n b_{ip} c'_{pj}.\end{aligned}$$ We need one more fact: for each $j$, $c_{ij}$ are simultaneously nonnegative or nonpositive for all $j=1,\dots,n$. This is proved for $B$ is skew symmetric by Theorem \[thm:pos\]. Then, using these facts and the induction hypothesis ${C'}^{T}G'=I$, the claim ${C''}^{T}G''=I$ can be easily verified. Theorem \[thm:CG\] appeared in [@Keller10; @Plamondon10b] implicitly. Namely, the claim is also a consequence of the following formula in [@Keller10 Corollaries 6.9 & 6.13] and [@Plamondon10b Proposition 3.6]: $$\begin{aligned} g'_{ij}&=[\mathrm{ind}_{T} T'_j:T_i] ,\\ c'_{ij}&=[\mathrm{ind}^{\mathrm{op}}_{T'}T_i:T'_j] ,\end{aligned}$$ where $T=\bigoplus_{i=1}^n T_i$ and $T'=\bigoplus_{i=1}^n T'_i$ are some objects in the ‘cluster category’ for $\mathcal{A}(B,x,y)$. We ask the reader to consult [@Plamondon10b] and the forthcoming paper [@Plamondon10c] for details. Now let us turn to the main statement of the section. For a pair of index sets $I\subset \tilde{I}$, suppose that there is a pair of skew symmetric matrices $B =(b_{ij})_{i,j\in I}$ and $\tilde{B} =(\tilde{b}_{ij})_{i,j\in \tilde{I}}$ such that $B=\tilde{B}\vert_I$ under the restriction of the index set $\tilde{I}$ to $I$. (In terms of quivers, $Q$ is a full subquiver of $\tilde{Q}$.) Then, we say that $B$ is the [$I$-restriction]{} of $\tilde{B}$ and $\tilde{B}$ is an [$\tilde{I}$-extension]{} of $B$. \[thm:r/e\] For $I\subset \tilde{I}$, assume that $B$ and $B'$ are skew symmetric matrices such that $B$ is the $I$-restriction of $\tilde{B}$ and $\tilde{B}$ is an $\tilde{I}$-extension of $B$. \(a) (Restriction) Suppose that an $I$-sequence $\mathbf{i}=(i_1,\dots,i_r)$ is a period of $(\tilde{B},\tilde{x},\tilde{y})$ in $\mathcal{A}(\tilde{B},\tilde{x},\tilde{y})$, Then, $\mathbf{i}$ is also a period of $(B,x,y)$ in $\mathcal{A}(B,x,y)$. \(b) (Extension) Suppose that an $I$-sequence $\mathbf{i}=(i_1,\dots,i_r)$ is a period of $(B,x,y)$ in $\mathcal{A}(B,x,y)$. Then, $\mathbf{i}$ is also a period of $(\tilde{B},\tilde{x},\tilde{y})$ in $\mathcal{A}(\tilde{B},\tilde{x},\tilde{y})$. \(a) This is a trivial part. In general, the seed $(B',x',y')=\mu_{\mathbf{i}}(B,x,y)$ is obtained from the seed $ (\tilde{B}',\tilde{x}',\tilde{y}') = \mu_{\mathbf{i}}(\tilde{B},\tilde{x},\tilde{y})$ by restricting the index set $\tilde{I}$ of $(\tilde{B}',\tilde{x}',\tilde{y}')$ to $I$ [and]{} specializing the ‘external’ initial cluster variables $x_j$ ($j\in \tilde{I}\setminus I$) to $1$ appearing in the ‘internal’ cluster variables $\tilde{x}'_i$ ($i\in I$). Thus, the periodicity of $(B,x,y)$ follows from the periodicity of $(\tilde{B},\tilde{x},\tilde{y})$. \(b) Set $(\tilde{B}',\tilde{x}', \tilde{y}'):=\mu_{\mathbf{i}} (\tilde{B},\tilde{x},\tilde{y})$. Thanks to Theorem \[thm:tropperiod\], it is enough to prove $$\begin{aligned} \label{eq:ytp} [\tilde{y}'_i]_{\mathbf{T}}= \tilde{y}_i \quad (i\in \tilde{I}).\end{aligned}$$ Since the ‘external’ coefficients $\tilde{y}'_j$ ($j\in \tilde{I}\setminus I$) do not influence each other under the mutation $\mu_{\mathbf{i}}$, it is enough to verify when there is [only one]{} external index $j\in \tilde{I}\setminus I$. Thus, we may assume $I=\{1,\dots,n\}$ and $\tilde{I}= \{1, \dots,n+1\}$. By the periodicity assumption and the exchange relations and , we know [a priori]{} the following periodicities hold. $$\begin{aligned} \label{eq:yp} [\tilde{y}'_i]_{\mathbf{T}}&= \tilde{y}_i\quad (i\in I),\\ \label{eq:xp} \tilde{x}'_{n+1}&=\tilde{x}_{n+1}.\end{aligned}$$ Let us show that and imply . We introduce the integers $c'_{ij}$ ($i,j\in \tilde{I}$) by $$\begin{aligned} \label{eq:cint} [\tilde{y}'_i]_{\mathbf{T}}&= \prod_{j=1}^{n+1} \tilde{y}_j^{c_{ji}'}.\end{aligned}$$ Let $\mathbf{g}'_i=(g'_{ji})_{j=1}^{n+1}$ be the $g$-vector for $\tilde{x}'_i$ ($i\in \tilde{I}$) in . Then, by Theorem \[thm:CG\] the transpose of $C'=(c'_{ij})_{i,j=1}^{n+1}$ and $G'=(g'_{ij})_{i,j=1}^{n+1}$ are inverse to each other, i.e., $$\begin{aligned} \label{eq:inv} {C'}^T G'=I.\end{aligned}$$ Meanwhile, the equalities and imply that ${C'}^T$ and $G'$ have the following form: $$\begin{aligned} {C'}^T= \left( \begin{array}{ccc\|c} 1&&&0\\ &\ddots&&\vdots\\ &&1&0\\ \hline &\cdots&&\\ \end{array} \right), \quad G'= \left( \begin{array}{ccc\|c} &\cdots&&0\\ \vdots&&\vdots&\vdots\\ &\cdots&&0\\ \hline &\cdots&&1\\ \end{array} \right).\end{aligned}$$ The condition further imposes that the matrix has the form $$\begin{aligned} {C'}^T= \left( \begin{array}{ccc\|c} 1&&&0\\ &\ddots&&\vdots\\ &&1&0\\ \hline \alpha_1&\cdots&\alpha_n&1\\ \end{array} \right), \quad G'= \left( \begin{array}{ccc\|c} 1&&&0\\ &\ddots&&\vdots\\ &&1&0\\ \hline \beta_1&\cdots&\beta_{n}&1\\ \end{array} \right).\end{aligned}$$ with $\alpha_i=-\beta_i$. Now we recall Theorem \[thm:pos\]. \(a) The Laurent monomial is either positive or negative. So we have $\alpha_i \geq 0$. \(b) For each $i$, $g'_{ij}$ ($j=1,\dots,n+1$) are simultaneously nonpositive or nonnegative. So we have $\beta_i \geq 0$. Thus, we have $\alpha_i=\beta_i=0$ and we conclude that $C'=G'=I$; therefore, holds. The extension theorem (b) was partially formulated by Keller [@Keller08b], where only the periodicity of $\tilde{B}$ was considered for the periodicities of seeds for ‘pairs of Dynkin diagrams’ studied in [@Keller10]. His result motivates us to formulate Theorem \[thm:r/e\] (b). A generalized version of the result of [@Keller08b] will be contained in [@Plamondon10c]. Theorem \[thm:r/e\] tells that one may assume without losing much generality that the components of a period $\mathbf{i}$ of a seed under study exhaust the index set $I$. It may be worth to mention that the involution property of a mutation $\mu_i$ is also regarded as the simplest case of Extension Theorem applied for the subquiver of type $A_1$ consisting of a single vertex $i$. T-systems and Y-systems {#sec:T} ======================= In this section, we define the T- and Y-systems associated with any period of an exchange matrix. We do it in two steps. First, we treat the special case when the period is ‘regular’. The corresponding T- and Y-systems are natural generalizations of the known ‘classic’ T- and Y-systems. Next, the notions of T- and Y-systems are further extended to the general case. The latter ones will be used in Section \[sec:dilog\]. We stress that in this section we do [not]{} assume the periodicity of seeds. Regular period -------------- Let $B$ a skew symmetrizable matrix with index set $I$. \[def:regular\] We say that a $\nu$-period $\mathbf{i}$ of $B$ is [regular]{} if it is a complete system of representatives of the $\nu$-orbits in $I$; in other words, it satisfies the following conditions: - All the components of $\mathbf{j}(\mathbf{i},\nu)=\mathbf{i} \,\|\, \nu(\mathbf{i})\,\|\, \cdots \,\|\, \nu^{g-1}(\mathbf{i})$ exhaust $I$, where $g$ is the order of $\nu$. - All the components of $\mathbf{i}$ belong to distinct $\nu$-orbits in $I$. \[ex:adm\] In the previous examples, we have the following regular $\nu$-periods of $B$ or $Q$. In Example \[ex:preperiod\], $\mathbf{i}_+\, \| \, \mathbf{i}_-$ is a regular period of $B$. In Example \[ex:G5\], $\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,1} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,4} $ is a regular $\nu$-period of $Q$. In Example \[ex:fordy\], $(1,2)$ is a regular $\rho^2$-period of $Q$. In Example \[ex:finite\] (a), $\mathbf{i}_+ \, \| \, \mathbf{i}_-$ is a regular period of $Q$. In Example \[ex:finite\] (b), $\mathbf{i}_+$ is a regular $\nu$-period of $Q$. In Example \[ex:affine\] (a), $\mathbf{i}_+$ is a regular $\nu$-period of $Q$. In Example \[ex:affine\] (b), $\mathbf{i}^{\bullet}_+ \, \| \, \mathbf{i}^{\circ}_+\, \| \, \mathbf{i}^{\bullet}_-$ is a regular $\nu$-period of $Q$. In Example \[ex:sine\], $\mathbf{i}^{\bullet}_+ \, \| \, (1 )\, \| \, \mathbf{i}^{\bullet}_-$ is a regular $\nu$-period of $Q$. Suppose that $\mathbf{i}=(i_1,\dots,i_r)$ is a $\nu$-period of $B$. Let us decompose $\mathbf{i}$ into $t$ parts as follows: $$\begin{aligned} \label{eq:slice} \mathbf{i}&=\mathbf{i}(0)\,\|\, \mathbf{i}(1)\,\|\, \cdots \,\|\, \mathbf{i}(t-1),\\ \mathbf{i}(p)&= (i(p)_1,\dots,i(p)_{r_p}), \quad \sum_{p=0}^{t-1} r_{p} = r.\end{aligned}$$ Let $B(p)$ ($p=0,\dots,t-1$) be the matrices defined by the sequence of mutations $$\begin{aligned} \label{eq:Bmutseq} B(0)=B \ \mathop{\longrightarrow}^{\mu_{\mathbf{i}(0)}} \ B(1) \ \mathop{\longrightarrow}^{\mu_{\mathbf{i}(1)}} \ \cdots \ \mathop{\longrightarrow}^{\mu_{\mathbf{i}(t-1)}} \ B(t)=\nu(B),\end{aligned}$$ where $\nu(B)=(b'_{ij})$ is the matrix defined by $b'_{\nu(i)\nu(j)}=b_{ij}$. A decomposition of a $\nu$-period $\mathbf{i}$ of $B$ is called a [slice]{} of $\mathbf{i}$ (of length $t$) if it satisfies the following condition: $$\begin{aligned} \label{eq:compat} b(p)_{i(p)_a, i(p)_b}=0 \quad (1\leq a,b \leq r_p)\quad \mbox{for any $p=0,\dots,t-1$.}\end{aligned}$$ For any $\mathbf{i}$, there is at least one slice of $\mathbf{i}$, i.e., the one $\mathbf{i}=( i_1)\,\|\, (i_2) \,\|\, \cdots \,\|\, ( i_r)$ of maximal length. In general, there may be several slices of $\mathbf{i}$. If $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ is a slice of a $\nu$-period of $B$, then, due to Lemma \[lem:order\], each composite mutation $\mu_{\mathbf{i}(p)}$ in does not depend on the order of the sequence $\mathbf{i}(p)$ and it is involutive. Furthermore, the sequence is extended to the infinite one $$\begin{aligned} \label{eq:Bmutseq1} \begin{split} &\hskip100pt \cdots \ \mathop{\longleftrightarrow}^{\mu_{\nu^{-1}(\mathbf{i}(t-2))}} \ B(-1) \ \mathop{\longleftrightarrow}^{\mu_{\nu^{-1}(\mathbf{i}(t-1))}} \\ &B(0) \ \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(0)}} \ B(1) \ \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(1)}} \ \cdots \ \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(t-2)}} \ B(t-1) \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(t-1)}} \\ &B(t) \ \mathop{\longleftrightarrow}^{\mu_{\nu(\mathbf{i}(0))}} B(t+1) \ \mathop{\longleftrightarrow}^{\mu_{\nu(\mathbf{i}(1))}} \ \cdots \ \mathop{\longleftrightarrow}^{\mu_{\nu(\mathbf{i}(t-2))}} \ B(2t-1) \mathop{\longleftrightarrow}^{\mu_{\nu(\mathbf{i}(t-1))}} \\ &B(2t) \ \mathop{\longleftrightarrow}^{\mu_{\nu^2(\mathbf{i}(0))}} \ \cdots, \end{split}\end{aligned}$$ where $B(nt)=\nu^n(B)$ for $n\in \mathbb{Z}$. In particular, $B(gt)=B$, where $g$ is the order of $\nu$. Thus, the sequence has a period $gt$ with respect to $u$. \[ex:slice\] For the regular $\nu$-periods of $B$ or $Q$ in Examples \[ex:adm\], we have the following slices. In Example \[ex:preperiod\], $\mathbf{i}_+\, \|\, \mathbf{i}_-$ itself is a slice of length 2, and $gt=2$. In Example \[ex:G5\], $ \mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,1} \,\|\, \mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,4} = (\mathbf{i}^{\bullet}_+\,\|\, \mathbf{i}^{\circ}_{+,1}) \,\|\, (\mathbf{i}^{\bullet}_-\,\|\, \mathbf{i}^{\circ}_{+,4}) $ is a slice of length 2, and $gt=10$. In Example \[ex:fordy\], $(1,2)=(1)\, \| \, (2) $ is a slice of length 2, and $gt=6$. In Example \[ex:finite\] (a), $\mathbf{i}_+\, \|\, \mathbf{i}_-$ itself is a slice of length 2, and $gt=2$. In Example \[ex:finite\] (b), $\mathbf{i}_+$ itself is a slice of length 1, and $gt=2$. In Example \[ex:affine\] (a), $\mathbf{i}_+$ itself is a slice of length 1, and $gt=2$. In Example \[ex:affine\] (b), $\mathbf{i}^{\bullet}_+ \, \| \, \mathbf{i}^{\circ}_+\, \| \, \mathbf{i}^{\bullet}_-= (\mathbf{i}^{\bullet}_+ \, \| \, \mathbf{i}^{\circ}_+)\, \| \, \mathbf{i}^{\bullet}_-$ is a slice of length 2, and $gt=4$. In Example \[ex:sine\], $ \mathbf{i}^{\bullet}_+ \, \| \, (1 )\, \| \, \mathbf{i}^{\bullet}_- = (\mathbf{i}^{\bullet}_+ \, \| \, (1 ))\, \| \, \mathbf{i}^{\bullet}_-$ is a slice of length 2, and $gt=12$. T- and Y-systems for regular period {#subsec:TY1} ----------------------------------- Here we introduce the T- and Y-systems for [regular]{} $\nu$-periods, which are especially important in applications. The T- and Y-systems for general $\nu$-periods will be treated in Section \[subsec:nonadm\]. Assume that $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ is a slice of a regular $\nu$-period of $B$. In view of , we set $(B(0),x(0),y(0)):=(B,x,y)$ (the initial seed of $\mathcal{A}(B,x,y)$), and consider the corresponding infinite sequence of mutations of [seeds]{} $$\begin{aligned} \label{eq:seedmutseq} \begin{split} \cdots \mathop{\longleftrightarrow}^{\mu_{\nu^{-1}(\mathbf{i}(t-2))}} \ &(B(-1),x(-1),y(-1)) \ \mathop{\longleftrightarrow}^{\mu_{\nu^{-1}(\mathbf{i}(t-1))}} \ (B(0),x(0),y(0)) \ \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(0)}} \\ &(B(1),x(1),y(1)) \ \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(1)}} \ (B(2),x(2),y(2)) \ \mathop{\longleftrightarrow}^{\mu_{\mathbf{i}(2)}} \ \cdots, \end{split}\end{aligned}$$ thereby introducing a family of clusters $x(u)$ ($u\in \mathbb{Z}$) and coefficients tuples $y(u)$ ($u\in \mathbb{Z}$). We define a subset $P_+$ of $I\times \mathbb{Z}$ by $(i,u)\in P_+$ if and only if $i$ is a component of $\nu^{m}(\mathbf{i}(k))$ for $u=mt+k$ ($m\in \mathbb{Z}, 0\leq k\leq t-1$). Plainly speaking, $(i,u)\in P_+$ is a [forward mutation point]{} in . Similarly, we define a subset $P_-$ of $I\times \mathbb{Z}$ by $(i,u)\in P_-$ if and only if $(i,u-1)\in P_+$, namely, $(i,u)\in P_-$ is a [backward mutation point]{} in . Below we mainly use $P_+$. (Alternatively, one may use $P_-$ throughout.) Let $g$ be the order of $\nu$. For each $i\in I$, let $g_i$ be the smallest positive integer such that $\nu^{g_i}(i)=i$. Therefore, $g_i$ is a divisor of $g$. Note that $(i,u)\in P_+$ if and only if $(i,u+tg_i)\in P_+$. It is convenient to define a subset $\tilde{P}_+$ of $I \times \frac{1}{2}\mathbb{Z}$ by $(i,u)\in \tilde{P}_+$ if and only if $(i,u+\frac{tg_i}{2})\in P_+$. Consequently, we have $$\begin{aligned} \textstyle (i,u) \in \tilde{P}_+ \Longleftrightarrow (i,u\pm\frac{tg_i}{2}) \in P_+.\end{aligned}$$ First, we explain what is the Y-system, in short. The sequence of mutations gives various relations among coefficients $y_i(u)$ ($(i,u)\in I \times \mathbb{Z}$) by the exchange relation . Then, thanks to the assumption (A1) in Definition \[def:regular\], all these coefficients are products of the ‘generating’ coefficients $y_i(u)$ and $1+y_i(u)$ ($(i,u)\in P_+$) and their inverses. Here we also used the fact that $y_i(u)=y_i(u-1)^{-1}$ for $(i,u)\in P_-$. Furthermore, these generating coefficients obey some relations, which are the Y-system. (340,75)(0,5) (0,0)[ (0,20)(150,20) (0,80)(150,80) (30,50)(120,50) (30,20)(30,22) (30,29)(30,31) (30,39)(30,41) (30,49)(30,51) (30,59)(30,61) (30,69)(30,71) (30,79)(30,80) (120,20)(120,22) (120,29)(120,31) (120,39)(120,41) (120,49)(120,51) (120,59)(120,61) (120,69)(120,71) (120,79)(120,80) (60,50)(60,65) (90,50)(90,35) (28,48)(32,52) (32,48)(28,52) (118,48)(122,52) (122,48)(118,52) (58,63)(62,67) (62,63)(58,67) (88,33)(92,37) (92,33)(88,37) (60,50) (90,50) (28,5)[$u$]{} (108,5)[$u+tg_i$]{} (18,47)[$i$]{} (48,62)[$j$]{} ]{} (190,0)[ (0,20)(150,20) (0,80)(150,80) (30,50)(120,50) (10,65)(60,65) (20,35)(90,35) (30,20)(30,22) (30,29)(30,31) (30,39)(30,41) (30,49)(30,51) (30,59)(30,61) (30,69)(30,71) (30,79)(30,80) (120,20)(120,22) (120,29)(120,31) (120,39)(120,41) (120,49)(120,51) (120,59)(120,61) (120,69)(120,71) (120,79)(120,80) (30,50)(30,65) (30,50)(30,35) (28,48)(32,52) (32,48)(28,52) (118,48)(122,52) (122,48)(118,52) (58,63)(62,67) (62,63)(58,67) (8,63)(12,67) (12,63)(8,67) (88,33)(92,37) (92,33)(88,37) (18,33)(22,37) (22,33)(18,37) (30,65) (30,35) (28,5)[$u$]{} (108,5)[$u+tg_i$]{} (18,47)[$i$]{} (-2,62)[$j$]{} ]{} Let us write down the relations explicitly. Take $(i,u)\in P_+$ and consider the mutation at $(i,u)$, where $y_i(u)$ is exchanged to $y_i(u+1)=y_i(u)^{-1}$. Then, for each $(j,v)\in P_+$ such that $v\in (u,u+tg_i)$ (i.e., $u <v < u+tg_i$), $y_i(v)$ are multiplied by factors $(1+y_j(v))^{-b_{ji}(v)}$ for $b_{ji}(v)<0$ and $(1+y_j(v)^{-1})^{-b_{ji}(v)}$ for $b_{ji}(v)>0$. The result coincides with the coefficient $y_i(u+tg_i)$ by the assumption (A2) in Definition \[def:regular\]. See Figure \[fig:diagram\]. In summary, we have the following relations: For $(i,u)\in P_+$, $$\begin{aligned} \label{eq:yi'} \textstyle y_i\left(u \right)y_i\left(u+tg_i \right) &= \frac{ \displaystyle \prod_{(j,v)\in P_+} (1+y_j(v))^{G'_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} (1+y_j(v)^{-1})^{G'_-(j,v;i,u)} },\\ \label{eq:G'} G'_{\pm}(j,v;i,u)&= \begin{cases} \mp b_{ji}(v) & v\in (u, u+tg_i), b_{ji}(v)\lessgtr 0\\ 0 & \mbox{otherwise}. \end{cases}\end{aligned}$$ Or, equivalently, for $(i,u)\in \tilde{P}_+$, $$\begin{aligned} \label{eq:yi} \textstyle y_i\left(u-\frac{tg_i}{2}\right)y_i\left(u+\frac{tg_i}{2}\right) &= \frac{ \displaystyle \prod_{(j,v)\in P_+} (1+y_j(v))^{G_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} (1+y_j(v)^{-1})^{G_-(j,v;i,u)} },\\ \label{eq:G} G_{\pm}(j,v;i,u)&= \begin{cases} \mp b_{ji}(v) & v\in (u-\frac{tg_i}{2}, u+\frac{tg_i}{2}), b_{ji}(v)\lessgtr 0\\ 0 & \mbox{otherwise}. \end{cases}\end{aligned}$$ We call the system of relations the [Y-system associated with a slice $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ of a regular $\nu$-period of $B$]{}. Next, we explain what is the T-system, in short. The sequence of mutations gives various relations among cluster variables $x_i(u)$ ($(i,u)\in I \times \mathbb{Z}$) by the exchange relation . Again, thanks to the assumption (A1) in Definition \[def:regular\], all these coefficients are represented by the ‘generating’ cluster variables $x_i(u)$ ($(i,u)\in P_+$). Furthermore, these generating cluster variables obey some relations, which are the T-system. Let us write down the relations explicitly. Take $(i,u)\in P_+$ and consider the mutation at $(i,u)$. Then, by and the assumption (A2) in Definition \[def:regular\], we have $$\begin{aligned} \label{eq:xi'} \begin{split} x_i(u)x_i(u+tg_i) &= \frac{y_i(u)}{1+y_i(u)} \prod_{(j,v)\in P_+} x_j(v)^{H'_+(j,v;i,u)}\\ &\quad + \frac{1}{1+y_i(u)} \prod_{(j,v)\in P_+} x_j(v)^{H'_-(j,v;i,u)}, \end{split}\\ H'_{\pm}(j,v;i,u)&= \begin{cases} \pm b_{ji}(u)& u\in (v- tg_j,v), b_{ji}(u)\gtrless 0\\ 0 & \mbox{otherwise}. \end{cases}\end{aligned}$$ See Figure \[fig:diagram\]. By introducing the ‘shifted cluster variables’ $\tilde{x}_i(u):=x_i(u+\frac{tg_i}{2})$ for $(i,u)\in \tilde{P}_+$, these relations can be written in a more ‘balanced’ form and become parallel to as follows: For $(i,u)\in P_+$, $$\begin{aligned} \label{eq:xi} \begin{split} \textstyle \tilde{x}_i(u-\frac{tg_i}{2})\tilde{x}_i(u+\frac{tg_i}{2}) &= \frac{y_i(u)}{1+y_i(u)} \prod_{(j,v)\in\tilde{P}_+} \tilde{x}_j(v)^{H_+(j,v;i,u)}\\ &\quad + \frac{1}{1+y_i(u)} \prod_{(j,v)\in \tilde{P}_+} \tilde{x}_j(v)^{H_-(j,v;i,u)}, \end{split}\\ \label{eq:tH} H_{\pm}(j,v;i,u)&= \begin{cases} \pm b_{ji}(u)& u\in (v- \frac{tg_j}{2},v+\frac{tg_j}{2}), b_{ji}(u)\gtrless 0\\ 0 & \mbox{otherwise}. \end{cases}\end{aligned}$$ We call the system of relations the [T-system (with coefficients) associated with a slice $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ of a regular $\nu$-period of $B$]{}. Let $\mathcal{A}(B,x)$ be the cluster algebra with trivial coefficients with initial seed $(B,x)$. Namely, we set every coefficient to be $1$ in the trivial semifield $\mathbf{1}=\{1\}$. Let $\pi_{\mathbf{1}}:\mathbb{P}_{\mathrm{univ}}(y) \rightarrow \mathbf{1}$ be the projection. Let $[x_i(u)]_{\mathbf{1}}$ be the image of $x_i(u)$ by the algebra homomorphism $\mathcal{A}(B,x,y)\rightarrow \mathcal{A}(B,x)$ induced from $\pi_{\mathbf{1}}$. By the specialization of , we have $$\begin{aligned} \label{eq:ti} \textstyle [\tilde{x}_i(u-\frac{tg_i}{2})]_{\mathbf{1}} [\tilde{x}_i(u+\frac{tg_i}{2})]_{\mathbf{1}} &= \prod_{(j,v)\in\tilde{P}_+} [\tilde{x}_j(v)]_{\mathbf{1}}^{H_+(j,v;i,u)} + \prod_{(j,v)\in \tilde{P}_+} [\tilde{x}_j(v)]_{\mathbf{1}}^{H_-(j,v;i,u)}.\end{aligned}$$ We call the system of relations the [T-system (without coefficients) associated with a slice $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \allowbreak\mathbf{i}(t-1)$ of a regular $\nu$-period of $B$]{}. Needless to say, the unbalanced form and may be also useful in some situation. \[rem:choice\] For a given regular $\nu$-period $\mathbf{i}$, different choices of slices of $\mathbf{i}$ give different Y-systems/T-systems. But they are easily identified by ‘change of variables’, so the choice of a slice is not essential. (See Proposition \[prop:choice\] for a more precise statement.) However, in view of the sequence , it is economical, and often natural, to use a slice [whose length $t$ is minimal]{} among all the other slices, as in Example \[ex:slice\]. One may think that the Y- and T-systems are the both sides of the coin in the sense that they directly determine each other as follows. \[prop:TY1\] Let $D=\mathrm{diag} (d_i)_{i\in I}$ be a diagonal matrix such that ${}^{t}(DB)=-DB$. Let $G_{\pm}(j,v;i,u)$ and $H_{\pm}(j,v;i,u)$ the ones in and , respectively. Then, the following relation holds for any $(j,v)\in P_+$ and $(i,u)\in \tilde{P}_+$. $$\begin{aligned} \label{eq:GH1} d_j G_{\pm} (j,v;i,u)= d_i H_{\pm} (i,u;j,v).\end{aligned}$$ In particular, if $B$ is skew symmetric, we have $$\begin{aligned} \label{eq:GH2} G_{\pm} (j,v;i,u)= H_{\pm} (i,u;j,v).\end{aligned}$$ We recall that any matrix $B'$ obtained from $B$ by mutation shares the same diagonalizing matrix $D$ with $B$ [@Fomin02 Proposition 4.5]. Then, by comparing and , we immediately obtain the claim. Examples {#subsec:exTY} -------- Let us write down the Y- and T-systems explicitly for the ones in Example \[ex:affine\]. \(a) $(X,\ell)=(A_4,4)$. Let $Q$, $\mathbf{i}_+$, and $\nu$ be the one therein. Then, $\mathbf{i}_+$ is a regular $\nu$-period of $Q$. We regard $\mathbf{i}_+$ as a slice of itself of length 1, and consider the associated Y- and T-systems. We use the index set $I=\{1,2,3,4\}\times \{1,2,3\}$ such that $(i,j)\in I$ corresponds to the vertex at the $i$th column (from the left) and the $j$th row (from the bottom). Thus, we have the data $t=1$, $g=2$, and $g_{(i,j)} =2$ for any $(i,j)\in I$. The condition for the forward mutation points is given by $$\begin{aligned} ((i,j),u)\in P_+ \ &\Longleftrightarrow \ \mbox{$i+j+u$ is even}.\end{aligned}$$ By writing $y_{(i,j)}(u)$ and $\tilde{x}_{(i,j)}(u)$ as $y_{i,j}(u)$ and $\tilde{x}_{i,j}(u)$, the resulting Y-system and T-system (without coefficients) are as follows. [Y-system:]{} For $((i,j),u)\in \tilde{P}_+$, $$\begin{aligned} \label{eq:YA} y_{i,j}(u-1)y_{i,j}(u+1) &= \frac{ (1+y_{i-1,j}(u))(1+y_{i+1,j}(u)) } { (1+y_{i,j-1}(u)^{-1})(1+y_{i,j+1}(u)^{-1}) },\end{aligned}$$ where $y_{0,j}(u)=y_{5,j}(u)=0$ and $y_{i,0}(u)^{-1}=y_{i,4}(u)^{-1}=0$ in the right hand side. [T-system:]{} For $((i,j),u)\in P_+$, $$\begin{aligned} \label{eq:TA} [\tilde{x}_{i,j}(u-1)]_{\mathbf{1}}[\tilde{x}_{i,j}(u+1)]_{\mathbf{1}} &= [\tilde{x}_{i-1,j}(u)]_{\mathbf{1}}[\tilde{x}_{i+1,j}(u)]_{\mathbf{1}} + [\tilde{x}_{i,j-1}(u)]_{\mathbf{1}}[\tilde{x}_{i,j+1}(u)]_{\mathbf{1}} ,\end{aligned}$$ where $[\tilde{x}_{0,j}(u)]_{\mathbf{1}} =[\tilde{x}_{5,j}(u)]_{\mathbf{1}}= [\tilde{x}_{i,0}(u)]_{\mathbf{1}} =[\tilde{x}_{i,4}(u)]_{\mathbf{1}}=1$ in the right hand side. These are the Y- and T-systems associated with the quantum affine algebras of type $A_4$ with level 4. The relation is also a special case of Hirota’s bilinear difference equation [@Hirota77], and it is one of the most studied difference equations in various view points. See [@Inoue10c] for more information. \(b) $(X,\ell)=(B_4,4)$. Let $Q$, $\mathbf{i}^{\bullet}_+$, $\mathbf{i}^{\bullet}_-$, $\mathbf{i}^{\circ}_+$, $\mathbf{i}^{\circ}_-$, and $\nu$ be the one therein. Then, $(\mathbf{i}^{\bullet}_+\, \| \, \mathbf{i}^{\circ}_+\,) \| \, \mathbf{i}^{\bullet}_-$ is a regular $\nu$-period of $Q$. We regard it as a slice of itself of length 2, and consider the associated Y- and T-systems. We use the index set $I$ which is the disjoint union of $\{1,2,3,5,6,7\}\times \{1,2,3\}$ and $\{4\} \times \{1,\dots,7\}$ such that $(i,j)\in I$ corresponds to the vertex at the $i$th column (from the left) and the $j$th row (from the bottom). Thus, we have the data $t=2$, $g=2$, and $g_{(i,j)} =2$ for $i\neq 4$ and $g_{(4,j)} =1$. The condition for the forward mutation points is given by $$\begin{aligned} ((i,j),u)\in P_+ \ &\Longleftrightarrow \ \begin{cases} (i,j)\in \mathbf{i}^{\bullet}_+ \sqcup \mathbf{i}^{\circ}_+ & u \equiv 0 \ (4)\\ (i,j)\in \mathbf{i}^{\bullet}_- & u \equiv 1,3 \ (4)\\ (i,j)\in \mathbf{i}^{\bullet}_+ \sqcup \mathbf{i}^{\circ}_- & u \equiv 2 \ (4).\\ \end{cases}\end{aligned}$$ The resulting Y-system and T-system (without coefficients) are as follows, where the ‘boundary terms’ in the right hand should be ignored as before. [Y-system:]{} For $((i,j),u)\in \tilde{P}_+$ with $i=1,2,6,7$, $$\begin{aligned} y_{i,j}(u-2)y_{i,j}(u+2) &= \frac{ (1+y_{i-1,j}(u))(1+y_{i+1,j}(u)) } { (1+y_{i,j-1}(u)^{-1})(1+y_{i,j+1}(u)^{-1}) },\end{aligned}$$ and, with $i=3,4,5$, $$\begin{aligned} \begin{split} y_{3,j}(u-2)y_{3,j}(u+2) &= \frac { \genfrac{}{}{0pt}{}{ \displaystyle (1+y_{2,j}(u))(1+y_{4,2j-1}(u))(1+y_{4,2j+1}(u)) } { \displaystyle \times (1+y_{4,2j}(u-1))(1+y_{4,2j}(u+1)) } } { (1+y_{3,j-1}(u)^{-1})(1+y_{3,j+1}(u)^{-1}) },\\ y_{4,2j}(u-1)y_{4,2j}(u+1) &= \begin{cases} \dfrac { 1+y_{3,j}(u) } { (1+y_{4,2j-1}(u)^{-1})(1+y_{4,2j+1}(u)^{-1}) }& u+2j\equiv 0\ (4) \\ \dfrac { 1+y_{5,j}(u) } { (1+y_{4,2j-1}(u)^{-1})(1+y_{4,2j+1}(u)^{-1}) }&u+2j\equiv 2\ (4),\\ \end{cases} \\ y_{4,2j+1}(u-1)y_{4,2j+1}(u+1) &= \frac { 1 } { (1+y_{4,2j}(u)^{-1})(1+y_{4,2j+2}(u)^{-1}) },\\ y_{5,j}(u-2)y_{5,j}(u+2) &= \frac { \genfrac{}{}{0pt}{}{ \displaystyle (1+y_{6,j}(u))(1+y_{4,2j-1}(u))(1+y_{4,2j+1}(u)) } { \displaystyle \times (1+y_{4,2j}(u-1))(1+y_{4,2j}(u+1)) } } { (1+y_{5,j-1}(u)^{-1})(1+y_{5,j+1}(u)^{-1}) }. \end{split}\end{aligned}$$ [T-system:]{} For $((i,j),u)\in P_+$ with $i=1,2,6,7$, $$\begin{aligned} [\tilde{x}_{i,j}(u-2)]_{\mathbf{1}}[\tilde{x}_{i,j}(u+2)]_{\mathbf{1}} &= [\tilde{x}_{i-1,j}(u)]_{\mathbf{1}}[\tilde{x}_{i+1,j}(u)]_{\mathbf{1}} + [\tilde{x}_{i,j-1}(u)]_{\mathbf{1}}[\tilde{x}_{i,j+1}(u)]_{\mathbf{1}} ,\end{aligned}$$ and, with $i=3,4,5$, $$\begin{aligned} \begin{split} &[\tilde{x}_{3,j}(u-2)]_{\mathbf{1}}[\tilde{x}_{3,j}(u+2)]_{\mathbf{1}} = [\tilde{x}_{2,j}(u)]_{\mathbf{1}}[\tilde{x}_{4,2j}(u)]_{\mathbf{1}} + [\tilde{x}_{3,j-1}(u)]_{\mathbf{1}}[\tilde{x}_{3,j+1}(u)]_{\mathbf{1}} ,\\ &[\tilde{x}_{4,2j}(u-1)]_{\mathbf{1}}[\tilde{x}_{4,2j}(u+1)]_{\mathbf{1}}\\ & = \begin{cases} [\tilde{x}_{5,j}(u-1)]_{\mathbf{1}}[\tilde{x}_{3,j}(u+1)]_{\mathbf{1}} + [\tilde{x}_{4,2j-1}(u)]_{\mathbf{1}}[\tilde{x}_{4,2j+1}(u)]_{\mathbf{1}} &u+2j\equiv 1\ (4)\\ [\tilde{x}_{3,j}(u-1)]_{\mathbf{1}}[\tilde{x}_{5,j}(u+1)]_{\mathbf{1}} + [\tilde{x}_{4,2j-1}(u)]_{\mathbf{1}}[\tilde{x}_{4,2j+1}(u)]_{\mathbf{1}} &u+2j\equiv 3\ (4),\\ \end{cases} \\ & [\tilde{x}_{4,2j+1}(u-1)]_{\mathbf{1}}[\tilde{x}_{4,2j+1}(u+1)]_{\mathbf{1}} \\ &= \begin{cases} [\tilde{x}_{5,j}(u)]_{\mathbf{1}}[\tilde{x}_{3,j+1}(u)]_{\mathbf{1}} + [\tilde{x}_{4,2j}(u)]_{\mathbf{1}}[\tilde{x}_{4,2j+2}(u)]_{\mathbf{1}} &u+2j\equiv 0\ (4)\\ [\tilde{x}_{3,j}(u)]_{\mathbf{1}}[\tilde{x}_{5,j+1}(u)]_{\mathbf{1}} + [\tilde{x}_{4,2j}(u)]_{\mathbf{1}}[\tilde{x}_{4,2j+2}(u)]_{\mathbf{1}} &u+2j\equiv 2\ (4),\\ \end{cases} \\ &[\tilde{x}_{5,j}(u-2)]_{\mathbf{1}}[\tilde{x}_{5,j}(u+2)]_{\mathbf{1}} = [\tilde{x}_{6,j}(u)]_{\mathbf{1}}[\tilde{x}_{4,2j}(u)]_{\mathbf{1}} + [\tilde{x}_{5,j-1}(u)]_{\mathbf{1}}[\tilde{x}_{5,j+1}(u)]_{\mathbf{1}} . \end{split}\end{aligned}$$ Under a suitable identification of variables, they coincide with the Y- and T-systems associated with the quantum affine algebras of type $B_4$ with level 4. See [@Inoue10a] for more information. Standalone versions of T- and Y-systems {#subsec:stand} --------------------------------------- Here we establish certain formal property concerning T- and Y-systems. We just introduced the T-systems (without coefficient, for simplicity) and Y-systems as relations inside a cluster algebra $\mathcal{A}(B,x)$ and a coefficient group $\mathcal{G}(B,y)$. On the other hand, usually these relations appear without their ‘ambient’ cluster algebras or coefficient groups. Therefore, to apply the cluster algebraic machinery of [@Fomin07], which has been proved to be so efficient and powerful, it is necessary to establish a precise connection between such ‘standalone’ T- and Y-systems and the ‘built-in’ T- and Y-systems inside cluster algebras. In fact, this has been repeatedly established case-by-case for each explicit example (e.g., [@Inoue10c; @Kuniba09; @Inoue10a; @Nakanishi10a; @Nakanishi10b]). Here we do it again, and hopeful for the last time, in a general setting. We continue to use the same notations as in Section \[subsec:TY1\]. First, we introduce the ring/group associated with the built-in T-system/Y-system in cluster algebra/coefficient group. \[def:BI\] (1) The [T-subalgebra $\mathcal{A}_{\mathbf{i}}(B,x)$ of $\mathcal{A}(B,x)$ associated with a slice $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ of a regular $\nu$-period of $B$]{} is the subring of $\mathcal{A}(B,x)$ generated by $[x_i(u)]_{\mathbf{1}}$ ($(i,u)\in P_+$), or equivalently, generated by $[\tilde{x}_i(u)]_{\mathbf{1}}$ ($(i,u\in \tilde{P}_+$). \(2) The [Y-subgroup $\mathcal{G}_{\mathbf{i}}(B,y)$ of $\mathcal{G}(B,y)$ associated with a slice $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ of a regular $\nu$-period of $B$]{} is the multiplicative subgroup of $\mathcal{G}(B,y)$ generated by $y_i(u)$ and $1+y_i(u)$ ($(i,u)\in P_+$). The following fact was casually mentioned in Remark \[rem:choice\]. \[prop:choice\] The algebra $\mathcal{A}_{\mathbf{i}}(B,x)$ and the group $\mathcal{G}_{\mathbf{i}}(B,y)$ depend only on $\mathbf{i}$ and do not depend on the choice of a slice of $\mathbf{i}$. Suppose we take two different slices of $\mathbf{i}$. Then, there is a natural bijection between the forward mutation points $(i,u) \leftrightarrow (i',u')$ for two choices. Then, we have $y_i(u)= y_{i'}(u')$ and $x_i(u)= x_{i'}(u')$ in $\mathcal{A}(B,x,y)$, thanks to the condition and Lemma \[lem:order\]. Next, we introduce the corresponding ring/group for the standalone T-system/Y-system. \[def:SA\] (1) Let $\EuScript{T}(B,\mathbf{i})$ be the commutative ring over $\mathbb{Z}$ with identity element, with generators $T_i(u)^{\pm 1}$ ($(i,u)\in \tilde{P}_+$) and relations , where $[\tilde{x}_i(u)]_{\mathbf{1}}$ is replaced with $T_i(u)$, together with $T_i(u) T_i(u)^{-1}=1$. Let $\EuScript{T}^{\circ}(B,\mathbf{i})$ be the subring of $\EuScript{T}(B,\mathbf{i})$ generated by $T_i(u)$ ($(i,u)\in \tilde{P}_+$). \(2) Let $\EuScript{Y}(B,\mathbf{i})$ be the semifield with generators $Y_i(u)$ ($(i,u)\in P_+$) and relations , where $y_i(u)$ is replaced with $Y_i(u)$. Let $\EuScript{Y}^{\circ}(B,\mathbf{i})$ be the multiplicative subgroup of $\EuScript{Y}(B,\mathbf{i})$ generated by $Y_i(u)$ and $1+Y_i(u)$ ($(i,u)\in P_+$). Two rings/groups defined above are isomorphic. \[thm:iso\] (1) The ring $\EuScript{T}^{\circ}(B,\mathbf{i})$ is isomorphic to $\mathcal{A}_{\mathbf{i}}(B,x)$ by the correspondence $T_i(u)\mapsto [\tilde{x}_i(u)]_{\mathbf{1}}$. \(2) The group $\EuScript{Y}^{\circ}(B,\mathbf{i})$ is isomorphic to $\mathcal{G}_{\mathbf{i}}(B,y)$ by the correspondence $Y_i(u)\mapsto y_i(u)$ and $1+Y_i(u)\mapsto 1+y_i(u)$. \(1) The map $\rho :T_i(u)\mapsto [\tilde{x}_i(u)]_{\mathbf{1}}$ is a ring homomorphism by definition. We can construct the inverse of $\rho$ as follows. For each $i\in I$, let $u_i\in \mathbb{Z}$ be the smallest nonnegative $u_i$ such that $(i,u_i)\in P_+$. We define a ring homomorphism $\phi:\mathbb{Z}[x_i^{\pm1}]_{i\in I} \rightarrow \EuScript{T}(B,\mathbf{i})$ by $x_i^{\pm1}\mapsto T_i(u_i-\frac{tg_i}{2})^{\pm1}$. Thus, we have $\phi:[\tilde{x}_i(u_i-\frac{tg_i}{2})]_{\mathbf{1}} = [x_i(u_i)]_{\mathbf{1}}= [x_i(0)]_{\mathbf{1}}\mapsto T_i(u_i-\frac{tg_i}{2})$. Furthermore, one can prove that $\phi:[\tilde{x}_i(u)]_{\mathbf{1}}\mapsto T_i(u)$ for any $(i,u)\in \tilde{P}_+$ by induction on the forward mutations for $u>u_i-\frac{tg_i}{2}$ and on the backward mutations for $u<u_i-\frac{tg_i}{2}$, using the common T-systems for the both sides. By the restriction of $\phi$ to $\mathcal{A}_{\mathbf{i}}(B,x)$, we obtain a ring homomorphism $\varphi: \mathcal{A}_{\mathbf{i}}(B,x)\rightarrow \EuScript{T}^{\circ}(B,\mathbf{i})$, which is the inverse of $\rho$. \(2) This is parallel to (1). The map $\rho :Y_i(u)\mapsto y_i(u)$, $1+Y_i(u)\mapsto 1+y_i(u)$ is a group homomorphism by definition. We can construct the inverse of $\rho$ as follows. For each $i\in I$, let $u_i\in \mathbb{Z}$ be the largest nonpositive $u_i$ such that $(i,u_i)\in P_+$. We define a semifield homomorphism $\phi:\mathbb{P}_{\mathrm{univ}}(y_i)_{i\in I} \rightarrow \EuScript{Y}(B,\mathbf{i})$ as follows. If $u_i=0$, then $\phi(y_i)=Y_i(0)$. If $u_i < 0$, we define $$\begin{aligned} \phi(y_i)= Y_i(u_i)^{-1} \frac{ \displaystyle \prod_{(j,v)} (1+Y_j(v))^{-b_{ji}(v)} } { \displaystyle \prod_{(j,v)} (1+Y_j(v)^{-1})^{b_{ji}(v)}, }\end{aligned}$$ where the product in the numerator is taken for $(j,v)\in P_+$ such that $u_i < v < 0$ and $b_{ji}(v)<0$, and the product in the denominator is taken for $(j,v)\in P_+$ such that $u_i < v < 0$ and $b_{ji}(v)>0$. Then, we have $\phi:y_i(u_i)\mapsto Y_i(u_i)$. Furthermore, one can prove that $\phi:y_i(u)\mapsto Y_i(u)$ for any $(i,u)\in P_+$ by induction on the forward mutations for $u>u_i$ and on the backward mutations for $u<u_i$, using the common Y-systems for the both sides. By the restriction of $\phi$ to $\mathcal{G}_{\mathbf{i}}(B,y)$, we obtain a group homomorphism $\varphi: \mathcal{G}_{\mathbf{i}}(B,y)\rightarrow \EuScript{Y}^{\circ}(B,\mathbf{i})$, which is the inverse of $\rho$. Aside from the direct connection between T- and Y-systems in Proposition \[prop:TY1\], there is an algebraic connection, which has been noticed since the inception of the original T- and Y-systems [@Klumper92; @Kuniba94a]. \[prop:GH\] Let $\EuScript{T}(B,\mathbf{i})$ be the ring in Definition \[def:SA\]. For each $(i,u)\in P_+$, we set $$\begin{aligned} \label{eq:YT1} Y_i(u):=\frac{ \displaystyle \prod_{(j,v)\in \tilde{P}_+} T_j(v)^{H_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in\tilde{P}_+} T_j(v)^{H_-(j,v;i,u)} }.\end{aligned}$$ Then, $Y_i(u)$ satisfies the Y-system in $\EuScript{T}(B,\mathbf{i})$ by replacing $y_i(u)$ with $Y_i(u)$. Note that is the ratio of the first and second terms in . Thanks to the isomorphism in Theorem \[thm:iso\], one can work in the localization of $\mathcal{A}_{\mathbf{i}}(B,x)$ by generators $[\tilde{x}_i(u)]_{\mathbf{1}}$ ($(i,u)\in\tilde{P}_+$), which is a subring of the ambient field $\mathbb{Q}(x)$. The claim is translated therein as follows: [For each $(i,u)\in P_+$, we set $$\begin{aligned} \bar{y}_i(u)&:= \frac{ \displaystyle \prod_{(j,v)\in\tilde{P}_+} [\tilde{x}_j(v)]_{\mathbf{1}}^{H_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in\tilde{P}_+} [\tilde{x}_j(v)]_{\mathbf{1}}^{H_-(j,v;i,u)} } = \frac{ \displaystyle \prod_{(j,v)\in P_+} [{x}_j(v)]_{\mathbf{1}}^{{H}'_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} [{x}_j(v)]_{\mathbf{1}}^{{H}'_-(j,v;i,u)} } =\prod_{j\in I} [x_j(u)]_{\mathbf{1}}^{b_{ji}(u)}.\end{aligned}$$ Then $\bar{y}_i(u)$ satisfies the Y-system by replacing $y_i(u)$ with $\bar{y}_i(u)$.]{} In fact, this claim is an immediate consequence of [@Fomin07 Proposition 3.9]. Some classic examples of T- and Y-systems are not always in the ‘straight form’ presented here, but represented by generators $T_{\bar{i}}(u)$ and $Y_{\bar{i}}(u)$ whose indices $\bar{i}$ belong to the orbit space $I/\nu$ of $I$ by $\nu$. For example, the T- and Y-systems for type $(X,\ell)= (B_4,4)$ in Section \[subsec:exTY\] [@Inoue10a], and the sine-Gordon T- and Y-systems for Example \[ex:sine\] [@Nakanishi10b] are such cases. In these examples, it is just a ‘change of notation’ for generators. However, this makes the reconstruction of the initial exchange matrix $B$ from given T- or Y-systems nontrivial, because [a priori]{} we only know $I/\nu$, and we have to find out true index set $I$ and $\nu$ with some guesswork. T- and Y-systems for general period {#subsec:nonadm} ----------------------------------- Conceptually, the notions of T- and Y-systems can be straightforwardly extended to general $\nu$-periods of $B$, though they become a little apart from the ‘classic’ T- and Y-systems. We will use them in Section \[sec:dilog\]. Let $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ be a slice of any (not necessarily regular) $\nu$-period $\mathbf{i}$ of $B$. One can still define the sequence of seeds $(B(u),x(u),y(u))$ ($u\in \mathbb{Z}$) and the forward mutation points $(i,u)\in P_+$ as in the regular case. Fix $i\in I$, and let $$\begin{aligned} \label{eq:for1} \dots ,(i,u),\, (i,u'),\, (i,u''),\, \dots \quad (\dots < u < u' < u''<\dots)\end{aligned}$$ be the sequence of the forward mutation points. (It may be empty for some $i$.) In general, if it is not empty, the sequence $\dots, u, u', u'',\dots$ is periodic for $u \rightarrow u + tg$, but it does [not necessarily]{} have the common difference. For each $(i,u)\in P_+$, let $(i,u+\lambda_+(i,u))$ and $(i,u-\lambda_-(i,u))$ be the nearest ones to $(i,u)$ in the sequence in the forward and backward directions, respectively; in other words, $(i,u-\lambda_-(i,u))$, $(i,u)$, $(i,u+\lambda_+(i,u))$ are three consecutive forward mutation points in . If $\mathbf{i}$ is regular, then $\lambda_{\pm}(i,u)=tg_i$, which is the common difference (therefore, called regular). In general, we have $0< \lambda_{\pm}(i,u)<tg$, $\lambda_{\pm}(i,u+tg)=\lambda_{\pm}(i,u)$, and $$\begin{aligned} \lambda_+(i,u-\lambda_-(i,u))=\lambda_-(i,u).\end{aligned}$$ Let $J(\mathbf{i},\nu)$ be the subset of $I$ consisting of all the components of $\mathbf{j}(\mathbf{i},\nu)$. Note that the condition (A1) in Definition \[def:regular\] means that $J(\mathbf{i},\nu)=I$. Using these notations, the relations and are generalized as follows. For $(i,u)\in P_+$, $$\begin{aligned} \label{eq:yi'2} \textstyle y_i\left(u \right)y_i\left(u+\lambda_+(i,u) \right) &= \frac{ \displaystyle \prod_{(j,v)\in P_+} (1+y_j(v))^{G'_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} (1+y_j(v)^{-1})^{G'_-(j,v;i,u)} },\\ \label{eq:G'2} G'_{\pm}(j,v;i,u)&= \begin{cases} \mp b_{ji}(v) & v\in (u, u+\lambda_+(i,u)), b_{ji}(v)\lessgtr 0\\ 0 & \mbox{otherwise}, \end{cases}\end{aligned}$$ and $$\begin{aligned} \label{eq:xi'3} \begin{split} x_i(u)x_i(u+\lambda_+(i,u)) &= \frac{y_i(u)}{1+y_i(u)} \prod_{j\in I\setminus J(\mathbf{i},\nu):\, b_{ji}>0} x_j^{b_{ji}(u)} \prod_{(j,v)\in P_+} x_j(v)^{H'_+(j,v;i,u)}\\ &\quad + \frac{1}{1+y_i(u)} \prod_{j\in I\setminus J(\mathbf{i},\nu):\, b_{ji}<0} x_j^{-b_{ji}(u)} \prod_{(j,v)\in P_+} x_j(v)^{H'_-(j,v;i,u)}, \end{split}\\ \label{eq:tH2} H'_{\pm}(j,v;i,u)&= \begin{cases} \pm b_{ji}(u)& u\in (v- \lambda_-(j,v),v), b_{ji}(u)\gtrless 0\\ 0 & \mbox{otherwise}. \end{cases}\end{aligned}$$ When a $\nu$-period $\mathbf{i}$ of $B$ satisfies the condition (A1) in Definition \[def:regular\], the relation slightly simplifies as $$\begin{aligned} \label{eq:xi'2} \begin{split} x_i(u)x_i(u+\lambda_+(i,u)) &= \frac{y_i(u)}{1+y_i(u)} \prod_{(j,v)\in P_+} x_j(v)^{H'_+(j,v;i,u)}\\ &\quad + \frac{1}{1+y_i(u)} \prod_{(j,v)\in P_+} x_j(v)^{H'_-(j,v;i,u)}. \end{split}\end{aligned}$$ Unfortunately, one cannot simultaneously rewrite and / into the balanced form similar to and . Therefore, we just regard and / as the Y- and T-systems associated with a slice $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ of a $\nu$-period of $B$. Accordingly, Proposition \[prop:TY1\] is generalized as follows. \[prop:TY1’\] Let $D=\mathrm{diag} (d_i)_{i\in I}$ be a diagonal matrix such that ${}^{t}(DB)=-DB$. Let $G'_{\pm}(j,v;i,u)$ and $H'_{\pm}(j,v;i,u)$ the ones in and , respectively. Then, the following relation holds for any $(j,v)\in P_+$ and $(i,u)\in \tilde{P}_+$. $$\begin{aligned} \label{eq:GH1'} d_j G'_{\pm} (j,v;i,u-\lambda_-(i,u))= d_i H'_{\pm} (i,u;j,v).\end{aligned}$$ In particular, if $B$ is skew symmetric, we have $$\begin{aligned} \label{eq:GH2'} G'_{\pm} (j,v;i,u-\lambda_-(i,u))= H'_{\pm} (i,u;j,v).\end{aligned}$$ This is immediate from and . The results in Section \[subsec:stand\] can be also generalized and/or modified straightforwardly. We leave it as an exercise for the reader. Dilogarithm identities {#sec:dilog} ====================== Now we are ready to work on the main subject of the paper, the dilogarithm identities associated with any period of a seed. They are natural generalizations of the results in the former examples [@Frenkel95; @Chapoton05; @Nakanishi09; @Inoue10a; @Inoue10b; @Nakanishi10b]. The subject has originated from the pioneering works by Bazhanov, Kirillov, and Reshetikhin [@Kirillov86; @Kirillov89; @Kirillov90; @Bazhanov90]. See also [@Kirillov95; @Zagier07; @Nahm07] for further background of the identities. Dilogarithm identities {#dilogarithm-identities} ---------------------- Here we concentrate on the case when the exchange matrix $B$ is [skew symmetric]{}. (See Section \[subsec:dilog2\] for the skew symmetrizable case.) Let $\mathbf{i}$ be any period of $(B,x,y)$. In particular, $\mathbf{i}$ is also a period of $B$. Let $\mathbf{i}=\mathbf{i}(0)\,\|\, \cdots \,\|\, \mathbf{i}(t-1)$ be any slice of $\mathbf{i}$. Then, the sequence of seeds $(B(u),x(u),y(u))$ ($u\in \mathbb{Z}$) and the forward mutation points $(i,u)\in P_+$ are defined as in Section \[sec:T\], and we have the associated Y-system . To emphasize the periodicity, we set $\Omega:=t$. Then, by the periodicity assumption, we have $$\begin{aligned} \label{eq:yperiod} y_i(u+\Omega)&=y_i(u).\end{aligned}$$ Accordingly, we define the ‘fundamental regions’ $S_+$ of forward mutation points by $$\begin{aligned} S_+&=\{ (i,u)\in P_+ \mid 0\leq u < \Omega\}.\end{aligned}$$ Thanks to Theorem \[thm:pos\], each Laurent monomial $[y_i(u)]_{\mathbf{T}}$ in $y$ ($(i,u)\in P_+$) is either positive or negative. Let $N_+$ (resp. $N_-$) be the total number of positive (resp. negative) monomials $[y_i(u)]_{\mathbf{T}}$ in the fundamental region $S_+$. Let $L(x)$ be the Rogers dilogarithm function [@Lewin81]. $$\begin{aligned} \label{eq:L0} L(x)=-\frac{1}{2}\int_{0}^x \left\{ \frac{\log(1-y)}{y}+ \frac{\log y}{1-y} \right\} dy \quad (0\leq x\leq 1).\end{aligned}$$ It satisfies the following properties. $$\begin{gathered} \label{eq:L01} L(0)=0, \quad L(1)=\frac{\pi^2}{6},\\ \label{eq:euler} L(x)+L(1-x)=\frac{\pi^2}{6} \quad (0\leq x\leq 1).\end{gathered}$$ The following is the main result of the paper. \[thm:DI\] Suppose that a family of real positive numbers $\{Y_i(u) \mid (i,u)\in P_+\}$ satisfies the Y-system by replacing $y_i(u)$ with $Y_i(u)$. Then, we have the identities. $$\begin{aligned} \label{eq:DI} \frac{6}{\pi^2} \sum_{ (i,u)\in S_+ } L\left( \frac{Y_i(u)}{1+Y_i(u)} \right) &=N_-,\\ \label{eq:DI'} \frac{6}{\pi^2} \sum_{ (i,u)\in S_+ } L\left( \frac{1}{1+Y_i(u)} \right) &=N_+.\end{aligned}$$ Two identities and are equivalent to each other due to . Proof of Theorem \[thm:DI\] --------------------------- We prove Theorem \[thm:DI\] using the method developed by [@Frenkel95; @Chapoton05; @Nakanishi09]. We start from a very general theorem on the ‘constancy property’ of dilogarithm sum by Frenkel-Szenes [@Frenkel95]. For any multiplicative abelian group $A$, let $A\otimes_{\mathbb{Z}} A$ be the additive abelian group with generators $f\otimes g$ ($f,g\in A$) and relations $$\begin{aligned} (fg)\otimes h = f\otimes h + g \otimes h, \quad f\otimes (gh)=f\otimes g + f\otimes h.\end{aligned}$$ It follows that we have $$\begin{aligned} 1\otimes h = h\otimes 1=0, \quad f^{-1}\otimes g= f\otimes g^{-1}=- f\otimes g.\end{aligned}$$ Let $S^2 A$ be the subgroup of $A\otimes_{\mathbb{Z}} A$ generated by $f\otimes f$ ($f\in A$). Define $\bigwedge^2 A$ be the quotient group of $A\otimes_{\mathbb{Z}} A$ by $S^2 A$. In $A\otimes_{\mathbb{Z}} A$ we use $\wedge$ instead of $\otimes$. Hence, $f\wedge g = -g \wedge f$ holds. Now let $\mathcal{I}$ be any open or closed interval of $\mathbb{R}$, and let $\EuScript{C}(\mathcal{I})$ be the multiplicative abelian group of all the differentiable functions $f(t)$ from $\mathcal{I}$ to the set of all the positive real numbers $\mathbb{R}_{+}$. \[thm:const\] Let $f_1(t),\dots,f_k(t)$ be differentiable functions from $\mathcal{I}$ to $(0,1)$. Suppose that they satisfy the following relation in $\bigwedge^2 \EuScript{C}(\mathcal{I})$. $$\begin{aligned} \label{eq:const1} \sum_{i=1}^k f_i(t) \wedge (1-f_i(t)) = 0 \quad \mbox{\rm (constancy condition)}.\end{aligned}$$ Then, the dilogarithm sum $\sum_{i=1}^k L(f_i(t))$ is constant with respect to $t\in \mathcal{I}$. We remark that the proof of the theorem by [@Frenkel95] is quite simple, just showing the derivative of the dilogarithm sum vanishes by the symmetry reason. Next we use the idea of Chapoton [@Chapoton05] to integrate the condition in the cluster algebra setting. We are going to prove the following claim. \[prop:const\] In $\bigwedge ^2 \mathbb{P}_{\mathrm{univ}}(y)$, the following relation holds. $$\begin{aligned} \label{eq:const2} \sum_{(i,u)\in S_+} \frac{y_i(u)}{1+y_i(u)}\wedge \frac{1}{1+y_i(u)}=0,\end{aligned}$$ or, equivalently, $$\begin{aligned} \label{eq:const3} \sum_{(i,u)\in S_+} y_i(u)\wedge (1+y_i(u))=0.\end{aligned}$$ It is clear that the relations and are equivalent. We call them the [constancy condition]{} for the dilogarithm identities and . Temporarily assuming Proposition \[prop:const\], we prove the following theorem, which is equivalent to Theorem \[thm:DI\]. Let $\mathbb{R}_+$ be the semifield of all the positive real numbers with the usual multiplication and addition. Then, for any semifield homomorphism $\varphi: \mathbb{P}_{\mathrm{univ}}(y) \rightarrow \mathbb{R}_+$, we have the identity, $$\begin{aligned} \label{eq:DI1} \frac{6}{\pi^2} \sum_{ (i,u)\in S_+ } L\left( \varphi \left( \frac{y_i(u)}{1+y_i(u)} \right) \right) &=N_-,\\ \label{eq:DI'1} \frac{6}{\pi^2} \sum_{ (i,u)\in S_+ } L\left( \varphi \left( \frac{1}{1+y_i(u)} \right) \right) &=N_+.\end{aligned}$$ We concentrate on the first case . The proof separates into two steps. [Step 1: Consistency.]{} Let us first show that the left hand side of is independent of the choice of $\varphi$. Note that any semifield homomorphism $\varphi$ is determined by the values $\varphi(y_i)$ ($i\in I$) of generators of $\mathbb{P}_{\mathrm{univ}}(y)$. Let $\varphi_0$ and $\varphi_1$ be different homomorphisms from $\mathbb{P}_{\mathrm{univ}}(y)$ to $\mathbb{R}_+$. Then we introduce the one parameter family of homomorphisms $\varphi_t: \mathbb{P}_{\mathrm{univ}}(y)\rightarrow \mathbb{R}_+$ ($t\in [0,1]$) interpolating $\varphi_0$ to $\varphi_1$ by $\varphi_t(y_i)= (1-t) \varphi_0(y_i)+ t \varphi_1(y_i)$. Set $f_{i,u}(t):= \varphi_t(y_i(u)/(1+y_i(u)))$ ($(i,u)\in S_+$) and regard them as a family of functions of $t$ in the interval $[0,1]$. Then, due to , they satisfy the constancy condition with $\mathcal{I}=[0,1]$. Then, by Theorem \[thm:const\], the left hand side of for $\varphi=\varphi_t$ is independent of $t$; in particular, it is the same for $\varphi=\varphi_0$ and $\varphi_1$. [Step 2: Evaluation at $0/\infty$ limit.]{} Since we established the constancy of the left hand side of , we evaluate it at a certain limit of $\varphi$ where [each value $\varphi(y_i(u)/(1+y_i(u)))$ $((i,u)\in S_+)$ goes to either 0 or $\infty$]{} ($0/\infty$ limit). Then, by , the value of the left hand side of is the total number of $(i,u)\in S_+$ such that the value $\varphi(y_i(u)/(1+y_i(u)))$ goes to $\infty$. It is not obvious that such a limit exists. However, simply take the one parameter family of the homomorphism $\varphi_t: \mathbb{P}_{\mathrm{univ}}\rightarrow \mathbb{R}_+$ ($t\in (0,\varepsilon)$) for some $\varepsilon>0$ defined by $\varphi_t(y_i)=t$ for any $i\in I$. Then, thanks to and the parts (a) and (c) of Conjecture \[conj:pos\]/Theorem \[thm:pos\], the limit $\lim_{t\rightarrow 0} \varphi_t$ is indeed a $0/\infty$ limit, and the total number of $(i,u)\in S_+$ such that the value $\varphi(y_i(u)/(1+y_i(u)))$ goes to $\infty$ is exactly $N_-$. In the above proof, Step 1 (constancy) is due to [@Frenkel95], and Step 2 (evaluation at $0/\infty$ limit) is due to [@Chapoton05]. First proof of Proposition \[prop:const\] ----------------------------------------- Now we only have to prove Proposition \[prop:const\]. We give two proofs in this and the next subsections. The first one here is a generalization of the former proofs for special cases [@Nakanishi09; @Inoue10a], which is a somewhat brute force proof with ‘change of indices’. To start, we claim that one can assume all the components of $\mathbf{i}$ exhaust $I$. In fact, if the condition does not hold, one may reduce the index set $I$ so that the condition holds, since doing that does not affect the Y-system . Note that the condition is nothing but the condition (A1) in Definition \[def:regular\] with $\nu=\mathrm{id}$. Thus, the accompanying T-system has the simplified form . Furthermore, by the periodicity assumption and Theorem \[thm:r/e\] (a), we have $$\begin{aligned} \label{eq:xperiod} x_i(u+\Omega)&=x_i(u).\end{aligned}$$ Now, let $F_i(u)$ be the $F$-polynomials at $(i,u)$. \[lem:F\] The following properties hold. \(a) Periodicity: $F_i(u+\Omega)=F_i(u)$. \(b) For $(i,u)\in {S}_+$, $$\begin{aligned} \label{eq:Fi} \begin{split} \textstyle F_i(u)F_i(u+\lambda_+(i,u)) &= \left[ \frac{y_i(u)}{1+y_i(u)} \right]_{\mathbf{T}} \prod_{(j,v)\in P_+} F_j(v)^{H'_+(j,v;i,u)}\\ &\quad + \left[ \frac{1}{1+y_i(u)} \right]_{\mathbf{T}} \prod_{(j,v)\in P_+} F_j(v)^{H'_-(j,v;i,u)}. \end{split}\end{aligned}$$ \(c) For $(i,u)\in{S}_+$, $$\begin{aligned} \label{eq:yi1} y_i(u)&= [y_i(u)]_{\mathbf{T}} \frac { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_-(j,v;i,u)} },\\ \label{eq:yi2} 1+y_i(u)&= [1+y_i(u)]_{\mathbf{T}} \frac { F_i(u)F_i(u+\lambda_+(i,u)) } { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_-(j,v;i,u)} }.\end{aligned}$$ \(a) This is obtained from the specialization of . (b) This is obtained from the specialization of . (c) The first equality is obtained by rewriting with . The second one is obtained from the first one and (b). We prove . We put and into the left hand side of , expand it, then, sum them up into three parts as follows. The first part consists of the terms only involving tropical coefficients, i.e., $$\begin{aligned} \label{eq:const4} \sum_{(i,u)\in S_+} [y_i(u)]_{\mathbf{T}}\wedge [1+y_i(u)]_{\mathbf{T}}.\end{aligned}$$ By Theorem \[thm:pos\], each monomial $[y_i(u)]_{\mathbf{T}}$ is either positive or negative. If it is positive, then $[y_i(u)]_{\mathbf{T}}\wedge [1+y_i(u)]_{\mathbf{T}}= [y_i(u)]_{\mathbf{T}}\wedge 1=0$. If it is negative, then, $[y_i(u)]_{\mathbf{T}}\wedge [1+y_i(u)]_{\mathbf{T}}= [y_i(u)]_{\mathbf{T}}\wedge [y_i(u)]_{\mathbf{T}}=0$. Therefore, the sum vanishes. The second part consists of the terms involving both tropical coefficients and $F$-polynomials. We separate them into five parts, $$\begin{aligned} \label{eq:yF} \begin{split} \sum_{(i,u)\in S_+} [y_{i}(u)]_{\mathrm{T}} \wedge \textstyle F_{i}(u), & %= %\sum_{(j,v)\in S_+} %\textstyle %[y_{j}(v)]_{\mathrm{T}} %\wedge %F_{j}(v), \\ \sum_{(i,u)\in S_+} \textstyle [y_{i}(u)]_{\mathrm{T}} \wedge F_{i}(u+\lambda_+(i,u)) &= \sum_{(i,u)\in S_+} \textstyle [y_{i}(u-\lambda_-(i,u))]_{\mathrm{T}} \wedge F_{i}(u), \\ -\sum_{(i,u)\in S_+} [y_{i}(u)]_{\mathrm{T}} \wedge \prod_{(j,v)\in P_+} & F_{j}(v)^{H'_-(j,v;i,u)}\\ &= \sum_{(i,u)\in S_+} \prod_{(j,v)\in P_+} [y_j(v)]_{\mathrm{T}}^{-H'_-(i,u;j,v)} \wedge F_{i}(u), \\ -\sum_{(i,u)\in S_+} [1+y_{i}(u)]_{\mathrm{T}} \wedge \prod_{(j,v)\in P_+ } & F_{j}(v)^{H'_+(j,v;i,u)}\\ &= \sum_{(i,u)\in S_+} \prod_{(j,v)\in P_+} [1+y_j(v)]_{\mathrm{T}}^{-H'_+(i,u;j,v)} \wedge F_{i}(u), \\ \sum_{(i,u)\in S_+} [1+y_{i}(u)]_{\mathrm{T}} \wedge \prod_{(j,v)\in P_+} & F_{j}(v)^{H'_-(j,v;i,u)}\\ &= \sum_{(i,u)\in S_+} \prod_{(j,v)\in P_+} [1+y_{j}(v)]_{\mathrm{T}}^{H'_-(i,u;j,v)} \wedge F_{i}(u), \end{split}\end{aligned}$$ where we changed indices and also used the periodicity of $[y_i(u)]_{\mathbf{T}}$, $F_i(u)$, $\lambda_-(i,u)$, and $H'_{\pm}(j,v;i,u)$. Recall the relation $H'_{\pm}(i,u;j,v)= G'_{\pm}(j,v;i,u-\lambda_-(i,u))$ in . Then, the sum of the above five terms vanishes due to the ‘tropical Y-system’ $$\begin{aligned} \textstyle [y_i\left(u-\lambda_-(i,u)\right)]_{\mathbf{T}} [y_i\left(u\right)]_{\mathbf{T}} &= \frac{ \displaystyle \prod_{(j,v)\in P_+} [1+y_j(v)]_{\mathbf{T}}^{G'_+(j,v;i,u-\lambda_-(i,u))} } { \displaystyle \prod_{(j,v)\in P_+} [1+y_j(v)^{-1}]_{\mathbf{T}}^{G'_-(j,v;i,u-\lambda_-(i,u))} },\end{aligned}$$ which is a specialization of . The third part consists of the terms involving only $F$-polynomials. It turns out that this part requires the most elaborated treatment. We separate them into three parts, $$\begin{aligned} \label{eq:FF1} \mathrm{(A)}&=\sum_{(i,u)\in S_+} \frac { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_-(j,v;i,u)} } \wedge \textstyle F_i(u),\\ \label{eq:FF2} \mathrm{(B)}&=\sum_{(i,u)\in S_+} \frac { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_+(j,v;i,u)} } { \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_-(j,v;i,u)} } \wedge \textstyle F_i(u+\lambda_+(i,u)),\\ \label{eq:FF3} \mathrm{(C)}&=\sum_{(i,u)\in S_+} \displaystyle \prod_{(j,v)\in P_+} F_j(v)^{H'_+(j,v;i,u)} \wedge \prod_{(j,v)\in P_+} F_j(v)^{-H'_-(j,v;i,u)}.\end{aligned}$$ Let us rewrite each term so that their cancellation becomes manifest. The first term (A) is rewritten as follows. $$\begin{aligned} \mathrm{(A)}&=\sum_{(i,u)\in S_+} \prod_{\scriptstyle (j,v)\in P_+ \atop \scriptstyle u\in (v-\lambda_-(j,v),v) } F_j(v)^{b_{ji}(u)} \wedge \textstyle F_i(u)\\ &=\sum_{\scriptstyle (i,u)\in S_+,\, (j,v)\in P_+ \atop \scriptstyle u\in (v-\lambda_-(j,v),v) } \textstyle b_{ji}(u) F_j(v) \wedge \textstyle F_i(u)\\ &=\sum_{\scriptstyle (i,u)\in S_+,\, (j,v)\in P_+ \atop \scriptstyle v\in (u-\lambda_-(i,u),u) } \textstyle b_{ji}(v) F_j(v) \wedge \textstyle F_i(u)\\ & = \frac{1}{2} \sum_{\scriptstyle (i,u)\in S_+,\, (j,v)\in P_+ \atop \scriptstyle (u-\lambda_-(i,u),u) \cap(v-\lambda_-(j,v),v)\neq \emptyset } \textstyle b_{ji}(\min(u,v)) F_j(v) \wedge \textstyle F_i(u).\end{aligned}$$ Here the third line is obtained from the second one by the exchange $(i,u)\leftrightarrow (j,v)$ of indices, the skew symmetric property $b_{ji}(u)=-b_{ij}(u)$, and the periodicity; the last line is obtained by averaging the second and the third ones; therefore, there is the factor $1/2$ in the front. We also mention that, in the last line, the pair $(i,u), (j,v)$ with $u=v$ does not contribute to the sum, because in that case we have $b_{ji}(u)=0$ due to the condition . Similarly, the second term (B) is rewritten as follows. $$\begin{aligned} \mathrm{(B)}&=\sum_{(i,u)\in S_+} \prod_{\scriptstyle (j,v)\in P_+ \atop \scriptstyle u\in (v-\lambda_-(j,v),v) } F_j(v)^{b_{ji}(u)} \wedge \textstyle F_i(u+\lambda_+(i,u))\\ &=\sum_{\scriptstyle (i,u)\in S_+,\, (j,v)\in P_+ \atop \scriptstyle u-\lambda_-(i,u)\in (v-\lambda_-(j,v),v) } \textstyle b_{ji}(u-\lambda_-(i,u)) F_j(v) \wedge \textstyle F_i(u)\\ & = \frac{1}{2} \sum_{\scriptstyle (i,u)\in S_+,\, (j,v)\in P_+ \atop \scriptstyle (u-\lambda_-(i,u),u) \cap(v-\lambda_-(j,v),v)\neq \emptyset } \textstyle b_{ji}(\max(u-\lambda_-(i,u), v-\lambda_-(j,v))) F_j(v) \wedge \textstyle F_i(u).\end{aligned}$$ The third term (C) is written as follows. $$\begin{aligned} \mathrm{(C)}&=\sum_{(i,u),(j,v)\in P_+} \Biggl( \sum_{\scriptstyle (k,w)\in S_+ \atop { \scriptstyle k\in (u-\lambda_-(i,u),u)\cap (v-\lambda_-(j,v),v)\atop \scriptstyle b_{jk}(w)>0,\, b_{ik}(w)<0 } } b_{jk}(w) b_{ik}(w) \Biggr) F_j(v) \wedge \textstyle F_i(u)\\ &=\sum_{(i,u),(j,v)\in P_+} \Biggl( \sum_{\scriptstyle (k,w)\in S_+ \atop { \scriptstyle k\in (u-\lambda_-(i,u),u)\cap (v-\lambda_-(j,v),v)\atop \scriptstyle b_{jk}(w)<0,\, b_{ik}(w)>0 } } -b_{jk}(w) b_{ik}(w) \Biggr) F_j(v) \wedge \textstyle F_i(u)\\ &=\frac{1}{2}\sum_{(i,u)\in S_+, (j,v)\in P_+} \Biggl( \sum_{\scriptstyle (k,w)\in P_+ \atop \scriptstyle k\in (u-\lambda_-(i,u),u)\cap (v-\lambda_-(j,v),v) }\\ &\hskip100pt -\frac{1}{2}\Bigl( b_{jk}(w) \|b_{ki}(w)\| + \|b_{jk}(w)\|b_{ki}(w)\Bigr) \Biggr) F_j(v) \wedge \textstyle F_i(u).\end{aligned}$$ Here the second line is obtained from the first one by the exchange $(i,u)\leftrightarrow (j,v)$ of indices; the last line is obtained by averaging the first and the second ones; therefore, there is the factor $1/2$ in the front. We also mention that, in the last line, $(k,w)$ with $b_{jk}(w)$ and $b_{ik}(w)$ having the same sign does not contribute to the second sum, because in that case $b_{jk}(w)\|b_{ki}(w)\|+\|b_{jk}(w)\|b_{ki}(w)=0$. Now the sum $\mathrm{(A)}+\mathrm{(B)}+\mathrm{(C)}$ cancel due to the mutation of matrices (see ) $$\begin{aligned} \begin{split} &\textstyle b_{ji}(\min(u,v)) = -b_{ji}(\max(u-\lambda_i(i,u), v-\lambda_j(j,v)))\\ &\quad + \sum_{\scriptstyle (k,w)\in P_+ \atop \scriptstyle k\in (u-\lambda_-(i,u),u)\cap (v-\lambda_-(j,v),v) } \frac{1}{2}\Bigl( b_{jk}(w) \|b_{ki}(w)\| + \|b_{jk}(w)\|b_{ki}(w)\Bigr) \end{split}\end{aligned}$$ for $(u-\lambda_-(i,u),u) \cap(v-\lambda_-(j,v),v)\neq \emptyset$. Local version of constancy condition and second proof of Proposition \[prop:const\] {#subsec:local} ------------------------------------------------------------------------------------- Here we give a ‘local’ version of Proposition \[prop:const\], thereby providing an alternative proof of Proposition \[prop:const\]. The formula here is parallel to the ones in [@Fock09 Lemma 6.1] and [@Fock07 Lemma 2.17] by Fock and Goncharov. In fact, we have reached the formula while trying to interpolate Proposition \[prop:const\] and their results. An important difference from their formulas is the use of $F$-polynomials in . Let $\mathcal{A}(B,x,y)$ be any cluster algebra. For each seed $(B',x',y')$, we set $$\begin{aligned} \label{eq:W} V'&:=\frac{1}{2}\sum_{i\in I} F'_i \wedge (y'_i [y'_i]_{\mathbf{T}}) =\sum_{i\in I} F'_i \wedge y'_i +\frac{1}{2}\sum_{i,j\in I} b'_{ij}F'_i\wedge F'_j,\end{aligned}$$ where $F'_i=F'_i(y)$ ($i\in I$) are the $F$-polynomials for $(B',x',y')$. \[prop:local\] Let $(B',x',y')$ and $(B'',x'',y'')$ be any seeds such that $(B'',x'',y'')=\mu_k(B',x',y')$. Then, we have the following relation in $\bigwedge ^2 \mathbb{P}_{\mathrm{univ}}(y)$. $$\begin{aligned} \label{eq:local} V''-V' = y_k' \wedge (1+y'_k).\end{aligned}$$ Our proof is parallel to the first proof of Proposition \[prop:const\]. We use the following property of the corresponding $F$-polynomials, which are derived as Lemma \[lem:F\]. \(a) For the above $k$, $$\begin{aligned} \label{eq:Fi2} \begin{split} \textstyle F'_kF''_k &= \left[ \frac{y'_k}{1+y'_k} \right]_{\mathbf{T}} \prod_{j:\, b'_{jk}>0} F'_j{}^{b'_{jk}} + \left[ \frac{1}{1+y'_k} \right]_{\mathbf{T}} \prod_{j:\, b'_{jk}<0} F'_j{}^{-b'_{jk}}. \end{split}\end{aligned}$$ \(b) For any $i\in I$, $$\begin{aligned} \label{eq:yi11} y'_i&= [y'_i]_{\mathbf{T}} \frac { \displaystyle \prod_{j:\, b'_{ji}>0} F'_j{}^{b'_{ji}} } { \displaystyle \prod_{j:\, b'_{ji}<0} F'_j{}^{-b'_{ji}} } = [y'_i]_{\mathbf{T}}\prod_{j\in I} F'_j{}^{b'_{ji}}.\end{aligned}$$ \(c) For the above $k$, $$\begin{aligned} \label{eq:yi22} 1+y'_k&= [1+y'_k]_{\mathbf{T}} \frac { F'_kF''_k } { \displaystyle \prod_{j:\, b'_{jk}<0} F'_j{}^{-b'_{jk}} }.\end{aligned}$$ We put and into the right hand side of , expand it, then, sum them up into three parts as before. The first part consists of the single term $[y'_k]_{\mathbf{T}}\wedge [1+y'_k]_{\mathbf{T}}$, which vanishes by the same reason as before. The second part consists of the terms involving both tropical coefficients and $F$-polynomials. We separate them into five parts as . By an easy calculation as before, they are summarized as follows. $$\begin{aligned} \sum_{i\in I} F''_i \wedge [y''_i]_{\mathbf{T}} - \sum_{i\in I} F'_i \wedge [y'_i]_{\mathbf{T}}.\end{aligned}$$ The third part consists of the terms involving only $F$-polynomials. We separate them into three parts as –. Then, they are summarized as follows. $$\begin{aligned} \frac{1}{2} \sum_{i\in I} F''_i \wedge \frac{y''_i}{[y''_i]_{\mathbf{T}}} - \frac{1}{2} \sum_{i\in I} F'_i \wedge \frac{y'_i}{[y'_i]_{\mathbf{T}}}.\end{aligned}$$ Therefore, we obtain the claim. Proposition \[prop:const\] is immediately obtained from Proposition \[prop:local\] as follows. For any $u\in \mathbb{Z}$, we set $$\begin{aligned} V(u)&=\frac{1}{2}\sum_{i\in I} F_i(u) \wedge ( y_i(u) [y_i(u)]_{\mathbf{T}}).\end{aligned}$$ Note that $V(0)=0$ because $F_i(0)=1$. Then, by Proposition \[prop:local\], we have, for $u>0$, $$\begin{aligned} V(u) = \sum_{ \scriptstyle (k,v)\in P_+ \atop \scriptstyle 0\leq v < u }y_k(v) \wedge (1+y_k(v)).\end{aligned}$$ Then, from the periodicity $V(\Omega)=V(0)=0$, we obtain . Skew symmetrizable case {#subsec:dilog2} ----------------------- For a [skew symmetrizable]{} matrix $B$, the dilogarithm identities should be slightly modified. Let $D=\mathrm{diag}(d_i)_{i\in I}$ be the (left) skew symmetrizer of $B$, namely, $d_i b_{ij} = - d_j b_{ji}$ holds. Let $d=\mathrm{LCM}(d_i)_{i\in I}$ and define $\tilde{d}_i= d/d_i \in \mathbb{N}$ ($i\in I$). Then, $\tilde{D}=\mathrm{diag}(\tilde{d}_i)_{i\in I}$ be the right skew symmetrizer of $B$, namely, $b_{ij}\tilde{d}_j = - b_{ji}\tilde{d}_i$ holds. Let $\tilde{N}_+$ (resp. $\tilde{N}_-$) be the total number of positive (resp. negative) monomials $[y_i(u)]_{\mathbf{T}}$ in the fundamental region $S_+$ [with multiplicity $\tilde{d}_i$]{}. Then, the identities and should be modified as $$\begin{aligned} \label{eq:DI5} \frac{6}{\pi^2} \sum_{ (i,u)\in S_+ } \tilde{d}_i L\left( \varphi \left( \frac{y_i(u)}{1+y_i(u)} \right) \right) &=\tilde{N}_-,\\ \label{eq:DI'6} \frac{6}{\pi^2} \sum_{ (i,u)\in S_+ } \tilde{d}_i L\left( \varphi \left( \frac{1}{1+y_i(u)} \right) \right) &=\tilde{N}_+,\end{aligned}$$ and and should be modified as $$\begin{aligned} \label{eq:W'} V':=\frac{1}{2}\sum_{i\in I} \tilde{d}_i F'_i \wedge (y'_i [y'_i]_{\mathbf{T}}) &=\sum_{i\in I} \tilde{d}_i F'_i \wedge y'_i +\frac{1}{2}\sum_{i,j\in I} b'_{ij} \tilde{d}_j F'_i\wedge F'_j,\\ \label{eq:local'} V''-V' &= \tilde{d}_k y_k' \wedge (1+y'_k).\end{aligned}$$ Unfortunately, for skew symmetrizable matrices one cannot yet prove , , and . 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--- abstract: 'Let $\Omega$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $\Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.' address: \| Department of Mathematics\ Lorestan University\ Khoramabad\ Iran author: - 'S.Morteza Mirafzal and Meysam Ziaee' title: \| A note on the automorphism group of\ the Hamming graph --- [^1] Introduction ============ Let $\Omega$ be a $m$-set, where $m>1$, is an integer. The $Hamming \ graph$ $H(n,m)$, has $\Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. This graph is very famous and much is known about it, for instance this graph is actually the Cartesian product of $n$ complete graphs $K_m$, that is, $K_m \Box \cdots \Box K_m$. In general, the connection between Hamming graphs and coding theory is of major importance. If $m=2$, then $H(n,m)=Q_n$, where $Q_n$ is the hypercube of dimension $n$. Since, the automorphism group of the hypercube $Q_n$ has been already determined \[10\], in the sequel, we assume that $m \geq 3$. Figure 1. displays $H(2,3)$ in the plane. Note that in this figure, we denote the vertex $(x,y)$ by $xy$. It follows from the definition of the Hamming graph $ H(n,m) $ that if $\theta \in \mbox{Sym}([n]$), where $ \Omega=[n]=\{1,\cdots, n \} $, then $$f_\theta : V( H(n,m) )\longrightarrow V( H(n,m)), f_\theta (x_1, ..., x_n ) = (x_{\theta (1)}, ..., x_{\theta (n)} ),$$ is an automorphism of the Hamming graph $ H(n,m), $ and the mapping $ \psi : \mbox{Sym} ([n]) \longrightarrow Aut ( H(n,m) )$, defined by this rule, $ \psi ( \theta ) = f_\theta $, is an injection. Therefore, the set $H= \{ f_\theta \ \|\ \theta \in \mbox{Sym}([n]) \} $, is a subgroup of $ Aut (H(n,m)) $, which is isomorphic with $\mbox{Sym}([n])$. Hence, we have $\mbox{Sym}([n]) \leq Aut ( H(n,m) )$. (-4.3,-2.44) rectangle (11.32,6.3); (0.64,4.)– (2.92,3.22); (2.92,3.22)– (5.38,3.96); (5.38,3.96)– (6.26,2.34); (6.26,2.34)– (5.54,-0.14); (5.54,-0.14)– (3.02,0.62); (3.02,0.62)– (0.76,-0.28); (0.76,-0.28)– (-0.28,2.18); (-0.28,2.18)– (0.64,4.); (0.64,4.)– (5.38,3.96); (5.38,3.96)– (5.54,-0.14); (5.54,-0.14)– (0.76,-0.28); (0.76,-0.28)– (0.64,4.); (2.92,3.22)– (2.4,1.64); (2.4,1.64)– (3.02,0.62); (-0.28,2.18)– (6.26,2.34); (2.92,3.22)– (3.02,0.62); (-0.28,2.18)– (2.4,1.64); (2.4,1.64)– (6.26,2.34); (0.66,-0.38) node\[anchor=north west\] [00]{}; (3.26,1.12) node\[anchor=north west\] [01]{}; (5.64,-0.24) node\[anchor=north west\] [02]{}; (-1.1,2.36) node\[anchor=north west\] [10]{}; (1.98,1.4) node\[anchor=north west\] [11]{}; (6.4,2.84) node\[anchor=north west\] [12]{}; (-0.22,4.32) node\[anchor=north west\] [20]{}; (2.48,3.98) node\[anchor=north west\] [21]{}; (5.68,4.24) node\[anchor=north west\] [22]{}; (-1.1,-1.12) node\[anchor=north west\] [Figure 1. The Hamming graph H(2,3)]{}; (0.64,4.) circle (1.5pt); (5.38,3.96) circle (1.5pt); (5.54,-0.14) circle (1.5pt); (0.76,-0.28) circle (1.5pt); (-0.28,2.18) circle (1.5pt); (6.26,2.34) circle (1.5pt); (2.92,3.22) circle (1.5pt); (3.02,0.62) circle (1.5pt); (2.4,1.64) circle (1.5pt); Let $A,B$, be non-empty sets. Let $Fun(A,B)$, be the set of functions from $A$ to $B$, in other words, $Fun(A,B)= \{f \ \| \ f: A \rightarrow B \}$. If $B$ is a group, then we can turn $Fun(A,B)$ into a group by defining a product, $(fg)(a)=f(a)g(a), \ \ f, g\in Fun(A,B), \ \ a\in A,$ where the product on the right of the equation is in $B$. If $f \in Fun([n], \mbox{Sym}([m]))$, then we define the mapping, $$A_f: V(\Gamma)\rightarrow V(\Gamma), \ \mbox{ by this rule}, \ A_f(x_1, \cdots ,x_n)=(f(1)(x_1), \cdots, f(n)(x_n)).$$ It is easy to show that the mapping $A_f$ is an automorphism of the Hamming graph $\Gamma =H(n,m)$, and hence the group, $F=\{ A_f \ \| \ f\in Fun([n], \mbox{Sym}([m])) \}$, is also a subgroup of the Hamming graph $\Gamma =H(n,m)$. Therefore, the subgroup which is generated by $H$ and $F$ in the group $Aut(\Gamma)$, namely, $W=<H,F> $ is a subgroup of $Aut(\Gamma)$. In this paper, we want to show that; $$Aut(H(n,m))=W=<H,F>=\mbox{Sym}(\Omega) wr_I \mbox{Sym}([n])$$There are various important families of graphs $\Gamma$, in which we know that for a particular group $G$, we have $G \leq Aut(\Gamma)$, but showing that in fact we have $G = Aut(\Gamma)$, is a difficult task. For example note to the following cases. \(1) The $Boolean\ lattice$ $BL_n, n \geq 1$, is the graph whose vertex set is the set of all subsets of $[n]= \{ 1,2,...,n \}$, where two subsets $x$ and $y$ are adjacent if and only if their symmetric difference has precisely one element. The $hypercube$ $Q_n$ is the graph whose vertex set is $ \{0,1 \}^n $, where two $n$-tuples are adjacent if they differ in precisely one coordinates. It is an easy task to show that $Q_n \cong BL_n $, and $ Q_n \cong Cay(\mathbb{Z}_{2}^n, S )$, where $\mathbb{Z}_{2}$ is the cyclic group of order 2, and $S=\{ e_i \ \| \ 1\leq i \leq n \}, $ where $e_i = (0, ..., 0, 1, 0, ..., 0)$, with 1 at the $i$th position. It is an easy task to show that the set $H= \{ f_\theta \ \|\ \theta \in \mbox{Sym}([n]) \} $, $ f_\theta (\{x_1, ..., x_n \}) = \{ \theta (x_1), ..., \theta (x_n) \}$ is a subgroup of $Aut(BL_n)$, and hence $H$ is a subgroup of the group $Aut(Q_n)$. We know that in every Cayley graph $\Gamma= Cay(G,S)$, the group $Aut(\Gamma)$ contains a subgroup isomorphic with the group $G$. Therefore, $\mathbb{Z}_{2}^n $ is a subgroup of $Aut(Q_n)$. Now, showing that $Aut(Q_n) = <\mathbb{Z}_{2}^n, \mbox{Sym}([n])>( \cong \mathbb{Z}_{2}^n \rtimes \mbox{Sym}([n]))$, is not an easy task \[10\]. \(2) Let $n,k \in \mathbb{ N}$ with $ k < \frac{n}{2} $ and Let $[n]=\{1,...,n\}$. The $Kneser\ graph$ $K(n,k)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n], \|v\|=k\}$ and two vertices $v$ and $w$ are adjacent if and only if $\|v\cap w\|$=0. The Kneser graph $K(n,k)$ is a vertex-transitive graph \[6\]. It is an easy task to show that the set $H= \{ f_\theta \ \|\ \theta \in \mbox{Sym}([n]) \} $, $ f_\theta (\{x_1, ..., x_k \}) = \{ \theta (x_1), ..., \theta (x_k) \}$, is a subgroup of $ Aut ( K(n,k) )$ \[6\]. But, showing that $$H= \{ f_\theta \ \| \ \theta \in \mbox{Sym}([n]) \}= Aut ( K(n,k) )$$ is not very easy \[6 chapter 7, 13\]. (3) Let $n,k \in \mathbb{ N}$ with $ k < n, $ and let $[n]=\{1,...,n\}$. The $Johnson\ graph$ $J(n,k)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n], \|v\|=k\}$ and two vertices $v$ and $w $ are adjacent if and only if $\|v\cap w\|=k-1$. The Johnson graph $J(n,k)$ is a vertex-transitive graph \[6\]. It is an easy task to show that the set $H= \{ f_\theta \ \| \ \theta \in \mbox{Sym}([n]) \} $, $f_\theta (\{x_1, ..., x_k \}) = \{ \theta (x_1), ..., \theta (x_k) \} $, is a subgroup of $ Aut( J(n,k) ) $ \[6\]. It has been shown that $Aut(J(n,k)) \cong \mbox{Sym}([n])$, if $ n\neq 2k, $ and $Aut(J(n,k)) \cong \mbox{Sym}([n]) \times \mathbb{Z}_2$, if $ n=2k$, where $\mathbb{Z}_2$ is the cyclic group of order 2 \[3,7,12\]. Preliminaries ============= In this paper, a graph $\Gamma=\Gamma(V,E) $ is considered as a simple undirected graph with vertex-set $V(\Gamma)=V $, and edge-set $ E(\Gamma)=E $. For all the terminology and notation not defined here, we follow \[1,2,5,6\]. The group of all permutations of a set $V$ is denoted by $\mbox{Sym}(V)$ or just $\mbox{Sym}(n)$ when $\|V\| =n $. A $permutation\ group$ $G$ on $V$ is a subgroup of $\mbox{Sym}(V)$. In this case we say that $G$ act on $V$. If $\Gamma$ is a graph with vertex set $V$, then we can view each automorphism as a permutation of $V$, and so $Aut(\Gamma)$ is a permutation group. Let $G$ act on $V$, we say that $G$ is $transitive$ (or $G$ acts $transitively$ on $V$), if there is just one orbit. This means that given any two elements $u$ and $v$ of $V$, there is an element $ \beta $ of $G$ such that $\beta (u)= v. $ Let $\Gamma, \Lambda $ be arbitrary graphs with vertex-set $V_{1},V_{2}$, respectively. An isomorphism from $\Gamma $ to $\Lambda $ is a bijection $\psi:V_{1}\longrightarrow V_{2}$ such that $\{x,y\}$ is an edge in $\Gamma $ if and only if $\{\psi(x),\psi(y)\}$ is an edge in $\Lambda$. An isomorphism from a graph $\Gamma $ to itself is called an automorphism of the graph $\Gamma$. The set of automorphisms of graph $\Gamma $ with the operation of composition of functions is a group, called the automorphism group of $\Gamma $ and denoted by $\mbox{Aut}(\Gamma)$. In most situations, it is difficult to determine the automorphism group of a graph, but there are various in the literature and some of the recent works appear in the references \[7,8,9,11,13,14,15,16,17\]. The graph $\Gamma $ is called $vertex$-$transitive,$ if $\mbox{Aut}(\Gamma) $ acts transitively on $V(\Gamma)$. In other words, given any vertices $u,v $ of $\Gamma$, there is an $f\in \mbox{Aut}(\Gamma)$ such that $f(u)=v$. For $v\in V (\Gamma )$ and $G = Aut(\Gamma )$, the $stabilizer\ subgroup$ $G_{v}$ is the subgroup of $G$ containing of all automorphisms which fix $v$. In the vertex-transitive case all stabilizer subgroups $G_{v}$ are conjugate in $G$, and consequently isomorphic, in this case, the index of $G_{v}$ in $G$ is given by the equation, $\|G : G_{v}\| = \|G\|\|G_{v}\| = \|V (\Gamma)\|$. If each stabilizer $G_{v}$ is the identity group, then every element of $G$, except the identity, does not fix any vertex and we say that $G$ act semiregularly on $V$. We say that $G$ act regularly on $V$ if and only if $G$ acts transitively and semiregularly on $V,$ and in this case we have $\|V \| = \|G\|. $ Let $N$ and $H$ be groups, and let $ \phi: H \rightarrow Aut(N)$ be a group homomorphism. In other words, the group $H$ acts on the group $N$, by this rule $n^h = \phi (h)(n)$, $n \in N, h \in H$. Note that in this case we have $ {(n_1 n_2)}^h={n_1}^h {n_2}^h$, $n_1,n_2 \in N$. The $semidirect \ product$ $N$ by $H$ which is denoted by $N \rtimes H$ is a group on the set $N \times H$= $\{ (n,h) \ \| \ n\in N, h\in H \}$, with the multiplication $(n,h)(n_1,h_1)=(n {(n_1)}^{-h}, hh_1) $. Note that the identity element of the group $ N \rtimes H$ is $(1_N, 1_H)$, and the inverse of the element $(n,h)$ is the element $ ({(n^{-1})}^{h}, {h}^{-1}) $. Main Results ============ Let $\Gamma $ be a connected graph with diameter $d$. Then we can partition the vertex-set $V(\Gamma) $ with respect to the distances of vertices from a fixed vertex. Let $v$ be a fixed vertex of the graph $\Gamma $. We denote the set of vertices at distance $i$ from $v$, by $\Gamma _{i}(v)$. Thus it is obvious that $\{v\}=\Gamma _{0}(v)$ and $ \Gamma_{1}(v) = N(v) $, the set of adjacent vertices to vertex $v$, and $V (\Gamma )$ is partitioned into the disjoint subsets $\Gamma_{0}(x), ..., \Gamma_{D}(x)$. If $\Gamma =H(n,m)$, then it is clear that two vertices are at distance $k$ if and only if they differ in exactly $k$ coordinates. Then the maximum distance occurs when the two vertices (regarded as ordered $n$-tuples) differ in all $n$ coordinates. Thus the diameter of $H(n,m)$ is equal to $n$.\ Let $m \geq 3$ and $\Gamma =H(n,m)$. Let $x\in V(\Gamma )$, $\Gamma_{i}=\Gamma_{i}(x)$ and $v\in \Gamma_{i}$. Then we have; $\displaystyle \bigcap _{w\in \Gamma_{i-1}\cap N(v)}(N(w)\cap \Gamma _{i})=\{v\} $. It is obvious that $v\in \displaystyle \bigcap _{w\in \Gamma_{i-1}\cap N(v)}(N(w)\cap \Gamma _{i}) $. Let $x=(x_{1},\cdots ,x_{n})$. Since the Hamming graph $H(n,m)$ is a distance-transitive graph \[3\], then we can assume that, $v=(x_{1},\cdots ,x_{n-i},y_{n-i+1},\cdots ,y_{n})$, where $y_{j}\in \mathbb{Z}_{m}-\{x_{j}\}$ for all $j=n-i+1,\cdots ,n$. Let $w\in \Gamma_{i-1}\cap N(v)$. Then $w,x$ differ in exactly $i-1$ coordinates and $w,v$ differ in exactly one coordinate. Note that, if in $v$ we change one of $x_{j}$s, where $j=1,\cdots, n-i$, then we obtain a vertex $u$ such that $d(u,x)\geq i+1$. Thus, $w$ has a form such as; $w=w_{r}=(x_{1},\cdots ,x_{n-i},y_{n-i+1},\cdots,y_{r-1},x_{r},y_{r+1},\cdots ,y_{n})$ We show that if $u\in \Gamma_{i}$ and $u\neq v$ and $u$ is adjacent to some $w_{r}$, then there is some $w_{p}$ such that $u$ is not adjacent to $w_{p}. $\ If $v\neq u\in \Gamma_{i}$ is adjacent to $w_{r}$ then $u$ has one of the following forms; $u_{1}=(x_{1},\cdots ,x_{n-i}, y_{n-i+1},\cdots ,y_{r-1},y,y_{r+1},\cdots ,y_{n})$, where $y \in \mathbb{Z}_{m} $, and $y\neq y_{r},x_{r}$ (note that since $m \geq 3$, hence there is such a $ y$). $u_{2}=(x_{1}\cdots,x_{j-1},y,x_{j+1},\cdots,x_{n-i},y_{n-i+1},\cdots,y_{r-1},x_{r},y_{r+1},\cdots,y_{n})$,where $y \in \mathbb{Z}_{m} $, and $y\neq x_{j}$.In the case (i), $u_{1}$ is not adjacent to $w_{t}$, for all possible $t$, $t\neq r$.\ In the case (ii), it is obvious that $u_{2}$ is also not adjacent to $w_{t}$ for all possible $t$, $t\neq r$.\ Our argument shows that if $ u\in \Gamma_{i}$, and $u\neq v$, then there is some $w_{r}$ such that $u$ is not adjacent to $w_{r}$, in other words $u \notin N(w_r)$. Thus we have; $\displaystyle \bigcap _{w\in N(v)\cap \Gamma_{i-1}}(N(w)\cap \Gamma _{i})=\{v\} $. Let $I=\{\gamma_1, ... ,\gamma_{n}\}$ be a set and $K$ be a group. Let $Fun(I, K)$ be the set of all functions from $I$ into $K$. We can turn $Fun(I, K)$ into a group by defining a product: $$(fg)(\gamma)=f(\gamma)g(\gamma), \ \ f, g\in Fun(I, K), \ \ \gamma\in I,$$ where the product on the right of the equation is in $K$. Since $I$ is finite, the group $Fun(I, K)$ is isomorphic to $K^{n}$ (the direct product of $n$ copies of $K$), by the isomorphism $f\mapsto (f(\gamma_1), ... , f(\gamma_{n}))$. Let $H$ be a group and assume that $H$ acts on the nonempty set $I$. Then, the wreath product of $K$ by $H$ with respect to this action is the semidirect product $Fun(I, K)\rtimes H$ where $H$ acts on the group $Fun(I, K)$, by the following rule, $$f^{x}(\gamma)=f(\gamma^{ x^{-1}}),\ f\in Fun(I, K), \gamma \in I, \, x \in H.$$ We denote this group by $K wr_{I}H$. Consider the wreath product $G=K wr_{I}H$. If $K$ acts on a set $\Delta$ then we can define an action of $G$ on $\Delta\times I$ by the following rule, $$(\delta, \gamma)^{(f, h)}=(\delta^{f(\gamma)}, \gamma^h), \ \ (\delta, \gamma)\in \Delta\times I,$$ where $(f, h)\in Fun(I, K)\rtimes H=K wr_{I}H$. It is clear that if $I, K$ and H, are finite sets, then $G=K wr_{I}H$, is a finite group, and we have $\|G\|= {\|K\|}^{\|I\|} \|H\|$.\ We have the following theorem \[4\]. Let $\Gamma $ be a graph with $n$ connected components $\Gamma_{1},\Gamma_{2},\cdots,\Gamma_{n}$, where $\Gamma_{i}$ is isomorphic to $\Gamma_1$ for all $i \in [n]= \{1,\cdots,n \}=I$. Then we have, $Aut(\Gamma)=Aut(\Gamma_1)wr_{I}\mbox{Sym}([n])$. Let $n \geq 2,\ m \geq 3$. Let $v$ be a vertex of the Hamming graph $H(n,m)$. Then, $\Gamma_{1} =< N(v) >$, the induced subgraph of $N(v)$ in $H(n,m)$, is isomorphic with $nK_{m-1}$, where $nK_{m-1}$ is the disjoint union of $n$ copies of the complete graph $K_{m-1}$. Let $v=(v_{1},\cdots,v_{n})$. Then, for all $i$, $i=1,\cdots,n$, there are $m-1$ elements $w_{j}$, $w_{j}\in \mathbb{Z}_{m}-\{v_{i}\}$. Let $x_{ij}=(v_{1},\cdots,v_{i-1},w_{j},v_{i+1},\cdots,v_{n})$, $1\leq i\leq n, 1\leq j\leq n-1$. Then, $N(v)=\{x_{ij}:\;\; 1\leq i\leq n,\; 1\leq j\leq m-1\}$. Let $x_{ij},x_{rs}$, be two vertices in $\Gamma_{1}=<N(v)>$, then $x_{ij}, x_{rs}$ are adjacent in $\Gamma_{1}$ if and only if $i=r$. Note that two vertices, $(v_{1}, ..., v_{i-1}, w_{j} , v_{i+1}, ..., v_{n})$ and $(v_{1}, ..., v_{i-1}, w_{s} , v_{i+1}, ..., v_{n})$ differ in only one coordinate. Therefore, for each $i=1,\cdots ,n$, there are $m-2$ vertices $w_{ir}$ in $ \Gamma_{1} $ which are adjacent to the vertex $w_{ij}$, where $r\neq j$. Now, it is obvious that the subgraph induced by the set $ \{ x_{ij}: 1 \leq j \leq m-1 \} $, is isomorphic with $K_{m-1}$, the complete graph of order $m-1$. Now, it is easy to see that, the subgraph induced by the set $ \{x_{ij}:\ i=1,\cdots,n,\ j=1,\cdots,m-1\}$, is isomorphic with $ nK_{m-1}$, the disjoint union of $n$ copies of the complete graph $K_{m-1}$. We now are ready to prove the main result of this paper. Let $n \geq 2,\ m \geq 3$, and $\Gamma=H(n,m)$ be a Hamming graph. Then Aut$(\Gamma)\cong \mbox{Sym}([n])wr_I \mbox{Sym}([m]) $, where $I= [n]= \{1,2,\cdots n \}$. Let $G = \mbox{Aut}(\Gamma)$. Let $x \in V = V (\Gamma)$, and $G_{x} = \{f \in G\; \|\;f(x) = x\}$ be the stabilizer subgroup of the vertex $x$ in Aut$(\Gamma)$. Let $< N(x) >= \Gamma_{1}$ be the induced subgroup of $N(x)$ in $\Gamma$. If $f\in G_{x}$ then $f_{\|N(x)}$, the restriction of $f$ to $N(x)$ is an automorphism of the graph $\Gamma_{1}$. We define the mapping $\psi : G_{x}\longrightarrow \mbox{Aut}(\Gamma_{1})$ by this rule, $\psi(f) = f_{\|N(x)}$. It is an easy task to show that $\psi$ is a group homomorphism. We show that $Ker(\psi)$ is the identity group. If $f \in Ker(\psi)$, then $f(x) = x$ and $f(w) = w$ for every $w \in N(x)$. Let $\Gamma_{i}$ be the set of vertices of $\Gamma$ which are at distance $i$ from the vertex $x$. Since, the diameter of the graph $\Gamma = H(n,m)$, is $n$, then $V = V (\Gamma) = \displaystyle \bigcup_{i=0}^{n}\Gamma_{i}$. We prove by induction on $i$, that $f(u) = u$ for every $u \in \Gamma_{i}$. Let $d(u, x)$ be the distance of the vertex $u$ from $x$. If $d(u, x) = 1$, then $u \in \Gamma_{1}$ and we have $f(u) = u$. Assume that $f(u) = u$, when $d(u, x) = i-1$. If $d(u, x) = i$, then by Lemma 1. $\{u\} =\displaystyle \bigcap_{w\in \Gamma_{i-1}\cap N(u)}(N(w)\cap \Gamma_{i})$. Note that if $w\in \Gamma_{i-1}$, then $d(w, x) = i - 1$, and hence $f(w) = w$. Therefore, $\{f(u)\} = \displaystyle \bigcap_{w\in \Gamma_{i-1}\cap N(u)}(N(f(w))\cap \Gamma_{i})=\displaystyle \bigcap_{w\in \Gamma_{i-1}\cap N(u)}(N(w)\cap \Gamma_{i})= u$. Thus, $f(u)=u$ for all $u\in V(\Gamma)$, hence we have $Ker(\psi)=\{1\} $. On the other hand, $\frac{G_{v}}{Ker(\psi)}\cong \psi(G_{v})\leq \mbox{Aut}(\Gamma_{1}) $, hence $G_{v} \cong \psi (G_{v})\leq \mbox{Aut}(\Gamma_{1})$. Thus, $\|G_{v}\|\leq \| \mbox{Aut}(\Gamma_{1})\|$.\ We know by Lemma 3. that $\Gamma_{1}\cong nK_{m-1}$. We know that, Aut$(K_{m-1})\cong \mbox{Sym}([m-1]) $. Then, by the above equation, we have; $\|G_{v}\|\leq \|\mbox{Aut}(\Gamma_{1})\|=\|\mbox{Sym}([m-1])wr_{I}\mbox{Sym}([n])\|=((m-1)!)^{n}n!$, where $I= [n]=\{1, \cdots, n \}$. Since $\Gamma = H(n,m)$ is a vertex-transitive graph, then we have $\|V (\Gamma)\| = \|G\| \|G_{v}\|$, and therefore; $$\|G\| =\|G_{v}\|\|V (\Gamma)\| \leq \|\mbox{Aut}(nK_{m-1})\|m^{n}=m^{n}((m-1)!)^{n}n!=(m!)^{n}n! \ \ \ \ \ ()$$ We have seen (in the introduction section of this paper) that if $\theta \in \mbox{Sym}([n]$), where $ \Omega=[n]=\{1,\cdots, n \} $, then $$f_\theta : V( H(n,m) )\longrightarrow V( H(n,m)), f_\theta (x_1, ..., x_n ) = (x_{\theta (1)}, ..., x_{\theta (n)}),$$ is an automorphism of the Hamming graph $ H(n,m), $ and the mapping $ \psi : \mbox{Sym} ([n]) \longrightarrow Aut ( H(n,m) )$, defined by this rule, $ \psi ( \theta ) = f_\theta $, is an injection. Therefore, the set $H= \{ f_\theta \ \|\ \theta \in \mbox{Sym}([n]) \} $, is a subgroup of $ Aut (( H(n,m) )) $, which is isomorphic with $\mbox{Sym}([n])$. Hence, we have $\mbox{Sym}([n]) \leq Aut ( H(n,m) )$. On the other hand, if $f \in Fun([n], \mbox{Sym}([m]))$, then we define the mapping; $A_f: V(\Gamma)\rightarrow V(\Gamma),$ by this rule, $ A_f(x_1, \cdots ,x_n)$=$(f(1)(x_1), \cdots, f(n)(x_n)).$ It is an easy task to show that the mapping $A_f$ is an automorphism of the Hamming graph $\Gamma$, and hence the group, $F=\{ A_f \ \| \ f\in Fun([n], ([m])) \}$, is a subgroup of the Hamming graph $\Gamma =H(n,m)$. Therefore, the subgroup which is generated by $H$ and $F$ is in the group $Aut(\Gamma)$, namely, $W=<H,F>$ is a subgroup of $Aut(\Gamma)$. Note that $W= \mbox{Sym}([m]) wr_I \mbox{Sym}([n])$, where $I=[n]=\{1,2,\cdots,n \}$. Since, the subgroup $W$ has $(m!)^{n}n!$ elements, then by $()$, we conclude that; $$Aut(\Gamma)=W= \mbox{Sym}([m]) wr_I \mbox{Sym}([n])$$ [99]{} Biggs, NL. (1993). Algebraic Graph Theory (Second edition). Cambridge Mathematical Library: Cambridge University Press. Bondy, JA., Murty, USR. (2008). Graph Theory. New York: Springer-Verlag. 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